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T-carrier

The T-carrier is a member of the series of carrier systems developed by AT&T Bell Laboratories for digital transmission of multiplexed telephone calls.

The first version, the Transmission System 1 (T1), was introduced in 1962 in the Bell System, and could transmit up to 24 telephone calls simultaneously over a single transmission line of copper wire. Subsequent specifications carried multiples of the basic T1 (1.544 Mbit/s) data rates, such as T2 (6.312 Mbit/s) with 96 channels, T3 (44.736 Mbit/s) with 672 channels, and others.

Although a T2 was defined as part of AT&T's T-carrier system, which defined five levels, T1 through T5, only the T1 and T3 were commonly in use.

Transmission System 1
The T-carrier is a hardware specification for carrying multiple time-division multiplexed (TDM) telecommunications channels over a single four-wire transmission circuit. It was developed by AT&T at Bell Laboratories ca. 1957 and first employed by 1962 for long-haul pulse-code modulation (PCM) digital voice transmission with the D1 channel bank.

The T-carriers are commonly used for trunking between switching centers in a telephone network, including to private branch exchange (PBX) interconnect points. It uses the same twisted pair copper wire that analog trunks used, employing one pair for transmitting, and another pair for receiving. Signal repeaters may be used for extended distance requirements.

Before the digital T-carrier system, carrier wave systems such as 12-channel carrier systems worked by frequency-division multiplexing; each call was an analog signal. A T1 trunk could transmit 24 telephone calls at a time, because it used a digital carrier signal called Digital Signal 1 (DS-1). DS-1 is a communications protocol for multiplexing the bitstreams of up to 24 telephone calls, along with two special bits: a framing bit (for frame synchronization) and a maintenance-signaling bit. T1's maximum data transmission rate is 1.544 megabits per second.

Outside of the United States, Canada, Japan, and South Korea, the E-carrier system is used. E-carrier is similar transmission system with higher capacity that is not directly compatible with the T-carrier.

Legacy
Existing frequency-division multiplexing carrier systems worked well for connections between distant cities, but required expensive modulators, demodulators and filters for every voice channel. In the late 1950s, Bell Labs sought cheaper terminal equipment for connections within metropolitan areas. Pulse-code modulation allowed sharing a coder and decoder among several voice trunks, so this method was chosen for the T1 system introduced into local use in 1961. In later decades, the cost of digital electronics declined to the point that an individual codec per voice channel became commonplace, but by then the other advantages of digital transmission had become entrenched.

The T1 format carried 24 pulse-code modulated, time-division multiplexed speech signals each encoded in 64 kbit/s streams, leaving 8 kbit/s of framing information which facilitates the synchronization and demultiplexing at the receiver. The T2 and T3 circuit channels carry multiple T1 channels multiplexed, resulting in transmission rates of 6.312 and 44.736 Mbit/s, respectively. A T3 line comprises 28 T1 lines, each operating at total signaling rate of 1.544 Mbit/s. It is possible to get a fractional T3 line, meaning a T3 line with some of the 28 lines turned off, resulting in a slower transfer rate but typically at reduced cost.

Supposedly, the 1.544 Mbit/s rate was chosen because tests by AT&T Long Lines in Chicago were conducted underground. The test site was typical of Bell System outside plant of the time in that, to accommodate loading coils, cable vault manholes were physically 2000 m apart, which determined the repeater spacing. The optimum bit rate was chosen empirically—the capacity was increased until the failure rate was unacceptable, then reduced to leave a margin. Companding allowed acceptable audio performance with only seven bits per PCM sample in this original T1/D1 system. The later D3 and D4 channel banks had an extended frame format, allowing eight bits per sample, reduced to seven every sixth sample or frame when one bit was "robbed" for signaling the state of the channel. The standard does not allow an all zero sample which would produce a long string of binary zeros and cause the repeaters to lose bit sync. However, when carrying data (Switched 56) there could be long strings of zeros, so one bit per sample is set to "1" (jam bit 7) leaving 7 bits × 8,000 frames per second for data.

A more detailed understanding of the development of the 1.544 Mbit/s rate and its division into channels is as follows. Given that the telephone system nominal voiceband (including guardband) is 4,000 Hz, the required digital sampling rate is 8,000 Hz (see Nyquist rate). Since each T1 frame contains 1 byte of voice data for each of the 24 channels, that system needs then 8,000 frames per second to maintain those 24 simultaneous voice channels. Because each frame of a T1 is 193 bits in length (24 channels × 8 bits per channel + 1 framing bit = 193 bits), 8,000 frames per second is multiplied by 193 bits to yield a transfer rate of 1.544 Mbit/s (8,000 × 193 = 1,544,000).

Initially, T1 used Alternate Mark Inversion (AMI) to reduce frequency bandwidth and eliminate the DC component of the signal. Later B8ZS became common practice. For AMI, each mark pulse had the opposite polarity of the previous one and each space was at a level of zero, resulting in a three level signal which carried only binary data. Similar 1970s British 23 channel systems at 1.536 megabaud were equipped with ternary signal repeaters, in anticipation of using a 3B2T or 4B3T code to increase the number of voice channels in the future. But in the 1980s, the systems were merely replaced with European standard ones. American T-carriers could only work in AMI or B8ZS mode.

The AMI or B8ZS signal allowed a simple error rate measurement. The D bank in the central office could detect a bit with the wrong polarity, or "bipolarity violation" and sound an alarm. Later systems could count the number of violations and reframes and otherwise measure signal quality and allow a more sophisticated alarm indication signal system.

The decision to use a 193-bit frame was made in 1958. To allow for the identification of information bits within a frame, two alternatives were considered. Assign (a) just one extra bit, or (b) additional eight bits per frame. The 8-bit choice is cleaner, resulting in a 200-bit frame, twenty-five 8-bit channels, of which 24 are traffic and one 8-bit channel available for operations, administration, and maintenance (OA&M). AT&T chose the single bit per frame not to reduce the required bit rate (1.544 vs 1.6 Mbit/s), but because AT&T Marketing worried that "if 8 bits were chosen for OA&M function, someone would then try to sell this as a voice channel and you wind up with nothing."

Soon after commercial success of T1 in 1962, the T1 engineering team realized the mistake of having only one bit to serve the increasing demand for housekeeping functions. They petitioned AT&T management to change to 8-bit framing. This was flatly turned down because it would make installed systems obsolete.

Having this hindsight, some ten years later, CEPT chose eight bits for framing the European E1, although, as feared, the extra channel is sometimes appropriated for voice or data.

Higher bandwidth carriers
In the 1970s, Bell Labs developed higher rate systems. T1C with a more sophisticated modulation scheme carried 3 Mbit/s, on those balanced pair cables that could support it. T-2 carried 6.312 Mbit/s, requiring a special low-capacitance cable with foam insulation. This was standard for Picturephone. T-4 and T-5 used coaxial cables, similar to the old L-carriers used by AT&T Long Lines. TD microwave radio relay systems were also fitted with high rate modems to allow them to carry a DS1 signal in a portion of their FM spectrum that had too poor quality for voice service. Later they carried DS3 and DS4 signals. During the 1980s companies such as RLH Industries, Inc. developed T1 over optical fiber. The industry soon developed and evolved with multiplexed T1 transmission schemes.

Digital signal cross-connect
DS1 signals are interconnected typically at Central Office locations at a common metallic cross-connect point known as a DSX-1. When a DS1 is transported over metallic outside plant cable, the signal travels over conditioned cable pairs known as a T1 span. A T1 span can have up to +-130 Volts of DC power superimposed on the associated four wire cable pairs to supply power to line or "Span" signal repeaters, and T1 NIU's (T1 Smartjacks). T1 span repeaters are typically engineered up to 6000 ft apart, depending on cable gauge, and at no more than 36 dB of loss before requiring a repeated span. There can be no cable bridge taps or Load Coils across any pairs.

T1 copper spans are being replaced by optical transport systems, but if a copper (Metallic) span is used, the T1 is typically carried over an HDSL encoded copper line. Four wire HDSL does not require as many repeaters as conventional T1 spans. Newer two wire HDSL (HDSL-2) equipment transports a full 1.544 Mbit/s T1 over a single copper wire pair up to approximately twelve thousand (12,000) feet (3.5 km), if all 24 gauge cable is used. HDSL-2 does not employ multiple repeaters as does conventional four wire HDSL, or newer HDSL-4 systems.

One advantage of HDSL is its ability to operate with a limited number of bridge taps, with no tap being closer than 500 ft from any HDSL transceiver. Both two or four wire HDSL equipment transmits and receives over the same cable wire pair, as compared to conventional T1 service that utilizes individual cable pairs for transmit or receive.

DS3 signals are rare except within buildings, where they are used for interconnections and as an intermediate step before being multiplexed onto a SONET circuit. This is because a T3 circuit can only go about 600 ft between repeaters. A customer who orders a DS3 usually receives a SONET circuit run into the building and a multiplexer mounted in a utility box. The DS3 is delivered in its familiar form, two coax cables (1 for send and 1 for receive) with BNC connectors on the ends.

Bit robbing
Twelve DS1 frames make up a single T1 Superframe (T1 SF). Each T1 Superframe is composed of two signaling frames. All T1 DS0 channels that employ in-band signaling will have its eighth bit over written, or "robbed" from the full 64 kbit/s DS0 payload, by either a logical ZERO or ONE bit to signify a circuit signaling state or condition. Hence robbed bit signaling will restrict a DS0 channel to a rate of only 56 kbit/s during two of the twelve DS1 frames that make up a T1 SF framed circuit. T1 SF framed circuits yield two independent signaling channels (A and B) T1 ESF framed circuits four signaling frames in a twenty four frame extended frame format that yield four independent signaling channels (A, B, C, and D).

Fifty-six kbit/s DS0 channels are associated with digital data service (DDS) services typically do not utilize the eighth bit of the DS0 as voice circuits that employ A&B out of band signaling. One exception is Switched 56 kbit/s DDS. In DDS, bit eight is used to identify DTE request to send (RTS) condition. With Switched 56 DDS, bit eight is pulsed (alternately set to logical ZERO and ONE) to transmit two state dial pulse signaling information between a SW56 DDS CSU/DSU, and a digital end office switch.

