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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.

- Alternative names
- Definition
- Attenuation constant
- Copper lines
- Optical fibre
- Phase constant
- Filters and two-port networks
- Cascaded networks
- See also
- Notes
- References
- External links

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.

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.

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

*α*, the real part, is called the attenuation constant*β*, the imaginary part, is called the phase constant

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, *iβ*, 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.

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.

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.

The attenuation constant for a particular propagation mode in an optical fiber is the real part of the axial propagation 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 2*π*.

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.

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

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.

- ↑ Pozar, David (2012).
*Microwave Engineering*(4th ed.). John Wiley &Sons. pp. 62–164. ISBN 978-0-470-63155-3. - ↑ 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. - ↑ Tremblay,R., Doyon,N., Beaudoin-Bertrand,J. (2016). "TE-TM Electromagnetic modes and states in quantum physics". arXiv: 1611.01472 [quant-ph].CS1 maint: multiple names: authors list (link)
- ↑ Matthaei et al, p49
- ↑ Matthaei et al pp51-52
- ↑ Matthaei et al pp37-38

In radio-frequency 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, distributing cable television signals, trunklines routing calls between telephone switching centres, computer network connections and high speed computer data buses.

**Synchrotron radiation** is the electromagnetic radiation emitted when charged particles are accelerated radially, e.g., when they are subject to an acceleration perpendicular to their velocity. It is produced, for example, in synchrotrons using bending magnets, undulators and/or wigglers. If the particle is non-relativistic, then the emission is called cyclotron emission. If, on the other hand, the particles are relativistic, sometimes referred to as ultrarelativistic, the emission is called synchrotron emission. Synchrotron radiation may be achieved artificially in synchrotrons or storage rings, or naturally by fast electrons moving through magnetic fields. The radiation produced in this way has a characteristic polarization and the frequencies generated can range over the entire electromagnetic spectrum which is also called continuum radiation.

The **Smith chart**, invented by Phillip H. Smith (1905–1987), is a graphical aid or nomogram designed for electrical and electronics engineers specializing in radio frequency (RF) engineering to assist in solving problems with transmission lines and matching circuits. The Smith chart can be used to simultaneously display multiple parameters including impedances, admittances, reflection coefficients, scattering parameters, noise figure circles, constant gain contours and regions for unconditional stability, including mechanical vibrations analysis. The Smith chart is most frequently used at or within the unity radius region. However, the remainder is still mathematically relevant, being used, for example, in oscillator design and stability analysis.

In physics, a **wave vector** is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important. Its magnitude is either the wavenumber or angular wavenumber of the wave, and its direction is ordinarily the direction of wave propagation.

In mathematics, the **Hamilton–Jacobi equation** (**HJE**) is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations, and is a special case of the Hamilton–Jacobi–Bellman equation. It is named for William Rowan Hamilton and Carl Gustav Jacob Jacobi.

In optics, an **ultrashort pulse** of light is an electromagnetic pulse whose time duration is of the order of a picosecond or less. Such pulses have a broadband optical spectrum, and can be created by mode-locked oscillators. They are commonly referred to as ultrafast events. Amplification of ultrashort pulses almost always requires the technique of chirped pulse amplification, in order to avoid damage to the gain medium of the amplifier.

The **rigid rotor** is a mechanical model of rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space requires three angles, known as Euler angles. A special rigid rotor is the *linear rotor* requiring only two angles to describe, for example of a diatomic molecule. More general molecules are 3-dimensional, such as water, ammonia, or methane.

The **Heaviside condition**, named for Oliver Heaviside (1850–1925), is the condition an electrical transmission line must meet in order for there to be no distortion of a transmitted signal. Also known as the **distortionless condition**, it can be used to improve the performance of a transmission line by adding loading to the cable.

The **Havriliak–Negami relaxation** is an empirical modification of the Debye relaxation model in electromagnetism. Unlike the Debye model, the Havriliak–Negami relaxation accounts for the asymmetry and broadness of the dielectric dispersion curve. The model was first used to describe the dielectric relaxation of some polymers, by adding two exponential parameters to the Debye equation:

The **Duffing equation**, named after Georg Duffing (1861–1944), is a non-linear second-order differential equation used to model certain damped and driven oscillators. The equation is given by

A **theoretical motivation for general relativity**, including the motivation for the geodesic equation and the Einstein field equation, can be obtained from special relativity by examining the dynamics of particles in circular orbits about the earth. A key advantage in examining circular orbits is that it is possible to know the solution of the Einstein Field Equation *a priori*. This provides a means to inform and verify the formalism.

