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Coupling

A coupling is a device used to connect two shafts together at their ends for the purpose of transmitting power. The primary purpose of couplings is to join two pieces of rotating equipment while permitting some degree of misalignment or end movement or both. In a more general context, a coupling can also be a mechanical device that serves to connect the ends of adjacent parts or objects. Couplings do not normally allow disconnection of shafts during operation, however there are torque-limiting couplings which can slip or disconnect when some torque limit is exceeded. Selection, installation and maintenance of couplings can lead to reduced maintenance time and maintenance cost.

Uses
Shaft couplings are used in machinery for several purposes. A primary function is to transfer power from one end to another end (ex: motor transfer power to pump through coupling).

Other common uses:

 * To alter the vibration characteristics of rotating units
 * To connect the driving and the driven part
 * To introduce protection
 * To reduce the transmission of shock loads from one shaft to another
 * To slip when overload occurs

Beam
A beam coupling, also known as helical coupling, is a flexible coupling for transmitting torque between two shafts while allowing for angular misalignment, parallel offset and even axial motion, of one shaft relative to the other. This design utilizes a single piece of material and becomes flexible by removal of material along a spiral path resulting in a curved flexible beam of helical shape. Since it is made from a single piece of material, the beam style coupling does not exhibit the backlash found in some multi-piece couplings. Another advantage of being an all machined coupling is the possibility to incorporate features into the final product while still keeping the single piece's integrity.

Changes to the lead of the helical beam provide changes to misalignment capabilities as well as other performance characteristics such as torque capacity and torsional stiffness. It is even possible to have multiple starts within the same helix.

The material used to manufacture the beam coupling also affects its performance and suitability for specific applications such as food, medical and aerospace. Materials are typically aluminum alloy and stainless steel, but they can also be made in acetal, maraging steel and titanium. The most common applications are attaching rotary encoders to shafts and motion control for robotics.

Beam couplings can be known by various names depending upon industry. These names include flexible coupling, flexible beam coupling, flexible shaft coupling, flexure, helical coupling, and shaft coupling.

The primary benefit to using a flexible beam coupling to join two rotating shafts is to reducing vibration and reaction loads which in turn will reduce overall wear and tear on machinery and prolong equipment life.

Bush pin flange
Bush pin flange coupling is used for slightly imperfect alignment of the two shafts.

This is modified form of the protected type flange coupling. This type of coupling has pins and it works with coupling bolts. The rubber or leather bushes are used over the pins. The coupling has two halves dissimilar in construction. The pins are rigidly fastened by nuts to one of the flange and kept loose on the other flange. This coupling is used to connect shafts which have a small parallel misalignment, angular misalignment or axial misalignment. In this coupling the rubber bushing absorbs shocks and vibration during its operations. This type of coupling is mostly used to couple electric motors and machines.

Constant velocity
There are various types of constant-velocity (CV) couplings: Rzeppa joint, Double cardan joint, and Thompson coupling.

Clamp or split-muff
In this coupling, the muff or sleeve is made into two halves parts of the cast iron and they are joined by means of mild steel studs or bolts. The advantages of this coupling is that assembling or disassembling of the coupling is possible without changing the position of the shaft. This coupling is used for heavy power transmission at moderate speed.

Diaphragm
Diaphragm couplings transmit torque from the outside diameter of a flexible plate to the inside diameter, across the spool or spacer piece, and then from inside to outside diameter. The deforming of a plate or series of plates from I.D. to O.D accomplishes the misalignment.

Disc
Disc couplings transmit torque from a driving to a driven bolt tangentially on a common bolt circle. Torque is transmitted between the bolts through a series of thin, stainless steel discs assembled in a pack. Misalignment is accomplished by deforming of the material between the bolts.

Elastic
An elastic coupling transmits torque or other load by means of an elastic component. One example is the coupling used to join a windsurfing rig (sail, mast, and components) to the sailboard. In windsurfing terminology it is usually called a "universal joint", but modern designs are usually based on a strong flexible material, and better technically described as an elastic coupling. They can be tendon or hourglass-shaped, and are constructed of a strong and durable elastic material. In this application, the coupling does not transmit torque, but instead transmits sail-power to the board, creating thrust (some portion of sail-power is also transmitted through the rider's body).

Flexible
Flexible couplings are usually used to transmit torque from one shaft to another when the two shafts are slightly misaligned. They can accommodate varying degrees of misalignment up to 1.5° and some parallel misalignment. They can also be used for vibration damping or noise reduction. In rotating shaft applications a flexible coupling can protect the driving and driven shaft components (such as bearings) from the harmful effects of conditions such as misaligned shafts, vibration, shock loads, and thermal expansion of the shafts or other components.

At first, flexible couplings separate into two essential groups, metallic and elastomeric. Metallic types utilize freely fitted parts that roll or slide against one another or, on the other hand, non-moving parts that bend to take up misalignment. Elastomeric types, then again, gain flexibility from resilient, non-moving, elastic or plastic elements transmitting torque between metallic hubs.

Gear
A gear coupling is a mechanical device for transmitting torque between two shafts that are not collinear. It consists of a flexible joint fixed to each shaft. The two joints are connected by a third shaft, called the spindle.

Each joint consists of a 1:1 gear ratio internal/external gear pair. The tooth flanks and outer diameter of the external gear are crowned to allow for angular displacement between the two gears. Mechanically, the gears are equivalent to rotating splines with modified profiles. They are called gears because of the relatively large size of the teeth.

Gear couplings and universal joints are used in similar applications. Gear couplings have higher torque densities than universal joints designed to fit a given space while universal joints induce lower vibrations. The limit on torque density in universal joints is due to the limited cross sections of the cross and yoke. The gear teeth in a gear coupling have high backlash to allow for angular misalignment. The excess backlash can contribute to vibration.

Gear couplings are generally limited to angular misalignments, i.e., the angle of the spindle relative to the axes of the connected shafts, of 4–5°. Universal joints are capable of higher misalignments.

Single joint gear couplings are also used to connect two nominally coaxial shafts. In this application the device is called a gear-type flexible, or flexible coupling. The single joint allows for minor misalignments such as installation errors and changes in shaft alignment due to operating conditions. These types of gear couplings are generally limited to angular misalignments of 1/4–1/2°.

Grid
A grid coupling is composed of two shaft hubs, a metallic grid spring, and a split cover kit. Torque is transmitted between the two coupling shaft hubs through the metallic grid spring element.

Like metallic gear and disc couplings, grid couplings have a high torque density. A benefit of grid couplings, over either gear or disc couplings, is the ability their grid coupling spring elements have to absorb and spread peak load impact energy over time. This reduces the magnitude of peak loads and offers some vibration dampening capability. A negative of the grid coupling design is that it generally is very limited in its ability to accommodate the misalignment.

Highly flexible
Highly flexible couplings are installed when resonance or torsional vibration might be an issue, since they are designed to eliminate torsional vibration problems and to balance out shock impacts.

They are used in installations where the systems require a high level of torsional flexibility and misalignment capacity. This type of coupling provides an effective damping of torsional vibrations, and high displacement capacity, which protects the drive. The design of the highly flexible elastic couplings makes assembly easier. These couplings also compensate shaft displacements (radial, axial and angular) and the torque is transmitted in shear. Depending on the size and stiffness of the coupling, the flexible part may be single- or multi-row.

Hirth joints
Hirth joints use tapered teeth on two shaft ends meshed together to transmit torque.

Jaw
Jaw coupling is also known as spider or Lovejoy coupling.

Magnetic
A magnetic coupling uses magnetic forces to transmit the power from one shaft to another without any contact. This allows for full medium separation. It can provide the ability to hermetically separate two areas whilst continuing to transmit mechanical power from one to the other making these couplings ideal for applications where prevention of cross-contamination is essential.

Oldham
An Oldham coupling has three discs, one coupled to the input, one coupled to the output, and a middle disc that is joined to the first two by tongue and groove. The tongue and groove on one side is perpendicular to the tongue and groove on the other. The middle disc rotates around its center at the same speed as the input and output shafts. Its center traces a circular orbit, twice per rotation, around the midpoint between input and output shafts. Often springs are used to reduce backlash of the mechanism. An advantage to this type of coupling, as compared to two universal joints, is its compact size. The coupler is named for John Oldham who invented it in Ireland, in 1821, to solve a problem in a paddle steamer design.

Rag joint
Rag joints are commonly used on automotive steering linkages and drive trains. When used on a drive train they are sometimes known as giubos.

Rigid
Rigid couplings are used when precise shaft alignment is required; any shaft misalignment will affect the coupling's performance as well as its life span, because rigid couplings do not have the ability to compensate for misalignment. Due to this, their application is limited, and they're typically used in applications involving vertical drivers.

Clamped or compression rigid couplings come in two parts and fit together around the shafts to form a sleeve. They offer more flexibility than sleeved models, and can be used on shafts that are fixed in place. They generally are large enough so that screws can pass all the way through the coupling and into the second half to ensure a secure hold. Flanged rigid couplings are designed for heavy loads or industrial equipment. They consist of short sleeves surrounded by a perpendicular flange. One coupling is placed on each shaft so the two flanges line up face to face. A series of screws or bolts can then be installed in the flanges to hold them together. Because of their size and durability, flanged units can be used to bring shafts into alignment before they are joined.

Sleeve, box, or muff
A sleeve coupling consists of a pipe whose bore is finished to the required tolerance based on the shaft size. Based on the usage of the coupling a keyway is made in the bore in order to transmit the torque by means of the key. Two threaded holes are provided in order to lock the coupling in position.

Sleeve couplings are also known as box couplings. In this case shaft ends are coupled together and abutted against each other which are enveloped by muff or sleeve.

A gib head sunk keys hold the two shafts and sleeve together (this is the simplest type of the coupling) It is made from the cast iron and very simple to design and manufacture. It consists of a hollow pipe whose inner diameter is same as diameter of the shafts. The hollow pipe is fitted over a two or more ends of the shafts with the help of the taper sunk key. A key and sleeve are useful to transmit power from one shaft to another shaft.

Tapered shaft lock
A tapered lock is a form of keyless shaft locking device that does not require any material to be removed from the shaft. The basic idea is similar to a clamp coupling but the moment of rotation is closer to the center of the shaft. An alternative coupling device to the traditional parallel key, the tapered lock removes the possibility of play due to worn keyways. It is more robust than using a key because maintenance only requires one tool and the self-centering balanced rotation means it lasts longer than a keyed joint would, but the downside is that it costs more.

