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Caesium standard

The caesium standard is a primary frequency standard in which the photon absorption by transitions between the two hyperfine ground states of caesium-133 atoms is used to control the output frequency. The first caesium clock was built by Louis Essen in 1955 at the National Physical Laboratory in the UK and promoted worldwide by Gernot M. R. Winkler of the United States Naval Observatory.

Caesium atomic clocks are one of the most accurate time and frequency standards, and serve as the primary standard for the definition of the second in the International System of Units (SI), the modern metric system. By definition, radiation produced by the transition between the two hyperfine ground states of caesium-133 (in the absence of external influences such as the Earth's magnetic field) has a frequency, $Δν_{Cs}$, of exactly $9,192,631,770 Hz$. That value was chosen so that the caesium second equaled, to the limit of measuring ability in 1960 when it was adopted, the existing standard ephemeris second based on the Earth's orbit around the Sun. Because no other measurement involving time had been as precise, the effect of the change was less than the experimental uncertainty of all existing measurements.

While the second is the only base unit to be explicitly defined in terms of the caesium standard, the majority of SI units have definitions that mention either the second, or other units defined using the second. Consequently, every base unit except the mole and every named derived unit except the coulomb, ohm, siemens, weber, gray, sievert, radian, and steradian have values that are implicitly defined by the properties of the caesium-133 hyperfine transition radiation. And of these, all but the mole, the coulomb, and the dimensionless radian and steradian are implicitly defined by the general properties of electromagnetic radiation.

Technical details
The official definition of the second was first given by the BIPM at the 13th General Conference on Weights and Measures in 1967 as: "The second is the duration of $9,192,631,770$ periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom." At its 1997 meeting the BIPM added to the previous definition the following specification: "This definition refers to a caesium atom at rest at a temperature of 0 K."

The BIPM restated this definition in its 26th conference (2018), "The second is defined by taking the fixed numerical value of the caesium frequency ∆Cs, the unperturbed ground-state hyperfine transition frequency of the caesium 133 atom, to be 9 192 631 770 when expressed in the unit Hz, which is equal to s–1."

The meaning of the preceding definition is as follows. The caesium atom has a ground state electron state with configuration [Xe] 6s1 and, consequently, atomic term symbol 2S1/2. This means that there is one unpaired electron and the total electron spin of the atom is 1/2. Moreover, the nucleus of caesium-133 has a nuclear spin equal to 7/2. The simultaneous presence of electron spin and nuclear spin leads, by a mechanism called hyperfine interaction, to a (small) splitting of all energy levels into two sub-levels. One of the sub-levels corresponds to the electron and nuclear spin being parallel (i.e., pointing in the same direction), leading to a total spin F equal to F = 7/2 + 1/2 = 4; the other sub-level corresponds to anti-parallel electron and nuclear spin (i.e., pointing in opposite directions), leading to a total spin F = 7/2 − 1/2 = 3. In the caesium atom it so happens that the sub-level lowest in energy is the one with F = 3, while the F = 4 sub-level lies energetically slightly above. When the atom is irradiated with electromagnetic radiation having an energy corresponding to the energetic difference between the two sub-levels the radiation is absorbed and the atom is excited, going from the F = 3 sub-level to the F = 4 one. After a small fraction of a second the atom will re-emit the radiation and return to its F = 3 ground state. From the definition of the second it follows that the radiation in question has a frequency of exactly $9.193 GHz$, corresponding to a wavelength of about 3.26 cm and therefore belonging to the microwave range.

This particular caesium resonance was agreed upon under la Convention du Mètre and remains to the present time as the official definition of the second for the world community.

Note that a common confusion involves the conversion from angular frequency ($$\omega$$) to frequency ($$f$$), or vice versa. Angular frequencies are conventionally given as s–1 in scientific literature, but here the units implicitly mean radians per second. In contrast, the unit Hz should be interpreted as cycles per second. The conversion formula is $$\omega = 2\pi f$$, which implies that 1 Hz corresponds to an angular frequency of approximately 6.28 radians per second (or 6.28 s–1 where radians is omitted for brevity by convention).

Parameters and significance in the second and other SI units
Suppose the caesium standard has the parameters:
 * Velocity: c

 * Energy/frequency: h

 * Time period: $Δt_{Cs}$

 * Frequency: $Δν_{Cs}$

 * Wavelength: $Δλ_{Cs}$

 * Photon energy: $ΔE_{Cs}$

 * Photon mass equivalent: $ΔM_{Cs}$

Time and frequency
The first set of units defined using the caesium standard were those relating to time, with the second being defined in 1967 as "the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom" meaning that:

 * 1 second, s, = 9,192,631,770 $Δt_{Cs}$

 * 1 hertz, Hz, = 1/s = $$Δν_{Cs}$⁄9,192,631,770$

 * 1 becquerel, Bq, = 1 nuclear decay/s = $1⁄9,192,631,770$ nuclear decays/$Δt_{Cs}$

This also linked the definitions of the derived units relating to force and energy (see below) and of the ampere, whose definition at the time made reference to the newton, to the caesium standard. Before 1967 the SI units of time and frequency were defined using the tropical year and before 1960 by the length of the mean solar day

Length
In 1983, the meter was, indirectly, defined in terms of the caesium standard with the formal definition "The metre is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second. This implied:

 * 1 metre, m, = $c s⁄299,792,458$ = $9,192,631,770⁄299,792,458$ c $Δt_{Cs}$ = $9,192,631,770⁄299,792,458$ $Δλ_{Cs}$

 * 1 radian, rad, = 1 m/m = $Δλ_{Cs}$/$Δλ_{Cs}$ = 1 (dimensionless unit of angle)

 * 1 steradian, sr, = 1 m2/m2 = $Δλ_{Cs}$2/$Δλ_{Cs}$2 = 1 (dimensionless unit of solid angle)

Between 1960 and 1983, the metre had been defined by the wavelength of a different transition frequency associated with the krypton-86 atom. This had a much higher frequency and shorter wavelength than the caesium standard, falling inside the visible spectrum. The first definition, used between 1889 and 1960, was by the international prototype meter.

Mass, energy, and force
Following the 2019 redefinition of the SI base units, electromagnetic radiation, in general, was explicitly defined to have the exact parameters:

 * c = 299,792,458 m/s

 * h = $6.626$ J s

The caesium-133 hyperfine transition radiation was explicitly defined to have frequency:

 * $Δν_{Cs}$ = 9,192,631,770 Hz

Though the above values for c and $Δν_{Cs}$ were already obviously implicit in the definitions of the metre and second. Together they imply:

 * $Δt_{Cs}$ = c $Δν_{Cs}$ = $1⁄$Δλ_{Cs}$$ m

 * $Δt_{Cs}$ = h $ΔE_{Cs}$ = 9,192,631,770 Hz × $s⁄9,192,631,770$ J s = $299,792,458⁄9,192,631,770$ J

 * $Δν_{Cs}$ = $6.626$ = $6.091$ = $$ΔM_{Cs}$⁄c^{2}$ kg

Notably, the wavelength has a fairly human-sized value of about 3.26 centimetres and the photon energy is surprisingly close to the average molecular kinetic energy per degree of freedom per kelvin. From these it follows that:

 * 1 kilogram, kg, = $6.091 J⁄89,875,517,873,681,764 m^{2}/s^{2}$ $ΔE_{Cs}$

 * 1 joule, J, = $6.091⁄8.988$ $ΔM_{Cs}$

 * 1 watt, W, = 1 J/s = $8.988⁄6.091$ $ΔE_{Cs}$ $ΔE_{Cs}$

 * 1 newton, N, = 1 J/m = $undefined⁄6.091$ $Δν_{Cs}$/$ΔE_{Cs}$

 * 1 pascal, Pa, = 1 N/m2 = $undefined⁄5.599$ $Δλ_{Cs}$/$ΔE_{Cs}$3

 * 1 gray, Gy, = 1 J/kg = $2.998⁄5.599$ $Δλ_{Cs}$/$ΔE_{Cs}$ = $2.694⁄4.732$

 * 1 sievert, Sv, = the ionizing radiation dose equivalent to 1 gray of gamma rays

Prior to the revision, between 1889 and 2019, the family of metric (and later SI) units relating to mass, force, and energy were somewhat notoriously defined by the mass of the International Prototype of the Kilogram (IPK), a specific object stored at the headquarters of the International Bureau of Weights and Measures in Paris, meaning that any change to the mass of that object would have resulted in a change to the size of the kilogram and of the many other units whose value at the time depended on that of the kilogram.

Temperature
From 1954 to 2019, the SI temperature scales were defined using the triple point of water and absolute zero. The 2019 revision replaced these with an assigned value for the Boltzmann constant, k, of $1⁄89,875,517,873,681,764$ J/K, implying:

 * 1 kelvin, K, = $c^{2}⁄89,875,517,873,681,764$ J/2 per degree of freedom = $1.381$ $ΔM_{Cs}$ per degree of freedom = $1.381$ $ΔE_{Cs}$ per degree of freedom

 * Temperature in degrees Celsius, °C, = temperature in kelvins - 273.15 = $1.381 × /2⁄6.091$

Amount of substance
The mole is an extremely large number of "elementary entities" (i.e. atoms, molecules, ions, etc). From 1969 to 2019, this number was 0.012 × the mass ratio between the IPK and a carbon 12 atom. The 2019 revision simplified this by assigning the Avogadro constant the exact value $1.381⁄1.218$ elementary entities per mole, thus, uniquely among the base units, the mole maintained its independence from the caesium standard:

 * 1 mole, mol, = $1.218 × kinetic energy per degree of freedom - 377.124 $ΔE_{Cs}$⁄1.381 $ΔE_{Cs}$$ elementary entities

 * 1 katal, kat, = 1 mol/s = $6.022$ elementary entities/$ΔE_{Cs}$

Electromagnetic units
Prior to the revision, the ampere was defined as the current needed to produce a force between 2 parallel wires 1 m apart of 0.2 μN per meter. The 2019 revision replaced this definition by giving the charge on the electron, e, the exact value $6.022$ coulombs. Somewhat incongruously, the coulomb is still considered a derived unit and the ampere a base unit, rather than vice versa. In any case, this convention entailed the following exact relationships between the SI electromagnetic units, elementary charge, and the caesium-133 hyperfine transition radiation:

