Propagation constant

The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a circuit, or a field vector such as electric field strength or flux density. The propagation constant itself measures the change per unit length, but it is otherwise dimensionless. In the context of two-port networks and their cascades, propagation constant measures the change undergone by the source quantity as it propagates from one port to the next.

The propagation constant's value is expressed logarithmically, almost universally to the base e, rather than the more usual base 10 that is used in telecommunications in other situations. The quantity measured, such as voltage, is expressed as a sinusoidal phasor. The phase of the sinusoid varies with distance which results in the propagation constant being a complex number, the imaginary part being caused by the phase change.

Alternative names

The term "propagation constant" is somewhat of a misnomer as it usually varies strongly with ω. It is probably the most widely used term but there are a large variety of alternative names used by various authors for this quantity. These include transmission parameter, transmission function, propagation parameter, propagation coefficient and transmission constant. If the plural is used, it suggests that α and β are being referenced separately but collectively as in transmission parameters, propagation parameters, etc. In transmission line theory, α and β are counted among the "secondary coefficients", the term secondary being used to contrast to the primary line coefficients. The primary coefficients are the physical properties of the line, namely R,C,L and G, from which the secondary coefficients may be derived using the telegrapher's equation. Note that in the field of transmission lines, the term transmission coefficient has a different meaning despite the similarity of name: it is the companion of the reflection coefficient.

Definition

The propagation constant, symbol , for a given system is defined by the ratio of the complex amplitude at the source of the wave to the complex amplitude at some distance x, such that,

Since the propagation constant is a complex quantity we can write:

where

That β does indeed represent phase can be seen from Euler's formula:

which is a sinusoid which varies in phase as θ varies but does not vary in amplitude because

The reason for the use of base e is also now made clear. The imaginary phase constant, , can be added directly to the attenuation constant, α, to form a single complex number that can be handled in one mathematical operation provided they are to the same base. Angles measured in radians require base e, so the attenuation is likewise in base e.

The propagation constant for copper (or any other conductor) lines can be calculated from the primary line coefficients by means of the relationship

where

, the series impedance of the line per unit length and,
, the shunt admittance of the line per unit length.

Attenuation constant

In telecommunications, the term attenuation constant, also called attenuation parameter or attenuation coefficient, is the attenuation of an electromagnetic wave propagating through a medium per unit distance from the source. It is the real part of the propagation constant and is measured in nepers per metre. A neper is approximately 8.7 dB. Attenuation constant can be defined by the amplitude ratio

The propagation constant per unit length is defined as the natural logarithmic of ratio of the sending end current or voltage to the receiving end current or voltage.

Copper lines

The attenuation constant for copper lines (or ones made of any other conductor) can be calculated from the primary line coefficients as shown above. For a line meeting the distortionless condition, with a conductance G in the insulator, the attenuation constant is given by

however, a real line is unlikely to meet this condition without the addition of loading coils and, furthermore, there are some frequency dependent effects operating on the primary "constants" which cause a frequency dependence of the loss. There are two main components to these losses, the metal loss and the dielectric loss.

The loss of most transmission lines are dominated by the metal loss, which causes a frequency dependency due to finite conductivity of metals, and the skin effect inside a conductor. The skin effect causes R along the conductor to be approximately dependent on frequency according to

Losses in the dielectric depend on the loss tangent (tan δ) of the material divided by the wavelength of the signal. Thus they are directly proportional to the frequency.

Optical fibre

The attenuation constant for a particular propagation mode in an optical fiber is the real part of the axial propagation constant.

Phase constant

In electromagnetic theory, the phase constant, also called phase change constant, parameter or coefficient is the imaginary component of the propagation constant for a plane wave. It represents the change in phase per unit length along the path travelled by the wave at any instant and is equal to the real part of the angular wavenumber of the wave. It is represented by the symbol β and is measured in units of radians per unit length.