The use of robbed-bit signaling in America has decreased significantly as a result of Signaling System No 7 (SS7) on inter-office dial trunks. With SS7, the full 64 kbit/s DS0 channel is available for use on a connection, and allows 64 kbit/s, and 128 kbit/s ISDN data calls to exist over a switched trunk network connection if the supporting T1 carrier entity is optioned B8ZS (Clear Channel Capable).

Carrier pricing
Carriers price DS1 lines in many different ways. However, most boil down to two simple components: local loop (the cost the local incumbent charges to transport the signal from the end user's central office, otherwise known as a CO, to the point of presence, otherwise known as a POP, of the carrier) and the port (the cost to access the telephone network or the Internet through the carrier's network). Typically, the port price is based upon access speed and yearly commitment level while the loop is based on geography. The farther the CO and POP, the more the loop costs.

The loop price has several components built into it, including the mileage calculation (performed in V/H coordinates, not standard GPS coordinates) and the telco piece. Each local Bell operating company—namely Verizon, AT&T Inc., and Qwest—charge T-carriers different price per mile rates. Therefore, the price calculation has two distance steps: geomapping and the determination of local price arrangements.

While most carriers utilize a geographic pricing model as described above, some Competitive Local Exchange Carriers (CLECs), such as TelePacific, Integra Telecom, tw telecom, Windstream, Level 3 Communications, and XO Communications offer national pricing.

Under this DS1 pricing model, a provider charges the same price in every geography it services. National pricing is an outgrowth of increased competition in the T-carrier market space and the commoditization of T-carrier products. Providers that have adopted a national pricing strategy may experience widely varying margins as their suppliers, the Bell operating companies (e.g., Verizon, AT&T Inc., and Qwest), maintain geographic pricing models, albeit at wholesale prices.

For voice DS1 lines, the calculation is mostly the same, except that the port (required for Internet access) is replaced by LDU (otherwise known as Long Distance Usage). Once the price of the loop is determined, only voice-related charges are added to the total. In short, the total price = loop + LDU x minutes used.

title: T-carrier


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Technical control facility

In telecommunication, a technical control facility (TCF) is a telecommunications facility, or a designated and specially configured part thereof, that (a) contains the equipment necessary for ensuring fast, reliable, and secure exchange of information, (b) typically includes distribution frames and associated panels, jacks, and switches and monitoring, test, conditioning, and orderwire equipment, and (c) allows telecommunications systems control personnel to exercise operational control of communications paths and facilities, make quality analyses of communications and communications channels, monitor operations and maintenance functions, recognize and correct deteriorating conditions, restore disrupted communications, provide requested on-call circuits, and take or direct such actions as may be required and practical to provide effective telecommunications services.

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

title: Technical control facility


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Telecommunications service

In telecommunication, a telecommunications service is a service provided by a telecommunications provider, or a specified set of user-information transfer capabilities provided to a group of users by a telecommunications system.

The telecommunications service user is responsible for the information content of the message. The telecommunications service provider has the responsibility for the acceptance, transmission, and delivery of the message.

For purposes of regulation by the Federal Communications Commission under the U.S. Communications Act of 1934 and Telecommunications Act of 1996, the definition of telecommunications service is "the offering of telecommunications for a fee directly to the public, or to such classes of users as to be effectively available directly to the public, regardless of the facilities used." Telecommunications, in turn, is defined as "the transmission, between or among points specified by the user, of information of the user’s choosing, without change in the form or content of the information as sent and received."

title: Telecommunications service


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Teleconference

A teleconference or telecon is a live exchange of information among several people remote from one another but linked by a telecommunications system. Terms such as audio conferencing, telephone conferencing, and phone conferencing are also sometimes used to refer to teleconferencing.

The telecommunications system may support the teleconference by providing one or more of the following: audio, video, and/or data services by one or more means, such as telephone, computer, telegraph, teletypewriter, radio, and television.

Internet teleconferencing
Internet teleconferencing includes internet telephone conferencing, videotelephony, web conferencing, virtual workplace, and augmented reality conferencing.

Internet telephony involves conducting a teleconference over the Internet or a wide area network. One key technology in this area is Voice over Internet Protocol (VOIP).

A working example of an augmented reality conferencing was demonstrated at the Salone di Mobile in Milano by AR+RFID Lab.

title: Teleconference


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Terminal adapter

A terminal adapter or TA is a device that connects a terminal device – a computer, a mobile communications device, or other – to a communications network.

ISDN
In ISDN terminology, the terminal adapter connects a terminal (computer) to the ISDN network. The TA therefore fulfills a similar function to the ones a modem has on the POTS network, and is therefore sometimes called an ISDN modem. The latter term, however, is partially misleading as there is no modulation or demodulation performed.

There are devices on the market that combine the functions of an ISDN TA with those of a classical modem (with an ISDN line interface). These combined TA/modems permit connections from both ISDN and analog-line/modem counterparts. In addition, a TA may contain an interface and codec for one or more analog telephone lines (aka a/b line), allowing an existing POTS installation to be upgraded to ISDN without changing phones.

Terminal adapters typically connect to a basic rate interface (S0, sometimes also U0). On the terminal side, the most popular interfaces are RS-232 serial and USB; others like V.35 or RS-449 are only of historical interest.

Devices connecting ISDN to a network (e.g. Ethernet) commonly include routing functionality; while they technically include a TA function, they are referred to as (ISDN) routers.

Mobile networks
In mobile networks, the terminal adapter is used by the terminal equipment to access the mobile termination, using AT commands (see Hayes command set).

In 2G (such as GSM or CDMA), the terminal adapter is a theoretically optional while in 3G (such as W-CDMA), the terminal adapter is mandatory and is part of the mobile termination.

Automation industry
In the automation industry, a terminal adapter is a passive device that converts a connector like the 8P8C (RJ-45) modular connector or 9 pin D-Sub into a terminal block to facilitate wiring. It is often used when daisy chain wiring is necessary on a multi-node serial communication network like RS-485 or RS-422.

title: Terminal adapter


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Terminal equipment

In telecommunication, the term terminal equipment has the following meanings:

 * Communications equipment at either end of a communications link, used to permit the stations involved to accomplish the mission for which the link was established.
 * In radio-relay systems, equipment used at points where data are inserted or derived, as distinct from equipment used only to relay a reconstituted signal.
 * Telephone and telegraph switchboards and other centrally located equipment at which communications circuits are terminated.

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Ternary signal

In telecommunication, a ternary signal is a signal that can assume, at any given instant, one of three states or significant conditions, such as power level, phase position, pulse duration, or frequency.

Examples of ternary signals are (a) a pulse that can have a positive, zero, or negative voltage value at any given instant (PAM-3), (b) a sine wave that can assume phases of 0°, 120°, or 240° relative to a clock pulse (3-PSK), and (c) a carrier signal that can assume any one of three different frequencies depending on three different modulation signal significant conditions (3-FM).

Some examples of PAM-3 line codes that use ternary signals are:
 * hybrid ternary code
 * bipolar encoding
 * MLT-3 encoding used in 100BASE-TX Ethernet
 * B3ZS
 * 4B3T used in some ISDN basic rate interface
 * 8B6T used in 100BASE-T4 Ethernet
 * return-to-zero
 * SOQPSK-TG uses ternary continuous phase modulation

3-PSK can be seen as falling between "binary phase-shift keying" (BPSK), which uses two phases, and "quadrature phase-shift keying" (QPSK), which uses four phases.

title: Ternary signal


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Thermodynamic temperature

Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.

Historically, thermodynamic temperature was defined by Lord Kelvin in terms of a macroscopic relation between thermodynamic work and heat transfer as defined in thermodynamics, but the kelvin was redefined by international agreement in 2019 in terms of phenomena that are now understood as manifestations of the kinetic energy of free motion of microscopic particles such as atoms, molecules, and electrons. From the thermodynamic viewpoint, for historical reasons, because of how it is defined and measured, this microscopic kinetic definition is regarded as an "empirical" temperature. It was adopted because in practice it can generally be measured more precisely than can Kelvin's thermodynamic temperature.

A thermodynamic temperature of zero is of particular importance for the third law of thermodynamics. By convention, it is reported on the Kelvin scale of temperature in which the unit of measurement is the kelvin (unit symbol: K). For comparison, a temperature of 295 K corresponds to 21.85 °C and 71.33 °F.

Overview
Thermodynamic temperature, as distinct from SI temperature, is defined in terms of a macroscopic Carnot cycle. Thermodynamic temperature is of importance in thermodynamics because it is defined in purely thermodynamic terms. SI temperature is conceptually far different from thermodynamic temperature. Thermodynamic temperature was rigorously defined historically long before there was a fair knowledge of microscopic particles such as atoms, molecules, and electrons.

The International System of Units (SI) specifies the international absolute scale for measuring temperature, and the unit of measure kelvin (unit symbol: K) for specific values along the scale. The kelvin is also used for denoting temperature intervals (a span or difference between two temperatures) as per the following example usage: "A 60/40 tin/lead solder is non-eutectic and is plastic through a range of 5 kelvins as it solidifies." A temperature interval of one degree Celsius is the same magnitude as one kelvin.

The magnitude of the kelvin was redefined in 2019 in relation to the physical property underlying thermodynamic temperature: the kinetic energy of atomic free particle motion. The redefinition fixed the Boltzmann constant at precisely $1.381 joules per kelvin$ (J/K).

The microscopic property that imbues material substances with a temperature can be readily understood by examining the ideal gas law, which relates, per the Boltzmann constant, how heat energy causes precisely defined changes in the pressure and temperature of certain gases. This is because monatomic gases like helium and argon behave kinetically like freely moving perfectly elastic and spherical billiard balls that move only in a specific subset of the possible motions that can occur in matter: that comprising the three translational degrees of freedom. The translational degrees of freedom are the familiar billiard ball-like movements along the X, Y, and Z axes of 3D space (see Fig. 1, below). This is why the noble gases all have the same specific heat capacity per atom and why that value is lowest of all the gases.