 * 1 coulomb, C, = $6.022⁄9.193$ e

 * 1 ampere, or amp, A, = 1 C/s = $1.602$ e $Δt_{Cs}$

 * 1 volt, V, = 1 J/C = $undefined⁄1.602$ $Δν_{Cs}$/e

 * 1 farad, F, = 1 C/V = $undefined⁄1.473$ e2/$ΔE_{Cs}$

 * 1 ohm, Ω, = 1 V/A = $1.602⁄6.091$ $ΔE_{Cs}$/$ΔE_{Cs}$ e2 = $6.091⁄2.567$ h/e2

 * 1 siemens, S, = 1/Ω = $2.36⁄6.091$ e2/h

 * 1 weber, Wb, = 1 V s = $2.36⁄6.091$ $Δν_{Cs}$ $ΔE_{Cs}$/e = $6.091⁄2.36$ h/e

 * 1 tesla, T, = 1 Wb/m2 = $1.602⁄6.626$ $Δt_{Cs}$ $ΔE_{Cs}$/e $Δt_{Cs}$2 = $1.602⁄6.626$ E/e c $Δλ_{Cs}$

 * 1 henry, H, = Ω s = $1.44⁄5.599$ h $Δλ_{Cs}$/e2

Optical units
From 1967 to 1979 the SI optical units, lumen, lux, and candela are defined using the Incandescent glow of platinum at its melting point. After 1979, the candela was defined as the luminous intensity of a monochromatic visible light source of frequency 540 THz (i.e $1.44⁄5.599$ that of the caesium standard) and radiant intensity $2.36⁄6.626$ watts per steradian. This linked the definition of the candela to the caesium standard and, until 2019, to the IPK. Unlike the units relating to mass, energy, temperature, amount of substance, and electromagnetism, the optical units were not massively redefined in 2019, though they were indirectly affected since their values depend on that of the watt, and hence of the kilogram. The frequency used to define the optical units has the parameters:

 * Frequency: 540 THz

 * Time period: $6000⁄1.02140353$ fs

 * Wavelength: $1⁄683$ μm

 * Photon energy: $50⁄27$ Hz × $14.9896229⁄27$ J s = $5.4$ J

 * luminous efficacy, KCD, = 683 lm/W

 * Luminous energy per photon, $$Q_\mathrm v$$, = $6.626$ J × 683 lm/W = $3.578$ lm s

This implies:

 * 1 lumen, lm, = $3.578$ $$Q_\mathrm v$$ $Δt_{Cs}$

 * 1 candela, cd, = 1 lm/sr = $2.444$ $$Q_\mathrm v$$ $Δν_{Cs}$/sr

 * 1 Lux, lx, = 1 lm/m2 = $undefined⁄2.247$ $$Q_\mathrm v$$ $Δν_{Cs}$/$Δν_{Cs}$2

Summary
The parameters of the caesium 133 hyperfine transition radiation expressed exactly in SI units are:

 * Frequency = 9,192,631,770 Hz

 * Time period = $undefined⁄2.247$

 * Wavelength = $8.988⁄1.898$ m

 * Photon energy = $s⁄9,192,631,770$ J

 * Photon mass equivalent = $299,792,458⁄9,192,631,770$ kg

If the 7 base units of the SI are expressed explicitly in terms of the SI defining constants, they are:

 * 1 second = $6.091$

 * 1 metre = $6.091⁄8.988$ c/$Δλ_{Cs}$

 * 1 kilogram = $9,192,631,770⁄$Δν_{Cs}$$ h $Δν_{Cs}$/c2

 * 1 ampere = $9,192,631,770⁄299,792,458$ e $Δν_{Cs}$

 * 1 kelvin = $8.988⁄6.091$ h $Δν_{Cs}$/k

 * 1 mole = $undefined⁄1.473$ elementary entities

 * 1 candela = $13.806⁄6.091$ h $Δν_{Cs}$2 KCD/sr

Ultimately, 6 of the 7 base units notably have values that depend on that of $Δν_{Cs}$, which appears far more often than any of the other defining constants.

title: Caesium standard


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Channel

Channel, channels, channeling, etc., may refer to:

Geography

 * Channel (geography), in physical geography, a landform consisting of the outline (banks) of the path of a narrow body of water.

Australia

 * Channel Country, region of outback Australia in Queensland and partly in South Australia, Northern Territory and New South Wales.
 * Channel Highway, a regional highway in Tasmania, Australia.

Europe

 * Channel Islands, an archipelago in the English Channel, off the French coast of Normandy
 * Channel Tunnel or Chunnel, a rail tunnel underneath the English Channel
 * English Channel, called simply "The Channel", the part of the Atlantic Ocean that separates Great Britain from northern France

North America

 * Channel Islands of California, a chain of eight islands located in the Pacific Ocean off the coast of Southern California, United States
 * Channel Lake, Illinois, a census-designated place in Lake County, Illinois, United States
 * Channels State Forest, a state forest in Virginia, United States
 * Channel, Channel-Port aux Basques, Newfoundland, Newfoundland and Labrador, Canada; a town district

Other places

 * Channel Glacier, a glacier in the Palmer Archipelago, Antarctica
 * Channel Rock (disambiguation), various rocks

Television

 * Channel Television, a division of British ITV
 * Channels TV (Niger) A company of Tele Sahel.

Other uses in arts, entertainment, and media

 * Channeler (The Wheel of Time), fictional characters in the Wheel of Time fantasy book series by Robert Jordan
 * Channeling (Rolemaster), a magic element in the Rolemaster role-playing game
 * Channels (band), fronted by former Jawbox singer/guitarist J. Robbins
 * Channels (film), a 2008 film
 * The Channel (film), a 2023 film
 * Pre-chorus, or build or channel, part of a song structure of popular music

Communications

 * Communication channel, a transmission medium used to convey information
 * Audio channel, a communications channel in a storage device used in operations such as multitrack recording and playback
 * Channel (broadcasting), a range of frequencies assigned for the operation of a television, radio, or other broadcast station
 * Television channel, a television station or its cable/satellite counterpart
 * Channel (digital image), the grayscale representation of a primary color in a digital image
 * Channel (programming), in computer science, a tool used for interprocess communication
 * Channel I/O, in computing, a high-performance input/output (I/O) architecture
 * Video blog, hosted by a video-sharing website or channel

Other uses in science and technology

 * Channel (semiconductor), a part of the structure of a field-effect transistor
 * Ion channel, a protein that allows ion flow through a cell membrane
 * Scattering channel, a concept in quantum mechanics
 * Strut channel, a standardized formed structural system used in the construction and electrical industries
 * Structural channel

Brands and enterprises

 * The Channel (nightclub), a music venue in Boston, Massachusetts, U.S.
 * Channel Express, a defunct British airline
 * Channel Home Centers, a defunct home improvement chain based in the northeastern United States

Business and legal terms

 * Channel (chart pattern), a pair of parallel trend lines that form a chart pattern for a stock or commodity
 * Distribution channel, a chain of intermediaries, each passing the product down the chain to the next organization, before it finally reaches the consumer
 * Legal channeling, the act of legally making one entity responsible for an event, and thereby dismissing other parties from liability
 * Marketing channel, set of activities necessary to transfer the ownership of goods from the point of production to the point of consumption

Sports

 * Channel (association football), football (soccer) terminology for particular areas of the pitch
 * Channel (horse), a racehorse
 * Channel Football Club, an Australian football club in Tasmania

Other uses

 * Channel catfish, a catfish species in North America
 * Meridian (Chinese medicine), or channels, a concept central to traditional Chinese medical techniques such as acupuncture, and to martial arts such as tai chi and qigong
 * Mediumship or channelling, communication with spirits

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Channel


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Channel noise level

In telecommunications, the term channel noise level has the following meanings:

 * 1) The ratio of the noise in the communication channel at any point in a transmission system to an arbitrary level chosen as a reference.
 * 2) The noise power spectral density in the frequency range of interest.
 * 3) The average noise power in the frequency range of interest.

title: Channel noise level


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Channel reliability

In telecommunication, channel reliability (ChR) is the percentage of time a communication channel was available for use in a specified period of scheduled availability.

Channel reliability is given by

 * $$ChR = 100 (1 - \frac{T_o}{T_s} ) = 100 \frac{T_a}{T_s} $$

where T o is the channel total outage time, T s is the channel total scheduled time, and T a is the channel total available time.

title: Channel reliability


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Channel service unit

In telecommunications, a channel service unit (CSU) is a line bridging device for use with T-carrier, which
 * is used to perform loopback testing;
 * may perform bit stuffing;
 * may also provide a framing and formatting pattern compatible with the network;
 * provides a barrier for electrical interference from either side of the unit; and
 * is the last signal regeneration point, on the loop side, coming from the central office, before the regenerated signal reaches a multiplexer or data terminal equipment (DTE).

Common varieties
CSUs can be categorized by the class of service they support (DS1, DS3, DDS, etc.) and by the capabilities within that class. For example, basic DS1 (T1) CSUs support loopback of each interface and will produce alarm indication signal to the provider's network interface device (NID) in the event of loss of signal from the customer-premises equipment (CPE). More advanced units will include internal monitors of the performance of the carrier in both directions and may have test pattern generation and monitor capabilities.

Common practice
CSUs are required by PSTN providers at digital interfaces that terminate in a DSU on the customer side. They are not used when the service terminates in a modem, such as the DSL family of service. The maintenance capabilities of the CSU provide important guidance as to whether the provider needs to dispatch a repairman to the customer location.

title: Channel service unit


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Character interval

Character interval: In a communications system, the total number of unit intervals required to transmit any given character, including synchronizing, information, error checking, or control characters, but not including signals that are not associated with individual characters.

An example of a time interval that is excluded when determining character interval is any time added between the end of a stop signal and the beginning of the next start signal to accommodate changing transmission conditions, such as a change in data signaling rate or buffering requirements. This added time is defined as a part of the intercharacter interval.

title: Character interval


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Characteristic impedance

The characteristic impedance or surge impedance (usually written Z0) of a uniform transmission line is the ratio of the amplitudes of voltage and current of a wave travelling in one direction along the line in the absence of reflections in the other direction. Equivalently, it can be defined as the input impedance of a transmission line when its length is infinite. Characteristic impedance is determined by the geometry and materials of the transmission line and, for a uniform line, is not dependent on its length. The SI unit of characteristic impedance is the ohm.

The characteristic impedance of a lossless transmission line is purely real, with no reactive component. Energy supplied by a source at one end of such a line is transmitted through the line without being dissipated in the line itself. A transmission line of finite length (lossless or lossy) that is terminated at one end with an impedance equal to the characteristic impedance appears to the source like an infinitely long transmission line and produces no reflections.

Transmission line model
The characteristic impedance $$Z(\omega)$$ of an infinite transmission line at a given angular frequency $$\omega$$ is the ratio of the voltage and current of a pure sinusoidal wave of the same frequency travelling along the line. This relation is also the case for finite transmission lines until the wave reaches the end of the line. Generally, a wave is reflected back along the line in the opposite direction. When the reflected wave reaches the source, it is reflected yet again, adding to the transmitted wave and changing the ratio of the voltage and current at the input, causing the voltage-current ratio to no longer equal the characteristic impedance. This new ratio including the reflected energy is called the input impedance.