Twin spring
A flexible coupling made from two counter-wound springs with a ball bearing in the center, which allows torque transfer from input to output shaft. Requires no lubrication to consistently run as it has no internal components.

Maintenance and failure
Coupling maintenance requires a regularly scheduled inspection of each coupling. It consists of:
 * Performing visual inspections
 * Checking for signs of wear or fatigue
 * Cleaning couplings regularly
 * Checking and changing lubricant regularly if the coupling is lubricated. This maintenance is required annually for most couplings and more frequently for couplings in adverse environments or in demanding operating conditions.
 * Documenting the maintenance performed on each coupling, along with the date.

Even with proper maintenance, however, couplings can fail. Underlying reasons for failure, other than maintenance, include:
 * Improper installation
 * Poor coupling selection
 * Operation beyond design capabilities.

External signs that indicate potential coupling failure include:
 * Abnormal noise, such as screeching, squealing or chattering
 * Excessive vibration or wobble
 * Failed seals indicated by lubricant leakage or contamination.

Balance
Couplings are normally balanced at the factory prior to being shipped, but they occasionally go out of balance in operation. Balancing can be difficult and expensive, and is normally done only when operating tolerances are such that the effort and the expense are justified. The amount of coupling unbalance that can be tolerated by any system is dictated by the characteristics of the specific connected machines and can be determined by detailed analysis or experience.

title: Coupling


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Cover

Cover or covers may refer to:

Packaging

 * Another name for a lid
 * Cover (philately), generic term for envelope or package
 * Album cover, the front of the packaging
 * Book cover or magazine cover
 * Book design
 * Back cover copy, part of copywriting
 * CD and DVD cover, CD and DVD packaging
 * Smartphone cover, a mobile phone accessory that protects a mobile phone

People

 * Cover (surname)

Albums

 * Cover
 * Cover (Tom Verlaine album), 1984
 * Cover (Joan as Policewoman album), 2009
 * Covered
 * Covered (Cold Chisel album), 2011
 * Covered (Macy Gray album), 2012
 * Covered (Robert Glasper album), 2015
 * Covers
 * Covers (Beni album), 2012
 * Covers (Regine Velasquez album), 2004
 * Covers (Placebo album), 2003
 * Covers (Show of Hands album), 2000
 * Covers (James Taylor album), 2008
 * Covers (Fayray album), 2005
 * Covers (Deftones album), 2011
 * Covers (Cat Power album), 2022
 * Covers, an album by Break of Reality

Extended plays

 * Covers (A Camp EP), 2009
 * Covers (Franz Ferdinand EP), 2009
 * Covers (Get Cape. Wear Cape. Fly EP), 2009
 * Covers (The Autumns EP), 2001
 * Covers (Young Statues EP), 2012

Other uses in music

 * Cover version, a new version of a previously released song
 * Cover, an understudy in opera

Films

 * Cover (film), a 2007 film directed by Bill Duke
 * Covers (film), an upcoming comedy film directed by Nisha Ganatra

Magazines

 * Cover Magazine (publication), a New York City arts monthly publication

Business

 * Cover (finance), repurchasing a short order made on the stock/equity, forex or futures markets
 * Cover (law), a remedy for the breach of a contract for the receipt of goods
 * Cover charge, an entry fee
 * Cover (hospitality)
 * Cover Corp., parent organization for VTuber idol company Hololive Production

Deception and concealment

 * Cover (telecommunications), a communications concealment technique
 * Cover, something fake used in a cover-up
 * Non-official cover, the identity assumed by an operative who takes a covert role in an organization without official ties to the government
 * Official cover, the identity assumed by an operative who takes a position in an organization with diplomatic ties

Mathematics, science and technology

 * Cover (algebra), the concept of an algebraic structure that maps onto another structure in structure-preserving fashion
 * Cover (topology), the mathematical concept of a collection of sets whose union contains each set as a subset
 * Cover, a pair in the covering relation of a partially ordered set, or the greater element in such a pair
 * Cover, in database theory, an equivalent set of constraints
 * Land cover, physical material on the surface of the earth
 * Concrete cover, in engineering, distance between reinforcement and the outer surface of element
 * Sedimentary cover, in geology, overlies a basement or crystalline basement

Military

 * Protection from enemy fire - see, for example, infantry tactics
 * Cover or covering fire, also known as suppressive fire
 * A term for any type of uniform hat:
 * Campaign cover or campaign hat, a broad-brimmed felt or straw hat
 * Utility cover, the United States Marine Corps cap

Sports

 * Cover (cricket), a region of the field with respect to the batsman in cricket
 * To cover, a term in sports betting regarding a game's point spread

Other uses

 * Cover letter, a letter of introduction accompanying another document
 * Cover, or covers, the top layer of bedding

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

title: Cover


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Critical angle

Critical angle may refer to:

 * Critical angle (optics), the angle of incidence above which total internal reflection occurs
 * Critical angle of attack, in aerodynamics; the angle of attack which produces the maximum lift coefficient
 * Critical angle of repose, in engineering; the steepest angle of descent of a slope when the material is on the verge of sliding

title: Critical angle


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Critical frequency

In telecommunication, the term critical frequency has the following meanings:

 * In radio propagation by way of the ionosphere, the limiting frequency at or below which a wave component is reflected by, and above which it penetrates through, an ionospheric layer.
 * At near vertical incidence, the limiting frequency at or below which incidence, the wave component is reflected by, and above which it penetrates through, an ionospheric layer.

Critical Frequency changes with time of day, atmospheric conditions and angle of fire of the radio waves by antenna.

The existence of the critical frequency is the result of electron limitation, i.e., the inadequacy of the existing number of free electrons to support reflection at higher frequencies.

In signal processing the critical frequency it is also another name for the Nyquist frequency.

Critical frequency is the highest magnitude of frequency above which the waves penetrate the ionosphere and below which the waves are reflected back from the ionosphere. It is denoted by "fc". Its value is not fixed and it depends upon the electron density of the ionosphere.

Critical frequency as a function of electron density
Critical frequency can be computed with the electron density given by:

$$f_\text{c} = 9\sqrt{N_\text{max}}$$

where Nmax is maximum electron density per m3 and fc is in Hz.

Critical frequency as a function of maximum usable frequency
Critical frequency can be computed by:

$$f_\text{c} = MUF*cos\theta$$

where MUF is maximum usable frequency and $$\theta$$ is the angle of incidence

Relationship with plasma frequency
The dependence of critical frequency with respect with electron density can be related through plasma oscillation concept particularly the 'Cold' Electrons mechanism.

$$\omega_{\mathrm{pe}} = \sqrt{\frac{n_\mathrm{e} e^{2}}{m^*\varepsilon_0}}, \left[\mathrm{rad/s}\right]$$

Using the electron charge $$e=1.602\cdot 10^{-19} Coulombs$$, electron mass $$m^*=9.10938356\cdot 10^{-31} kilograms$$ and permittivity of free space $$\epsilon_o=8.854187817\cdot 10^{-12}A^2s^4m^{-3}kg^{-1}$$gives,

$$\omega_{\mathrm{pe}} = 2\pi f=56.415 \sqrt{n_e}$$

and solving for the frequency,

$$f_\text{c} = 8.979\sqrt{N_\text{max}} \approx 9\sqrt{N_\text{max}}$$

Relationship with index of refraction
The index of refraction has the formula $$n=\frac{c}{v}$$which shows dependence in wavelength. The result that the force due to the polarization field in an ionized gas of low concentration is canceled by the effect of collisions between ions and electrons is re‐established in a simple manner that clearly displays the physical basis for the effect. Because of this cancellation the Sellmeyer formula, determines the relation between the electron number density, N, and the index of refraction, n, in the ionosphere when collisions are neglected.

$$n^2-1=\frac{-Ne^2}{\epsilon_o m \omega^2}$$.

Using the default values for electron charge $$e$$, permittivity of free space and electron mass $$\epsilon_o$$, and changing angular velocity $$\omega$$with respect to frequency $$f$$this yields to

$$n^2-1=\frac{3182.607N}{(2\pi f)^2}$$

and solving for the refraction index n,

$$n=\sqrt{1-\frac{80.616N}{f^2}} \approx \sqrt{1-\frac{81N}{f^2}} $$

Critical frequency and the F layer of the ionosphere

 * All long-distance HF radio communications use HF radio signals that are obliquely incident on the ionosphere. If the HF frequency is above the critical frequency, the radio signals pass through the ionosphere at an angle instead of head-on.
 * The critical frequency changes continuously and the F layer of the ionosphere is mostly responsible for the reflection of radio waves back to Earth.
 * The other layers(D) interact in other ways - absorption of frequency and during the day, the D Layers forms, and the F layer splits into F1 and F2 layers.
 * Because of changing the Ionosphere during day and night, during daytime higher frequency bands under critical Frequency work best, but during nighttime the lower frequency bands work best.
 * The D layer is present during the day and it is a good absorber of radio waves, increasing losses, Higher frequencies are absorbed less, so higher frequencies tends to perform better during daytime.
 * The actual F2-Layer Critical Frequency Map link which refreshes every five minutes can be seen in this website http://www.spacew.com/www/fof2.html
 * The Ionosphere and the Practical Maximum Usable Frequencies (MUFs) Map link which refreshes every five minutes can be seen in this website http://www.sws.bom.gov.au/HF_Systems/6/9/1

title: Critical frequency


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Crosstalk (disambiguation)

Crosstalk refers to any signal or circuit unintentionally affecting another signal or circuit.

Crosstalk may also refer to:

Science and computing

 * Crosstalk (biology)
 * Crosstalk Mk.4, a communications application for PCs.