Molecules (two or more chemically bound atoms), however, have internal structure and therefore have additional internal degrees of freedom (see Fig. 3, below), which makes molecules absorb more heat energy for any given amount of temperature rise than do the monatomic gases. Heat energy is born in all available degrees of freedom; this is in accordance with the equipartition theorem, so all available internal degrees of freedom have the same temperature as their three external degrees of freedom. However, the property that gives all gases their pressure, which is the net force per unit area on a container arising from gas particles recoiling off it, is a function of the kinetic energy borne in the freely moving atoms' and molecules' three translational degrees of freedom.

Fixing the Boltzmann constant at a specific value, along with other rule making, had the effect of precisely establishing the magnitude of the unit interval of SI temperature, the kelvin, in terms of the average kinetic behavior of the noble gases. Moreover, the starting point of the thermodynamic temperature scale, absolute zero, was reaffirmed as the point at which zero average kinetic energy remains in a sample; the only remaining particle motion being that comprising random vibrations due to zero-point energy.

Absolute zero of temperature
Temperature scales are numerical. The numerical zero of a temperature scale is not bound to the absolute zero of temperature. Nevertheless, some temperature scales have their numerical zero coincident with the absolute zero of temperature. Examples are the International SI temperature scale, the Rankine temperature scale, and the thermodynamic temperature scale. Other temperature scales have their numerical zero far from the absolute zero of temperature. Examples are the Fahrenheit scale and the Celsius scale.

At the zero point of thermodynamic temperature, absolute zero, the particle constituents of matter have minimal motion and can become no colder. Absolute zero, which is a temperature of zero kelvins (0 K), precisely corresponds to −273.15 °C and −459.67 °F. Matter at absolute zero has no remaining transferable average kinetic energy and the only remaining particle motion is due to an ever-pervasive quantum mechanical phenomenon called ZPE (zero-point energy). While scientists are achieving temperatures ever closer to absolute zero, they can not fully achieve a state of zero temperature. However, even if scientists could remove all kinetic thermal energy from matter, quantum mechanical zero-point energy (ZPE) causes particle motion that can never be eliminated. Encyclopædia Britannica Online defines zero-point energy as the "vibrational energy that molecules retain even at the absolute zero of temperature". ZPE is the result of all-pervasive energy fields in the vacuum between the fundamental particles of nature; it is responsible for the Casimir effect and other phenomena. See Zero Point Energy and Zero Point Field. See also Solid Helium  by the University of Alberta's Department of Physics to learn more about ZPE's effect on Bose–Einstein condensates of helium.

Although absolute zero (T = 0) is not a state of zero molecular motion, it is the point of zero temperature and, in accordance with the Boltzmann constant, is also the point of zero particle kinetic energy and zero kinetic velocity. To understand how atoms can have zero kinetic velocity and simultaneously be vibrating due to ZPE, consider the following thought experiment: two T = 0 helium atoms in zero gravity are carefully positioned and observed to have an average separation of 620 pm between them (a gap of ten atomic diameters). It is an "average" separation because ZPE causes them to jostle about their fixed positions. Then one atom is given a kinetic kick of precisely 83 yoctokelvins (1 yK = $1 K$). This is done in a way that directs this atom's velocity vector at the other atom. With 83 yK of kinetic energy between them, the 620 pm gap through their common barycenter would close at a rate of 719 pm/s and they would collide after 0.862 second. This is the same speed as shown in the Fig. 1 animation above. Before being given the kinetic kick, both T = 0 atoms had zero kinetic energy and zero kinetic velocity because they could persist indefinitely in that state and relative orientation even though both were being jostled by ZPE. At T = 0, no kinetic energy is available for transfer to other systems.

Note too that absolute zero serves as the baseline atop which thermodynamics and its equations are founded because they deal with the exchange of thermal energy between "systems" (a plurality of particles and fields modeled as an average). Accordingly, one may examine ZPE-induced particle motion within a system that is at absolute zero but there can never be a net outflow of thermal energy from such a system. Also, the peak emittance wavelength of black-body radiation shifts to infinity at absolute zero; indeed, a peak no longer exists and black-body photons can no longer escape. Because of ZPE, however, virtual photons are still emitted at T = 0. Such photons are called "virtual" because they can't be intercepted and observed. Furthermore, this zero-point radiation has a unique zero-point spectrum. However, even though a T = 0 system emits zero-point radiation, no net heat flow Q out of such a system can occur because if the surrounding environment is at a temperature greater than T = 0, heat will flow inward, and if the surrounding environment is at 'T = 0, there will be an equal flux of ZP radiation both inward and outward. A similar Q equilibrium exists at T = 0 with the ZPE-induced spontaneous emission of photons (which is more properly called a stimulated emission in this context). The graph at upper right illustrates the relationship of absolute zero to zero-point energy. The graph also helps in the understanding of how zero-point energy got its name: it is the vibrational energy matter retains at the zero-kelvin point. Derivation of the classical electromagnetic zero-point radiation spectrum via a classical thermodynamic operation involving van der Waals forces, Daniel C. Cole, Physical Review A, 42 (1990) 1847. Though the atoms in, for instance, a container of liquid helium that was precisely at absolute zero would still jostle slightly due to zero-point energy, a theoretically perfect heat engine with such helium as one of its working fluids could never transfer any net kinetic energy (heat energy) to the other working fluid and no thermodynamic work could occur.

Temperature is generally expressed in absolute terms when scientifically examining temperature's interrelationships with certain other physical properties of matter such as its volume or pressure (see Gay-Lussac's law), or the wavelength of its emitted black-body radiation. Absolute temperature is also useful when calculating chemical reaction rates (see Arrhenius equation). Furthermore, absolute temperature is typically used in cryogenics and related phenomena like superconductivity, as per the following example usage: "Conveniently, tantalum's transition temperature (T$c$) of 4.4924 kelvin is slightly above the 4.2221 K boiling point of helium."

Boltzmann constant
The Boltzmann constant and its related formulas describe the realm of particle kinetics and velocity vectors whereas ZPE (zero-point energy) is an energy field that jostles particles in ways described by the mathematics of quantum mechanics. In atomic and molecular collisions in gases, ZPE introduces a degree of chaos, i.e., unpredictability, to rebound kinetics; it is as likely that there will be less ZPE-induced particle motion after a given collision as more. This random nature of ZPE is why it has no net effect upon either the pressure or volume of any bulk quantity (a statistically significant quantity of particles) of gases. However, in temperature T = 0 condensed matter; e.g., solids and liquids, ZPE causes inter-atomic jostling where atoms would otherwise be perfectly stationary. Inasmuch as the real-world effects that ZPE has on substances can vary as one alters a thermodynamic system (for example, due to ZPE, helium won't freeze unless under a pressure of at least 2.5 MPa (25 bar)), ZPE is very much a form of thermal energy and may properly be included when tallying a substance's internal energy.

Rankine scale
Though there have been many other temperature scales throughout history, there have been only two scales for measuring thermodynamic temperature which have absolute zero as their null point (0): The Kelvin scale and the Rankine scale.

Throughout the scientific world where modern measurements are nearly always made using the International System of Units, thermodynamic temperature is measured using the Kelvin scale. The Rankine scale is part of English engineering units and finds use in certain engineering fields, particularly in legacy reference works. The Rankine scale uses the degree Rankine (symbol: °R) as its unit, which is the same magnitude as the degree Fahrenheit (symbol: °F).

A unit increment of one kelvin is exactly 1.8 times one degree Rankine; thus, to convert a specific temperature on the Kelvin scale to the Rankine scale, $x$ K = 1.8 $x$ °R, and to convert from a temperature on the Rankine scale to the Kelvin scale, $x$ °R = $x$/1.8 K. Consequently, absolute zero is "0" for both scales, but the melting point of water ice (0 °C and 273.15 K) is 491.67 °R.

To convert temperature intervals (a span or difference between two temperatures), the formulas from the preceding paragraph are applicable; for instance, an interval of 5 kelvin is precisely equal to an interval of 9 degrees Rankine.

Modern redefinition of the kelvin
For 65 years, between 1954 and the 2019 redefinition of the SI base units, a temperature interval of one kelvin was defined as $1⁄273.16$ the difference between the triple point of water and absolute zero. The 1954 resolution by the International Bureau of Weights and Measures (known by the French-language acronym BIPM), plus later resolutions and publications, defined the triple point of water as precisely 273.16 K and acknowledged that it was "common practice" to accept that due to previous conventions (namely, that 0 °C had long been defined as the melting point of water and that the triple point of water had long been experimentally determined to be indistinguishably close to 0.01 °C), the difference between the Celsius scale and Kelvin scale is accepted as 273.15 kelvins; which is to say, 0 °C corresponds to 273.15 kelvins. The net effect of this as well as later resolutions was twofold: 1) they defined absolute zero as precisely 0 K, and 2) they defined that the triple point of special isotopically controlled water called Vienna Standard Mean Ocean Water occurred at precisely 273.16 K and 0.01 °C. One effect of the aforementioned resolutions was that the melting point of water, while very close to 273.15 K and 0 °C, was not a defining value and was subject to refinement with more precise measurements.

The 1954 BIPM standard did a good job of establishing—within the uncertainties due to isotopic variations between water samples—temperatures around the freezing and triple points of water, but required that intermediate values between the triple point and absolute zero, as well as extrapolated values from room temperature and beyond, to be experimentally determined via apparatus and procedures in individual labs. This shortcoming was addressed by the International Temperature Scale of 1990, or ITS90, which defined 13 additional points, from 13.8033 K, to 1,357.77 K. While definitional, ITS90 had—and still has—some challenges, partly because eight of its extrapolated values depend upon the melting or freezing points of metal samples, which must remain exceedingly pure lest their melting or freezing points be affected—usually depressed.

The 2019 redefinition of the SI base units was primarily for the purpose of decoupling much of the SI system's definitional underpinnings from the kilogram, which was the last physical artifact defining an SI base unit (a platinum/iridium cylinder stored under three nested bell jars in a safe located in France) and which had highly questionable stability. The solution required that four physical constants, including the Boltzmann constant, be definitionally fixed.