The input impedance of an infinite line is equal to the characteristic impedance since the transmitted wave is never reflected back from the end. Equivalently: The characteristic impedance of a line is that impedance which, when terminating an arbitrary length of line at its output, produces an input impedance of equal value. This is so because there is no reflection on a line terminated in its own characteristic impedance.

Applying the transmission line model based on the telegrapher's equations as derived below, the general expression for the characteristic impedance of a transmission line is: $$Z_0 = \sqrt{ \frac{R + j\omega L}{G + j\omega C}\, }$$ where $R$ is the resistance per unit length, considering the two conductors to be in series,

$L$ is the inductance per unit length,

$G$ is the conductance of the dielectric per unit length,

$C$ is the capacitance per unit length,

$j$ is the imaginary unit, and

$\omega$ is the angular frequency. This expression extends to DC by letting $$\omega$$ tend to 0.

A surge of energy on a finite transmission line will see an impedance of $$Z_0$$ prior to any reflections returning; hence surge impedance is an alternative name for characteristic impedance. Although an infinite line is assumed, since all quantities are per unit length, the “per length” parts of all the units cancel, and the characteristic impedance is independent of the length of the transmission line.

The voltage and current phasors on the line are related by the characteristic impedance as: $$\frac{V_{(+)}}{I_{(+)}} = Z_\text{0} = -\frac{V_{(-)}}{I_{(-)}}$$ where the subscripts (+) and (−) mark the separate constants for the waves traveling forward (+) and backward (−).

Using the telegrapher's equation
The differential equations describing the dependence of the voltage and current on time and space are linear, so that a linear combination of solutions is again a solution. This means that we can consider solutions with a time dependence $$\ e^{j \omega t}\ ; $$ doing so is functionally equivalent of solving for the Fourier coefficients for voltage and current amplitudes, at some fixed angular frequency $$\ \omega ~.$$ Doing so causes the time dependence to factor out, leaving an ordinary differential equation for the coefficients, which will be phasors, dependent on position (space) only. Moreover, the parameters can be generalized to be frequency-dependent.

Let $$ V(x,t) \equiv V(x)\ e^{+j \omega t} $$ and $$ I(x,t) \equiv I(x)\ e^{+j \omega t} $$

Take the positive direction for $$\ V\ $$ and $$\ I\ $$ in the loop to be clockwise.

We find that $$\ \operatorname{d}V = -\left( R + j\ \omega L \right) I\ \operatorname{d}x = -Z'\ I\ \operatorname{d}x $$ and $$\ \operatorname{d}I = -\left( G + j\ \omega C \right) V\ \operatorname{d}x = -Y'\ V\ \operatorname{d}x $$ or $$\ \frac{\ \operatorname{d}V\ }{\ \operatorname{d}x\ } = -Z'\ I \qquad \text{ and } \qquad\frac{\ \operatorname{d}I\ }{\ \operatorname{d}x\ } = -Y'\ V$$ where $$\ Z' \equiv R + j\ \omega L \qquad \text{ and } \qquad Y' \equiv G + j\ \omega C ~.$$

These two first-order equations are easily uncoupled by a second differentiation, with the results: $$\ \frac{\ \operatorname{d}^2 V\ }{\ \operatorname{d}x^2} = Z' Y'\ V\ $$ and $$\ \frac{\ \operatorname{d}^2 I\ }{\ \operatorname{d}x^2} = Z' Y'\ I\ $$

Notice that both $$\ V\ $$ and $$\ I\ $$ satisfy the same equation.

Since $$\ Z' Y'\ $$ is independent of $$\ x\ $$ and $$\ t\ ,$$ it can be represented by a single constant $$\ -k^2 ~.$$ (The minus sign is included for later convenience.) That is: $$\ -k^2 \equiv Z'\, Y'\ $$ so $$\ j k = \pm \sqrt{Z'\, Y'\ }\ $$

We can write the above equation as $$\ k = \pm \omega \sqrt{ \left( L - j\ \frac{\ R\ }{ \omega } \right) \left(C - j\ \frac{\ G\ }{ \omega } \right)\ } = \pm \omega\sqrt{L\ C\ } \sqrt{ \left( 1 - j\ \frac{ R }{\ \omega L\ } \right) \left( 1 - j\ \frac{ G }{\ \omega C\ }\right)\ }\ $$ which is correct for any transmission line in general. And for typical transmission lines, that are carefully built from wire with low loss resistance $$\ R\ $$ and small insulation leakage conductance $$\ G\ ;$$ further, used for high frequencies, the inductive reactance $$\ \omega L\ $$ and the capacitive admittance $$\ \omega C\ $$ will both be large, so the constant $$\ k\ $$ is very close to being a real number: $$\ k \approx \pm \omega \sqrt{L C\;} ~.$$

With this definition of $$\ k\ ,$$ the position- or $\ x$-dependent part will appear as $$\ \pm j\ k\ x\ $$ in the exponential solutions of the equation, similar to the time-dependent part $$\ +j\ \omega\ t\ $$ so the solution reads $$ V(x) = v_{(+)}\ e^{-j k x} + v_{(-)}\ e^{+j k x}$$ where $$\ v_{(+)}\ $$ and $$\ v_{(-)}\ $$ are the constants of integration for the forward moving (+) and backward moving (−) waves, as in the prior section. When we recombine the time-dependent part we obtain the full solution: $$\ V(x,t) ~ = ~ V(x)\ e^{+j \omega t} ~ = ~ v_{(+)}\ e^{-j k x + j \omega t} + v_{(-)} e^{+j k x + j \omega t} ~.$$

Since the equation for $$\ I\ $$ is the same form, it has a solution of the same form: $$\ I(x) = i_{(+)}\ e^{-j k x} + i_{(-)}\ e^{+j k x}\ ,$$ where $$\ i_{(+)}\ $$ and $$\ i_{(-)}\ $$ are again constants of integration.

The above equations are the wave solution for $$V$$ and $$I$$. In order to be compatible, they must still satisfy the original differential equations, one of which is $$\ \frac{\ \operatorname{d}V\ }{\ \operatorname{d}x\ } = -Z' I ~.$$

Substituting the solutions for $$\ V\ $$ and $$\ I\ $$ into the above equation, we get $$\ \frac{ \operatorname{d} }{\ \operatorname{d}x\ }\left[ v_{(+)}\ e^{-j\ k\ x} + v_{(-)}\ e^{+j\ k\ x} \right] = -(R + j\ \omega L)\left[\ i_{(+)}\ e^{-j\ k\ x} + i_{(-)}\ e^{+j\ k\ x} \right] $$ or $$\ -j\ k\ v_{(+)}\ e^{-j\ k\ x} + j k\ v_{(-)}\ e^{+j\ k\ x} = -(R + j\omega L)\ i_{(+)}\ e^{-j k x} - \left( R + j\ \omega L \right)\ i_{(-)} \ e^{+j k x} $$

Isolating distinct powers of $$\ e\ $$ and combining identical powers, we see that in order for the above equation to hold for all possible values of $$\ x\ $$ we must have:

 * For the co-efficients of $$\ e^{-j\ k\ x}\ $$: $$\ -j\ k\ v_{(+)} = - \left( R + j\ \omega L \right)\ i_{(+)}\ $$
 * For the co-efficients of $$\ e^{+j\ k\ x}\ $$: $$\ +j\ k\ v_{(-)} = -\left( R + j\ \omega L \right)\ i_{(-)}\ $$

Since $\ j\ k = \sqrt{ \left(R + j\ \omega L \right) \left(G + j\ \omega C \right)\ }\ $ $$\ \begin{align} +\frac{v_{(+)}}{i_{(+)}} &= \frac{\ R + j\ \omega L\ }{ j\ k } = \sqrt{ \frac{\ R + j\ \omega L\ }{G + j\ \omega C}\ } \equiv Z_0 \\[1ex] -\frac{v_{(-)}}{i_{(-)}} &= \frac{\ R + j\ \omega L\ }{ j\ k } = \sqrt{ \frac{\ R + j\ \omega L\ }{G + j\ \omega C}\ } \equiv Z_0 \end{align}\ $$ hence, for valid solutions require $$\ v_{(+)} = +Z_0\ i_{(+)} \quad \text{ and } \quad v_{(-)} = -Z_0\ i_{(-)}\ $$

It can be seen that the constant $$\ Z_0\ ,$$ defined in the above equations has the dimensions of impedance (ratio of voltage to current) and is a function of primary constants of the line and operating frequency. It is called the “characteristic impedance” of the transmission line, and conventionally denoted by $$\ Z_0 ~.$$ $$ Z_0 \quad = \quad \sqrt{\frac{\ R + j\ \omega L\ }{ G + j\ \omega C }\ } \quad = \quad \sqrt{\ \frac{\ L\ }{ C }\ } \sqrt{\frac{\ 1 - j\ \left(\frac{ R }{\ \omega L\ }\right)\, }{\ 1 - j\ \left(\frac{ G }{\ \omega C\ } \right)\ }\ } $$ which holds generally, for any transmission line. For well-functioning transmission lines, with either $$\ R\ $$ and $$\ G\ $$ both very small, or with $$\ \omega\ $$ very high, or all of the above, we get $$ Z_0 \approx \sqrt{\frac{\ L\ }{ C } \ } $$ hence the characteristic impedance is typically very close to being a real number. Manufacturers make commercial cables to approximate this condition very closely over a wide range of frequencies.

Intuition
Consider an infinite ladder network consisting of a series impedance $$\ Z\ $$ and a shunt admittance $$\ Y ~.$$ Let its input impedance be $$\ Z_\mathrm{IT} ~.$$ If a new pair of impedance $$\ Z\ $$ and admittance $$\ Y\ $$ is added in front of the network, its input impedance $$\ Z_\mathrm{IT}\ $$ remains unchanged since the network is infinite. Thus, it can be reduced to a finite network with one series impedance $$\ Z\ $$ and two parallel impedances $$\ 1 / Y\ $$ and $$\ Z_\text{IT} ~.$$ Its input impedance is given by the expression

 * $$\ Z_\mathrm{IT} = Z + \left( \frac{\ 1\ }{ Y } \parallel Z_\mathrm{IT} \right)\ $$

which is also known as its iterative impedance. Its solution is:

 * $$\ Z_\mathrm{ IT } = {Z \over 2} \pm \sqrt { {Z^2 \over 4} + {Z \over Y} }\ $$

For a transmission line, it can be seen as a limiting case of an infinite ladder network with infinitesimal impedance and admittance at a constant ratio. Taking the positive root, this equation simplifies to:

 * $$\ Z_\mathrm{IT} = \sqrt{ \frac{\ Z\ }{ Y }\ }\ $$

Derivation
Using this insight, many similar derivations exist in several books  and are applicable to both lossless and lossy lines.

