Secondary Line Constants or Quaternary Constants, Quaternary Parameters

In the field of telecommunications, "secondary line constants" likely refer to parameters used to describe the characteristics of transmission lines, which are used to transmit signals (such as telephone, data, or video) over long distances. These parameters help engineers and technicians analyze and design efficient and reliable communication systems. The primary line constants are the resistance, inductance, conductance, and capacitance per unit length of the transmission line.

However, the term “secondary line constants” is not commonly used. It is normally known as “quaternary parameters” or “quaternary constants” used in telecommunication line analysis. These parameters extend the analysis of transmission lines beyond the primary parameters by including additional effects, such as radiation and shunt capacitance. Quaternary parameters are also used to model the behavior of transmission lines at higher frequencies.

Here are some terms related to the characterization of transmission lines in telecommunications:

Propagation Constant

The propagation constant, often denoted as γ (gamma), is a fundamental parameter used to describe the behavior of electromagnetic waves as they propagate through various mediums, such as transmission lines, waveguides, or optical fibers. It is a complex number that combines both the phase constant (β) and the attenuation constant (α). The propagation constant provides insights into how signals change in amplitude and phase as they travel along a transmission medium.

The propagation constant can be mathematically expressed as:

γ = α + jβ


α (alpha) represents the attenuation constant, which measures the rate of amplitude decay of the signal as it travels through the medium. It is a real number and is usually measured in Nepers per unit length or decibels per unit length.

β (beta) represents the phase constant, which determines the phase shift experienced by the signal as it propagates through the medium. It is an imaginary number and is usually measured in radians per unit length.

The magnitude of the propagation constant (γ) gives the overall rate of signal decay, while the argument or phase angle of the propagation constant (arg(γ)) gives the phase shift experienced by the signal.

In telecommunications and transmission line theory, the propagation constant is used to analyze the behavior of signals traveling along the line. It helps engineers understand how signals attenuate and phase shift as they move along the transmission medium. By manipulating the propagation constant, engineers can design and optimize transmission systems for various applications, such as minimizing signal loss, reducing distortion, and maximizing signal quality.

Propagation Constant Formula

Here y stands for admittance and z stands for the impedance of the transmission line

Attenuation Constant (α)

The attenuation constant, denoted as α (alpha), is a fundamental parameter used to describe the attenuation or loss of energy that occurs as electromagnetic waves propagate through a medium. It is an essential concept in various fields, including telecommunications, signal processing, and optics. The attenuation constant quantifies the decrease in amplitude (or intensity) of a signal as it travels over a certain distance within the medium.

Here's an expanded explanation of the attenuation constant:


The attenuation constant measures the rate at which the amplitude of an electromagnetic wave decreases as it propagates through a medium. It is usually expressed in units of Nepers per unit length (Np/m) or decibels per unit length (dB/m).

Physical Interpretation

When electromagnetic waves travel through a medium, they interact with the material, which can lead to energy absorption and scattering. As a result, the energy carried by the wave is gradually reduced, leading to a decrease in amplitude. The attenuation constant quantifies how rapidly this energy reduction occurs.

Mathematical Representation

The attenuation constant (α) is a real number. It is often used in the exponential form of the signal attenuation equation:

A(z) = A0 * e^(-α * z)


A(z) is the amplitude of the signal at a distance of z along the medium. 
A0 is the initial amplitude of the signal. 
α is the attenuation constant. 
z is the distance traveled within the medium.


The attenuation constant is typically measured in Nepers per unit length (Np/m) or decibels per unit length (dB/m). These units help quantify the amount of attenuation that occurs over a specific distance.


In telecommunications and signal transmission, understanding the attenuation constant is crucial for designing communication systems that maintain signal quality over long distances. It also plays a role in determining the maximum achievable data rates and the effective range of communication links.

In summary, the attenuation constant describes how rapidly the amplitude of an electromagnetic wave decreases as it propagates through a medium. It is a key parameter in the analysis and design of communication systems and helps engineers ensure reliable and efficient signal transmission over various distances and mediums.


α = R/2z Neper/km

1 Neper = 8.686 αβ

Phase Constant (β)

The phase constant is a crucial concept in the study of wave propagation, particularly in fields such as physics, engineering, and telecommunications. It is an essential parameter used to describe the phase relationship between two waveforms or the phase shift experienced by a wave as it travels through a medium. The phase constant plays a significant role in understanding how waves interfere, combine, and propagate.


The phase constant (often denoted as β or φ) is a quantity that represents the angular phase change experienced by a waveform as it propagates over a given distance or through a specific medium. It is typically measured in radians per unit length (rad/m) or degrees per unit length (°/m).

