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The propagation constant of a sinusoidal
electromagnetic wave In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) ...
is a measure of the change undergone by the
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
and phase of the wave as it propagates in a given direction. The quantity being measured can be the
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to ...
, the current in a circuit, or a field vector such as electric field strength or
flux density Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ...
. The propagation constant itself measures the change
per unit length Reciprocal length or inverse length is a quantity or measurement used in several branches of science and mathematics. As the reciprocal of length, common units used for this measurement include the reciprocal metre or inverse metre (symbol: m&minus ...
, but it is otherwise dimensionless. In the context of
two-port networks A two-port network (a kind of four-terminal network or quadripole) is an electrical network ( circuit) or device with two ''pairs'' of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them sati ...
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
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
ically, almost universally to the base '' e'', rather than the more usual base 10 that is used in
telecommunication Telecommunication is the transmission of information by various types of technologies over wire, radio, optical, or other electromagnetic systems. It has its origin in the desire of humans for communication over a distance greater than that ...
s in other situations. The quantity measured, such as voltage, is expressed as a sinusoidal
phasor In physics and engineering, a phasor (a portmanteau of phase vector) is a complex number representing a sinusoidal function whose amplitude (''A''), angular frequency (''ω''), and initial phase (''θ'') are time-invariant. It is related to ...
. The phase of the sinusoid varies with distance which results in the propagation constant being a
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
, 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 In physics and electrical engineering the reflection coefficient is a parameter that describes how much of a wave is reflected by an impedance discontinuity in the transmission medium. It is equal to the ratio of the amplitude of the reflected ...
.


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, :\frac=e^ Since the propagation constant is a complex quantity we can write: :\gamma = \alpha +i \beta \, 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 Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
: :e^=\cos+i\sin\,\! which is a sinusoid which varies in phase as ''θ'' varies but does not vary in amplitude because :\left, e^\=\sqrt=1 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 conducting lines can be calculated from the primary line coefficients by means of the relationship :\gamma=\sqrt where :Z=R+i\omega L\,\!, the series impedance of the line per unit length and, :Y=G+i\omega C\,\!, the shunt
admittance In electrical engineering, admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the reciprocal of impedance, analogous to how conductance & resistance are defined. The SI unit of admittanc ...
of the line per unit length.


Plane wave

The propagation factor of a plane wave traveling in a linear media in the x direction is given by P = e^ where * \gamma = \alpha + i\beta = \sqrt\; * x = distance traveled in the x direction * \alpha = attenuation constant in the units of
neper The neper (symbol: Np) is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. The unit's name is derived from the name of John Napier, the inventor of logarithms. A ...
s/meter * \beta = phase constant in the units of
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
s/meter * \omega= frequency in radians/second * \sigma = conductivity of the media * \varepsilon = \varepsilon' - i\varepsilon'' \; = complex permitivity of the media * \mu = \mu' - i\mu'' \; = complex permeability of the media * i=\sqrt The sign convention is chosen for consistency with propagation in lossy media. If the attenuation constant is positive, then the wave amplitude decreases as the wave propagates in the ''x'' direction.
Wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
,
phase velocity The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...
, and skin depth have simple relationships to the components of the propagation constant: \lambda = \frac \beta \qquad v_p = \frac \omega \beta \qquad \delta = \frac 1 \alpha


Attenuation constant

In
telecommunication Telecommunication is the transmission of information by various types of technologies over wire, radio, optical, or other electromagnetic systems. It has its origin in the desire of humans for communication over a distance greater than that ...
s, the term attenuation constant, also called attenuation parameter or attenuation coefficient, is the attenuation of an electromagnetic wave propagating through a
medium Medium may refer to: Science and technology Aviation * Medium bomber, a class of war plane * Tecma Medium, a French hang glider design Communication * Media (communication), tools used to store and deliver information or data * Medium ...
per unit distance from the source. It is the real part of the propagation constant and is measured in
neper The neper (symbol: Np) is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. The unit's name is derived from the name of John Napier, the inventor of logarithms. A ...
s per metre. A neper is approximately 8.7  dB. Attenuation constant can be defined by the amplitude ratio :\left, \frac\=e^ The propagation constant per unit length is defined as the natural logarithm of the ratio of the sending end current or voltage to the receiving end current or voltage.