Assigning the Boltzmann constant a precisely defined value had no practical effect on modern thermometry except for the most exquisitely precise measurements. Before the redefinition, the triple point of water was exactly 273.16 K and 0.01 °C and the Boltzmann constant was experimentally determined to be $1.381 J$, where the "(51)" denotes the uncertainty in the two least significant digits (the 03) and equals a relative standard uncertainty of 0.37 ppm. Afterwards, by defining the Boltzmann constant as exactly $1.381 J$, the 0.37 ppm uncertainty was transferred to the triple point of water, which became an experimentally determined value of $273.16 K$ ($0.01 °C$). That the triple point of water ended up being exceedingly close to 273.16 K after the SI redefinition was no accident; the final value of the Boltzmann constant was determined, in part, through clever experiments with argon and helium that used the triple point of water for their key reference temperature.

Notwithstanding the 2019 redefinition, water triple-point cells continue to serve in modern thermometry as exceedingly precise calibration references at 273.16 K and 0.01 °C. Moreover, the triple point of water remains one of the 14 calibration points comprising ITS90, which spans from the triple point of hydrogen (13.8033 K) to the freezing point of copper (1,357.77 K), which is a nearly hundredfold range of thermodynamic temperature.

Nature of kinetic energy, translational motion, and temperature
The thermodynamic temperature of any bulk quantity of a substance (a statistically significant quantity of particles) is directly proportional to the mean average kinetic energy of a specific kind of particle motion known as translational motion. These simple movements in the three X, Y, and Z–axis dimensions of space means the particles move in the three spatial degrees of freedom. This particular form of kinetic energy is sometimes referred to as kinetic temperature. Translational motion is but one form of heat energy and is what gives gases not only their temperature, but also their pressure and the vast majority of their volume. This relationship between the temperature, pressure, and volume of gases is established by the ideal gas law's formula $pV = nRT$ and is embodied in the gas laws.

Though the kinetic energy borne exclusively in the three translational degrees of freedom comprise the thermodynamic temperature of a substance, molecules, as can be seen in Fig. 3, can have other degrees of freedom, all of which fall under three categories: bond length, bond angle, and rotational. All three additional categories are not necessarily available to all molecules, and even for molecules that can experience all three, some can be "frozen out" below a certain temperature. Nonetheless, all those degrees of freedom that are available to the molecules under a particular set of conditions contribute to the specific heat capacity of a substance; which is to say, they increase the amount of heat (kinetic energy) required to raise a given amount of the substance by one kelvin or one degree Celsius.

The relationship of kinetic energy, mass, and velocity is given by the formula Ek = $1⁄2$mv. Accordingly, particles with one unit of mass moving at one unit of velocity have precisely the same kinetic energy, and precisely the same temperature, as those with four times the mass but half the velocity.

The extent to which the kinetic energy of translational motion in a statistically significant collection of atoms or molecules in a gas contributes to the pressure and volume of that gas is a proportional function of thermodynamic temperature as established by the Boltzmann constant (symbol: kB). The Boltzmann constant also relates the thermodynamic temperature of a gas to the mean kinetic energy of an individual particles' translational motion as follows: $$\tilde{E} = \frac{3}{2} k_\text{B} T$$ where:
 * $$ \tilde{E}$$ is the mean kinetic energy for an individual particle
 * kB = $1.381 J/K$
 * T is the thermodynamic temperature of the bulk quantity of the substance

While the Boltzmann constant is useful for finding the mean kinetic energy in a sample of particles, it is important to note that even when a substance is isolated and in thermodynamic equilibrium (all parts are at a uniform temperature and no heat is going into or out of it), the translational motions of individual atoms and molecules occurs across a wide range of speeds (see animation in Fig. 1 above). At any one instant, the proportion of particles moving at a given speed within this range is determined by probability as described by the Maxwell–Boltzmann distribution. The graph shown here in Fig. 2 shows the speed distribution of 5500 K helium atoms. They have a most probable speed of 4.780 km/s (0.2092 s/km). However, a certain proportion of atoms at any given instant are moving faster while others are moving relatively slowly; some are momentarily at a virtual standstill (off the x–axis to the right). This graph uses inverse speed for its x–axis so the shape of the curve can easily be compared to the curves in Fig. 5 below. In both graphs, zero on the x–axis represents infinite temperature. Additionally, the x and y–axis on both graphs are scaled proportionally.

High speeds of translational motion
Although very specialized laboratory equipment is required to directly detect translational motions, the resultant collisions by atoms or molecules with small particles suspended in a fluid produces Brownian motion that can be seen with an ordinary microscope. The translational motions of elementary particles are very fast and temperatures close to absolute zero are required to directly observe them. For instance, when scientists at the NIST achieved a record-setting cold temperature of 700 nK (billionths of a kelvin) in 1994, they used optical lattice laser equipment to adiabatically cool cesium atoms. They then turned off the entrapment lasers and directly measured atom velocities of 7 mm per second to in order to calculate their temperature. Formulas for calculating the velocity and speed of translational motion are given in the following footnote.The rate of translational motion of atoms and molecules is calculated based on thermodynamic temperature as follows: $$\tilde{v} = \sqrt{\frac {\frac{k_\text{B}}{2} \cdot T}{\frac{m}{2}}}$$ where
 * $$\tilde{v}$$ is the vector-isolated mean velocity of translational particle motion in m/s
 * kB (Boltzmann constant) = $2.42$
 * T is the thermodynamic temperature in kelvins
 * m is the molecular mass of substance in kg/particle

In the above formula, molecular mass, m, in kg/particle is the quotient of a substance's molar mass (also known as atomic weight, atomic mass, relative atomic mass, and unified atomic mass units) in g/mol or daltons divided by $1.381 J/K$ (which is the Avogadro constant times one thousand). For diatomic molecules such as H2, N2, and O2, multiply atomic weight by two before plugging it into the above formula.

The mean speed (not vector-isolated velocity) of an atom or molecule along any arbitrary path is calculated as follows: $$\tilde{s} = \tilde{v} \cdot \sqrt{3}$$ where $$\tilde{s}$$ is theundefinedmean speed of translational particle motion in m/s.

Noteundefinedthat the mean energy of the translational motions of a substance's constituent particles correlates to their mean speed, not velocity. Thus, substituting $$\tilde{s}$$ for v in the classic formula for kinetic energy, $E_{k} = 1⁄2mv$ produces precisely the same value as does $E_{mean} = 3/2k_{B}T$ (as shown in the section titled The nature of kinetic energy, translational motion, and temperature).undefined Noteundefinedtoo that the Boltzmann constant and its related formulas establish that absolute zero is the point of both zero kinetic energy of particle motion and zero kinetic velocity (see also Note 1 above).

It is neither difficult to imagine atomic motions due to kinetic temperature, nor distinguish between such motions and those due to zero-point energy. Consider the following hypothetical thought experiment, as illustrated in Fig. 2.5 at left, with an atom that is exceedingly close to absolute zero. Imagine peering through a common optical microscope set to 400 power, which is about the maximum practical magnification for optical microscopes. Such microscopes generally provide fields of view a bit over 0.4 mm in diameter. At the center of the field of view is a single levitated argon atom (argon comprises about 0.93% of air) that is illuminated and glowing against a dark backdrop. If this argon atom was at a beyond-record-setting one-trillionth of a kelvin above absolute zero, and was moving perpendicular to the field of view towards the right, it would require 13.9 seconds to move from the center of the image to the 200-micron tick mark; this travel distance is about the same as the width of the period at the end of this sentence on modern computer monitors. As the argon atom slowly moved, the positional jitter due to zero-point energy would be much less than the 200-nanometer (0.0002 mm) resolution of an optical microscope. Importantly, the atom's translational velocity of 14.43 microns per second constitutes all its retained kinetic energy due to not being precisely at absolute zero. Were the atom precisely at absolute zero, imperceptible jostling due to zero-point energy would cause it to very slightly wander, but the atom would perpetually be located, on average, at the same spot within the field of view. This is analogous to a boat that has had its motor turned off and is now bobbing slightly in relatively calm and windless ocean waters; even though the boat randomly drifts to and fro, it stays in the same spot in the long term and makes no headway through the water. Accordingly, an atom that was precisely at absolute zero would not be "motionless", and yet, a statistically significant collection of such atoms would have zero net kinetic energy available to transfer to any other collection of atoms. This is because regardless of the kinetic temperature of the second collection of atoms, they too experience the effects of zero-point energy. Such are the consequences of statistical mechanics and the nature of thermodynamics.

Internal motions of molecules and internal energy
As mentioned above, there are other ways molecules can jiggle besides the three translational degrees of freedom that imbue substances with their kinetic temperature. As can be seen in the animation at right, molecules are complex objects; they are a population of atoms and thermal agitation can strain their internal chemical bonds in three different ways: via rotation, bond length, and bond angle movements; these are all types of internal degrees of freedom. This makes molecules distinct from monatomic substances (consisting of individual atoms) like the noble gases helium and argon, which have only the three translational degrees of freedom (the X, Y, and Z axis). Kinetic energy is stored in molecules' internal degrees of freedom, which gives them an internal temperature. Even though these motions are called "internal", the external portions of molecules still move—rather like the jiggling of a stationary water balloon. This permits the two-way exchange of kinetic energy between internal motions and translational motions with each molecular collision. Accordingly, as internal energy is removed from molecules, both their kinetic temperature (the kinetic energy of translational motion) and their internal temperature simultaneously diminish in equal proportions. This phenomenon is described by the equipartition theorem, which states that for any bulk quantity of a substance in equilibrium, the kinetic energy of particle motion is evenly distributed among all the active degrees of freedom available to the particles. Since the internal temperature of molecules are usually equal to their kinetic temperature, the distinction is usually of interest only in the detailed study of non-local thermodynamic equilibrium (LTE) phenomena such as combustion, the sublimation of solids, and the diffusion of hot gases in a partial vacuum.

The kinetic energy stored internally in molecules causes substances to contain more heat energy at any given temperature and to absorb additional internal energy for a given temperature increase. This is because any kinetic energy that is, at a given instant, bound in internal motions, is not contributing to the molecules' translational motions at that same instant. This extra kinetic energy simply increases the amount of internal energy that substance absorbs for a given temperature rise. This property is known as a substance's specific heat capacity.