From the definition of (angular) wavenumber for TEM waves:

For a transmission line, the Heaviside condition of the telegrapher's equation tells us that the wavenumber must be proportional to frequency for the transmission of the wave to be undistorted in the time domain. This includes, but is not limited to, the ideal case of a lossless line. The reason for this condition can be seen by considering that a useful signal is composed of many different wavelengths in the frequency domain. For there to be no distortion of the waveform, all these waves must travel at the same velocity so that they arrive at the far end of the line at the same time as a group. Since wave phase velocity is given by

it is proved that β is required to be proportional to ω. In terms of primary coefficients of the line, this yields from the telegrapher's equation for a distortionless line the condition

However, practical lines can only be expected to approximately meet this condition over a limited frequency band.

In particular, the phase constant is not always equivalent to the wavenumber . Generally speaking, the following relation

is tenable to the TEM wave (transverse electromagnetic wave) which travels in free space or TEM-devices such as the coaxial cable and two parallel wires transmission lines. Nevertheless, it is invalid to the TE wave (transverse electric wave) and TM wave (transverse magnetic wave). For example,[1] in a hollow waveguide where the TEM wave cannot exist but TE and TM waves can propagate,

Here is the cutoff frequency.In a rectangular waveguide, the cutoff frequency is

where the integers are the mode numbers, and a and b the lengths of the sides of the rectangle. For TE modes, (but is not allowed), while for TM modes . The phase velocity equals

The phase constant is also an important concept in quantum mechanics because the momentum of a quantum is directly proportional to it,[2] [3] i.e.

where ħ is called the reduced Planck constant (pronounced "h-bar"). It is equal to the Planck constant divided by 2π.

Filters and two-port networks

The term propagation constant or propagation function is applied to filters and other two-port networks used for signal processing. In these cases, however, the attenuation and phase coefficients are expressed in terms of nepers and radians per network section rather than per unit length. Some authors[4] make a distinction between per unit length measures (for which "constant" is used) and per section measures (for which "function" is used).

The propagation constant is a useful concept in filter design which invariably uses a cascaded section topology. In a cascaded topology, the propagation constant, attenuation constant and phase constant of individual sections may be simply added to find the total propagation constant etc.

Cascaded networks

Adding propagation constant of filters
Three networks with arbitrary propagation constants and impedances connected in cascade. The Zi terms represent image impedance and it is assumed that connections are between matching image impedances.

The ratio of output to input voltage for each network is given by[5]

The terms are impedance scaling terms[6] and their use is explained in the image impedance article.

The overall voltage ratio is given by

Thus for n cascaded sections all having matching impedances facing each other, the overall propagation constant is given by

See also

The concept of penetration depth is one of many ways to describe the absorption of electromagnetic waves. For the others, and their interrelationships, see the article: Mathematical descriptions of opacity.

Notes

  1. ^ Pozar, David (2012). Microwave Engineering (4th ed.). John Wiley &Sons. pp. 62–164. ISBN 978-0-470-63155-3.
  2. ^ Wang,Z.Y. (2016). "Generalized momentum equation of quantum mechanics". Optical and Quantum Electronics. 48 (2): 1–9. doi:10.1007/s11082-015-0261-8.
  3. ^ Tremblay,R., Doyon,N., Beaudoin-Bertrand,J. (2016). "TE-TM Electromagnetic modes and states in quantum physics". arXiv:1611.01472. Bibcode:2016arXiv161101472T.CS1 maint: Multiple names: authors list (link)
  4. ^ Matthaei et al, p49
  5. ^ Matthaei et al pp51-52
  6. ^ Matthaei et al pp37-38

References

External links

Absorption (electromagnetic radiation)

In physics, absorption of electromagnetic radiation is the way in which the energy of a photon is taken up by matter, typically the electrons of an atom. Thus, the electromagnetic energy is transformed into internal energy of the absorber, for example thermal energy. The reduction in intensity of a light wave propagating through a medium by absorption of a part of its photons is often called attenuation. Usually, the absorption of waves does not depend on their intensity (linear absorption), although in certain conditions (usually, in optics), the medium changes its transparency dependently on the intensity of waves going through, and saturable absorption (or nonlinear absorption) occurs.