Arts and entertainment

 * Cross talk, a style of comedy used in performance by double acts
 * Xiangsheng, also translated as "crosstalk", a traditional Chinese comedic monologue or dialogue
 * Crosstalk (novel), a 2016 novel by American science fiction author Connie Willis
 * Crosstalk (film), a 1982 science fiction thriller film
 * Cross Talk, a 1980 album by the Pretty Things
 * Crosstalk: American Speech Music, a compilation album by the produced by Mendi + Keith Obadike
 * CrossTalk (TV series), a television program on the Russian-based international broadcaster RT
 * Cross Talk, a defunct radio show hosted by Indian RJ Balaji

title: Crosstalk (disambiguation)


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Cryptochannel

In telecommunication, a cryptochannel is a complete system of crypto-communications between two or more holders or parties. It includes: (a) the cryptographic aids prescribed; (b) the holders thereof; (c) the indicators or other means of identification; (d) the area or areas in which effective; (e) the special purpose, if any, for which provided; and (f) pertinent notes as to distribution, usage, etc. A cryptochannel is analogous to a radio circuit.

title: Cryptochannel


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Crystal oscillator

A crystal oscillator is an electronic oscillator circuit that uses a piezoelectric crystal as a frequency-selective element. The oscillator frequency is often used to keep track of time, as in quartz wristwatches, to provide a stable clock signal for digital integrated circuits, and to stabilize frequencies for radio transmitters and receivers. The most common type of piezoelectric resonator used is a quartz crystal, so oscillator circuits incorporating them became known as crystal oscillators. However, other piezoelectric materials including polycrystalline ceramics are used in similar circuits.

A crystal oscillator relies on the slight change in shape of a quartz crystal under an electric field, a property known as inverse piezoelectricity. A voltage applied to the electrodes on the crystal causes it to change shape; when the voltage is removed, the crystal generates a small voltage as it elastically returns to its original shape. The quartz oscillates at a stable resonant frequency, behaving like an RLC circuit, but with a much higher Q factor (less energy loss on each cycle of oscillation). Once a quartz crystal is adjusted to a particular frequency (which is affected by the mass of electrodes attached to the crystal, the orientation of the crystal, temperature and other factors), it maintains that frequency with high stability.

Quartz crystals are manufactured for frequencies from a few tens of kilohertz to hundreds of megahertz. As of 2003, around two billion crystals are manufactured annually. Most are used for consumer devices such as wristwatches, clocks, radios, computers, and cellphones. However, in applications where small size and weight is needed crystals can be replaced by thin-film bulk acoustic resonators, specifically if ultra-high frequency (more than roughly 1.5 GHz) resonance is needed. Quartz crystals are also found inside test and measurement equipment, such as counters, signal generators, and oscilloscopes.

Terminology

A crystal oscillator is an electric oscillator type circuit that uses a piezoelectric resonator, a crystal, as its frequency-determining element. Crystal is the common term used in electronics for the frequency-determining component, a wafer of quartz crystal or ceramic with electrodes connected to it. A more accurate term for "crystal" is piezoelectric resonator. Crystals are also used in other types of electronic circuits, such as crystal filters. Piezoelectric resonators are sold as separate components for use in crystal oscillator circuits. They are also often incorporated in a single package with the crystal oscillator circuit.

History

Piezoelectricity was discovered by Jacques and Pierre Curie in 1880. Paul Langevin first investigated quartz resonators for use in sonar during World War I. The first crystal-controlled oscillator, using a crystal of Rochelle salt, was built in 1917 and patented in 1918 by Alexander M. Nicholson at Bell Telephone Laboratories, although his priority was disputed by Walter Guyton Cady. Cady built the first quartz crystal oscillator in 1921. Other early innovators in quartz crystal oscillators include G. W. Pierce and Louis Essen.

Quartz crystal oscillators were developed for high-stability frequency references during the 1920s and 1930s. Prior to crystals, radio stations controlled their frequency with tuned circuits, which could easily drift off frequency by 3–4 kHz. Since broadcast stations were assigned frequencies only 10 kHz (Americas) or 9 kHz (elsewhere) apart, interference between adjacent stations due to frequency drift was a common problem. In 1925, Westinghouse installed a crystal oscillator in its flagship station KDKA, and by 1926, quartz crystals were used to control the frequency of many broadcasting stations and were popular with amateur radio operators. In 1928, Warren Marrison of Bell Telephone Laboratories developed the first quartz-crystal clock. With accuracies of up to 1 second in 30 years (30 ms/y, or 0.95 ns/s), quartz clocks replaced precision pendulum clocks as the world's most accurate timekeepers until atomic clocks were developed in the 1950s. Using the early work at Bell Labs, AT&T eventually established their Frequency Control Products division, later spun off and known today as Vectron International.

A number of firms started producing quartz crystals for electronic use during this time. Using what are now considered primitive methods, about 100,000 crystal units were produced in the United States during 1939. Through World War II crystals were made from natural quartz crystal, virtually all from Brazil. Shortages of crystals during the war caused by the demand for accurate frequency control of military and naval radios and radars spurred postwar research into culturing synthetic quartz, and by 1950 a hydrothermal process for growing quartz crystals on a commercial scale was developed at Bell Laboratories. By the 1970s virtually all crystals used in electronics were synthetic.

In 1968, Juergen Staudte invented a photolithographic process for manufacturing quartz crystal oscillators while working at North American Aviation (now Rockwell) that allowed them to be made small enough for portable products like watches.

Although crystal oscillators still most commonly use quartz crystals, devices using other materials are becoming more common, such as ceramic resonators.

Principle
A crystal is a solid in which the constituent atoms, molecules, or ions are packed in a regularly ordered, repeating pattern extending in all three spatial dimensions.

Almost any object made of an elastic material could be used like a crystal, with appropriate transducers, since all objects have natural resonant frequencies of vibration. For example, steel is very elastic and has a high speed of sound. It was often used in mechanical filters before quartz. The resonant frequency depends on size, shape, elasticity, and the speed of sound in the material. High-frequency crystals are typically cut in the shape of a simple rectangle or circular disk. Low-frequency crystals, such as those used in digital watches, are typically cut in the shape of a tuning fork. For applications not needing very precise timing, a low-cost ceramic resonator is often used in place of a quartz crystal.

When a crystal of quartz is properly cut and mounted, it can be made to distort in an electric field by applying a voltage to an electrode near or on the crystal. This property is known as inverse piezoelectricity. When the field is removed, the quartz generates an electric field as it returns to its previous shape, and this can generate a voltage. The result is that a quartz crystal behaves like an RLC circuit, composed of an inductor, capacitor and resistor, with a precise resonant frequency.

Quartz has the further advantage that its elastic constants and its size change in such a way that the frequency dependence on temperature can be very low. The specific characteristics depend on the mode of vibration and the angle at which the quartz is cut (relative to its crystallographic axes). Therefore, the resonant frequency of the plate, which depends on its size, does not change much. This means that a quartz clock, filter or oscillator remains accurate. For critical applications the quartz oscillator is mounted in a temperature-controlled container, called a crystal oven, and can also be mounted on shock absorbers to prevent perturbation by external mechanical vibrations.

Electrical model
A quartz crystal can be modeled as an electrical network with low-impedance (series) and high-impedance (parallel) resonance points spaced closely together. Mathematically, using the Laplace transform, the impedance of this network can be written as:

 * $$Z(s) = \left( {\frac{1}{s\cdot C_1}+s\cdot L_1+R_1} \right) \left\| \left( {\frac{1}{s\cdot C_0}} \right) \right. ,$$

or

 * $$\begin{align}

Z(s) &= \frac{s^2 + s\frac{R_1}{L_1} + {\omega_\mathrm{s}}^2}{\left(s \cdot C_0\right)\left[s^2 + s\frac{R_1}{L_1} + {\omega_\mathrm{p}}^2\right]} \\[2pt] \Rightarrow \omega_\mathrm{s} &= \frac{1}{\sqrt{L_1 \cdot C_1}}, \quad \omega_\mathrm{p} = \sqrt{\frac{C_1 + C_0}{L_1 \cdot C_1 \cdot C_0}} = \omega_\mathrm{s} \sqrt{1 + \frac{C_1}{C_0}} \approx \omega_\mathrm{s} \left(1 + \frac{C_1}{2 C_0}\right) \quad \left(C_0 \gg C_1\right) \end{align}$$

where $$s$$ is the complex frequency ($$s=j\omega$$), $$\omega_\mathrm{s}$$ is the series resonant angular frequency, and $$\omega_\mathrm{p}$$ is the parallel resonant angular frequency.

Adding capacitance across a crystal causes the (parallel) resonant frequency to decrease. Adding inductance across a crystal causes the (parallel) resonant frequency to increase. These effects can be used to adjust the frequency at which a crystal oscillates. Crystal manufacturers normally cut and trim their crystals to have a specified resonant frequency with a known "load" capacitance added to the crystal. For example, a crystal intended for a 6 pF load has its specified parallel resonant frequency when a 6.0 pF capacitor is placed across it. Without the load capacitance, the resonant frequency is higher.

Resonance modes
A quartz crystal provides both series and parallel resonance. The series resonance is a few kilohertz lower than the parallel one. Crystals below 30 MHz are generally operated between series and parallel resonance, which means that the crystal appears as an inductive reactance in operation, this inductance forming a parallel resonant circuit with externally connected parallel capacitance. Any small additional capacitance in parallel with the crystal pulls the frequency lower. Moreover, the effective inductive reactance of the crystal can be reduced by adding a capacitor in series with the crystal. This latter technique can provide a useful method of trimming the oscillatory frequency within a narrow range; in this case inserting a capacitor in series with the crystal raises the frequency of oscillation. For a crystal to operate at its specified frequency, the electronic circuit has to be exactly that specified by the crystal manufacturer. Note that these points imply a subtlety concerning crystal oscillators in this frequency range: the crystal does not usually oscillate at precisely either of its resonant frequencies.

Crystals above 30 MHz (up to >200 MHz) are generally operated at series resonance where the impedance appears at its minimum and equal to the series resistance. For these crystals the series resistance is specified (<100 Ω) instead of the parallel capacitance. To reach higher frequencies, a crystal can be made to vibrate at one of its overtone modes, which occur near multiples of the fundamental resonant frequency. Only odd numbered overtones are used. Such a crystal is referred to as a 3rd, 5th, or even 7th overtone crystal. To accomplish this, the oscillator circuit usually includes additional LC circuits to select the desired overtone.