Different molecules absorb different amounts of internal energy for each incremental increase in temperature; that is, they have different specific heat capacities. High specific heat capacity arises, in part, because certain substances' molecules possess more internal degrees of freedom than others do. For instance, room-temperature nitrogen, which is a diatomic molecule, has five active degrees of freedom: the three comprising translational motion plus two rotational degrees of freedom internally. Not surprisingly, in accordance with the equipartition theorem, nitrogen has five-thirds the specific heat capacity per mole (a specific number of molecules) as do the monatomic gases. Another example is gasoline (see table showing its specific heat capacity). Gasoline can absorb a large amount of heat energy per mole with only a modest temperature change because each molecule comprises an average of 21 atoms and therefore has many internal degrees of freedom. Even larger, more complex molecules can have dozens of internal degrees of freedom.

Diffusion of thermal energy: entropy, phonons, and mobile conduction electrons

Heat conduction is the diffusion of thermal energy from hot parts of a system to cold parts. A system can be either a single bulk entity or a plurality of discrete bulk entities. The term bulk in this context means a statistically significant quantity of particles (which can be a microscopic amount). Whenever thermal energy diffuses within an isolated system, temperature differences within the system decrease (and entropy increases).

One particular heat conduction mechanism occurs when translational motion, the particle motion underlying temperature, transfers momentum from particle to particle in collisions. In gases, these translational motions are of the nature shown above in Fig. 1. As can be seen in that animation, not only does momentum (heat) diffuse throughout the volume of the gas through serial collisions, but entire molecules or atoms can move forward into new territory, bringing their kinetic energy with them. Consequently, temperature differences equalize throughout gases very quickly—especially for light atoms or molecules; convection speeds this process even more.

Translational motion in solids, however, takes the form of phonons (see Fig. 4 at right). Phonons are constrained, quantized wave packets that travel at the speed of sound of a given substance. The manner in which phonons interact within a solid determines a variety of its properties, including its thermal conductivity. In electrically insulating solids, phonon-based heat conduction is usually inefficient and such solids are considered thermal insulators (such as glass, plastic, rubber, ceramic, and rock). This is because in solids, atoms and molecules are locked into place relative to their neighbors and are not free to roam.

Metals however, are not restricted to only phonon-based heat conduction. Thermal energy conducts through metals extraordinarily quickly because instead of direct molecule-to-molecule collisions, the vast majority of thermal energy is mediated via very light, mobile conduction electrons. This is why there is a near-perfect correlation between metals' thermal conductivity and their electrical conductivity. Conduction electrons imbue metals with their extraordinary conductivity because they are delocalized (i.e., not tied to a specific atom) and behave rather like a sort of quantum gas due to the effects of zero-point energy (for more on ZPE, see Note 1 below). Furthermore, electrons are relatively light with a rest mass only $6.022$ that of a proton. This is about the same ratio as a .22 Short bullet (29 grains or 1.88 g) compared to the rifle that shoots it. As Isaac Newton wrote with his third law of motion,

"Law #3: All forces occur in pairs, and these two forces are equal in magnitude and opposite in direction."

However, a bullet accelerates faster than a rifle given an equal force. Since kinetic energy increases as the square of velocity, nearly all the kinetic energy goes into the bullet, not the rifle, even though both experience the same force from the expanding propellant gases. In the same manner, because they are much less massive, thermal energy is readily borne by mobile conduction electrons. Additionally, because they're delocalized and very fast, kinetic thermal energy conducts extremely quickly through metals with abundant conduction electrons.

Diffusion of thermal energy: black-body radiation
Thermal radiation is a byproduct of the collisions arising from various vibrational motions of atoms. These collisions cause the electrons of the atoms to emit thermal photons (known as black-body radiation). Photons are emitted anytime an electric charge is accelerated (as happens when electron clouds of two atoms collide). Even individual molecules with internal temperatures greater than absolute zero also emit black-body radiation from their atoms. In any bulk quantity of a substance at equilibrium, black-body photons are emitted across a range of wavelengths in a spectrum that has a bell curve-like shape called a Planck curve (see graph in Fig. 5 at right). The top of a Planck curve (the peak emittance wavelength) is located in a particular part of the electromagnetic spectrum depending on the temperature of the black-body. Substances at extreme cryogenic temperatures emit at long radio wavelengths whereas extremely hot temperatures produce short gamma rays (see Table of common temperatures).

Black-body radiation diffuses thermal energy throughout a substance as the photons are absorbed by neighboring atoms, transferring momentum in the process. Black-body photons also easily escape from a substance and can be absorbed by the ambient environment; kinetic energy is lost in the process.

As established by the Stefan–Boltzmann law, the intensity of black-body radiation increases as the fourth power of absolute temperature. Thus, a black-body at 824 K (just short of glowing dull red) emits 60 times the radiant power as it does at 296 K (room temperature). This is why one can so easily feel the radiant heat from hot objects at a distance. At higher temperatures, such as those found in an incandescent lamp, black-body radiation can be the principal mechanism by which thermal energy escapes a system.

Table of thermodynamic temperatures
The table below shows various points on the thermodynamic scale, in order of increasing temperature.

Heat of phase changes
The kinetic energy of particle motion is just one contributor to the total thermal energy in a substance; another is phase transitions, which are the potential energy of molecular bonds that can form in a substance as it cools (such as during condensing and freezing). The thermal energy required for a phase transition is called latent heat. This phenomenon may more easily be grasped by considering it in the reverse direction: latent heat is the energy required to break chemical bonds (such as during evaporation and melting). Almost everyone is familiar with the effects of phase transitions; for instance, steam at 100 °C can cause severe burns much faster than the 100 °C air from a hair dryer. This occurs because a large amount of latent heat is liberated as steam condenses into liquid water on the skin.

Even though thermal energy is liberated or absorbed during phase transitions, pure chemical elements, compounds, and eutectic alloys exhibit no temperature change whatsoever while they undergo them (see Fig. 7, below right). Consider one particular type of phase transition: melting. When a solid is melting, crystal lattice chemical bonds are being broken apart; the substance is transitioning from what is known as a more ordered state to a less ordered state. In Fig. 7, the melting of ice is shown within the lower left box heading from blue to green.

At one specific thermodynamic point, the melting point (which is 0 °C across a wide pressure range in the case of water), all the atoms or molecules are, on average, at the maximum energy threshold their chemical bonds can withstand without breaking away from the lattice. Chemical bonds are all-or-nothing forces: they either hold fast, or break; there is no in-between state. Consequently, when a substance is at its melting point, every joule of added thermal energy only breaks the bonds of a specific quantity of its atoms or molecules, converting them into a liquid of precisely the same temperature; no kinetic energy is added to translational motion (which is what gives substances their temperature). The effect is rather like popcorn: at a certain temperature, additional thermal energy can't make the kernels any hotter until the transition (popping) is complete. If the process is reversed (as in the freezing of a liquid), thermal energy must be removed from a substance.

As stated above, the thermal energy required for a phase transition is called latent heat. In the specific cases of melting and freezing, it's called enthalpy of fusion or heat of fusion. If the molecular bonds in a crystal lattice are strong, the heat of fusion can be relatively great, typically in the range of 6 to 30 kJ per mole for water and most of the metallic elements. If the substance is one of the monatomic gases, (which have little tendency to form molecular bonds) the heat of fusion is more modest, ranging from 0.021 to 2.3 kJ per mole. Relatively speaking, phase transitions can be truly energetic events. To completely melt ice at 0 °C into water at 0 °C, one must add roughly 80 times the thermal energy as is required to increase the temperature of the same mass of liquid water by one degree Celsius. The metals' ratios are even greater, typically in the range of 400 to 1200 times. And the phase transition of boiling is much more energetic than freezing. For instance, the energy required to completely boil or vaporize water (what is known as enthalpy of vaporization) is roughly 540 times that required for a one-degree increase.

Water's sizable enthalpy of vaporization is why one's skin can be burned so quickly as steam condenses on it (heading from red to green in Fig. 7 above); water vapors (gas phase) are liquefied on the skin with releasing a large amount of energy (enthalpy) to the environment including the skin, resulting in skin damage. In the opposite direction, this is why one's skin feels cool as liquid water on it evaporates (a process that occurs at a sub-ambient wet-bulb temperature that is dependent on relative humidity); the water evaporation on the skin takes a large amount of energy from the environment including the skin, reducing the skin temperature. Water's highly energetic enthalpy of vaporization is also an important factor underlying why solar pool covers (floating, insulated blankets that cover swimming pools when the pools are not in use) are so effective at reducing heating costs: they prevent evaporation. (In other words, taking energy from water when it is evaporated is limited.) For instance, the evaporation of just 20 mm of water from a 1.29-meter-deep pool chills its water 8.4 degrees Celsius (15.1 °F).

Internal energy
The total energy of all translational and internal particle motions, including that of conduction electrons, plus the potential energy of phase changes, plus zero-point energy of a substance comprise the internal energy of it.

Internal energy at absolute zero
As a substance cools, different forms of internal energy and their related effects simultaneously decrease in magnitude: the latent heat of available phase transitions is liberated as a substance changes from a less ordered state to a more ordered state; the translational motions of atoms and molecules diminish (their kinetic energy or temperature decreases); the internal motions of molecules diminish (their internal energy or temperature decreases); conduction electrons (if the substance is an electrical conductor) travel somewhat slower; and black-body radiation's peak emittance wavelength increases (the photons' energy decreases). When particles of a substance are as close as possible to complete rest and retain only ZPE (zero-point energy)-induced quantum mechanical motion, the substance is at the temperature of absolute zero (T = 0).

Note that whereas absolute zero is the point of zero thermodynamic temperature and is also the point at which the particle constituents of matter have minimal motion, absolute zero is not necessarily the point at which a substance contains zero internal energy; one must be very precise with what one means by internal energy. Often, all the phase changes that can occur in a substance, will have occurred by the time it reaches absolute zero. However, this is not always the case. Notably, T = 0 helium remains liquid at room pressure (Fig. 9 at right) and must be under a pressure of at least 25 bar to crystallize. This is because helium's heat of fusion (the energy required to melt helium ice) is so low (only 21 joules per mole) that the motion-inducing effect of zero-point energy is sufficient to prevent it from freezing at lower pressures.