Attenuation (disambiguation)

Attenuation is the gradual loss in intensity of any kind of flux through a medium, including:

Acoustic attenuation, the loss of sound energy in a viscous medium

Anelastic attenuation factor, a way to describe attenuation of seismic energy in the EarthAttenuation (or verb attenuate) may also refer to:

Attenuation (botany)

Attenuation (brewing), the percent of sugar converted to alcohol and carbon dioxide by the yeast in brewing

Attenuation coefficient, a basic quantity used in calculations of the penetration of materials by quantum particles or other energy beams

Mass attenuation coefficient, a measurement of how strongly a chemical species or substance absorbs or scatters light at a given wavelength, per unit mass

Regression attenuation or Regression dilution, a cause of statistical bias

The process of producing an attenuated vaccine by reducing the virulence of a pathogen

Attenuation constant, the real part of the propagation constant

Attenuator (genetics), form of regulation in prokaryotic cells.

Bloch wave – MoM method

Bloch wave – MoM is a first principles technique for determining the photonic band structure of triply-periodic electromagnetic media such as photonic crystals. It is based on the 3-dimensional spectral domain method (Kastner [1987]), specialized to triply-periodic media. This technique uses the method of moments (MoM) in combination with a Bloch wave expansion of the electromagnetic field to yield a matrix eigenvalue equation for the propagation bands. The eigenvalue is the frequency (for a given propagation constant) and the eigenvector is the set of current amplitudes on the surface of the scatterers. Bloch wave - MoM is similar in principle to the Plane wave expansion method, but since it additionally employs the method of moments to produce a surface integral equation, it is significantly more efficient both in terms of the number of unknowns and the number of plane waves needed for good convergence.

Bloch wave - MoM is the extension to 3 dimensions of the spectral domain MoM method commonly used for analyzing 2D periodic structures such as frequency selective surfaces (FSS). In both cases, the field is expanded as a set of eigenfunction modes (either a Bloch wave in 3D or a discrete plane wave - aka Floquet mode - spectrum in 2D), and an integral equation is enforced on the surface of the scatterers in each unit cell. In the FSS case, the unit cell is 2-dimensional and in the photonic crystal case, the unit cell is 3-dimensional.

Evanescent field

In electromagnetics, an evanescent field, or evanescent wave, is an oscillating electric and/or magnetic field that does not propagate as an electromagnetic wave but whose energy is spatially concentrated in the vicinity of the source (oscillating charges and currents). Even when there in fact is an electromagnetic wave produced (e.g., by a transmitting antenna), one can still identify as an evanescent field the component of the electric or magnetic field that cannot be attributed to the propagating wave observed at a distance of many wavelengths (such as the far field of a transmitting antenna).

A hallmark of an evanescent field is that there is no net energy flow in that region. Since the net flow of electromagnetic energy is given by the average Poynting vector, this means that the Poynting vector in these regions, as averaged over a complete oscillation cycle, is zero.

Loading coil

A loading coil or load coil is an inductor that is inserted into an electronic circuit to increase its inductance. The term originated in the 19th century for inductors used to prevent signal distortion in long-distance telegraph transmission cables. The term is also used for inductors in radio antennas, or between the antenna and its feedline, to make an electrically short antenna resonant at its operating frequency.

The concept of loading coils was discovered by Oliver Heaviside in studying the problem of slow signalling speed of the first transatlantic telegraph cable in the 1860s. He concluded additional inductance was required to prevent amplitude and time delay distortion of the transmitted signal. The mathematical condition for distortion-free transmission is known as the Heaviside condition. Previous telegraph lines were overland or shorter and hence had less delay, and the need for extra inductance was not as great. Submarine communications cables are particularly subject to the problem, but early 20th century installations using balanced pairs were often continuously loaded with iron wire or tape rather than discretely with loading coils, which avoided the sealing problem.

Loading coils are historically also known as Pupin coils after Mihajlo Pupin, especially when used for the Heaviside condition and the process of inserting them is sometimes called pupinization.