Temperature effects
A crystal's frequency characteristic depends on the shape or "cut" of the crystal. A tuning-fork crystal is usually cut such that its frequency dependence on temperature is quadratic with the maximum around 25 °C. This means that a tuning-fork crystal oscillator resonates close to its target frequency at room temperature, but slows when the temperature either increases or decreases from room temperature. A common parabolic coefficient for a 32 kHz tuning-fork crystal is −0.04 ppm/°C2:

 * $$f = f_0\left[1 - 0.04~\text{ppm}/^\circ\text{C}^2 \cdot (T - T_0)^2\right].$$

In a real application, this means that a clock built using a regular 32 kHz tuning-fork crystal keeps good time at room temperature, but loses 2 minutes per year at 10 °C above or below room temperature and loses 8 minutes per year at 20 °C above or below room temperature due to the quartz crystal.

Crystal oscillator circuits
The crystal oscillator circuit sustains oscillation by taking a voltage signal from the quartz resonator, amplifying it, and feeding it back to the resonator. The rate of expansion and contraction of the quartz is the resonant frequency, and is determined by the cut and size of the crystal. When the energy of the generated output frequencies matches the losses in the circuit, an oscillation can be sustained.

An oscillator crystal has two electrically conductive plates, with a slice or tuning fork of quartz crystal sandwiched between them. During startup, the controlling circuit places the crystal into an unstable equilibrium, and due to the positive feedback in the system, any tiny fraction of noise is amplified, ramping up the oscillation. The crystal resonator can also be seen as a highly frequency-selective filter in this system: it only passes a very narrow subband of frequencies around the resonant one, attenuating everything else. Eventually, only the resonant frequency is active. As the oscillator amplifies the signals coming out of the crystal, the signals in the crystal's frequency band becomes stronger, eventually dominating the output of the oscillator. The narrow resonance band of the quartz crystal filters out the unwanted frequencies.

The output frequency of a quartz oscillator can be either that of the fundamental resonance or of a multiple of that resonance, called a harmonic frequency. Harmonics are an exact integer multiple of the fundamental frequency. But, like many other mechanical resonators, crystals exhibit several modes of oscillation, usually at approximately odd integer multiples of the fundamental frequency. These are termed "overtone modes", and oscillator circuits can be designed to excite them. The overtone modes are at frequencies which are approximate, but not exact odd integer multiples of that of the fundamental mode, and overtone frequencies are therefore not exact harmonics of the fundamental.

High frequency crystals are often designed to operate at third, fifth, or seventh overtones. Manufacturers have difficulty producing crystals thin enough to produce fundamental frequencies over 30 MHz. To produce higher frequencies, manufacturers make overtone crystals tuned to put the 3rd, 5th, or 7th overtone at the desired frequency, because they are thicker and therefore easier to manufacture than a fundamental crystal that would produce the same frequency—although exciting the desired overtone frequency requires a slightly more complicated oscillator circuit. A fundamental crystal oscillator circuit is simpler and more efficient and has more pullability than a third overtone circuit. Depending on the manufacturer, the highest available fundamental frequency may be 25 MHz to 66 MHz. A major reason for the wide use of crystal oscillators is their high Q factor. A typical Q value for a quartz oscillator ranges from 104 to 106, compared to perhaps 102 for an LC oscillator. The maximum Q for a high stability quartz oscillator can be estimated as Q = 1.6 × 107/f, where f is the resonant frequency in megahertz.

One of the most important traits of quartz crystal oscillators is that they can exhibit very low phase noise. In many oscillators, any spectral energy at the resonant frequency is amplified by the oscillator, resulting in a collection of tones at different phases. In a crystal oscillator, the crystal mostly vibrates in one axis, therefore only one phase is dominant. This property of low phase noise makes them particularly useful in telecommunications where stable signals are needed, and in scientific equipment where very precise time references are needed.

Environmental changes of temperature, humidity, pressure, and vibration can change the resonant frequency of a quartz crystal, but there are several designs that reduce these environmental effects. These include the TCXO, MCXO, and OCXO which are defined below. These designs, particularly the OCXO, often produce devices with excellent short-term stability. The limitations in short-term stability are due mainly to noise from electronic components in the oscillator circuits. Long-term stability is limited by aging of the crystal.

Due to aging and environmental factors (such as temperature and vibration), it is difficult to keep even the best quartz oscillators within one part in 1010 of their nominal frequency without constant adjustment. For this reason, atomic oscillators are used for applications requiring better long-term stability and accuracy.

Spurious frequencies

For crystals operated at series resonance or pulled away from the main mode by the inclusion of a series inductor or capacitor, significant (and temperature-dependent) spurious responses may be experienced. Though most spurious modes are typically some tens of kilohertz above the wanted series resonance their temperature coefficient is different from the main mode and the spurious response may move through the main mode at certain temperatures. Even if the series resistances at the spurious resonances appear higher than the one at wanted frequency a rapid change in the main mode series resistance can occur at specific temperatures when the two frequencies are coincidental. A consequence of these activity dips is that the oscillator may lock at a spurious frequency at specific temperatures. This is generally minimized by ensuring that the maintaining circuit has insufficient gain to activate unwanted modes.

Spurious frequencies are also generated by subjecting the crystal to vibration. This modulates the resonant frequency to a small degree by the frequency of the vibrations. SC-cut (Stress Compensated) crystals are designed to minimize the frequency effect of mounting stress and they are therefore less sensitive to vibration. Acceleration effects including gravity are also reduced with SC-cut crystals as is frequency change with time due to long term mounting stress variation. There are disadvantages with SC-cut shear mode crystals, such as the need for the maintaining oscillator to discriminate against other closely related unwanted modes and increased frequency change due to temperature when subject to a full ambient range. SC-cut crystals are most advantageous where temperature control at their temperature of zero temperature coefficient (turnover) is possible, under these circumstances an overall stability performance from premium units can approach the stability of Rubidium frequency standards.

Commonly used crystal frequencies
Crystals can be manufactured for oscillation over a wide range of frequencies, from a few kilohertz up to several hundred megahertz. Many applications call for a crystal oscillator frequency conveniently related to some other desired frequency, so hundreds of standard crystal frequencies are made in large quantities and stocked by electronics distributors. For example 3.579545 MHz crystals, which are made in large quantities for NTSC color television receivers, are popular for many non-television applications uses too. Using frequency dividers, frequency multipliers and phase-locked loop circuits, it is practical to derive a wide range of frequencies from one reference frequency.

Quartz
The most common material for oscillator crystals is quartz. At the beginning of the technology, natural quartz crystals were used but now synthetic crystalline quartz grown by hydrothermal synthesis is predominant due to higher purity, lower cost and more convenient handling. One of the few remaining uses of natural crystals is for pressure transducers in deep wells. During World War II and for some time afterwards, natural quartz was considered a strategic material by the USA. Large crystals were imported from Brazil. Raw "lascas", the source material quartz for hydrothermal synthesis, are imported to USA or mined locally by Coleman Quartz. The average value of as-grown synthetic quartz in 1994 was 60 USD/kg.

Types
Two types of quartz crystals exist: left-handed and right-handed. The two differ in their optical rotation but they are identical in other physical properties. Both left and right-handed crystals can be used for oscillators, if the cut angle is correct. In manufacture, right-handed quartz is generally used. The SiO4 tetrahedrons form parallel helices; the direction of twist of the helix determines the left- or right-hand orientation. The helixes are aligned along the c-axis and merged, sharing atoms. The mass of the helixes forms a mesh of small and large channels parallel to the c-axis. The large ones are large enough to allow some mobility of smaller ions and molecules through the crystal.

Quartz exists in several phases. At 573 °C at 1 atmosphere (and at higher temperatures and higher pressures) the α-quartz undergoes quartz inversion, transforms reversibly to β-quartz. The reverse process however is not entirely homogeneous and crystal twinning occurs. Care must be taken during manufacturing and processing to avoid phase transformation. Other phases, e.g. the higher-temperature phases tridymite and cristobalite, are not significant for oscillators. All quartz oscillator crystals are the α-quartz type.

Quality
Infrared spectrophotometry is used as one of the methods for measuring the quality of the grown crystals. The wavenumbers 3585, 3500, and 3410 cm−1 are commonly used. The measured value is based on the absorption bands of the OH radical and the infrared Q value is calculated. The electronic grade crystals, grade C, have Q of 1.8 million or above; the premium grade B crystals have Q of 2.2 million, and special premium grade A crystals have Q of 3.0 million. The Q value is calculated only for the z region; crystals containing other regions can be adversely affected. Another quality indicator is the etch channel density; when the crystal is etched, tubular channels are created along linear defects. For processing involving etching, e.g. the wristwatch tuning fork crystals, low etch channel density is desirable. The etch channel density for swept quartz is about 10–100 and significantly more for unswept quartz. Presence of etch channels and etch pits degrades the resonator's Q and introduces nonlinearities.

Production
Quartz crystals can be grown for specific purposes.

Crystals for AT-cut are the most common in mass production of oscillator materials; the shape and dimensions are optimized for high yield of the required wafers. High-purity quartz crystals are grown with especially low content of aluminium, alkali metal and other impurities and minimal defects; the low amount of alkali metals provides increased resistance to ionizing radiation. Crystals for wrist watches, for cutting the tuning fork 32768 Hz crystals, are grown with very low etch channel density.

Crystals for SAW devices are grown as flat, with large X-size seed with low etch channel density.

Special high-Q crystals, for use in highly stable oscillators, are grown at constant slow speed and have constant low infrared absorption along the entire Z axis. Crystals can be grown as Y-bar, with a seed crystal in bar shape and elongated along the Y axis, or as Z-plate, grown from a plate seed with Y-axis direction length and X-axis width. The region around the seed crystal contains a large number of crystal defects and should not be used for the wafers.

Crystals grow anisotropically; the growth along the Z axis is up to 3 times faster than along the X axis. The growth direction and rate also influences the rate of uptake of impurities. Y-bar crystals, or Z-plate crystals with long Y axis, have four growth regions usually called +X, −X, Z, and S. The distribution of impurities during growth is uneven; different growth areas contain different levels of contaminants. The Z regions are the purest, the small occasionally present S regions are less pure, the +X region is yet less pure, and the -X region has the highest level of impurities. The impurities have a negative impact on radiation hardness, susceptibility to twinning, filter loss, and long and short term stability of the crystals. Different-cut seeds in different orientations may provide other kinds of growth regions. The growth speed of the −X direction is slowest due to the effect of adsorption of water molecules on the crystal surface; aluminium impurities suppress growth in two other directions. The content of aluminium is lowest in Z region, higher in +X, yet higher in −X, and highest in S; the size of S regions also grows with increased amount of aluminium present. The content of hydrogen is lowest in Z region, higher in +X region, yet higher in S region, and highest in −X. Aluminium inclusions transform into color centers with gamma-ray irradiation, causing a darkening of the crystal proportional to the dose and level of impurities; the presence of regions with different darkness reveals the different growth regions.