A further complication is that many solids change their crystal structure to more compact arrangements at extremely high pressures (up to millions of bars, or hundreds of gigapascals). These are known as solid–solid phase transitions wherein latent heat is liberated as a crystal lattice changes to a more thermodynamically favorable, compact one.

The above complexities make for rather cumbersome blanket statements regarding the internal energy in T = 0 substances. Regardless of pressure though, what can be said is that at absolute zero, all solids with a lowest-energy crystal lattice such those with a closest-packed arrangement (see Fig. 8, above left) contain minimal internal energy, retaining only that due to the ever-present background of zero-point energy. undefined One can also say that for a given substance at constant pressure, absolute zero is the point of lowest enthalpy (a measure of work potential that takes internal energy, pressure, and volume into consideration). Lastly, it is always true to say that all T = 0 substances contain zero kinetic thermal energy. undefined

Practical applications for thermodynamic temperature
Thermodynamic temperature is useful not only for scientists, it can also be useful for lay-people in many disciplines involving gases. By expressing variables in absolute terms and applying Gay-Lussac's law of temperature/pressure proportionality, solutions to everyday problems are straightforward; for instance, calculating how a temperature change affects the pressure inside an automobile tire. If the tire has a cold gage pressure of 200 kPa, then its absolute pressure is 300 kPa. Room temperature ("cold" in tire terms) is 296 K. If the tire temperature is 20 °C hotter (20 kelvins), the solution is calculated as $20.8 J⋅mol^{–1}⋅K^{–1}$ = 6.8% greater thermodynamic temperature and absolute pressure; that is, an absolute pressure of 320 kPa, which is a gage pressure of 220 kPa.

Relationship to ideal gas law
The thermodynamic temperature is closely linked to the ideal gas law and its consequences. It can be linked also to the second law of thermodynamics. The thermodynamic temperature can be shown to have special properties, and in particular can be seen to be uniquely defined (up to some constant multiplicative factor) by considering the efficiency of idealized heat engines. Thus the ratio T2/T1 of two temperatures T1 and T2 is the same in all absolute scales.

Strictly speaking, the temperature of a system is well-defined only if it is at thermal equilibrium. From a microscopic viewpoint, a material is at thermal equilibrium if the quantity of heat between its individual particles cancel out. There are many possible scales of temperature, derived from a variety of observations of physical phenomena.

Loosely stated, temperature differences dictate the direction of heat between two systems such that their combined energy is maximally distributed among their lowest possible states. We call this distribution "entropy". To better understand the relationship between temperature and entropy, consider the relationship between heat, work and temperature illustrated in the Carnot heat engine. The engine converts heat into work by directing a temperature gradient between a higher temperature heat source, TH, and a lower temperature heat sink, TC, through a gas filled piston. The work done per cycle is equal in magnitude to net heat taken up, which is sum of the heat qH taken up by the engine from the high-temperature source, plus the waste heat given off by the engine, qC < 0. The efficiency of the engine is the work divided by the heat put into the system or$$\textrm{Efficiency} = \frac {|w_\text{cy}|}{q_\text{H}} = \frac{q_\text{H}+q_\text{C}}{q_\text{H}} = 1 + \frac{q_\text{C}}{q_\text{H}}= 1 - \frac{|q_\text{C}|}{|q_\text{H}|} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$where $$w_\text{cy}$$ is the work done per cycle. Thus the efficiency depends only on |qC| / |qH|.

Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient. Thus, any reversible heat engine operating between temperatures T1 and T2 must have the same efficiency, that is to say, the efficiency is the function of only temperatures $$\frac{|q_\text{C}|}{|q_\text{H}|} = f(T_\text{H},T_\text{C}). \ \ \ \ \ \ \ \ (2)$$

In addition, a reversible heat engine operating between a pair of thermal reservoirs at temperatures T1 and T3 must have the same efficiency as one consisting of two cycles, one between T1 and another (intermediate) temperature T2, and the second between T2 andT3. If this were not the case, then energy (in the form of q) will be wasted or gained, resulting in different overall efficiencies every time a cycle is split into component cycles; clearly a cycle can be composed of any number of smaller cycles as an engine design choice, and any reversible engine between the same reservoir at T1 and T3 must be equally efficient regardless of the engine design.

If we choose engines such that work done by the one cycle engine and the two cycle engine are same, then the efficiency of each heat engine is written as the below. $$\eta _1 = 1 - \frac{|q_3|}{|q_1|} = 1 - f(T_1, T_3)$$, $$\eta _2 = 1 - \frac{|q_2|}{|q_1|} = 1 - f(T_1, T_2)$$, $$\eta _3 = 1 - \frac{|q_3|}{|q_2|} = 1 - f(T_2, T_3)$$.

Here, the engine 1 is the one cycle engine, and the engines 2 and 3 make the two cycle engine where there is the intermediate reservoir at T2. We also have used the fact that the heat $$q_2$$ passes through the intermediate thermal reservoir at $$T_2$$ without losing its energy. (I.e., $$q_2$$ is not lost during its passage through the reservoir at $$T_2$$.) This fact can be proved by the following. $$\begin{align} & { {\eta }_{2}}=1-\frac{|{ {q}_{2} } |}{|{ {q}_{1} }|}\to |{ {w}_{2} }|=|{ {q}_{1} }|-|{ {q}_{2} }|,\\ & { {\eta }_{3}}=1-\frac{|{ {q}_{3} }|}{|{ {q}_{2} }^{*}|}\to |{ {w}_{3} }|=|{ {q}_{2} }^{*}|-|{ {q}_{3} }|,\\ & |{ {w}_{2} }|+|{ {w}_{3} }|=(|{ {q}_{1} }|-|{{q}_{2}}|)+(|{{q}_{2}}^{*}|-|{{q}_{3}}|),\\ & { {\eta}_{1}}=1-\frac{|{ {q}_{3} }|}{|{ {q}_{1} }|}=\frac{(|{ {w}_{2} }|+|{ {w}_{3} }|)}{|{ {q}_{1} }|}=\frac{(|{ {q}_{1} }|-|{ {q}_{2} }|)+(|{ {q}_{2}} ^{*}|-|{ {q}_{3} }|)}{|{ {q}_{1} }|}.\\ \end{align}$$

In order to have the consistency in the last equation, the heat $$q_2$$ flown from the engine 2 to the intermediate reservoir must be equal to the heat $$q_2^*$$ flown out from the reservoir to the engine 3.

With this understanding of q1, q2 and q3, mathematically, $$f(T_1,T_3) = \frac{|q_3|}{|q_1|} = \frac{|q_2| |q_3|} {|q_1| |q_2|} = f(T_1,T_2)f(T_2,T_3).$$

But since the first function is NOT a function of T2, the product of the final two functions MUST result in the removal of T2 as a variable. The only way is therefore to define the function $f$ as follows: $$f(T_1,T_2) = \frac{g(T_2)}{g(T_1)}$$ and $$f(T_2,T_3) = \frac{g(T_3)}{g(T_2)}$$ so that $$f(T_1,T_3) = \frac{g(T_3)}{g(T_1)} = \frac{|q_3|}{|q_1|}.$$

I.e. the ratio of heat exchanged is a function of the respective temperatures at which they occur. We can choose any monotonic function for our $$g(T)$$; it is a matter of convenience and convention that we choose $$g(T) = T$$. Choosing then one fixed reference temperature (i.e. triple point of water), we establish the thermodynamic temperature scale.

Such a definition coincides with that of the ideal gas derivation; also it is this definition of the thermodynamic temperature that enables us to represent the Carnot efficiency in terms of TH and TC, and hence derive that the (complete) Carnot cycle is isentropic: $$\frac{|q_\text{C}|}{|q_\text{H}|} = f(T_\text{H},T_\text{C}) = \frac{T_\text{C}}{T_\text{H}}. \ \ \ \ \ \ \ \ \ \ \ (3)$$

Substituting this back into our first formula for efficiency yields a relationship in terms of temperature: $$\textrm{Efficiency} = 1 + \frac{q_\text{C}}{q_\text{H}}= 1 - \frac{|q_\text{C}|}{|q_\text{H}|} = 1 - \frac{T_\text{C}}{T_\text{H}}. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)$$

Note that for $T_{C} = 0$ the efficiency is 100% and that efficiency becomes greater than 100% for $T_{C} < 0$, which is unrealistic. Subtracting 1 from the right hand side of the Equation (4) and the middle portion gives $$\frac {q_\text{C}}{q_\text{H}} = - \frac{T_\text{C}}{T_\text{H}}$$ and thus $$\frac {q_\text{H}}{T_\text{H}} + \frac{q_\text{C}}{T_\text{C}} = 0.$$

The generalization of this equation is the Clausius theorem, which proposes the existence of a state function $$S$$ (i.e., a function which depends only on the state of the system, not on how it reached that state) defined (up to an additive constant) by

where the subscript rev indicates heat transfer in a reversible process. The function $$S$$ is the entropy of the system, mentioned previously, and the change of $$S$$ around any cycle is zero (as is necessary for any state function). The Equation 5 can be rearranged to get an alternative definition for temperature in terms of entropy and heat (to avoid a logic loop, we should first define entropy through statistical mechanics): $$T = \frac{dq_\mathrm{rev}}{dS}.$$

For a constant-volume system (so no mechanical work $$W$$) in which the entropy $$S$$ is a function $$S(E)$$ of its internal energy $$E$$, $$dE=dq_{rev} $$ and the thermodynamic temperature $$T$$ is therefore given by $$\frac{1}{T} = \frac{dS}{dE},$$ so that the reciprocal of the thermodynamic temperature is the rate of change of entropy with respect to the internal energy at the constant volume.