Physical Interpretation

When a wave travels through a medium, it encounters changes in the properties of that medium, which can cause a phase shift in the waveform. The phase constant quantifies the rate of this phase change per unit length. It describes how the phase of the waveform evolves as it moves through the medium.

Mathematical Representation

The phase constant is usually introduced in the mathematical representation of wave equations. For sinusoidal waves, the phase change of a wave as a function of distance (z) is given by:

φ(z) = β * z,

- φ(z) is the phase change at a distance z. 
- β is the phase constant.


  1. Wave Propagation: In various wave phenomena, such as electromagnetic waves, acoustic waves, and seismic waves, the phase constant helps describe the spatial variation of phase along the direction of propagation.
  2. Antenna Design: In radio frequency and microwave engineering, the phase constant is important for antenna design and analysis. It influences the radiation pattern and performance of antennas.
  3. Optical Fiber Communication: In optical fiber systems, the phase constant affects the phase shift of light as it travels through the fiber. This is crucial for maintaining signal integrity in long-distance communication.
  4. Interference and Superposition: When two or more waves interfere or superimpose, their phase difference determines whether they reinforce or cancel each other.
  5. Propagation in Transmission Lines: In telecommunications and electrical engineering, phase constant is used to analyze signal propagation along transmission lines.
  6. Sound Waves: In acoustics, the phase constant helps describe the phase relationship between different frequencies and the interference patterns of sound waves.

In summary, the phase constant is a fundamental parameter that characterizes the phase change experienced by a wave as it propagates through a medium. It is an essential concept for understanding wave behavior, interference, and the interaction of waves with different materials and structures.

Phase Velocity

Phase velocity is a fundamental concept in wave propagation and is particularly important in fields such as physics, engineering, and telecommunications. It describes the speed at which the phase of a wave travels through a medium, carrying information and forming patterns of constructive and destructive interference. Understanding phase velocity is essential for analyzing and designing various wave-based systems and applications.

Here's an expanded explanation of phase velocity:


Phase velocity is the speed at which the phase of a wave travels in a medium. It represents how quickly the peaks and troughs of a wave propagate through space as the wave moves forward. Phase velocity is commonly denoted as "v_phase" and is typically measured in meters per second (m/s) or other appropriate units depending on the context.

Physical Interpretation

When a wave travels through a medium, individual points along the wave oscillate with a certain frequency. The phase velocity characterizes how fast these points (or phase fronts) move forward in space, effectively determining how quickly the wave appears to propagate.

Mathematical Representation

The phase velocity (v_phase) of a wave is related to its frequency (f) and wavelength (λ) by the equation:

v_phase = f * λ,


  • v_phase is the phase velocity.
  • f is the frequency of the wave.
  • λ is the wavelength of the wave.


  1. Telecommunications: In the field of telecommunications, phase velocity plays a crucial role in determining the speed at which information travels through transmission lines, fiber-optic cables, and other communication mediums.
  2. Waveguides and Antennas: For electromagnetic waves, phase velocity influences the design and performance of waveguides and antennas, impacting factors like signal propagation and radiation patterns.
  3. Sound Waves: In acoustics, phase velocity affects the speed at which sound waves propagate through a medium, influencing phenomena like Doppler shift and the perception of sound.
  4. Optics: In optics, the phase velocity is important for understanding the behavior of light in different materials and optical systems, including lenses and prisms.
  5. Seismic Waves: Phase velocity is used to analyze seismic waves in geophysics, helping researchers understand the structure of the Earth's interior.
  6. Quantum Mechanics: In the realm of quantum mechanics, phase velocity is connected to wavefunctions and has implications for the behavior of particles described by quantum waves.

Important Note: While phase velocity is a key concept, it is important to distinguish it from group velocity, which represents the speed at which the envelope (or modulation) of a wave packet travels. In some cases, phase velocity can exceed the speed of light in a medium, but this does not violate the principle of relativity since it doesn't represent the speed of information transfer.

In summary, the phase velocity is a fundamental parameter that describes how the phase of a wave progresses through a medium. It has wide-ranging applications across various scientific and engineering disciplines and is essential for understanding and optimizing wave-based systems and technologies.


Velocity Factor

Velocity factor, also known as the propagation velocity factor or wave propagation factor, is a term used in the field of telecommunications and electronics, particularly in the context of transmission lines. It describes the speed at which signals or waves travel through a transmission line compared to their speed in free space. The velocity factor is an important parameter for accurately designing and analyzing transmission lines and ensuring efficient signal propagation.