The dominant type of defect of concern in quartz crystals is the substitution of an Al(III) for a Si(IV) atom in the crystal lattice. The aluminium ion has an associated interstitial charge compensator present nearby, which can be a H+ ion (attached to the nearby oxygen and forming a hydroxyl group, called Al−OH defect), Li+ ion, Na+ ion, K+ ion (less common), or an electron hole trapped in a nearby oxygen atom orbital. The composition of the growth solution, whether it is based on lithium or sodium alkali compounds, determines the charge compensating ions for the aluminium defects. The ion impurities are of concern as they are not firmly bound and can migrate through the crystal, altering the local lattice elasticity and the resonant frequency of the crystal. Other common impurities of concern are e.g. iron(III) (interstitial), fluorine, boron(III), phosphorus(V) (substitution), titanium(IV) (substitution, universally present in magmatic quartz, less common in hydrothermal quartz), and germanium(IV) (substitution). Sodium and iron ions can cause inclusions of acnite and elemeusite crystals. Inclusions of water may be present in fast-grown crystals; interstitial water molecules are abundant near the crystal seed. Another defect of importance is the hydrogen containing growth defect, when instead of a Si−O−Si structure, a pair of Si−OH HO−Si groups is formed; essentially a hydrolyzed bond. Fast-grown crystals contain more hydrogen defects than slow-grown ones. These growth defects source as supply of hydrogen ions for radiation-induced processes and forming Al-OH defects. Germanium impurities tend to trap electrons created during irradiation; the alkali metal cations then migrate towards the negatively charged center and form a stabilizing complex. Matrix defects can also be present; oxygen vacancies, silicon vacancies (usually compensated by 4 hydrogens or 3 hydrogens and a hole), peroxy groups, etc. Some of the defects produce localized levels in the forbidden band, serving as charge traps; Al(III) and B(III) typically serve as hole traps while electron vacancies, titanium, germanium, and phosphorus atoms serve as electron traps. The trapped charge carriers can be released by heating; their recombination is the cause of thermoluminescence.

The mobility of interstitial ions depends strongly on temperature. Hydrogen ions are mobile down to 10 K, but alkali metal ions become mobile only at temperatures around and above 200 K. The hydroxyl defects can be measured by near-infrared spectroscopy. The trapped holes can be measured by electron spin resonance. The Al−Na+ defects show as an acoustic loss peak due to their stress-induced motion; the Al−Li+ defects do not form a potential well so are not detectable this way. Some of the radiation-induced defects during their thermal annealing produce thermoluminescence; defects related to aluminium, titanium, and germanium can be distinguished.

Swept crystals are crystals that have undergone a solid-state electrodiffusion purification process. Sweeping involves heating the crystal above 500 °C in a hydrogen-free atmosphere, with a voltage gradient of at least 1 kV/cm, for several hours (usually over 12). The migration of impurities and the gradual replacement of alkali metal ions with hydrogen (when swept in air) or electron holes (when swept in vacuum) causes a weak electric current through the crystal; decay of this current to a constant value signals the end of the process. The crystal is then left to cool, while the electric field is maintained. The impurities are concentrated at the cathode region of the crystal, which is cut off afterwards and discarded. Swept crystals have increased resistance to radiation, as the dose effects are dependent on the level of alkali metal impurities; they are suitable for use in devices exposed to ionizing radiation, e.g. for nuclear and space technology. Sweeping under vacuum at higher temperatures and higher field strengths yields yet more radiation-hard crystals. The level and character of impurities can be measured by infrared spectroscopy. Quartz can be swept in both α and β phase; sweeping in β phase is faster, but the phase transition may induce twinning. Twinning can be mitigated by subjecting the crystal to compression stress in the X direction, or an AC or DC electric field along the X axis while the crystal cools through the phase transformation temperature region.

Sweeping can also be used to introduce one kind of an impurity into the crystal. Lithium, sodium, and hydrogen swept crystals are used for, e.g., studying quartz behavior.

Very small crystals for high fundamental-mode frequencies can be manufactured by photolithography.

Crystals can be adjusted to exact frequencies by laser trimming. A technique used in the world of amateur radio for slight decrease of the crystal frequency may be achieved by exposing crystals with silver electrodes to vapors of iodine, which causes a slight mass increase on the surface by forming a thin layer of silver iodide; such crystals however had problematic long-term stability. Another method commonly used is electrochemical increase or decrease of silver electrode thickness by submerging a resonator in lapis lazuli dissolved in water, citric acid in water, or water with salt, and using the resonator as one electrode, and a small silver electrode as the other.

By choosing the direction of current one can either increase or decrease the mass of the electrodes. Details were published in "Radio" magazine (3/1978) by UB5LEV.

Raising frequency by scratching off parts of the electrodes is not advised as this may damage the crystal and lower its Q factor. Capacitor trimmers can be also used for frequency adjustment of the oscillator circuit.

Other materials
Some other piezoelectric materials than quartz can be employed. These include single crystals of lithium tantalate, lithium niobate, lithium borate, berlinite, gallium arsenide, lithium tetraborate, aluminium phosphate, bismuth germanium oxide, polycrystalline zirconium titanate ceramics, high-alumina ceramics, silicon-zinc oxide composite, or dipotassium tartrate. Some materials may be more suitable for specific applications. An oscillator crystal can be also manufactured by depositing the resonator material on the silicon chip surface. Crystals of gallium phosphate, langasite, langanite and langatate are about 10 times more pullable than the corresponding quartz crystals, and are used in some VCXO oscillators.

Stability
The frequency stability is determined by the crystal's Q. It is inversely dependent on the frequency, and on the constant that is dependent on the particular cut. Other factors influencing Q are the overtone used, the temperature, the level of driving of the crystal, the quality of the surface finish, the mechanical stresses imposed on the crystal by bonding and mounting, the geometry of the crystal and the attached electrodes, the material purity and defects in the crystal, type and pressure of the gas in the enclosure, interfering modes, and presence and absorbed dose of ionizing and neutron radiation.

The stability of AT cut crystals decreases with increasing frequency. For more accurate higher frequencies it is better to use a crystal with lower fundamental frequency, operating at an overtone.

A badly designed oscillator circuit may suddenly begin oscillating on an overtone. In 1972, a train in Fremont, California crashed due to a faulty oscillator. An inappropriate value of the tank capacitor caused the crystal in a control board to be overdriven, jumping to an overtone, and causing the train to speed up instead of slowing down.

Temperature
Temperature influences the operating frequency; various forms of compensation are used, from analog compensation (TCXO) and microcontroller compensation (MCXO) to stabilization of the temperature with a crystal oven (OCXO). The crystals possess temperature hysteresis; the frequency at a given temperature achieved by increasing the temperature is not equal to the frequency on the same temperature achieved by decreasing the temperature. The temperature sensitivity depends primarily on the cut; the temperature compensated cuts are chosen as to minimize frequency/temperature dependence. Special cuts can be made with linear temperature characteristics; the LC cut is used in quartz thermometers. Other influencing factors are the overtone used, the mounting and electrodes, impurities in the crystal, mechanical strain, crystal geometry, rate of temperature change, thermal history (due to hysteresis), ionizing radiation, and drive level.

Crystals tend to suffer anomalies in their frequency/temperature and resistance/temperature characteristics, known as activity dips. These are small downward frequency or upward resistance excursions localized at certain temperatures, with their temperature position dependent on the value of the load capacitors.

Mechanical stress
Mechanical stresses also influence the frequency. The stresses can be induced by mounting, bonding, and application of the electrodes, by differential thermal expansion of the mounting, electrodes, and the crystal itself, by differential thermal stresses when there is a temperature gradient present, by expansion or shrinkage of the bonding materials during curing, by the air pressure that is transferred to the ambient pressure within the crystal enclosure, by the stresses of the crystal lattice itself (nonuniform growth, impurities, dislocations), by the surface imperfections and damage caused during manufacture, and by the action of gravity on the mass of the crystal; the frequency can therefore be influenced by position of the crystal. Other dynamic stress inducing factors are shocks, vibrations, and acoustic noise. Some cuts are less sensitive to stresses; the SC (Stress Compensated) cut is an example. Atmospheric pressure changes can also introduce deformations to the housing, influencing the frequency by changing stray capacitances.

Atmospheric humidity influences the thermal transfer properties of air, and can change electrical properties of plastics by diffusion of water molecules into their structure, altering the dielectric constants and electrical conductivity.

Other factors influencing the frequency are the power supply voltage, load impedance, magnetic fields, electric fields (in case of cuts that are sensitive to them, e.g., SC cuts), the presence and absorbed dose of γ-particles and ionizing radiation, and the age of the crystal.

Aging
Crystals undergo slow gradual change of frequency with time, known as aging. There are many mechanisms involved. The mounting and contacts may undergo relief of the built-in stresses. Molecules of contamination either from the residual atmosphere, outgassed from the crystal, electrodes or packaging materials, or introduced during sealing the housing can be adsorbed on the crystal surface, changing its mass; this effect is exploited in quartz crystal microbalances. The composition of the crystal can be gradually altered by outgassing, diffusion of atoms of impurities or migrating from the electrodes, or the lattice can be damaged by radiation. Slow chemical reactions may occur on or in the crystal, or on the inner surfaces of the enclosure. Electrode material, e.g. chromium or aluminium, can react with the crystal, creating layers of metal oxide and silicon; these interface layers can undergo changes in time. The pressure in the enclosure can change due to varying atmospheric pressure, temperature, leaks, or outgassing of the materials inside. Factors outside of the crystal itself are e.g. aging of the oscillator circuitry (and e.g. change of capacitances), and drift of parameters of the crystal oven. External atmosphere composition can also influence the aging; hydrogen can diffuse through nickel housing. Helium can cause similar issues when it diffuses through glass enclosures of rubidium standards.