History
1702–1703: Guillaume Amontons (1663–1705) published two papers that may be used to credit him as being the first researcher to deduce the existence of a fundamental (thermodynamic) temperature scale featuring an absolute zero. He made the discovery while endeavoring to improve upon the air thermometers in use at the time. His J-tube thermometers comprised a mercury column that was supported by a fixed mass of air entrapped within the sensing portion of the thermometer. In thermodynamic terms, his thermometers relied upon the volume / temperature relationship of gas under constant pressure. His measurements of the boiling point of water and the melting point of ice showed that regardless of the mass of air trapped inside his thermometers or the weight of mercury the air was supporting, the reduction in air volume at the ice point was always the same ratio. This observation led him to posit that a sufficient reduction in temperature would reduce the air volume to zero. In fact, his calculations projected that absolute zero was equivalent to −240 °C—only 33.15 degrees short of the true value of −273.15 °C. Amonton's discovery of a one-to-one relationship between absolute temperature and absolute pressure was rediscovered a century later and popularized within the scientific community by Joseph Louis Gay-Lussac. Today, this principle of thermodynamics is commonly known as Gay-Lussac's law but is also known as Amonton's law.

1742: Anders Celsius (1701–1744) created a "backwards" version of the modern Celsius temperature scale. In Celsius's original scale, zero represented the boiling point of water and 100 represented the melting point of ice. In his paper Observations of two persistent degrees on a thermometer, he recounted his experiments showing that ice's melting point was effectively unaffected by pressure. He also determined with remarkable precision how water's boiling point varied as a function of atmospheric pressure. He proposed that zero on his temperature scale (water's boiling point) would be calibrated at the mean barometric pressure at mean sea level.

1744: Coincident with the death of Anders Celsius, the famous botanist Carl Linnaeus (1707–1778) effectively reversed Celsius's scale upon receipt of his first thermometer featuring a scale where zero represented the melting point of ice and 100 represented water's boiling point. The custom-made linnaeus-thermometer, for use in his greenhouses, was made by Daniel Ekström, Sweden's leading maker of scientific instruments at the time. For the next 204 years, the scientific and thermometry communities worldwide referred to this scale as the centigrade scale. Temperatures on the centigrade scale were often reported simply as degrees or, when greater specificity was desired, degrees centigrade. The symbol for temperature values on this scale was °C (in several formats over the years). Because the term centigrade was also the French-language name for a unit of angular measurement (one-hundredth of a right angle) and had a similar connotation in other languages, the term "centesimal degree" was used when very precise, unambiguous language was required by international standards bodies such as the International Bureau of Weights and Measures (Bureau international des poids et mesures) (BIPM). The 9th CGPM (General Conference on Weights and Measures (Conférence générale des poids et mesures) and the CIPM (International Committee for Weights and Measures (Comité international des poids et mesures) formally adopted degree Celsius (symbol: °C) in 1948.

1777: In his book Pyrometrie (Berlin: Haude & Spener, 1779) completed four months before his death, Johann Heinrich Lambert (1728–1777), sometimes incorrectly referred to as Joseph Lambert, proposed an absolute temperature scale based on the pressure/temperature relationship of a fixed volume of gas. This is distinct from the volume/temperature relationship of gas under constant pressure that Guillaume Amontons discovered 75 years earlier. Lambert stated that absolute zero was the point where a simple straight-line extrapolation reached zero gas pressure and was equal to −270 °C.

Circa 1787: Notwithstanding the work of Guillaume Amontons 85 years earlier, Jacques Alexandre César Charles (1746–1823) is often credited with discovering, but not publishing, that the volume of a gas under constant pressure is proportional to its absolute temperature. The formula he created was $V_{1}/T_{1} = V_{2}/T_{2}$.

1802: Joseph Louis Gay-Lussac (1778–1850) published work (acknowledging the unpublished lab notes of Jacques Charles fifteen years earlier) describing how the volume of gas under constant pressure changes linearly with its absolute (thermodynamic) temperature. This behavior is called Charles's Law and is one of the gas laws. His are the first known formulas to use the number 273 for the expansion coefficient of gas relative to the melting point of ice (indicating that absolute zero was equivalent to −273 °C).

1848: William Thomson, (1824–1907) also known as Lord Kelvin, wrote in his paper, On an Absolute Thermometric Scale, of the need for a scale whereby infinite cold (absolute zero) was the scale's zero point, and which used the degree Celsius for its unit increment.

Like Gay-Lussac, Thomson calculated that absolute zero was equivalent to −273 °C on the air thermometers of the time. This absolute scale is known today as the kelvin thermodynamic temperature scale. It's noteworthy that Thomson's value of −273 was actually derived from 0.00366, which was the accepted expansion coefficient of gas per degree Celsius relative to the ice point. The inverse of −0.00366 expressed to five significant digits is −273.22 °C which is remarkably close to the true value of −273.15 °C.

In the paper he proposed to define temperature using idealized heat engines. In detail, he proposed that, given three heat reservoirs at temperatures $$T_A, T_B, T_C$$, if two reversible heat engines (Carnot engine), one working between $$T_A, T_B$$ and another between $$ T_B, T_C$$, can produce the same amount of mechanical work $$W$$ by letting the same amount of heat $$Q$$ pass through, then define $$T_A - T_B = T_B - T_C$$.

Note that like Carnot, Kelvin worked under the assumption that heat is conserved ("the conversion of heat (or caloric) into mechanical effect is probably impossible"), and if heat $$Q$$ goes into the heat engine, then heat $$Q$$ must come out.

1851: William Thomson, realizing after Joule's experiments that heat is not a conserved quantity, but is convertible with mechanical work, modified his scale in An Account of Carnot's Theory of the Motive Power of Heat. In this work, he defined as follows: "Given two heat reservoirs $T_A, T_B$, and a reversible heat engine working between them, such that if during an engine cycle, heat $Q_A$ moves into the engine, and heat $Q_B$ comes out of the engine, then $\frac{T_A}{T_B} = \frac{Q_A}{Q_B}$."The above definition fixes the ratios between absolute temperatures, but it does not fix a scale for absolute temperature. For the scale, Thomson proposed to use the Celsius degree, that is, $$\frac{1}{100}$$ the interval between the freezing and the boiling point of water.

1859: Macquorn Rankine (1820–1872) proposed a thermodynamic temperature scale similar to William Thomson's but which used the degree Fahrenheit for its unit increment, that is, $$\frac{1}{180}$$ the interval between the freezing and the boiling point of water. This absolute scale is known today as the Rankine thermodynamic temperature scale.

1877–1884: Ludwig Boltzmann (1844–1906) made major contributions to thermodynamics through an understanding of the role that particle kinetics and black body radiation played. His name is now attached to several of the formulas used today in thermodynamics.

Circa 1930s: Gas thermometry experiments carefully calibrated to the melting point of ice and boiling point of water showed that absolute zero was equivalent to −273.15 °C.

1948: Resolution 3 of the 9th CGPM (Conférence Générale des Poids et Mesures, also known as the General Conference on Weights and Measures) fixed the triple point of water at precisely 0.01 °C. At this time, the triple point still had no formal definition for its equivalent kelvin value, which the resolution declared "will be fixed at a later date". The implication is that if the value of absolute zero measured in the 1930s was truly −273.15 °C, then the triple point of water (0.01 °C) was equivalent to 273.16 K. Additionally, both the CIPM (Comité international des poids et mesures, also known as the International Committee for Weights and Measures) and the CGPM formally adopted the name Celsius for the degree Celsius and the Celsius temperature scale. According to The Oxford English Dictionary (OED), the term "Celsius's thermometer" had been used at least as early as 1797. Further, the term "The Celsius or Centigrade thermometer" was again used in reference to a particular type of thermometer at least as early as 1850. The OED also cites this 1928 reporting of a temperature: "My altitude was about 5,800 metres, the temperature was 28° Celsius". However, dictionaries seek to find the earliest use of a word or term and are not a useful resource as regards the terminology used throughout the history of science. According to several writings of Dr. Terry Quinn CBE FRS, Director of the BIPM (1988–2004), including Temperature Scales from the early days of thermometry to the 21st century (150 kB PDF, here) as well as Temperature (2nd Edition / 1990 / Academic Press / 0125696817), the term Celsius in connection with the centigrade scale was not used whatsoever by the scientific or thermometry communities until after the CIPM and CGPM adopted the term in 1948. The BIPM wasn't even aware that degree Celsius was in sporadic, non-scientific use before that time. It's also noteworthy that the twelve-volume, 1933 edition of OED did not even have a listing for the word Celsius (but did have listings for both centigrade and centesimal in the context of temperature measurement). The 1948 adoption of Celsius accomplished three objectives:
 * 1) All common temperature scales would have their units named after someone closely associated with them; namely, Kelvin, Celsius, Fahrenheit, Réaumur and Rankine.
 * 2) Notwithstanding the important contribution of Linnaeus who gave the Celsius scale its modern form, Celsius's name was the obvious choice because it began with the letter C. Thus, the symbol °C that for centuries had been used in association with the name centigrade could continue to be used and would simultaneously inherit an intuitive association with the new name.
 * 3) The new name eliminated the ambiguity of the term centigrade, freeing it to refer exclusively to the French-language name for the unit of angular measurement.

1954: Resolution 3 of the 10th CGPM gave the kelvin scale its modern definition by choosing the triple point of water as its upper defining point (with no change to absolute zero being the null point) and assigning it a temperature of precisely 273.16 kelvins (what was actually written 273.16 degrees Kelvin at the time). This, in combination with Resolution 3 of the 9th CGPM, had the effect of defining absolute zero as being precisely zero kelvins and −273.15 °C.

1967/1968: Resolution 3 of the 13th CGPM renamed the unit increment of thermodynamic temperature kelvin, symbol K, replacing degree absolute, symbol °K. Further, feeling it useful to more explicitly define the magnitude of the unit increment, the 13th CGPM also decided in Resolution 4 that "The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water".

2005: The CIPM (Comité International des Poids et Mesures, also known as the International Committee for Weights and Measures) affirmed that for the purposes of delineating the temperature of the triple point of water, the definition of the kelvin thermodynamic temperature scale would refer to water having an isotopic composition defined as being precisely equal to the nominal specification of Vienna Standard Mean Ocean Water.