Here, we follow an approach posted by Tim Healy. The line is modeled by a series of differential segments with differential series elements $$\ \left( R\ \operatorname{d}x,\ L\ \operatorname{d}x \right)\ $$ and shunt elements $$\ \left(C\ \operatorname{d}x,\ G\ \operatorname{d}x\ \right)\ $$ (as shown in the figure at the beginning of the article). The characteristic impedance is defined as the ratio of the input voltage to the input current of a semi-infinite length of line. We call this impedance $$\ Z_0 ~.$$ That is, the impedance looking into the line on the left is $$\ Z_0 ~.$$ But, of course, if we go down the line one differential length $$\ \operatorname{d}x\ ,$$ the impedance into the line is still $$\ Z_0 ~.$$ Hence we can say that the impedance looking into the line on the far left is equal to $$\ Z_0\ $$ in parallel with $$\ C\ \operatorname{d}x\ $$ and $$\ G\ \operatorname{d}x\ ,$$ all of which is in series with $$\ R\ \operatorname{d}x\ $$ and $$\ L\ \operatorname{d}x ~.$$ Hence: $$\begin{align} Z_0 &= (R + j\ \omega L)\ \operatorname{d}x + \frac{ 1 }{\ (G + j \omega C)\ \operatorname{d}x + \frac{1}{\ Z_0\ }\ } \\[1ex] Z_0 &= (R + j\ \omega L)\ \operatorname{d}x + \frac{\ Z_0\ }{Z_0\ (G + j \omega C)\ \operatorname{d}x + 1\, } \\[1ex] Z_0 + Z_0^2\ (G + j\ \omega C)\ \operatorname{d}x &= (R + j\ \omega L)\ \operatorname{d}x + Z_0\ (G + j\ \omega C)\ \operatorname{d}x\ (R + j\ \omega L)\ \operatorname{d}x + Z_0 \end{align} $$

The added $$\ Z_0\ $$ terms cancel, leaving $$\ Z_0^2\ (G + j\ \omega C)\ \operatorname{d}x = \left( R + j\ \omega L \right)\ \operatorname{d}x + Z_0\ \left( G + j\ \omega C \right)\ \left( R + j\ \omega L \right)\ \left( \operatorname{d}x \right)^2 $$

The first-power $$\ \operatorname{d}x\ $$ terms are the highest remaining order. Dividing out the common factor of $$\ \operatorname{d}x\ ,$$ and dividing through by the factor $$\ \left( G + j\ \omega C \right)\ ,$$ we get $$\ Z_0^2 = \frac{ \left( R + j\ \omega L \right) }{\ \left( G + j\ \omega C \right)\ } + Z_0\ \left( R + j\ \omega L \right)\ \operatorname{d}x ~.$$

In comparison to the factors whose $$\ \operatorname{d}x\ $$ divided out, the last term, which still carries a remaining factor $$\ \operatorname{d}x\ ,$$ is infinitesimal relative to the other, now finite terms, so we can drop it. That leads to $$\ Z_0 = \pm \sqrt{\frac{\ R + j\ \omega L\ }{G + j\ \omega C}\ } ~.$$

Reversing the sign $±$ applied to the square root has the effect of reversing the direction of the flow of current.

Lossless line
The analysis of lossless lines provides an accurate approximation for real transmission lines that simplifies the mathematics considered in modeling transmission lines. A lossless line is defined as a transmission line that has no line resistance and no dielectric loss. This would imply that the conductors act like perfect conductors and the dielectric acts like a perfect dielectric. For a lossless line, $R$ and $G$ are both zero, so the equation for characteristic impedance derived above reduces to: $$Z_0 = \sqrt{\frac{L}{C}\,}\,.$$

In particular, $$Z_0$$ does not depend any more upon the frequency. The above expression is wholly real, since the imaginary term $j$ has canceled out, implying that $$Z_0$$ is purely resistive. For a lossless line terminated in $$Z_0$$, there is no loss of current across the line, and so the voltage remains the same along the line. The lossless line model is a useful approximation for many practical cases, such as low-loss transmission lines and transmission lines with high frequency. For both of these cases, $R$ and $G$ are much smaller than $ωL$ and $ωC$, respectively, and can thus be ignored.

The solutions to the long line transmission equations include incident and reflected portions of the voltage and current: $$\begin{align} V &= \frac{V_r + I_r Z_c}{2} e^{\gamma x} + \frac{V_r - I_r Z_c}{2} e^{-\gamma x} \\[1ex] I &= \frac{V_r/Z_c + I_r}{2} e^{\gamma x} - \frac{V_r/Z_c - I_r}{2} e^{-\gamma x} \end{align}$$ When the line is terminated with its characteristic impedance, the reflected portions of these equations are reduced to 0 and the solutions to the voltage and current along the transmission line are wholly incident. Without a reflection of the wave, the load that is being supplied by the line effectively blends into the line making it appear to be an infinite line. In a lossless line this implies that the voltage and current remain the same everywhere along the transmission line. Their magnitudes remain constant along the length of the line and are only rotated by a phase angle.

Surge impedance loading
In electric power transmission, the characteristic impedance of a transmission line is expressed in terms of the surge impedance loading (SIL), or natural loading, being the power loading at which reactive power is neither produced nor absorbed: $$\mathit{SIL} = \frac{{V_\mathrm{LL}}^2}{Z_0}$$ in which $$V_\mathrm{LL}$$ is the RMS line-to-line voltage in volts.

Loaded below its SIL, the voltage at the load will be greater than the system voltage. Above it, the load voltage is depressed. The Ferranti effect describes the voltage gain towards the remote end of a very lightly loaded (or open ended) transmission line. Underground cables normally have a very low characteristic impedance, resulting in an SIL that is typically in excess of the thermal limit of the cable.

Practical examples
The characteristic impedance of coaxial cables (coax) is commonly chosen to be 50 Ω for RF and microwave applications. Coax for video applications is usually 75 Ω for its lower loss.

title: Characteristic impedance


text:
Chip

Chip may refer to:

Food

 * Chip (snack), thin sliced food, cooked until crunchy
 * Potato chip (US) or crisp (UK)
 * Chips (hot food) (UK) or French fries (US), fried potato batons
 * Game chips, thin chip/French fries
 * Chocolate chip

Sports and gaming

 * Chip (association football), a type of football shot
 * Chip (golf), a type of golf shot
 * Casino token, often referred to as a chip

People and fictional characters

 * Chip (name), a list of people and fictional characters with either the given name or nickname
 * Chip (rapper), English hip hop recording artist born Jahmaal Noel Fyffe in 1990
 * King Chip (born 1986), stage name of American hip hop rapper Charles Jawanzaa Worth, formerly known as Chip tha Ripper
 * Chip Douglas, American songwriter, musician, and record producer Douglas Hatlelid (born 1942)
 * Chip Taylor, stage name of American songwriter James Wesley Voight (born 1940)
 * Chip Fairway, a ring name of American professional wrestler Brett J. Keen (1972–2011)

Biology

 * Chromatin immunoprecipitation (ChIP), a molecular biology method to map DNA sites
 * STUB1, a human gene also known as CHIP (C terminus of HSC70-Interacting Protein)
 * Clonal hematopoiesis of indeterminate potential (CHIP)

Computing

 * Chip, a monolithic integrated circuit without its packaging, a microchip
 * An integrated circuit mounted on a surface mount chip carrier
 * Chip (CDMA), the fundamental unit of transmission in CDMA
 * CHIP (computer), Linux dev board built by Next Thing Co. and marketed as a miniature computer
 * CHIP (programming language), based on Prolog
 * CHIP-8, a video game programming language in the 1970s
 * ChucK for iPhone/iPad (ChiP), a programming language used for music synthesis
 * Connected Home over IP (CHIP), a protocol for home automation

Finance

 * Chip (stock market), description of stock of a particular quality
 * Clearing House Interbank Payments System (CHIPS), a United States private clearing house for large-value transactions

Organizations and programs

 * California Highway Patrol (CHiP), a law enforcement agency of California
 * CHIP Holding, a German-based publishing house
 * Children's Health Insurance Program (CHIP), a US federal government program
 * Community Housing and Infrastructure Program (CHIP), an Australian government funding program, along with its Community Development Employment Projects
 * Masonic Child Identification Programs, a charitable initiative by North American Masonic lodges

Other uses

 * Chip, one of the mascots of the University of Colorado at Boulder
 * Chip language, spoken in Nigeria
 * Swarf, also known as chips, the debris or waste resulting from metalworking operations
 * CHIP (magazine), a German computer magazine
 * Chip-India, Indian edition of German computer magazine CHIP

title: Chip


text:
Chirping

Chirping may refer to:
 * Bird vocalization
 * Chirping, the act of signaling with chirps, signals in which the frequency increases / decreases with time
 * Chirping, pulse compression by linear frequency modulation
 * Trash-talk in ice hockey

title: Chirping


text:
Circuit

Circuit may refer to:

Electrical engineering

 * Electrical circuit, a complete electrical network with a closed-loop giving a return path for current
 * Analog circuit, uses continuous signal levels
 * Balanced circuit, paths are impedance-matched
 * Circuit analysis, the process of finding the voltages across, and the currents through, every component in an electrical circuit
 * Circuit diagram, a graphical representation of an electrical circuit
 * Digital circuit, uses discrete signal levels
 * Electronic circuit, contains "active" (nonlinear) electronic components capable of performing amplification, computation, and data transfer
 * Asynchronous circuit, or self-timed circuit, a sequential digital logic circuit that is not governed by a clock circuit or global clock signal
 * Integrated circuit, a set of electronic circuits on a small "chip" of semiconductor material
 * Mixed-signal integrated circuit, contains both analog and digital signals
 * Synchronous circuit, a digital circuit in which the changes in the state of memory elements are synchronized by a clock signal
 * Printed circuit board, on which electronic components are supported and connected using copper tracks on a non-conductive substrate
 * Series and parallel circuits, two ways in which electrical components may be interconnected
 * Simple filters, including:
 * LC circuit or tank circuit, consisting of an inductance and a capacitance
 * RC circuit, comprises a resistance and a capacitance
 * RL circuit, comprises a resistance and an inductance
 * RLC circuit, comprises a resistance, an inductance, and a capacitance
 * Telecommunication circuit, on which information is transmitted
 * Nonlinear circuit, a circuit with nonlinear elements.