Gold is a favored electrode material for low-aging resonators; its adhesion to quartz is strong enough to maintain contact even at strong mechanical shocks, but weak enough to not support significant strain gradients (unlike chromium, aluminium, and nickel). Gold also does not form oxides; it adsorbs organic contaminants from the air, but these are easy to remove. However, gold alone can undergo delamination; a layer of chromium is therefore sometimes used for improved binding strength. Silver and aluminium are often used as electrodes; however both form oxide layers with time that increases the crystal mass and lowers frequency. Silver can be passivated by exposition to iodine vapors, forming a layer of silver iodide. Aluminium oxidizes readily but slowly, until about 5 nm thickness is reached; increased temperature during artificial aging does not significantly increase the oxide forming speed; a thick oxide layer can be formed during manufacture by anodizing. Exposition of silver-plated crystal to iodine vapors can also be used in amateur conditions for lowering the crystal frequency slightly; the frequency can also be increased by scratching off parts of the electrodes, but that carries risk of damage to the crystal and loss of Q.

A DC voltage bias between the electrodes can accelerate the initial aging, probably by induced diffusion of impurities through the crystal. Placing a capacitor in series with the crystal and a several-megaohm resistor in parallel can minimize such voltages.

Aging decreases logarithmically with time, the largest changes occurring shortly after manufacture. Artificially aging a crystal by prolonged storage at 85 to 125 °C can increase its long-term stability.

Mechanical damage
Crystals are sensitive to shock. The mechanical stress causes a short-term change in the oscillator frequency due to the stress-sensitivity of the crystal, and can introduce a permanent change of frequency due to shock-induced changes of mounting and internal stresses (if the elastic limits of the mechanical parts are exceeded), desorption of contamination from the crystal surfaces, or change in parameters of the oscillator circuit. High magnitudes of shocks may tear the crystals off their mountings (especially in the case of large low-frequency crystals suspended on thin wires), or cause cracking of the crystal. Crystals free of surface imperfections are highly shock-resistant; chemical polishing can produce crystals able to survive tens of thousands of g.

Crystals have no inherent failure mechanisms; some have operated in devices for decades. Failures may be, however, introduced by faults in bonding, leaky enclosures, corrosion, frequency shift by aging, breaking the crystal by too high mechanical shock, or radiation-induced damage when non-swept quartz is used. Crystals can be also damaged by overdriving.

Frequency fluctuations
Crystals suffer from minor short-term frequency fluctuations as well. The main causes of such noise are e.g. thermal noise (which limits the noise floor), phonon scattering (influenced by lattice defects), adsorption/desorption of molecules on the surface of the crystal, noise of the oscillator circuits, mechanical shocks and vibrations, acceleration and orientation changes, temperature fluctuations, and relief of mechanical stresses. The short-term stability is measured by four main parameters: Allan variance (the most common one specified in oscillator data sheets), phase noise, spectral density of phase deviations, and spectral density of fractional frequency deviations. The effects of acceleration and vibration tend to dominate the other noise sources; surface acoustic wave devices tend to be more sensitive than bulk acoustic wave (BAW) ones, and the stress-compensated cuts are even less sensitive. The relative orientation of the acceleration vector to the crystal dramatically influences the crystal's vibration sensitivity. Mechanical vibration isolation mountings can be used for high-stability crystals.

Phase noise plays a significant role in frequency synthesis systems using frequency multiplication; a multiplication of a frequency by N increases the phase noise power by N2. A frequency multiplication by 10 times multiplies the magnitude of the phase error by 10 times. This can be disastrous for systems employing PLL or FSK technologies.

Magnetic fields have little effect on the crystal itself, as quartz is diamagnetic; eddy currents or AC voltages can however be induced into the circuits, and magnetic parts of the mounting and housing may be influenced.

After the power-up, the crystals take several seconds to minutes to "warm up" and stabilize their frequency. The oven-controlled OCXOs require usually 3–10 minutes for heating up to reach thermal equilibrium; the oven-less oscillators stabilize in several seconds as the few milliwatts dissipated in the crystal cause a small but noticeable level of internal heating.

Drive level
The crystals have to be driven at the appropriate drive level. Low-frequency crystals, especially flexural-mode ones, may fracture at too high drive levels. The drive level is specified as the amount of power dissipated in the crystal. The appropriate drive levels are about 5 μW for flexural modes up to 100 kHz, 1 μW for fundamental modes at 1–4 MHz, 0.5 μW for fundamental modes 4–20 MHz and 0.5 μW for overtone modes at 20–200 MHz. Too low drive level may cause problems with starting the oscillator. Low drive levels are better for higher stability and lower power consumption of the oscillator. Higher drive levels, in turn, reduce the impact of noise by increasing the signal-to-noise ratio.

Crystal cuts
The resonator plate can be cut from the source crystal in many different ways. The orientation of the cut influences the crystal's aging characteristics, frequency stability, thermal characteristics, and other parameters. These cuts operate at bulk acoustic wave (BAW); for higher frequencies, surface acoustic wave (SAW) devices are employed.

Image of several crystal cuts

The letter ‘T’ in the cut name marks a temperature-compensated cut – a cut oriented in a way that the temperature coefficients of the lattice are minimal; the FC and SC cuts are also temperature-compensated.

The high frequency cuts are mounted by their edges, usually on springs; the stiffness of the spring has to be optimal, as if it is too stiff, mechanical shocks could be transferred to the crystal and cause it to break, and too little stiffness may allow the crystal to collide with the inside of the package when subjected to a mechanical shock, and break. Strip resonators, usually AT cuts, are smaller and therefore less sensitive to mechanical shocks. At the same frequency and overtone, the strip has less pullability, higher resistance, and higher temperature coefficient.

The low frequency cuts are mounted at the nodes where they are virtually motionless; thin wires are attached at such points on each side between the crystal and the leads. The large mass of the crystal suspended on the thin wires makes the assembly sensitive to mechanical shocks and vibrations.

The crystals are usually mounted in hermetically sealed glass or metal cases, filled with a dry and inert atmosphere, usually vacuum, nitrogen, or helium. Plastic housings can be used as well, but those are not hermetic and another secondary sealing has to be built around the crystal.

Several resonator configurations are possible, in addition to the classical way of directly attaching leads to the crystal. E.g. the BVA resonator (Boîtier à Vieillissement Amélioré, Enclosure with Improved Aging), developed in 1976; the parts that influence the vibrations are machined from a single crystal (which reduces the mounting stress), and the electrodes are deposited not on the resonator itself but on the inner sides of two condenser discs made of adjacent slices of the quartz from the same bar, forming a three-layer sandwich with no stress between the electrodes and the vibrating element. The gap between the electrodes and the resonator act as two small series capacitors, making the crystal less sensitive to circuit influences. The architecture eliminates the effects of the surface contacts between the electrodes, the constraints in the mounting connections, and the issues related to ion migration from the electrodes into the lattice of the vibrating element. The resulting configuration is rugged, resistant to shock and vibration, resistant to acceleration and ionizing radiation, and has improved aging characteristics. AT cut is usually used, though SC cut variants exist as well. BVA resonators are often used in spacecraft applications.

In the 1930s to 1950s, it was fairly common for people to adjust the frequency of the crystals by manual grinding. The crystals were ground using a fine abrasive slurry, or even a toothpaste, to increase their frequency. A slight decrease by 1–2 kHz when the crystal was overground was possible by marking the crystal face with a pencil lead, at the cost of a lowered $c$ ;.

The frequency of the crystal is slightly adjustable ("pullable") by modifying the attached capacitances. A varactor, a diode with capacitance depending on applied voltage, is often used in voltage-controlled crystal oscillators, VCXO. The crystal cuts are usually AT or rarely SC, and operate in fundamental mode; the amount of available frequency deviation is inversely proportional to the square of the overtone number, so a third overtone has only one-ninth of the pull-ability of the fundamental mode. SC cuts, while more stable, are significantly less pullable.

Circuit notations and abbreviations
On electrical schematic diagrams, crystals are designated with the class letter Y (Y1, Y2, etc.). Oscillators, whether they are crystal oscillators or others, are designated with the class letter G (G1, G2, etc.). Crystals may also be designated on a schematic with X or XTAL, or a crystal oscillator with XO.

Crystal oscillator types and their abbreviations:
 * ATCXO — Analog temperature controlled crystal oscillator
 * CDXO — Calibrated dual crystal oscillator
 * DTCXO — Digital temperature compensated crystal oscillator
 * EMXO — Evacuated miniature crystal oscillator
 * GPSDO — Global positioning system disciplined oscillator
 * MCXO — Microcomputer-compensated crystal oscillator
 * OCVCXO — oven-controlled voltage-controlled crystal oscillator
 * OCXO — Oven-controlled crystal oscillator
 * RbXO — Rubidium crystal oscillators (RbXO), a crystal oscillator (can be an MCXO) synchronized with a built-in rubidium standard which is run only occasionally to save power
 * TCVCXO — Temperature-compensated voltage-controlled crystal oscillator
 * TCXO — Temperature-compensated crystal oscillator
 * TMXO – Tactical miniature crystal oscillator
 * TSXO — Temperature-sensing crystal oscillator, an adaptation of the TCXO
 * VCTCXO — Voltage-controlled temperature-compensated crystal oscillator
 * VCXO — Voltage-controlled crystal oscillator

title: Crystal oscillator


text:
Curve-fitting compaction

Curve-fitting compaction is data compaction accomplished by replacing data to be stored or transmitted with an analytical expression.

Examples of curve-fitting compaction consisting of discretization and then interpolation are:
 * Breaking of a continuous curve into a series of straight line segments and specifying the slope, intercept, and range for each segment
 * Using a mathematical expression, such as a polynomial or a trigonometric function, and a single point on the corresponding curve instead of storing or transmitting the entire graphic curve or a series of points on it.

title: Curve-fitting compaction


text:
Customer office terminal

In telecommunications, the term customer office terminal has the following meanings:

 * 1. Termination equipment that (a) is located on the customer premises and (b) performs a function that may be integrated into the common carrier equipment.

Note: An example of a customer office terminal is a stand-alone multiplexer located on the customer premises.
 * 2. The digital loop carrier (DLC) multiplexing function that is near the exchange termination (ET) when provided by a stand-alone multiplexer.