Conductive lines

The attenuation constant for conductive lines 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 :\alpha=\sqrt\,\! however, a real line is unlikely to meet this condition without the addition of
loading coils 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 c ...
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 Skin effect is the tendency of an alternating electric current (AC) to become distributed within a conductor such that the current density is largest near the surface of the conductor and decreases exponentially with greater depths in the co ...
inside a conductor. The skin effect causes R along the conductor to be approximately dependent on frequency according to :R \propto \sqrt 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. :\alpha_d=


Optical fibre

The attenuation constant for a particular propagation mode in an
optical fiber An optical fiber, or optical fibre in Commonwealth English, is a flexible, transparent fiber made by drawing glass ( silica) or plastic to a diameter slightly thicker than that of a human hair Hair is a protein filament that grows ...
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 in lossless media: :k = \frac = \beta For a
transmission line In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmi ...
, the
Heaviside condition 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 u ...
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 In electronics, acoustics, and related fields, the waveform of a signal is the shape of its graph as a function of time, independent of its time and magnitude scales and of any displacement in time.David Crecraft, David Gorham, ''Electro ...
, 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 The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...
is given by :v_p = \frac = \frac = \frac, 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 :\beta = \omega \sqrt, where ''L'' and ''C'' are, respectively, the inductance and capacitance per unit length of the line. However, practical lines can only be expected to approximately meet this condition over a limited frequency band. In particular, the phase constant \beta is not always equivalent to the
wavenumber In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
k. Generally speaking, the following relation : \beta = k is tenable to the TEM wave (transverse electromagnetic wave) which travels in free space or TEM-devices such as the
coaxial cable Coaxial cable, or coax (pronounced ) is a type of electrical cable consisting of an inner conductor surrounded by a concentric conducting shield, with the two separated by a dielectric ( insulating material); many coaxial cables also have a p ...
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, in a hollow waveguide where the TEM wave cannot exist but TE and TM waves can propagate, :k=\frac :\beta=k\sqrt Here \omega_ is the
cutoff frequency In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced ( attenuated or reflected) rather tha ...
. In a rectangular waveguide, the cutoff frequency is : \omega_ = c \sqrt, where m,n \ge 0 are the mode numbers for the rectangle's sides of length a and b respectively. For TE modes, m,n \ge 0 (but m = n = 0 is not allowed), while for TM modes m,n \ge 1 . The phase velocity equals :v_p=\frac=\frac>c The phase constant is also an important concept in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
because the
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
p of a
quantum In physics, a quantum (plural quanta) is the minimum amount of any physical entity ( physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizat ...
is directly proportional to it, i.e. : p = \hbar \beta where is called the
reduced Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...
(pronounced "h-bar"). It is equal to the
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
divided by .


Filters and two-port networks

The term propagation constant or propagation function is applied to
filters Filter, filtering or filters may refer to: Science and technology Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component that ...
and other two-port networks used for
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
. 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 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 mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
. 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

The ratio of output to input voltage for each network is given by :\frac=\sqrte^ :\frac=\sqrte^ :\frac=\sqrte^ The terms \sqrt are impedance scaling termsMatthaei et al pp37-38 and their use is explained in the image impedance article. The overall voltage ratio is given by :\frac=\frac\cdot\frac\cdot\frac=\sqrte^ Thus for ''n'' cascaded sections all having matching impedances facing each other, the overall propagation constant is given by :\gamma_\mathrm=\gamma_1 + \gamma_2 + \gamma_3 + \cdots + \gamma_n


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. *
Propagation speed The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...


Notes


References

* . * Matthaei, Young, Jones ''Microwave Filters, Impedance-Matching Networks, and Coupling Structures'' McGraw-Hill 1964.


External links

* * * {{Cite journal , last =Janezic , first = Michael D. , author2=Jeffrey A. Jargon , title = Complex Permittivity determination from Propagation Constant measurements , journal = IEEE Microwave and Guided Wave Letters , volume = 9 , issue = 2 , pages = 76–78 , date =February 1999 , url =http://www.eeel.nist.gov/dylan_papers/MGWL99.pdf , doi = 10.1109/75.755052 , access-date =2 February 2011 Free PDF download is available. There is an updated version dated August 6, 2002. Filter theory Physical quantities Telecommunication theory Electromagnetism Electromagnetic radiation Analog circuits Image impedance filters