Mathematics and computer science

 * Circuit (computer theory), a theoretical structure simulating electrical and data paths
 * Boolean circuit, a mathematical model for digital logic circuits
 * Integer circuit, a mathematical object of computational complexity
 * Circuit complexity, a branch of computational complexity theory
 * Cycle (graph theory), a closed path, with no other repeated vertices than the starting and ending vertices
 * Circuit of a matroid

Other sciences

 * Biological neural network, in neuroscience
 * Hydraulic circuit, in fluid mechanics
 * Magnetic circuit, in physics, one or more closed loop paths containing a magnetic flux
 * Monetary circuit theory, a heterodox theory of monetary economics, also called circuitism
 * Pneumatic circuit, in fluid mechanics
 * Synthetic biological circuit, in synthetic biology

Arts, entertainment, and media

 * Circuit (film), a 2001 gay-themed film set in the world of gay circuit parties
 * Mario Kart: Super Circuit, a 2001 Game Boy Advance game and the third game in the Mario Kart series
 * Circuit (film character), a character from the Indian film series Munna Bhai
 * Circuit, a character from Power Rangers Time Force

Church

 * Circuit (LCMS), a local grouping of congregations in the Lutheran Church–Missouri Synod
 * Methodist Circuit, a form of church governance in Methodism

Government and law

 * Circuit (administrative division), an administrative country subdivision in East Asia
 * Circuit court, the name of court systems in several common law jurisdictions

Sports

 * Circuit training, a form of high-intensity aerobics
 * ITF Pro Circuit, pro tennis tours of the International Tennis Federation
 * ITF Men's Circuit
 * ITF Women's Circuit

Transportation and racing

 * Circuit (airfield), also called pattern, a standard path followed by aircraft when taking off or landing
 * Circuit, a race track or one complete traverse of a track
 * Formula One circuits (see list of Formula One circuits)
 * Kart circuit, a race track designed for kart racing

Other uses

 * Circuit (software), a collaboration service software by Unify
 * Circuit, slang for hangouts or events frequented by a given social circle
 * Circuit party, a gay dance event

title: Circuit


text:
Circuit noise level

At any point in a transmission system, the ratio of the circuit noise at that point to an arbitrary level chosen as a reference.

The circuit noise level is usually expressed in dBrn0, signifying the reading of a circuit noise meter, or in dBa0, signifying circuit noise meter reading adjusted to represent an interfering effect under specified conditions.

title: Circuit noise level


text:
Circuit reliability

Circuit reliability (also time availability) (CiR) is the percentage of time an electronic circuit was available for use in a specified period of scheduled availability. Circuit reliability is given by where T o is the circuit total outage time, Ts is the circuit total scheduled time, and T a is the circuit total available time. $$T_s = T_a + T_o$$

In addition, circuit reliability is the expected lifespan of operation of a functioning system under nominal conditions.

title: Circuit reliability


text:
Circuit restoration

In telecommunication, circuit restoration is the process by which a communications circuit is established between two users after disruption or loss of the original circuit. The loss may be widespread due to a natural disaster like an ice storm or hurricane, or local by being cut underground in construction or damaged in a thunderstorm or car accident.

Circuit restoration is usually performed in accordance with planned procedures and priorities. Restoration may be effected automatically, such as by switching to a hot standby, or manually, such as by manual patching.

title: Circuit restoration


text:
Circuit switching

Circuit switching is a method of implementing a telecommunications network in which two network nodes establish a dedicated communications channel (circuit) through the network before the nodes may communicate. The circuit guarantees the full bandwidth of the channel and remains connected for the duration of the communication session. The circuit functions as if the nodes were physically connected as with an electrical circuit.

Circuit switching originated in analog telephone networks where the network created a dedicated circuit between two telephones for the duration of a telephone call. It contrasts with message switching and packet switching used in modern digital networks in which the trunklines between switching centers carry data between many different nodes in the form of data packets without dedicated circuits.

Description
The defining example of a circuit-switched network is the early analog telephone network. When a call is made from one telephone to another, switches within the telephone exchanges create a continuous wire circuit between the two telephones, for as long as the call lasts.

In circuit switching, the bit delay is constant during a connection (as opposed to packet switching, where packet queues may cause varying and potentially indefinitely long packet transfer delays). No circuit can be degraded by competing users because it is protected from use by other callers until the circuit is released and a new connection is set up. Even if no actual communication is taking place, the channel remains reserved and protected from competing users.

While circuit switching is commonly used for connecting voice circuits, the concept of a dedicated path persisting between two communicating parties or nodes can be extended to signal content other than voice. The advantage of using circuit switching is that it provides for continuous transfer without the overhead associated with packets, making maximal use of available bandwidth for that communication. One disadvantage is that it can be relatively inefficient because unused capacity guaranteed to a connection cannot be used by other connections on the same network. In addition, calls cannot be established or will be dropped if the circuit is broken.

The call
For call setup and control (and other administrative purposes), it is possible to use a separate dedicated signalling channel from the end node to the network. ISDN is one such service that uses a separate signalling channel while plain old telephone service (POTS) does not.

The method of establishing the connection and monitoring its progress and termination through the network may also utilize a separate control channel as in the case of links between telephone exchanges which use CCS7 packet-switched signalling protocol to communicate the call setup and control information and use TDM to transport the actual circuit data.

Early telephone exchanges were a suitable example of circuit switching. The subscriber would ask the operator to connect to another subscriber, whether on the same exchange or via an inter-exchange link and another operator. The result was a physical electrical connection between the two subscribers' telephones for the duration of the call. The copper wire used for the connection could not be used to carry other calls at the same time, even if the subscribers were in fact not talking and the line was silent.

Alternatives
In circuit switching, a route and its associated bandwidth is reserved from source to destination, making circuit switching relatively inefficient since capacity is reserved whether or not the connection is in continuous use. Circuit switching contrasts with message switching and packet switching. Both of these methods can make better use of available network bandwidth between multiple communication sessions under typical conditions in data communication networks.

Message switching routes messages in their entirety, one hop at a time, that is, store and forward of the entire message. Packet switching divides the data to be transmitted into packets transmitted through the network independently. Instead of being dedicated to one communication session at a time, network links are shared by packets from multiple competing communication sessions, resulting in the loss of the quality of service guarantees that are provided by circuit switching.

Packet switching can be based on connection-oriented communication or connection-less communication. That is, based on virtual circuits or datagrams.

Virtual circuits use packet switching technology that emulates circuit switching, in the sense that the connection is established before any packets are transferred, and packets are delivered in order.

Connection-less packet switching divides the data to be transmitted into packets, called datagrams, transmitted through the network independently. Each datagram is labeled with its destination and a sequence number for ordering related packets, precluding the need for a dedicated path to help the packet find its way to its destination. Each datagram is dispatched independently and each may be routed via a different path. At the destination, the original message is reordered based on the packet number to reproduce the original message. As a result, datagram packet switching networks do not require a circuit to be established and allow many pairs of nodes to communicate concurrently over the same channel.

Multiplexing multiple telecommunications connections over the same physical conductor has been possible for a long time, but each channel on the multiplexed link was either dedicated to one call at a time, or it was idle between calls.

Examples of circuit-switched networks

 * Public switched telephone network (PSTN)
 * B channel of ISDN
 * Circuit Switched Data (CSD) and High-Speed Circuit-Switched Data (HSCSD) service in cellular systems such as GSM
 * Datakit
 * X.21 (Used in the German DATEX-L and Scandinavian DATEX circuit switched data network)
 * Optical mesh network

title: Circuit switching


text:
Circular polarization

In electrodynamics, circular polarization of an electromagnetic wave is a polarization state in which, at each point, the electromagnetic field of the wave has a constant magnitude and is rotating at a constant rate in a plane perpendicular to the direction of the wave.

In electrodynamics, the strength and direction of an electric field is defined by its electric field vector. In the case of a circularly polarized wave, the tip of the electric field vector, at a given point in space, relates to the phase of the light as it travels through time and space. At any instant of time, the electric field vector of the wave indicates a point on a helix oriented along the direction of propagation. A circularly polarized wave can rotate in one of two possible senses: right-handed circular polarization (RHCP) in which the electric field vector rotates in a right-hand sense with respect to the direction of propagation, and left-handed circular polarization (LHCP) in which the vector rotates in a left-hand sense.

Circular polarization is a limiting case of elliptical polarization. The other special case is the easier-to-understand linear polarization. All three terms were coined by Augustin-Jean Fresnel, in a memoir read to the French Academy of Sciences on 9 December 1822. Fresnel had first described the case of circular polarization, without yet naming it, in 1821.

The phenomenon of polarization arises as a consequence of the fact that light behaves as a two-dimensional transverse wave.

Circular polarization occurs when the two orthogonal electric field component vectors are of equal magnitude and are out of phase by exactly 90°, or one-quarter wavelength.

Characteristics
In a circularly polarized electromagnetic wave, the individual electric field vectors, as well as their combined vector, have a constant magnitude, and with changing phase angle. Given that this is a plane wave, each vector represents the magnitude and direction of the electric field for an entire plane that is perpendicular to the optical axis. Specifically, given that this is a circularly polarized plane wave, these vectors indicate that the electric field, from plane to plane, has a constant strength while its direction steadily rotates. Refer to these two images in the plane wave article to better appreciate this dynamic. This light is considered to be right-hand, clockwise circularly polarized if viewed by the receiver. Since this is an electromagnetic wave, each electric field vector has a corresponding, but not illustrated, magnetic field vector that is at a right angle to the electric field vector and proportional in magnitude to it. As a result, the magnetic field vectors would trace out a second helix if displayed.

Circular polarization is often encountered in the field of optics and, in this section, the electromagnetic wave will be simply referred to as light.

The nature of circular polarization and its relationship to other polarizations is often understood by thinking of the electric field as being divided into two components that are perpendicular to each other. The vertical component and its corresponding plane are illustrated in blue, while the horizontal component and its corresponding plane are illustrated in green. Notice that the rightward (relative to the direction of travel) horizontal component leads the vertical component by one quarter of a wavelength, a 90° phase difference. It is this quadrature phase relationship that creates the helix and causes the points of maximum magnitude of the vertical component to correspond with the points of zero magnitude of the horizontal component, and vice versa. The result of this alignment are select vectors, corresponding to the helix, which exactly match the maxima of the vertical and horizontal components.

To appreciate how this quadrature phase shift corresponds to an electric field that rotates while maintaining a constant magnitude, imagine a dot traveling clockwise in a circle. Consider how the vertical and horizontal displacements of the dot, relative to the center of the circle, vary sinusoidally in time and are out of phase by one quarter of a cycle. The displacements are said to be out of phase by one quarter of a cycle because the horizontal maximum displacement (toward the left) is reached one quarter of a cycle before the vertical maximum displacement is reached. Now referring again to the illustration, imagine the center of the circle just described, traveling along the axis from the front to the back. The circling dot will trace out a helix with the displacement toward our viewing left, leading the vertical displacement. Just as the horizontal and vertical displacements of the rotating dot are out of phase by one quarter of a cycle in time, the magnitude of the horizontal and vertical components of the electric field are out of phase by one quarter of a wavelength.

The next pair of illustrations is that of left-handed, counterclockwise circularly polarized light when viewed by the receiver. Because it is left-handed, the rightward (relative to the direction of travel) horizontal component is now lagging the vertical component by one quarter of a wavelength, rather than leading it.

Waveplate
To convert circularly polarized light to the other handedness, one can use a half-waveplate. A half-waveplate shifts a given linear component of light one half of a wavelength relative to its orthogonal linear component.