Marcatili's method

Marcatili’s method is an approximate analytical method that describes how light propagates through rectangular dielectric optical waveguides . It was published by Enrique Marcatili in 1969.Optical dielectric waveguides guide electromagnetic waves in the optical spectrum (light). This type of waveguide consists of dielectric materials (e.g., glass, silicon, indium phosphide, etc). The core of the waveguide has a higher index of refraction than its surrounding and the light is guided due to total internal reflection. In a ray description, the light zig-zags between the walls.

The geometry of the waveguide dictates the light to propagate with specific velocities and specific distributions of the electric and magnetic fields, known as modes. For rectangular waveguides, these modes cannot be computed analytically. This can be done either using a numerical mode solver, or using an approximate method such as Marcatili’s method.

Mathematical descriptions of opacity

When an electromagnetic wave travels through a medium in which it gets attenuated (this is called an "opaque" or "attenuating" medium), it undergoes exponential decay as described by the Beer–Lambert law. However, there are many possible ways to characterize the wave and how quickly it is attenuated. This article describes the mathematical relationships among:

attenuation coefficient;

penetration depth and skin depth;

complex angular wavenumber and propagation constant;

complex refractive index;

complex electric permittivity;

AC conductivity (susceptance).Note that in many of these cases there are multiple, conflicting definitions and conventions in common use. This article is not necessarily comprehensive or universal.

Oracle Developer Studio

Oracle Developer Studio, formerly named Oracle Solaris Studio, Sun Studio, Sun WorkShop, Forte Developer, and SunPro Compilers, is Oracle Corporation's flagship software development product for the Solaris and Linux operating systems. It includes optimizing C, C++, and Fortran compilers, libraries, and performance analysis and debugging tools, for Solaris on SPARC and x86 platforms, and Linux on x86/x64 platforms, including multi-core systems.

Oracle Developer Studio is downloadable and usable at no charge; however, there are many security and functionality patch updates which are only available with a support contract from Oracle.Version 12.4 adds support for the C++11 language standard. All C++11 features are supported except for concurrency and atomic operations, and user-defined literals. Version 12.6 supports the C++14 language standard.

Primary line constants

The primary line constants are parameters that describe the characteristics of conductive transmission lines, such as pairs of copper wires, in terms of the physical electrical properties of the line. The primary line constants are only relevant to transmission lines and are to be contrasted with the secondary line constants, which can be derived from them, and are more generally applicable. The secondary line constants can be used, for instance, to compare the characteristics of a waveguide to a copper line, whereas the primary constants have no meaning for a waveguide.

The constants are conductor resistance and inductance, and insulator capacitance and conductance, which are by convention given the symbols R, L, C, and G respectively. The constants are enumerated in terms of per unit length. The circuit representation of these elements requires a distributed element model and consequently calculus must be used to analyse the circuit. The analysis yields a system of two first order, simultaneous linear partial differential equations which may be combined to derive the secondary constants of characteristic impedance and propagation constant.

A number of special cases have particularly simple solutions and important practical applications. Low loss cable requires only L and C to be included in the analysis, useful for short lengths of cable. Low frequency applications, such as twisted pair telephone lines, are dominated by R and C only. High frequency applications, such as RF co-axial cable, are dominated by L and C. Lines loaded to prevent distortion need all four elements in the analysis, but have a simple, elegant solution.

Prism coupler

A prism coupler is a prism designed to couple a substantial fraction of the power contained in a beam of light (e.g., a laser beam) into a thin film to be used as a waveguide without the need for precision polishing of the edge of the film, without the need for sub-micrometer alignment precision of the beam and the edge of the film, and without the need for matching the numerical aperture of the beam to the film. Using a prism coupler, a beam coupled into a thin film can have a diameter hundreds of times the thickness of the film. Invention of the coupler contributed to the initiation of a field of study known as integrated optics.

Quarter-wave impedance transformer

A quarter-wave impedance transformer, often written as λ/4 impedance transformer, is a transmission line or waveguide used in electrical engineering of length one-quarter wavelength (λ), terminated with some known impedance. It presents at its input the dual of the impedance with which it is terminated.