**Image impedance** is a concept used in electronic network design and analysis and most especially in filter design. The term *image impedance* applies to the impedance seen looking into a port of a network. Usually a two-port network is implied but the concept can be extended to networks with more than two ports. The definition of image impedance for a two-port network is the impedance, *Z*_{i 1}, seen looking into port 1 when port 2 is terminated with the image impedance, *Z*_{i 2}, for port 2. In general, the image impedances of ports 1 and 2 will not be equal unless the network is symmetrical with respect to the ports.

**Constant k filters**, also **k-type filters**, are a type of electronic filter designed using the image method. They are the original and simplest filters produced by this methodology and consist of a ladder network of identical sections of passive components. Historically, they are the first filters that could approach the ideal filter frequency response to within any prescribed limit with the addition of a sufficient number of sections. However, they are rarely considered for a modern design, the principles behind them having been superseded by other methodologies which are more accurate in their prediction of filter response.

**m-derived filters** or **m-type filters** are a type of electronic filter designed using the image method. They were invented by Otto Zobel in the early 1920s. This filter type was originally intended for use with telephone multiplexing and was an improvement on the existing constant k type filter. The main problem being addressed was the need to achieve a better match of the filter into the terminating impedances. In general, all filters designed by the image method fail to give an exact match, but the m-type filter is a big improvement with suitable choice of the parameter m. The m-type filter section has a further advantage in that there is a rapid transition from the cut-off frequency of the pass band to a pole of attenuation just inside the stop band. Despite these advantages, there is a drawback with m-type filters; at frequencies past the pole of attenuation, the response starts to rise again, and m-types have poor stop band rejection. For this reason, filters designed using m-type sections are often designed as composite filters with a mixture of k-type and m-type sections and different values of m at different points to get the optimum performance from both types.

Filters designed using the image impedance methodology suffer from a peculiar flaw in the theory. The predicted characteristics of the filter are calculated assuming that the filter is terminated with its own image impedances at each end. This will not usually be the case; the filter will be terminated with fixed resistances. This causes the filter response to deviate from the theoretical. This article explains how the effects of **image filter end terminations** can be taken into account.

When an electromagnetic wave travels through a medium in which it gets attenuated, 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:

In the fields of nonlinear optics and fluid dynamics, **modulational instability** or **sideband instability** is a phenomenon whereby deviations from a periodic waveform are reinforced by nonlinearity, leading to the generation of spectral-sidebands and the eventual breakup of the waveform into a train of pulses.

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.

**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, 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 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.

A **frequency-selective surface** (**FSS**) is any thin, repetitive surface designed to reflect, transmit or absorb electromagnetic fields based on the frequency of the field. In this sense, an FSS is a type of optical filter or metal-mesh optical filters in which the filtering is accomplished by virtue of the regular, periodic pattern on the surface of the FSS. Though not explicitly mentioned in the name, FSS's also have properties which vary with incidence angle and polarization as well - these are unavoidable consequences of the way in which FSS's are constructed. Frequency-selective surfaces have been most commonly used in the radio frequency region of the electromagnetic spectrum and find use in applications as diverse as the aforementioned microwave oven, antenna radomes and modern metamaterials. Sometimes frequency selective surfaces are referred to simply as periodic surfaces and are a 2-dimensional analog of the new periodic volumes known as photonic crystals.

This article incorporates public domain material from the General Services Administration document "Federal Standard 1037C".. - Matthaei, Young, Jones
*Microwave Filters, Impedance-Matching Networks, and Coupling Structures*McGraw-Hill 1964.

- "Propagation constant" (Online). Microwave Encyclopedia. 2011. Retrieved February 2, 2011.
- Paschotta, Dr. Rüdiger (2011). "Propagation Constant" (Online). Encyclopedia of Laser Physics and Technology. Retrieved 2 February 2011.
- Janezic, Michael D.; Jeffrey A. Jargon (February 1999). "Complex Permittivity determination from Propagation Constant measurements" (PDF).
*IEEE Microwave and Guided Wave Letters*.**9**(2): 76–78. doi:10.1109/75.755052 . Retrieved 2 February 2011. Free PDF download is available. There is an updated version dated August 6, 2002.

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