Reflection
The handedness of polarized light is reversed reflected off a surface at normal incidence. Upon such reflection, the rotation of the plane of polarization of the reflected light is identical to that of the incident field. However, with propagation now in the opposite direction, the same rotation direction that would be described as "right-handed" for the incident beam, is "left-handed" for propagation in the reverse direction, and vice versa. Aside from the reversal of handedness, the ellipticity of polarization is also preserved (except in cases of reflection by a birefringent surface).

Note that this principle only holds strictly for light reflected at normal incidence. For instance, right circularly polarized light reflected from a dielectric surface at grazing incidence (an angle beyond the Brewster angle) will still emerge as right-handed, but elliptically, polarized. Light reflected by a metal at non-normal incidence will generally have its ellipticity changed as well. Such situations may be solved by decomposing the incident circular (or other) polarization into components of linear polarization parallel and perpendicular to the plane of incidence, commonly denoted p and s respectively. The reflected components in the p and s linear polarizations are found by applying the Fresnel coefficients of reflection, which are generally different for those two linear polarizations. Only in the special case of normal incidence, where there is no distinction between p and s, are the Fresnel coefficients for the two components identical, leading to the above property.

Conversion to linear polarization
Circularly polarized light can be converted into linearly polarized light by passing it through a quarter-waveplate. Passing linearly polarized light through a quarter-waveplate with its axes at 45° to its polarization axis will convert it to circular polarization. In fact, this is the most common way of producing circular polarization in practice. Note that passing linearly polarized light through a quarter-waveplate at an angle other than 45° will generally produce elliptical polarization.

Handedness conventions

Circular polarization may be referred to as right-handed or left-handed, and clockwise or anti-clockwise, depending on the direction in which the electric field vector rotates. Unfortunately, two opposing historical conventions exist.

From the point of view of the source
Using this convention, polarization is defined from the point of view of the source. When using this convention, left- or right-handedness is determined by pointing one's left or right thumb from the source, in the  direction that the wave is propagating, and matching the curling of one's fingers to the direction of the temporal rotation of the field at a given point in space. When determining if the wave is clockwise or anti-clockwise circularly polarized, one again takes the point of view of the source, and while looking from the source and in the  direction of the wave's propagation, one observes the direction of the field's temporal rotation.

Using this convention, the electric field vector of a left-handed circularly polarized wave is as follows: $$ \left( E_x ,\, E_y ,\, E_z \right) \propto \left(\cos \frac{2 \pi}{\lambda} \left(c t - z \right),\, -\sin \frac{2 \pi}{\lambda} \left(c t - z \right),\, 0 \right). $$

As a specific example, refer to the circularly polarized wave in the first animation. Using this convention, that wave is defined as right-handed because when one points one's right thumb in the same direction of the wave's propagation, the fingers of that hand curl in the same direction of the field's temporal rotation. It is considered clockwise circularly polarized because, from the point of view of the source, looking in the same direction of the wave's propagation, the field rotates in the clockwise direction. The second animation is that of left-handed or anti-clockwise light, using this same convention.

This convention is in conformity with the Institute of Electrical and Electronics Engineers (IEEE) standard and, as a result, it is generally used in the engineering community.

Quantum physicists also use this convention of handedness because it is consistent with their convention of handedness for a particle's spin.

Radio astronomers also use this convention in accordance with an International Astronomical Union (IAU) resolution made in 1973.

From the point of view of the receiver
In this alternative convention, polarization is defined from the point of view of the receiver. Using this convention, left- or right-handedness is determined by pointing one's left or right thumb the source,  the direction of propagation, and then matching the curling of one's fingers to the temporal rotation of the field.

When using this convention, in contrast to the other convention, the defined handedness of the wave matches the handedness of the screw type nature of the field in space. Specifically, if one freezes a right-handed wave in time, when one curls the fingers of one's right hand around the helix, the thumb will point in the direction of progression for the helix, given the sense of rotation. Note that, in the context of the nature of all screws and helices, it does not matter in which direction you point your thumb when determining its handedness.

When determining if the wave is clockwise or anti-clockwise circularly polarized, one again takes the point of view of the receiver and, while looking the source,  the direction of propagation, one observes the direction of the field's temporal rotation.

Just as in the other convention, right-handedness corresponds to a clockwise rotation, and left-handedness corresponds to an anti-clockwise rotation.

Many optics textbooks use this second convention. It is also used by SPIE as well as the International Union of Pure and Applied Chemistry (IUPAC).

Uses of the two conventions
As stated earlier, there is significant confusion with regards to these two conventions. As a general rule, the engineering, quantum physics, and radio astronomy communities use the first convention, in which the wave is observed from the point of view of the source. In many physics textbooks dealing with optics, the second convention is used, in which the light is observed from the point of view of the receiver.

To avoid confusion, it is good practice to specify "as defined from the point of view of the source" or "as defined from the point of view of the receiver" when discussing polarization matters.

The archive of the US Federal Standard 1037C proposes two contradictory conventions of handedness.

Note that the IEEE defines RHCP and LHCP the opposite as those used by physicists. The IEEE 1979 Antenna Standard will show RHCP on the South Pole of the Poincare Sphere. The IEEE defines RHCP using the right hand with thumb pointing in the direction of transmit, and the fingers showing the direction of rotation of the E field with time. The rationale for the opposite conventions used by Physicists and Engineers is that Astronomical Observations are always done with the incoming wave traveling toward the observer, where as for most engineers, they are assumed to be standing behind the transmitter watching the wave traveling away from them. This article is not using the IEEE 1979 Antenna Standard and is not using the +t convention typically used in IEEE work.

FM radio
FM broadcast radio stations sometimes employ circular polarization to improve signal penetration into buildings and vehicles. It is one example of what the International Telecommunication Union refers to as "mixed polarization", i.e. radio emissions that include both horizontally- and vertically-polarized components. In the United States, Federal Communications Commission regulations state that horizontal polarization is the standard for FM broadcasting, but that "circular or elliptical polarization may be employed if desired".

Dichroism
Circular dichroism (CD) is the differential absorption of left- and right-handed circularly polarized light. Circular dichroism is the basis of a form of spectroscopy that can be used to determine the optical isomerism and secondary structure of molecules.

In general, this phenomenon will be exhibited in absorption bands of any optically active molecule. As a consequence, circular dichroism is exhibited by most biological molecules, because of the dextrorotary (e.g., some sugars) and levorotary (e.g., some amino acids) molecules they contain. Noteworthy as well is that a secondary structure will also impart a distinct CD to its respective molecules. Therefore, the alpha helix, beta sheet and random coil regions of proteins and the double helix of nucleic acids have CD spectral signatures representative of their structures.

Also, under the right conditions, even non-chiral molecules will exhibit magnetic circular dichroism — that is, circular dichroism induced by a magnetic field.

Luminescence
Circularly polarized luminescence (CPL) can occur when either a luminophore or an ensemble of luminophores is chiral. The extent to which emissions are polarized is quantified in the same way it is for circular dichroism, in terms of the dissymmetry factor, also sometimes referred to as the anisotropy factor. This value is given by:

 * $$ g_{em} \ =\ 2\left ( {\theta_\mathrm{left} - \theta_\mathrm{right} \over \theta_\mathrm{left} + \theta_\mathrm{right} } \right )   $$

where $$ \theta_\mathrm{left} $$ corresponds to the quantum yield of left-handed circularly polarized light, and $$ \theta_\mathrm{right} $$ to that of right-handed light. The maximum absolute value of gem, corresponding to purely left- or right-handed circular polarization, is therefore 2. Meanwhile, the smallest absolute value that gem can achieve, corresponding to linearly polarized or unpolarized light, is zero.

Mathematical description
The classical sinusoidal plane wave solution of the electromagnetic wave equation for the electric and magnetic fields is:
 * $$\begin{align}

\mathbf{E} ( \mathbf{r}, t ) &= \left| \,\mathbf{E}\, \right| \mathrm{Re} \left\{ \mathbf{Q} \left|\psi\right\rangle \exp \left[ i \left( kz - \omega t \right) \right] \right\} \\ \mathbf{B} ( \mathbf{r}, t ) &= \dfrac{1}{c} \hat{ \mathbf{z} } \times \mathbf{E} ( \mathbf{r}, t ) \end{align}$$

where k is the wavenumber;
 * $$ \omega = c k$$

is the angular frequency of the wave; $$ \mathbf{Q} = \left [ \hat{ \mathbf{x}}, \hat{\mathbf{y}} \right ] $$ is an orthogonal $$ 2 \times 2$$ matrix whose columns span the transverse x-y plane; and $$ c $$ is the speed of light.

Here,
 * $$ \left| \,\mathbf{E}\, \right| $$

is the amplitude of the field, and
 * $$ |\psi\rangle \ \stackrel{\mathrm{def}}{=}\ \begin{pmatrix} \psi_x \\ \psi_y  \end{pmatrix} =  \begin{pmatrix} \cos\theta \exp \left ( i \alpha_x \right )  \\ \sin\theta \exp \left ( i \alpha_y \right )  \end{pmatrix}  $$

is the normalized Jones vector in the x-y plane.

If $$ \alpha_y  $$ is rotated by $$  \pi / 2  $$ radians with respect to $$  \alpha_x  $$ and the x amplitude equals the y amplitude, the wave is circularly polarized. The Jones vector is:
 * $$ |\psi\rangle = {1\over \sqrt{2}} \begin{pmatrix} 1  \\ \pm i \end{pmatrix} \exp \left ( i \alpha_x \right )  $$

where the plus sign indicates left circular polarization, and the minus sign indicates right circular polarization. In the case of circular polarization, the electric field vector of constant magnitude rotates in the x-y plane.