Velocity factor (VF) is a dimensionless ratio that compares the speed of signal propagation in a transmission line to the speed of light in a vacuum. It represents the fraction of the speed of light that the signal travels through the transmission line. The velocity factor is denoted by the symbol "VF" and is often expressed as a decimal or a percentage.

Physical Interpretation

When an electrical signal travels through a transmission line, it encounters the effects of the line's geometry, dielectric material, and other factors. These effects can cause the signal to propagate at a speed that is less than the speed of light in free space. The velocity factor quantifies this reduction in speed and reflects the delay introduced by the transmission line.

Mathematical Representation

The velocity factor is calculated using the formula:

VF = c / v


VF is the velocity factor.
c is the speed of light in a vacuum (approximately 299,792,458 meters per second).
v is the speed of signal propagation in the transmission line.


  • Transmission Line Design: The velocity factor is essential for accurately designing transmission lines, such as coaxial cables and waveguides. It affects the electrical length of the line and the time it takes for signals to travel from one end to the other.
  • Signal Delay: The velocity factor determines the amount of time it takes for a signal to propagate through a transmission line. This is important for synchronization, timing, and signal integrity in communication systems.
  • Antenna Matching: The velocity factor influences the impedance matching between transmission lines and antennas. Accurate knowledge of the velocity factor helps optimize signal transfer and minimize signal reflection.
  • RF and Microwave Systems: In RF and microwave engineering, the velocity factor affects the phase and group velocities of signals, which are crucial for designing filters, resonators, and other components.
  • Delay Lines: The velocity factor is used in designing delay lines for applications such as phased array antennas and time-delay circuits.
  • Time-Domain Reflectometry (TDR): The velocity factor is considered when interpreting time-domain reflectometry measurements, which are used to locate faults and impedance mismatches in transmission lines.

In summary, the velocity factor is a key parameter in the design and analysis of transmission lines, influencing signal propagation, delay, and impedance matching. By understanding the velocity factor of a transmission line, engineers can ensure accurate signal transmission, minimize signal degradation, and optimize the performance of communication and electronic systems.


Wavelength is a fundamental concept in wave physics that describes the spatial period of a wave, indicating the distance between successive peaks (crests) or troughs of the wave. It plays a critical role in understanding various types of waves, including electromagnetic waves, sound waves, water waves, and seismic waves. Wavelength is a key parameter used to characterize waves and their behavior.


Wavelength (often denoted as λ, lambda) is the distance between two consecutive points that are in phase along a wave. In other words, it is the spatial period of the wave, representing the distance over which the wave pattern repeats itself. Wavelength is typically measured in units such as meters (m), centimeters (cm), nanometers (nm), or other appropriate length units, depending on the scale of the phenomenon.

Physical Interpretation

Imagine a wave moving through a medium. The wavelength is the distance between two adjacent points that are experiencing the same phase of oscillation. For example, in an electromagnetic wave, it is the distance between two consecutive peaks (or troughs). In a sound wave, it is the distance between two successive compressions or rarefactions of the medium.

Mathematical Representation:
The relationship between wavelength (λ), wave speed (v), and frequency (f) is given by the equation:

λ = v / f


λ is the wavelength.
v is the wave speed.
f is the frequency of the wave.

This equation demonstrates that wavelength is inversely proportional to frequency. Higher frequencies correspond to shorter wavelengths, and lower frequencies correspond to longer wavelengths, assuming the wave speed remains constant.


  • Electromagnetic Waves: In the context of electromagnetic radiation, such as light, radio waves, and microwaves, wavelength determines the color of light and influences how different frequencies of light interact with matter.
  • Sound Waves: In acoustics, wavelength affects the pitch of sound. Shorter wavelengths correspond to higher-pitched sounds, and longer wavelengths correspond to lower-pitched sounds.
  • Wave Behavior: Wavelength is essential for understanding wave interference, diffraction, and resonance. It determines how waves interact with obstacles and openings.
  • Spectroscopy: In science and astronomy, the measurement of the wavelengths of emitted or absorbed light helps identify elements and molecules and provides insights into the composition of celestial bodies.
  • Seismic Waves: Wavelength is used to analyze seismic waves, helping scientists study the Earth's interior and monitor earthquakes.

In summary, wavelength is a fundamental property of waves that describes the distance between successive identical points in a wave pattern. It is a critical parameter in understanding wave behavior, interactions, and applications across a wide range of scientific and engineering disciplines.

λ = 2π/β    meter

Quaternary Parameters (or Constants)

These include parameters such as shunt capacitance (C') and series conductance (G') that account for additional effects in high-frequency transmission line analysis.