Note: This function may be integrated into the ET.

title: Customer office terminal


text:
Customer-premises equipment

In telecommunications, a customer-premises equipment or customer-provided equipment (CPE) is any terminal and associated equipment located at a subscriber's premises and connected with a carrier's telecommunication circuit at the demarcation point ("demarc"). The demarc is a point established in a building or complex to separate customer equipment from the equipment located in either the distribution infrastructure or central office of the communications service provider.

CPE generally refers to devices such as telephones, routers, network switches, residential gateways (RG), set-top boxes, fixed mobile convergence products, home networking adapters and Internet access gateways that enable consumers to access providers' communication services and distribute them in a residence or enterprise with a local area network (LAN).

A CPE can be an active equipment, as the ones mentioned above, or passive equipment such as analog telephone adapters (ATA) or xDSL-splitters. This includes key telephone systems and most private branch exchanges. Excluded from the CPE category are overvoltage protection equipment and pay telephones. Other types of materials that are necessary for the delivery of the telecommunication service, but are not defined as equipment, such as manuals and cable packages, and cable adapters are instead referred to as CPE-peripherals.

CPE can refer to devices purchased by the subscriber, or to those provided by the operator or service provider.

History
The two phrases, "customer-premises equipment" and "customer-provided equipment", reflect the history of this equipment.

Under the Bell System monopoly in the United States (post Communications Act of 1934), the Bell System owned the telephones, and one could not attach privately owned or supplied devices to the network, or to the station apparatus. Telephones were located on customers' premises, hence, customer-premises equipment. In the U.S. Federal Communications Commission (FCC) proceeding the Second Computer Inquiry, the FCC ruled that telecommunications carriers could no longer bundle CPE with telecommunications service, uncoupling the market power of the telecommunications service monopoly from the CPE market, and creating a competitive CPE market.

With the gradual breakup of the Bell monopoly, starting with Hush-A-Phone v. United States [1956], which allowed some non-Bell owned equipment to be connected to the network (a process called interconnection), equipment on customers' premises became increasingly owned by customers. Indeed, subscribers were eventually permitted to purchase telephones – hence, customer-provided equipment.

In the pay-TV industry many operators and service providers offer subscribers a set-top box with which to receive video services, in return for a monthly fee. As offerings have evolved to include multiple services [voice and data] operators have increasingly given consumers the opportunity to rent or buy additional devices like access modems, internet gateways and video extenders that enable them to access multiple services, and distribute them to a range of consumer electronics devices in the home.

Hybrid devices
The growth of multiple system operators, offering triple or quad-play services, required the development of hybrid CPE to make it easy for subscribers to access voice, video and data services. The development of this technology was led by Pay TV operators looking for a way to deliver video services via both traditional broadcast and broadband IP networks. Spain's Telefonica was the first operator to launch a hybrid broadcast and broadband TV service in 2003 with its Movistar TV DTT/IPTV offering, while Polish satellite operator 'n' was the first to offer its subscribers a Three-way hybrid (or Tri-brid) broadcast and broadband TV service, which launched in 2009

Set-back box
The term set-back box is used in the digital TV industry to describe a piece of consumer hardware that enables them to access both linear broadcast and internet-based video content, plus a range of interactive services like Electronic Programme Guides (EPG), Pay Per View (PPV) and video on demand (VOD) as well as internet browsing, and view them on a large screen television set. Unlike standard set-top boxes, which sit on top of or below the TV, a set-back box has a smaller form factor to enable it to be mounted to the rear of the display panel flat panel TV, hiding it from view.

Residential gateway
A residential gateway is a networking device used to connect devices in the home to the Internet or other wide area network (WAN). It is an umbrella term, used to cover multi-function networking appliances used in homes, which may combine a DSL modem or cable modem, a network switch, a consumer-grade router, and a wireless access point. In the past, such functions were provided by separate devices, but in recent years technological convergence has enabled multiple functions to be merged into a single device.

One of the first home gateway devices to be launched was selected by Telecom Italia to enable the operator to offer triple play services in 2002: Along with a SIP VoIP handset for making voice calls, it enabled subscribers to access voice, video and data services over a 10MB symmetrical ADSL fiber connection.

Virtual gateway
The virtual gateway concept enables consumers to access video and data services and distribute them around their homes using software rather than hardware. The first virtual gateway was introduced in 2010 by Advanced Digital Broadcast at the IBC exhibition in Amsterdam. The ADB Virtual Gateway uses software that resides within the middleware and is based on open standards, including DLNA home networking and the DTCP-IP standard, to ensure that all content, including paid-for encrypted content like Pay TV services, can only be accessed by secure CE devices.

Broadband
A subscriber unit, or SU is a broadband radio that is installed at a business or residential location to connect to an access point to send/receive high speed data wired or wirelessly. Devices commonly referred to as a subscriber unit include cable modems, access gateways, home networking adapters and mobile phones.

WAN
CPE may also refer to any devices that terminate a WAN circuit, such as an ISDN, E-carrier/T-carrier, DSL, or metro Ethernet. This includes any customer-owned hardware at the customer's site: routers, firewalls, network switches, PBXs, VoIP gateways, sometimes CSU/DSU and modems.

Application areas
 * Connected home
 * Pay TV
 * Over-the-top Media Services
 * Broadband
 * Voice over IP
 * Fixed–mobile convergence [FMC]

Other uses

 * Cellular carriers may sometimes internally refer to cellular phones a customer has purchased without a subsidy or from a third party as "customer provided equipment."
 * It is also notable that the fully qualified domain name and the PTR record of DSL and cable lines connected to a residence will often contain 'cpe'.

title: Customer-premises equipment


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Customer service unit

In telecommunication, a customer service unit (CSU) is a device that provides an accessing arrangement at a user location to either switched or point-to-point, data-conditioned circuits at a specifically established data signaling rate.

A CSU provides local loop equalization, transient protection, isolation, and central office loop-back testing capability.

title: Customer service unit


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Cutback technique

In telecommunications, a cutback technique is a destructive technique for determining certain optical fiber transmission characteristics, such as attenuation and bandwidth.

Procedure
The measurement technique consists of:
 * 1) performing the desired measurements on a long length of the fiber under test,
 * 2) cutting the fiber under test at a point near the launching end,
 * 3) repeating the measurements on the short length of fiber, and
 * 4) subtracting the results obtained on the short length to determine the results for the residual long length.

The cut should be made to retain 1 meter or more of the fiber, in order to establish equilibrium mode distribution conditions for the second measurement. In a multimode fiber, the lack of an equilibrium mode distribution could introduce errors in the measurement due to output coupling effects. In a single-mode fiber, measuring a shorter cutback fiber could result in significant transmission of cladding modes (light carried in the cladding rather than the core of the optical fiber), distorting the measurement. The errors introduced will result in conservative results (i.e., higher transmission losses and lower bandwidths) than would be realized under equilibrium conditions.

Benefits
The benefit of this technique is that it allows measurement of the fiber characteristics without introducing errors due to variation in the launch conditions. For example, the coupling efficiency of the light source is kept consistent between the initial and the cutback measurements.

Several characteristics may be determined using the same test fiber.

Attenuation measurement
Since the attenuation is defined as proportional to the logarithm of the ratio between $$P(x)$$ and $$P(y)$$, where $$P$$ is the power at point $$x$$ and $$y$$ respectively. Using the cutback technique, the power transmitted through a fiber of known length is measured and compared with the same measurement for the same fiber cut to a length of $$2m$$ approximately.

Related techniques
A variation of the cutback technique is the substitution method, in which measurements are made on a full-length of fiber, and then on a short length of fiber having the same characteristics (core size, numerical aperture), with the results from the short length being subtracted to give the results for the full length.

title: Cutback technique


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Cutoff frequency

In physics and electrical engineering, a cutoff frequency, corner frequency, or break frequency is a boundary in a system's frequency response at which energy flowing through the system begins to be reduced (attenuated or reflected) rather than passing through.

Typically in electronic systems such as filters and communication channels, cutoff frequency applies to an edge in a lowpass, highpass, bandpass, or band-stop characteristic – a frequency characterizing a boundary between a passband and a stopband. It is sometimes taken to be the point in the filter response where a transition band and passband meet, for example, as defined by a half-power point (a frequency for which the output of the circuit is approximately −3.01 dB of the nominal passband value). Alternatively, a stopband corner frequency may be specified as a point where a transition band and a stopband meet: a frequency for which the attenuation is larger than the required stopband attenuation, which for example may be 30 dB or 100 dB.

In the case of a waveguide or an antenna, the cutoff frequencies correspond to the lower and upper cutoff wavelengths.

Electronics
In electronics, cutoff frequency or corner frequency is the frequency either above or below which the power output of a circuit, such as a line, amplifier, or electronic filter has fallen to a given proportion of the power in the passband. Most frequently this proportion is one half the passband power, also referred to as the 3 dB point since a fall of 3 dB corresponds approximately to half power. As a voltage ratio this is a fall to $ \sqrt{1/2} \ \approx \ 0.707$ of the passband voltage. Other ratios besides the 3 dB point may also be relevant, for example see below. Far from the cutoff frequency in the transition band, the rate of increase of attenuation (roll-off) with logarithm of frequency is asymptotic to a constant. For a first-order network, the roll-off is −20 dB per decade (approximately −6 dB per octave.)

Single-pole transfer function example
The transfer function for the simplest low-pass filter, $$H(s) = \frac {1}{1+\alpha s},$$ has a single pole at $s = −1/α$. The magnitude of this function in the $jω$ plane is $$\left | H(j\omega) \right | = \left | \frac {1}{1+\alpha j \omega} \right | =\sqrt{ \frac {1}{1 + \alpha^2\omega^2}}.$$

At cutoff $$\left | H(j\omega_ \mathrm c) \right | = \frac {1}{\sqrt{2}} = \sqrt{ \frac {1}{1 + \alpha^2\omega_\mathrm c ^2}}.$$

Hence, the cutoff frequency is given by $$\omega_ \mathrm c = \frac {1}{\alpha}.$$

Where $s$ is the s-plane variable, $ω$ is angular frequency and $j$ is the imaginary unit.

Chebyshev filters
Sometimes other ratios are more convenient than the 3 dB point. For instance, in the case of the Chebyshev filter it is usual to define the cutoff frequency as the point after the last peak in the frequency response at which the level has fallen to the design value of the passband ripple. The amount of ripple in this class of filter can be set by the designer to any desired value, hence the ratio used could be any value.