If basis vectors are defined such that:
 * $$ |\mathrm{R}\rangle \ \stackrel{\mathrm{def}}{=}\  {1\over \sqrt{2}}\begin{pmatrix} 1  \\ -i \end{pmatrix}  $$

and:
 * $$ |\mathrm{L}\rangle \ \stackrel{\mathrm{def}}{=}\  {1\over \sqrt{2}}\begin{pmatrix} 1  \\ i \end{pmatrix}  $$

then the polarization state can be written in the "R-L basis" as:
 * $$ |\psi\rangle = \psi_\mathrm{R} |\mathrm{R}\rangle + \psi_\mathrm{L} |\mathrm{L}\rangle $$

where:
 * $$\begin{align}

\psi_\mathrm{R} ~&\stackrel{\mathrm{def}}{=}~ \frac{1}{\sqrt{2}} \left( \cos\theta + i\sin\theta \exp \left( i \delta \right) \right) \exp \left( i \alpha_x \right) \\ \psi_\mathrm{L} ~&\stackrel{\mathrm{def}}{=}~ \frac{1}{\sqrt{2}} \left( \cos\theta - i\sin\theta \exp \left( i \delta \right) \right) \exp \left( i \alpha_x \right) \end{align}$$

and:

 * $$ \delta= \alpha_y - \alpha_x. $$

Antennas
A number of different types of antenna elements can be used to produce circularly polarized (or nearly so) radiation; following Balanis, one can use dipole elements: "'... two crossed dipoles provide the two orthogonal field components.... If the two dipoles are identical, the field intensity of each along zenith ... would be of the same intensity. Also, if the two dipoles were fed with a 90° degree time-phase difference (phase quadrature), the polarization along zenith would be circular.... One way to obtain the 90° time-phase difference between the two orthogonal field components, radiated respectively by the two dipoles, is by feeding one of the two dipoles with a transmission line which is 1/4 wavelength longer or shorter than that of the other,' p.80;" or helical elements: "'To achieve circular polarization [in axial or end-fire mode] ... the circumference C of the helix must be ... with C/wavelength = 1 near optimum, and the spacing about S = wavelength/4,' p.571;" or patch elements: "'... circular and elliptical polarizations can be obtained using various feed arrangements or slight modifications made to the elements.... Circular polarization can be obtained if two orthogonal modes are excited with a 90° time-phase difference between them. This can be accomplished by adjusting the physical dimensions of the patch.... For a square patch element, the easiest way to excite ideally circular polarization is to feed the element at two adjacent edges.... The quadrature phase difference is obtained by feeding the element with a 90° power divider,' p.859."

In quantum mechanics
In the quantum mechanical view, light is composed of photons. Polarization is a manifestation of the spin angular momentum of light. More specifically, in quantum mechanics, the direction of spin of a photon is tied to the handedness of the circularly polarized light, and the spin of a beam of photons is similar to the spin of a beam of particles, such as electrons.

In nature

Only a few mechanisms in nature are known to systematically produce circularly polarized light. In 1911, Albert Abraham Michelson discovered that light reflected from the golden scarab beetle Chrysina resplendens is preferentially left-polarized. Since then, circular polarization has been measured in several other scarab beetles such as Chrysina gloriosa, as well as some crustaceans such as the mantis shrimp. In these cases, the underlying mechanism is the molecular-level helicity of the chitinous cuticle.

The bioluminescence of the larvae of fireflies is also circularly polarized, as reported in 1980 for the species Photuris lucicrescens and Photuris versicolor. For fireflies, it is more difficult to find a microscopic explanation for the polarization, because the left and right lanterns of the larvae were found to emit polarized light of opposite senses. The authors suggest that the light begins with a linear polarization due to inhomogeneities inside aligned photocytes, and it picks up circular polarization while passing through linearly birefringent tissue.

Circular polarization has been detected in light reflected from leaves and photosynthetic microbes.

Water-air interfaces provide another source of circular polarization. Sunlight that gets scattered back up towards the surface is linearly polarized. If this light is then totally internally reflected back down, its vertical component undergoes a phase shift. To an underwater observer looking up, the faint light outside Snell's window therefore is (partially) circularly polarized.

Weaker sources of circular polarization in nature include multiple scattering by linear polarizers, as in the circular polarization of starlight, and selective absorption by circularly dichroic media.

Radio emission from pulsars can be strongly circularly polarized.

Two species of mantis shrimp have been reported to be able to detect circular polarized light.

title: Circular polarization


text:
Circulator

In electrical engineering, a circulator is a passive, non-reciprocal three- or four-port device that only allows a microwave or radio-frequency (RF) signal to exit through the port directly after the one it entered. Optical circulators have similar behavior. Ports are where an external waveguide or transmission line, such as a microstrip line or a coaxial cable, connects to the device. For a three-port circulator, a signal applied to port 1 only comes out of port 2; a signal applied to port 2 only comes out of port 3; a signal applied to port 3 only comes out of port 1. An ideal three-port circulator thus has the following scattering matrix:

 * $$S = \begin{pmatrix}

0 & 0 & 1\\ 1 & 0 & 0 \\  0 & 1 & 0 \end{pmatrix}$$

Theory of operation
Microwave circulators rely on the anisotropic and non-reciprocal properties of magnetized microwave ferrite material. Microwave electromagnetic waves propagating in magnetized ferrite interact with electron spins in the ferrite and are consequently influenced by the microwave magnetic permeability of the ferrite. This permeability is mathematically described by a linear vector operator, also known as a tensor. In the case of magnetized ferrite, the permeability tensor is the Polder tensor. The permeability is a function of the direction of microwave propagation relative to the direction of static magnetization of the ferrite material. Hence, microwave signals propagating in different directions in the ferrite experience different magnetic permeabilities.

In the CGS system, the Polder tensor is

 * $$B = \begin{bmatrix} \mu & j \kappa & 0 \\ -j \kappa & \mu & 0 \\ 0 & 0 & 1 \end{bmatrix} H$$

where (neglecting damping)

 * $$\mu = 1 + \frac{\omega_0 \omega_m}{\omega_0^2 - \omega^2} $$

 * $$\kappa = \frac{\omega \omega_m}{{\omega_0}^2 - \omega^2}$$

 * $$\omega_0 = \gamma H_0 \ $$
 * $$\omega_m = \gamma M \ $$

$$\gamma = 1.40 \cdot g \,\, $$ MHz / Oe  is the effective gyromagnetic ratio and $$g$$, the so-called effective g-factor, is a ferrite material constant typically in the range of 1.5 - 2.6, depending on the particular ferrite material. $$\omega$$ is the frequency of the RF/microwave signal propagating through the ferrite, $$H_0$$ is the internal magnetic bias field, and $$M$$ is the magnetization of the ferrite material.

In junction circulators and differential phase shift circulators, microwave signal propagation is usually orthogonal to the static magnetic bias field in the ferrite. This is the so-called transverse field case. The microwave propagation constants for this case, neglecting losses are

 * $$\Gamma_+ = j\omega\sqrt{\mu_0 \epsilon}\,\sqrt{\frac{\mu^2 - \kappa^2} {\mu}}$$

 * $$\Gamma_- = j\omega\sqrt{\mu_0 \epsilon}$$

where $$\mu_0$$ is the Permeability of Free Space and $$\epsilon$$ is the Absolute permittivity of the ferrite material. In a circulator, these propagation constants describe waves having Elliptical polarization that would propagate in the direction of the static magnetic bias field, which is through the thickness of the ferrite. The plus and minus subscripts of the propagation constants indicate opposite wave polarizations.

Types
Microwave circulators fall into two main classes: differential phase shift circulators and junction circulators, both of which are based on cancellation of waves propagating over two different paths in or near magnetized ferrite material. Waveguide circulators may be of either type, while more compact devices based on stripline are usually of the junction type. Two or more junction circulators can be combined in a single component to give four or more ports. Typically permanent magnets produce a static magnetic bias in the microwave ferrite material. Ferrimagnetic garnet crystal is used in optical circulators.

Stripline junction circulators

A stripline junction circulator contains a resonator, which is located at the central junction of the striplines. This resonator may have any shape that has three-fold Rotational symmetry, such as a disk, hexagon, or triangle. An RF/microwave signal entering a circulator port is connected via a stripline to the resonator, where energy is coupled into two counter-rotating circular modes formed by the elliptically polarized waves. These circular modes have different phase velocities which can cause them to combine constructively or destructively at a given port. This produces an anti-node at one port (port 2 if the signal is incident upon port 1) and a node or null at another port (port 3 if the microwave energy is coupled from port 1 to port 2 and not reflected back into port 2).

If losses are neglected for simplification, the counter-rotating modes must differ in phase by an integer multiple of $$2\pi$$ for signal propagation from port 1 to port 2 (or from port 2 to port 3, or from port 3 to port 1):

 * $$2\Gamma_-l - \Gamma_+l = 2m\pi$$

and similarly, for the remaining port (port 3 if signal propagation is from port 1 to port 2) to be nulled,

 * $$-\Gamma_-l + 2\Gamma_+l = (2n - 1)\pi$$

where $$l$$ is the path length between adjacent ports and $$m$$ and $$n$$ are integers. Solving the two preceding equations simultaneously, for proper circulation the necessary conditions are

 * $$\Gamma_-l = \frac{4m + 2n - 1}{3}\pi$$

and

 * $$\Gamma_+l = \frac{2m + 4n - 2}{3}\pi$$

Each of the two counter-rotating modes has its own resonant frequency. The two resonant frequencies are known as the split frequencies. The circulator operating frequency is set between the two split frequencies.

These circulator types operate based on faraday rotation. Wave cancellation occurs when waves propagate with and against the circulator's direction of circulation. An incident wave arriving at any port is split equally into two waves. They propagate in each direction around the circulator with different phase velocities. When they arrive at the output port they have different phase relationships and thus combine accordingly. This combination of waves propagating at different phase velocities is how junction circulators fundamentally operate.

The geometry of a stripline junction circulator comprises two ferrite disks or triangles separated by a stripline center conductor and sandwiched between two parallel ground planes. A stripline circulator is essentially a stripline center conductor sandwich on ferrite, between ground planes. That is, there is one ferrite disk above the stripline circuit and one ferrite disk below the stripline circuit. Stripline circulators do not have to be constructed with disk- or triangle-shaped ferrites; the ferrites can have almost any shape that has three-way symmetry. This is also true of the resonator (the center junction portion of the center conductor)- it can be any shape that has three-way symmetry, although there are electrical considerations.

The ferrites are magnetized through their thicknesses, i.e., the static magnetic bias field is perpendicular to the plane of the device and the direction of signal propagation is transverse to the direction of the static magnetic field. Both ferrites are in the same static ad RF magnetic fields. The two ferrites can be thought of as one continuous ferrite with an embedded stripline center conductor. For practical manufacturing reasons, the center conductor is not generally embedded in ferrite, so two discrete ferrites are used. The static magnetic bias field is typically provided by permanent magnets that are located external to the circulator ground planes. Magnetic shielding incorporated into the circulator design prevents detuning or partial demagnetization of the circulator in the presence of external magnetic fields or ferrous materials, and protects nearby devices from the effects of the circulator's static magnetic field.

Waveguide junction circulators

A waveguide junction circulator contains a magnetized ferrite resonator, which is located at the junction of three waveguides. In contrast with a stripline junction circulator, the ferrite itself is the resonator, rather than the metal central portion of a stripline center conductor. The ferrite resonator may have any shape that has three-fold Rotational symmetry, such as a cylinder or Triangular prism. The resonator is often just one ferrite, but it is sometimes composed of two or more ferrites, which may be coupled to each other, in various geometrical configurations. The geometry of the resonator is influenced by electrical and thermal performance considerations. Waveguide junction circulators function in much the same way as stripline junction circulators, and their basic theory of operation is the same.