It is a similar concept to a stub; but, whereas a stub is terminated in a short (or open) circuit and the length is chosen so as to produce the required impedance, the λ/4 transformer is the other way around; it is a pre-determined length and the termination is designed to produce the required impedance.

The relationship between the characteristic impedance, Z0, input impedance, Zin and load impedance, ZL is:

Radiation mode

For an optical fiber or waveguide, a radiation mode or unbound mode is a mode which is not confined by the fiber core. Such a mode has fields that are transversely oscillatory everywhere external to the waveguide, and exists even at the limit of zero wavelength.

Specifically, a radiation mode is one for which

where β is the imaginary part of the axial propagation constant, integer l is the azimuthal index of the mode, n(r) is the refractive index at radius r, a is the core radius, and k is the free-space wave number, k = 2π/λ, where λ is the wavelength. Radiation modes correspond to refracted rays in the terminology of geometric optics.

Telegrapher's equations

The telegrapher's equations (or just telegraph equations) are a pair of coupled, linear differential equations that describe the voltage and current on an electrical transmission line with distance and time. The equations come from Oliver Heaviside who in the 1880s developed the transmission line model, which is described in this article. The model demonstrates that the electromagnetic waves can be reflected on the wire, and that wave patterns can appear along the line. The theory applies to transmission lines of all frequencies including high-frequency transmission lines (such as telegraph wires and radio frequency conductors), audio frequency (such as telephone lines), low frequency (such as power lines) and direct current.

Transmission function

Transmission function can refer to

transfer function

propagation constant

Transmission line

In communications and electronic engineering, a transmission line is a specialized cable or other structure designed to conduct alternating current of radio frequency, that is, currents with a frequency high enough that their wave nature must be taken into account. Transmission lines are used for purposes such as connecting radio transmitters and receivers with their antennas (they are then called feed lines or feeders), distributing cable television signals, trunklines routing calls between telephone switching centres, computer network connections and high speed computer data buses.

This article covers two-conductor transmission line such as parallel line (ladder line), coaxial cable, stripline, and microstrip. Some sources also refer to waveguide, dielectric waveguide, and even optical fibre as transmission line, however these lines require different analytical techniques and so are not covered by this article; see Waveguide (electromagnetism).

Transverse mode

A transverse mode of electromagnetic radiation is a particular electromagnetic field pattern of radiation measured in a plane perpendicular (i.e., transverse) to the propagation direction of the beam. Transverse modes occur in radio waves and microwaves confined to a waveguide, and also in light waves in an optical fiber and in a laser's optical resonator.Transverse modes occur because of boundary conditions imposed on the wave by the waveguide. For example, a radio wave in a hollow metal waveguide must have zero tangential electric field amplitude at the walls of the waveguide, so the transverse pattern of the electric field of waves is restricted to those that fit between the walls. For this reason, the modes supported by a waveguide are quantized. The allowed modes can be found by solving Maxwell's equations for the boundary conditions of a given waveguide.

Wave propagation

Wave propagation is any of the ways in which waves travel.

With respect to the direction of the oscillation relative to the propagation direction, we can distinguish between longitudinal wave and transverse waves.

For electromagnetic waves, propagation may occur in a vacuum as well as in a material medium. Other wave types cannot propagate through a vacuum and need a transmission medium to exist.

Waveguide

A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting expansion to one dimension or two. There is a similar effect in water waves constrained within a canal, or guns that have barrels which restrict hot gas expansion to maximize energy transfer to their bullets. Without the physical constraint of a waveguide, wave amplitudes decrease according to the inverse square law as they expand into three dimensional space.

There are different types of waveguides for each type of wave. The original and most common meaning is a hollow conductive metal pipe used to carry high frequency radio waves, particularly microwaves.

The geometry of a waveguide reflects its function. Slab waveguides confine energy in one dimension, fiber or channel waveguides in two dimensions. The frequency of the transmitted wave also dictates the shape of a waveguide: an optical fiber guiding high-frequency light will not guide microwaves of a much lower frequency.

Some naturally occurring structures can also act as waveguides. The SOFAR channel layer in the ocean can guide the sound of whale song across enormous distances.

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