Radio communications
In radio communication, skywave communication is a technique in which radio waves are transmitted at an angle into the sky and reflected back to Earth by layers of charged particles in the ionosphere. In this context, the term cutoff frequency refers to the maximum usable frequency, the frequency above which a radio wave fails to reflect off the ionosphere at the incidence angle required for transmission between two specified points by reflection from the layer.

Waveguides
The cutoff frequency of an electromagnetic waveguide is the lowest frequency for which a mode will propagate in it. In fiber optics, it is more common to consider the cutoff wavelength, the maximum wavelength that will propagate in an optical fiber or waveguide. The cutoff frequency is found with the characteristic equation of the Helmholtz equation for electromagnetic waves, which is derived from the electromagnetic wave equation by setting the longitudinal wave number equal to zero and solving for the frequency. Thus, any exciting frequency lower than the cutoff frequency will attenuate, rather than propagate. The following derivation assumes lossless walls. The value of c, the speed of light, should be taken to be the group velocity of light in whatever material fills the waveguide.

For a rectangular waveguide, the cutoff frequency is $$ \omega_{c} = c \sqrt{\left(\frac{m \pi}{a}\right)^2 + \left(\frac{n \pi}{b}\right) ^2}, $$ where $$m,n \ge 0$$ are the mode numbers for the rectangle's sides of length $$a$$ and $$b$$ respectively. For TE modes, $$ m,n \ge 0$$ (but $$ m = n = 0$$ is not allowed), while for TM modes $$ m,n \ge 1 $$.

The cutoff frequency of the TM01 mode (next higher from dominant mode TE11) in a waveguide of circular cross-section (the transverse-magnetic mode with no angular dependence and lowest radial dependence) is given by $$ \omega_{c} = c \frac{\chi_{01}}{r} = c \frac{2.4048}{r},$$ where $$r$$ is the radius of the waveguide, and $$\chi_{01}$$ is the first root of $$J_{0}(r)$$, the Bessel function of the first kind of order 1.

The dominant mode TE11 cutoff frequency is given by $$ \omega_{c} = c \frac{\chi_{11}}{r} = c \frac{1.8412}{r}$$

However, the dominant mode cutoff frequency can be reduced by the introduction of baffle inside the circular cross-section waveguide. For a single-mode optical fiber, the cutoff wavelength is the wavelength at which the normalized frequency is approximately equal to 2.405.

Mathematical analysis
The starting point is the wave equation (which is derived from the Maxwell equations), $$ \left(\nabla^2-\frac{1}{c^2}\frac{\partial^2}{\partial{t}^2}\right)\psi(\mathbf{r},t)=0, $$ which becomes a Helmholtz equation by considering only functions of the form $$ \psi(x,y,z,t) = \psi(x,y,z)e^{i \omega t}. $$ Substituting and evaluating the time derivative gives $$ \left(\nabla^2 + \frac{\omega^2}{c^2}\right) \psi(x,y,z) = 0. $$ The function $$ \psi $$ here refers to whichever field (the electric field or the magnetic field) has no vector component in the longitudinal direction - the "transverse" field. It is a property of all the eigenmodes of the electromagnetic waveguide that at least one of the two fields is transverse. The z axis is defined to be along the axis of the waveguide.

The "longitudinal" derivative in the Laplacian can further be reduced by considering only functions of the form $$ \psi(x,y,z,t) = \psi(x,y)e^{i \left(\omega t - k_{z} z \right)}, $$ where $$k_z$$ is the longitudinal wavenumber, resulting in $$ \left(\nabla_{T}^2 - k_{z}^2 + \frac{\omega^2}{c^2}\right) \psi(x,y,z) = 0, $$ where subscript T indicates a 2-dimensional transverse Laplacian. The final step depends on the geometry of the waveguide. The easiest geometry to solve is the rectangular waveguide. In that case, the remainder of the Laplacian can be evaluated to its characteristic equation by considering solutions of the form $$ \psi(x,y,z,t) = \psi_{0}e^{i \left(\omega t - k_{z} z - k_{x} x - k_{y} y\right)}. $$ Thus for the rectangular guide the Laplacian is evaluated, and we arrive at $$ \frac{\omega^2}{c^2} = k_x^2 + k_y^2 + k_z^2 $$ The transverse wavenumbers can be specified from the standing wave boundary conditions for a rectangular geometry cross-section with dimensions $a$ and $b$: $$ k_{x} = \frac{n \pi}{a},$$ $$ k_{y} = \frac{m \pi}{b},$$ where $n$ and $m$ are the two integers representing a specific eigenmode. Performing the final substitution, we obtain $$ \frac{\omega^2}{c^2} = \left(\frac{n \pi}{a}\right)^2 + \left(\frac{m \pi}{b}\right)^2 + k_{z}^2,$$ which is the dispersion relation in the rectangular waveguide. The cutoff frequency $$\omega_{c}$$ is the critical frequency between propagation and attenuation, which corresponds to the frequency at which the longitudinal wavenumber $$k_{z}$$ is zero. It is given by $$ \omega_{c} = c \sqrt{\left(\frac{n \pi}{a}\right)^2 + \left(\frac{m \pi}{b}\right)^2}$$ The wave equations are also valid below the cutoff frequency, where the longitudinal wave number is imaginary. In this case, the field decays exponentially along the waveguide axis and the wave is thus evanescent.

title: Cutoff frequency


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Data access arrangement

The term data access arrangement (DAA) has the following meanings:
 * 1) In public switched telephone networks, a single item or group of items at the customer side of the network interface device for data transmission purposes, including all equipment that may affect the characteristics of the interface.
 * 2) A data circuit-terminating equipment (DCE) supplied or approved by a common carrier that permits a DCE or data terminal equipment (DTE) to be attached to the common carrier network.

Data access arrangements are an integral part of all modems built for the public telephone network. In view of mixed voice and data access, DAAs are more generally referred to as direct access arrangements.

Requirement for DAAs
While DAA now describes an integral component of a device that connects to the telephone network, during the 60s and 70s it described a separate device mandated by the Bell System, connected between the telephone line and non-Bell equipment, typically a modem.

Following the Carterfone decision, which required Bell to allow customers to attach any non-harmful equipment to their network, Bell mandated that subscribers use PCAs/DAAs - purchased exclusively from Western Electric - to ensure the network was protected. These devices were not required for Bell-provided equipment, only equipment made by independent manufacturers.

At the time, some subscribers believed that the DAA was a scheme by AT&T to penalize and discourage use of non-Bell modems and recover lost profits from hardware sales, and the FCC began investigations into the legality of the practice. Subscribers also became frustrated when Bell failed to deliver DAAs in a timely fashion after the ruling, leading to the use of unauthorized third-party DAAs.

There were two main varieties of DAA described by AT&T: manual and automatic. A manual DAA required a call to be initiated (or answered) as normal, at which point it could then be connected to the third-party device, while an automatic DAA allowed an attached device to be connected without human intervention, important for receiving modem use.

In 1975, the FCC implemented Part 68 of the FCC Rules, which granted permission for direct connection of any equipment to the telephone network given compliance with specific electrical requirements. This technically eliminated the need for DAAs, although the first modem that didn't require a separate DAA was not marketed until 1977 when a court ruled that Part 68 was legal.

title: Data access arrangement


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Data bank

In database management and information architecture, a data bank or databank is a repository of information about one or more subjects, that is, a database which is organized in a way that facilitates local or remote information retrieval and is able to process many continual queries over a long period of time. A data bank may also refer to an organization primarily concerned with the construction and maintenance of such a database.

A data bank may be either centralized or decentralized, though most usage of this term refers to centralized storage and retrieval of information, by way of analogy to a monetary bank. The data in a data bank can be anything from scientific information like global temperature readings, and governmental information like census statistics, to financial-system records like credit card transactions, or the inventory available from various suppliers.

In computing, the term databank is also obsolete (1960s through 1970s) computer jargon for database itself, and is frequently used in that sense in materials written during the same time period.

title: Data bank


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Data compaction

In telecommunication, data compaction is the reduction of the number of data elements, bandwidth, cost, and time for the generation, transmission, and storage of data without loss of information by eliminating unnecessary redundancy, removing irrelevancy, or using special coding.

Examples of data compaction methods are the use of fixed-tolerance bands, variable-tolerance bands, slope-keypoints, sample changes, curve patterns, curve fitting, variable-precision coding, frequency analysis, and probability analysis.

Simply squeezing noncompacted data into a smaller space, for example by increasing packing density by transferring images from newsprint to microfilm or by transferring data on punched cards onto magnetic tape, is not data compaction.

Everyday examples
The use of acronyms in texting is an everyday example. The number of bits required to transmit and store "WYSIWYG" (What You See Is What You Get) is reduced from its expanded equivalent (7 characters vs 28). The representation of Mersenne primes is another example. The largest known is over 17 million digits long but it is represented as M57885161 in a much more compacted form.

title: Data compaction


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Data element

In metadata, the term data element is an atomic unit of data that has precise meaning or precise semantics. A data element has:
 * 1) An identification such as a data element name
 * 2) A clear data element definition
 * 3) One or more representation terms
 * 4) Optional enumerated values Code (metadata)
 * 5) A list of synonyms to data elements in other metadata registries Synonym ring

Data elements usage can be discovered by inspection of software applications or application data files through a process of manual or automated Application Discovery and Understanding. Once data elements are discovered they can be registered in a metadata registry.

In telecommunication, the term data element has the following components:

 * 1) A named unit of data that, in some contexts, is considered indivisible and in other contexts may consist of data items.
 * 2) A named identifier of each of the entities and their attributes that are represented in a database.
 * 3) A basic unit of information built on standard structures having a unique meaning and distinct units or values.
 * 4) In electronic record-keeping, a combination of characters or bytes referring to one separate item of information, such as name, address, or age.

In the areas of databases and data systems more generally a data element is a concept forming part of a data model. As an element of data representation, a collection of data elements forms a data structure.

In practice
In practice, data elements (fields, columns, attributes, etc.) are sometimes "overloaded", meaning a given data element will have multiple potential meanings. While a known bad practice, overloading is nevertheless a very real factor or barrier to understanding what a system is doing.

title: Data element