The internal geometry of a waveguide junction circulator comprises a junction of three waveguides, the ferrite resonator, and impedance matching structures. Many of these circulators contain pedestals located in the central junction, on which the ferrite resonator is located. These pedestals effectively reduce the height of the waveguide, reducing its characteristic impedance in the resonator region to optimize electrical performance. The reduced-height waveguide sections leading from the resonator to the full-height waveguides serve as impedance transformers.

The ferrite resonator is magnetized through its height, i.e., the static magnetic bias field is perpendicular to the plane of the device and the direction of signal propagation is transverse to the direction of the static magnetic field. The static magnetic bias field is typically provided by permanent magnets that are external to the waveguide junction.

E-Field Plots Showing Electromagnetic Wave Propagation in Waveguide Junction Circulators

Microstrip junction circulators

The microstrip junction circulator is another widely-used form of circulator that utilizes the microstrip transmission line topology. A microstrip circulator consists primarily of a circuit pattern on a ferrite substrate. The circuit is typically formed using thick-film or thin-film metallization processes, often including photolithography. The ferrite substrate is sometimes bonded to a ferrous metal carrier, which serves to improve the efficiency of the magnetic circuit, increase the mechanical strength of the circulator, and protect the ferrite from thermal expansion mismatches between it and the surface to which the circulator is mounted. A permanent magnet that is bonded to the circuit face of the ferrite substrate provides the static magnetic bias to the ferrite. Microstrip circulators function in the same way as stripline junction circulators, and their basic theory of operation is the same. In comparison with stripline circulators, electrical performance of microstrip circulators is somewhat reduced because of radiation and dispersion effects.

The performance disadvantages of microstrip circulators are offset by their relative ease of integration with other planar circuitry. The electrical connections of these circulators to adjacent circuitry are typically made using wire bonds or ribbon bonds. Another advantage of microstrip circulators is their smaller size and correspondingly lower mass than stripline circulators. Despite this advantage, microstrip circulators are often the largest components in microwave modules.

Self-biased junction circulators

Self-biased junction circulators are unique in that they do not utilize permanent magnets that are separate from the microwave ferrite. The elimination of external magnets significantly reduces the size and weight of the circulator compared to electrically-equivalent microstrip junction circulators for similar applications.

Monolithic ferrites that are used for self-biased circulators are M-type uniaxial (single magnetic axis) hexagonal ferrites that have been optimized to have low microwave losses. In contrast with the magnetically soft (low-coercivity) ferrites used in other circulators, the hexagonal ferrites used for self-biased circulators are magnetically hard (high-coercivity) materials. These ferrites are essentially ceramic permanent magnets. In addition to their high magnetic remanence, these ferrites have very large magnetic anisotropy fields, enabling circulator operation up to high microwave frequencies.

Because of their thin, planar shape, self-biased circulators can be conveniently integrated with other planar circuitry. Integration of self-biased circulators with semiconductor wafers has been demonstrated at KA-band and V-band frequencies.

Lumped-element circulators

Lumped-element circulators are small-size devices that are typically used at frequencies in the HF through UHF bands. In a junction circulator, the size of the ferrite(s) is proportional to signal wavelength, but in a lumped-element circulator, the ferrite can be smaller because there is no such wavelength proportionality.

In a lumped-element circulator, conductors are wrapped around the ferrite, forming what is typically a woven mesh. The conductor strips are insulated from each other by thin dielectric layers. In some circulators, the mesh is in the form of traces on a printed wiring board with metallized vias to make connections between layers. The conductive strips can be thought of as non-reciprocally coupled inductors. Impedance matching circuitry and broad-banding circuitry in lumped-element circulators are often constructed using discrete ceramic capacitors and air-core inductors.

This class of circulator offers a considerable size reduction compared with the junction circulators. On the other hand, lumped-element circulators generally have lower RF power handling capacity than equivalent junction devices and are more complex from a mechanical perspective. The discrete lumped-element inductors and capacitors can be less stable when exposed to vibration or mechanical shocks than the simple distributed impedance transformers in a stripline junction circulator.

Switching circulators

Switching circulators are similar to other junction circulators, and their microwave theory of operation is the same, except that their direction of circulation can be electronically controlled.

Junction circulators use permanent magnets to provide the static magnetic bias for the ferrite(s). However, switching circulators typically rely on the remanent magnetization of the ferrite itself. The ferrites that are used in switching circulators have square magnetic hysteresis loops and often sub-Oersted coercivities. Such a ferrite material requires a relatively small magnetic field and low energy level to flip its magnetic polarity. This is distinctly advantageous for a switching circulator, but the absence of permanent magnets would be a disadvantage of a non-switching junction circulator that must retain its magnetic bias despite exposures to the potentially demagnetizing effects of stray magnetic fields, nearby ferrous materials, and temperature variations.

The magnetization polarity of the ferrite, and hence the direction of circulation of a switching circulator, is controlled using a magnetizing coil that loops through the ferrite. The coil is connected to electronic driver circuitry that sends current pulses of the correct polarity through the magnetizing coil to magnetize the ferrite in the polarity to provide the desired direction of circulation.

Differential phase shift circulators

Differential phase shift circulators are mainly used in high power microwave applications. They are usually built from rectangular waveguide components. These circulators are 4-port devices having circulation in the sequence 1 - 2 - 3 - 4 - 1, with ports numbered as shown in the schematic. There are various feasible circulator architectures, the most common of which utilizes a magic tee hybrid coupler, a quadrature hybrid coupler, and two oppositely-magnetized differential phase shifters.

A differential phase shifter provides non-reciprocal transmission phase shift. That is, the forward phase shift is different from the phase shift in the reverse transmission direction. It is this difference in phase shifts that enables the non-reciprocal behavior of the circulator. A differential phase shifter consists of one or more ferrite slabs, usually positioned on the broad wall(s) of the waveguide. Permanent magnets located outside the waveguide provide static magnetic bias field to the ferrite(s). The ferrite-loaded waveguide is another example of a transverse-field device as described in. Different microwave propagation constants corresponding to different directions of signal propagation give rise to different phase velocities and hence, different transmission phase shifts.

Depending on which circulator port an incident signal enters, phase shift relationships in the hybrid couplers and the differential phase shifts cause signals to combine at one other port and cancel at each of the remaining two ports. Differential phase shift circulators are often used as 3-port circulators by connecting one circulator port to a reflectionless termination, or they can be used as isolators by terminating two circulator ports.

Non-ferrite circulators
Though ferrite circulators can provide good "forward" signal circulation while suppressing greatly the "reverse" circulation, their major shortcomings, especially at low frequencies, are the bulky sizes and the narrow bandwidths.

Early work on non-ferrite circulators includes active circulators using transistors that are non-reciprocal in nature. In contrast to ferrite circulators which are passive devices, active circulators require power. Major issues associated with transistor-based active circulators are the power limitation and the signal-to-noise degradation, which are critical when it is used as a duplexer for sustaining the strong transmit power and clean reception of the signal from the antenna.

Varactors offer one solution. One study employed a structure similar to a time-varying transmission line with the effective nonreciprocity triggered by a one-direction propagating carrier pump. This is like an AC-powered active circulator. The research claimed to be able to achieve positive gain and low noise for receiving path and broadband nonreciprocity. Another study used resonance with nonreciprocity triggered by angular-momentum biasing, which more closely mimics the way that signals passively circulate in a ferrite circulator.

In 1964, Mohr presented and experimentally demonstrated a circulator based on transmission lines and switches. In April, 2016 a research team significantly extended this concept, presenting an integrated circuit circulator based on N-path filter concepts. It offers the potential for full-duplex communication (transmitting and receiving at the same time with a single shared antenna over a single frequency). The device uses capacitors and a clock and is much smaller than conventional devices.

Isolator
When one port of a three-port circulator is terminated in a matched load, it can be used as an isolator, since a signal can travel in only one direction between the remaining ports. An isolator is used to shield equipment on its input side from the effects of conditions on its output side; for example, to prevent a microwave source being detuned by a mismatched load.

Duplexer
In radar, circulators are used as a type of duplexer, to route signals from the transmitter to the antenna and from the antenna to the receiver, without allowing signals to pass directly from transmitter to receiver. The alternative type of duplexer is a transmit-receive switch (TR switch) that alternates between connecting the antenna to the transmitter and to the receiver. The use of chirped pulses and a high dynamic range may lead to temporal overlap of the sent and received pulses, however, requiring a circulator for this function.

Reflection amplifier
A reflection amplifier is a type of microwave amplifier circuit utilizing negative differential resistance diodes such as tunnel diodes and Gunn diodes. Negative differential resistance diodes can amplify signals, and often perform better at microwave frequencies than two-port devices. However, since the diode is a one-port (two terminal) device, a nonreciprocal component is needed to separate the outgoing amplified signal from the incoming input signal. By using a 3-port circulator with the signal input connected to one port, the biased diode connected to a second, and the output load connected to the third, the output and input can be uncoupled.

title: Circulator


text:
Cladding

Cladding is an outer layer of material covering another. It may refer to the following:

 * Cladding (boiler), the layer of insulation and outer wrapping around a boiler shell
 * Cladding (construction), materials applied to the exterior of buildings
 * Wall cladding, exterior material applied to the walls of a building
 * Copper cladding, applying copper to the exterior of buildings
 * Rainscreen cladding, an exterior wall detail to create a capillary break and to allow drainage and evaporation of water
 * Cladding (fiber optics), fiber optics property to contain light in the core of the fiber by total internal reflection
 * Cladding (metalworking), a bonding together of dissimilar metals
 * Cladding (nuclear fuel), the outer layer of the fuel rods, standing between the coolant and the nuclear fuel

title: Cladding


text:
Clearing

Clearing or The Clearing may refer to:

Arts and media

 * Clearing (Fred Frith album), 2001
 * Clearing (Hyd album), 2022
 * The Clearing (film), a 2004 drama film
 * The Clearing (EP), a 2006 EP by Weatherbox
 * The Clearing (Sleep for Sleepers album), 2009
 * The Clearing (Locrian album), 2011
 * The Clearing (Homeland), an episode of the American television series Homeland
 * The Clearing (podcast), a 2019 true crime podcast
 * The Clearing (TV series), 2023

Ecology

 * Clearing (forest), a tract of land with few or no trees in the middle of a wooded area
 * Clearing (geography), the process by which vegetation is permanently removed
 * Deforestation, the clearing away of trees to make farmland

Economics and finance

 * Clearing (finance), the process of settling a transaction after committing to it
 * Market clearing, the matching of supply and demand via price movement

Other uses

 * Clearing, Chicago, a community area in Illinois, U.S.
 * The Clearing (Ellison Bay, Wisconsin), also known as The Clearing Folk School
 * Clearing (telecommunications), the disconnecting of a call
 * Yarn clearing, in textile industry
 * Clearing, a practice in Scientology
 * Clearing, a process used by the UK's Universities and Colleges Admissions Service (UCAS)

title: Clearing