2019: In November 2018, the 26th General Conference on Weights and Measures (CGPM) changed the definition of the Kelvin by fixing the Boltzmann constant to $1/undefined$ when expressed in the unit J/K. This change (and other changes in the definition of SI units) was made effective on the 144th anniversary of the Metre Convention, 20 May 2019.

title: Thermodynamic temperature


text:
Third-order intercept point

In telecommunications, a third-order intercept point (IP3 or TOI) is a specific figure of merit associated with the more general third-order intermodulation distortion (IMD3), which is a measure for weakly nonlinear systems and devices, for example receivers, linear amplifiers and mixers. It is based on the idea that the device nonlinearity can be modeled using a low-order polynomial, derived by means of Taylor series expansion. The third-order intercept point relates nonlinear products caused by the third-order nonlinear term to the linearly amplified signal, in contrast to the second-order intercept point that uses second-order terms.

The intercept point is a purely mathematical concept and does not correspond to a practically occurring physical power level. In many cases, it lies far beyond the damage threshold of the device.

Definitions
Two different definitions for intercept points are in use:

 * Based on harmonics: The device is tested using a single input tone. The nonlinear products caused by n-th-order nonlinearity appear at n times the frequency of the input tone.
 * Based on intermodulation products: The device is fed with two sine tones one at $$f_1$$ and one at $$f_2$$. When you cube the sum of these sine waves you will get sine waves at various frequencies including $$(2f_2-f_1)$$ and $$(2f_1-f_2)$$.  If $$f_1$$ and $$f_2$$ are large but very close together then $$(2f_2-f_1)$$ and $$(2f_1-f_2)$$ will be very close to $$f_1$$ and $$f_2$$. This two-tone approach has the advantage that it is not restricted to broadband devices and is commonly used for radio receivers.

The intercept point is obtained graphically by plotting the output power versus the input power both on logarithmic scales (e.g., decibels). Two curves are drawn; one for the linearly amplified signal at an input tone frequency, one for a nonlinear product. On a logarithmic scale, the function xn translates into a straight line with slope of n. Therefore, the linearly amplified signal will exhibit a slope of 1. A third-order nonlinear product will increase by 3 dB in power when the input power is raised by 1 dB.

Both curves are extended with straight lines of slope 1 and n (3 for a third-order intercept point). The point where the curves intersect is the intercept point. It can be read off from the input or output power axis, leading to input (IIP3) or output (OIP3) intercept point respectively.

Input and output intercept point differ by the small-signal gain of the device.

Practical considerations
The concept of intercept point is based on the assumption of a weakly nonlinear system, meaning that higher-order nonlinear terms are small enough to be negligible. In practice, the weakly nonlinear assumption may not hold for the upper end of the input power range, be it during measurement or during use of the amplifier. As a consequence, measured or simulated data will deviate from the ideal slope of n. The intercept point according to its basic definition should be determined by drawing the straight lines with slope 1 and n through the measured data at the smallest possible power level (possibly limited towards lower power levels by instrument or device noise). It is a frequent mistake to derive intercept points by either changing the slope of the straight lines, or fitting them to points measured at too high power levels. In certain situations such a measure can be useful, but it is not an intercept point according to definition. Its value depends on the measurement conditions that need to be documented, whereas the IP according to definition is mostly unambiguous; although there is some dependency on frequency and tone spacing, depending on the physics of the device under test.

One of the useful applications of third-order intercept point is as a rule-of-thumb measure to estimate nonlinear products. When comparing systems or devices for linearity, a higher intercept point is better. It can be seen that the spacing between two straight lines with slopes of 3 and 1 closes with slope 2.

For example, assume a device with an input-referred third-order intercept point of 10 dBm is driven with a test signal of −5 dBm. This power is 15 dB below the intercept point, therefore nonlinear products will appear at approximately 2×15 dB below the test signal power at the device output (in other words, 3×15 dB below the output-referred third-order intercept point).

A rule of thumb that holds for many linear radio-frequency amplifiers is that the 1 dB compression point point falls approximately 10 dB below the third-order intercept point.

Theory

The third-order intercept point (TOI) is a property of the device transfer function O (see diagram). This transfer function relates the output signal voltage level to the input signal voltage level. We assume a "linear" device having a transfer function whose small-signal form may be expressed in terms of a power series containing only odd terms, making the transfer function an odd function of input signal voltage, i.e., O(−s) = −O(s). Where the signals passing through the actual device are modulated sinusoidal voltage waveforms (e.g., RF amplifier), device nonlinearities can be expressed in terms of how they affect individual sinusoidal signal components. For example, say the input voltage signal is the sine wave

 * $$s(t) = V \cos(\omega t),$$

and the device transfer function produces an output of the form

 * $$O(s) = G s - D_3 s^3 + \ldots,$$

where G is the amplifier gain, and D3 is cubic distortion. We may substitute the first equation into the second and, using the trigonometric identity

 * $$\cos^3(x) = \frac{3}{4}\cos(x) + \frac{1}{4}\cos(3x),$$

we obtain the device output voltage waveform as

 * $$O(s(t)) = \left(G V - \frac{3}{4} D_3 V^3\right) \cos(\omega t) - \frac{1}{4} D_3 V^3 \cos(3 \omega t).$$

The output waveform contains the original waveform, cos(ωt), plus a new harmonic term, cos(3ωt), the third-order term. The coefficient of the cos(ωt) harmonic has two terms, one that varies linearly with V and one that varies with the cube of V. In fact, the coefficient of cos(ωt) has nearly the same form as the transfer function, except for the factor $3⁄4$ on the cubic term. In other words, as signal level V is increased, the level of the cos(ωt) term in the output eventually levels off, similar to how the transfer function levels off. Of course, the coefficients of the higher-order harmonics will increase (with increasing V) as the coefficient of the cos(ωt) term levels off (the power has to go somewhere).

If we now restrict our attention to the portion of the cos(ωt) coefficient that varies linearly with V, and then ask ourselves, at what input voltage level V will the coefficients of the first- and third-order terms have equal magnitudes (i.e., where the magnitudes intersect), we find that this happens when

 * $$V^2 = \frac{4 G}{3 D_3},$$

which is the third-order intercept point (TOI). So, we see that the TOI input power level is simply 4/3 times the ratio of the gain and the cubic distortion term in the device transfer function. The smaller the cubic term is in relation to the gain, the more linear the device is, and the higher the TOI is. The TOI, being related to the magnitude squared of the input voltage waveform, is a power quantity, typically measured in milliwatts (mW). The TOI is always beyond operational power levels because the output power saturates before reaching this level.

The TOI is closely related to the amplifier's "1 dB compression point", which is defined as that point at which the total coefficient of the cos(ωt) term is 1 dB below the linear portion of that coefficient. We can relate the 1 dB compression point to the TOI as follows. Since 1 dB = 20 log10 1.122, we may say, in a voltage sense, that the 1 dB compression point occurs when

 * $$1.122 \left(G V - \frac{3}{4}D_3 V^3\right) = G V,$$

or

 * $$V^2 = 0.10875 \times \frac{4 G}{3 D_3},$$

or

 * $$V^2 = 0.10875 \times \mathrm{TOI}.$$

In a power sense (V2 is a power quantity), a factor of 0.10875 corresponds to −9.636 dB, so by this approximate analysis, the 1 dB compression point occurs roughly 9.6 dB below the TOI.

Recall: decibel figure = 10 dB × log10(power ratio) = 20 dB × log10(voltage ratio).

title: Third-order intercept point


text:
Threshold

Threshold may refer to:

Biology

 * Threshold (reference value)
 * Absolute threshold
 * Absolute threshold of hearing
 * Action potential
 * Aerobic threshold
 * Anaerobic threshold
 * Dark adaptation threshold
 * Epidemic threshold
 * Flicker fusion threshold
 * Masking threshold
 * Odor detection threshold
 * Renal threshold
 * Seizure threshold
 * Sensory threshold
 * Threshold expression
 * Threshold limit value
 * Threshold model
 * Threshold of pain
 * Threshold potential

Other science-related

 * Ecological threshold
 * Error threshold (evolution)
 * Extinction threshold
 * Lasing threshold
 * Percolation threshold
 * Polygyny threshold model
 * Threshold cryptosystem
 * Threshold displacement energy
 * Threshold energy
 * Threshold graph
 * Threshold knowledge
 * Threshold model
 * Threshold voltage
 * Thresholding (image processing)
 * Transparency threshold

Architecture

 * Threshold (architecture), the sill of a door

Literature

 * Threshold (1990 novel), a science fiction novel by Chris and Janet Morris
 * Threshold (Sara Douglass novel), a fantasy novel
 * Threshold (Palmer novel), a science fiction novel by David R. Palmer
 * Threshold, the first volume of the collected short fiction of Roger Zelazny
 * Threshold (DC Comics), a comic book published by DC Comics
 * Threshold, a science-fiction novel by Caitlín R. Kiernan
 * Threshold, a Christian novel by Bill Myers (author)
 * Threshold, a science-fiction novel by Ben Mezrich
 * Threshold Editions, a conservative imprint

Media

 * Threshold (1981 film)
 * Threshold, a 2008 film directed by Deepak Rauniyar
 * Threshold (TV series), an American science fiction drama TV series
 * "Threshold" (Stargate SG-1), an episode of the TV series
 * "Threshold" (Star Trek: Voyager), an episode of the TV series
 * Threshold Entertainment, a Hollywood Intellectual Property Management and Production Company
 * Threshold Podcast, a podcast focused on long-form reporting of climate justice topics

Music

 * Threshold (band), a UK progressive metal band
 * Threshold House, a record label created by Coil
 * Threshold Records, a record label created by The Moody Blues

Albums

 * Threshold (album), a 2006 album by HammerFall
 * Thresholds (album), a 1992 album by Nocturnus
 * Threshold, a 1997 album by Erik Norlander

Songs

 * "Threshold", by Pocahaunted, 2010
 * "Threshold", by Slayer on the album God Hates Us All, 2001
 * "Threshold", by The Steve Miller Band from Book of Dreams, 1977
 * "Threshold", by Turnover from Peripheral Vision, 2015
 * "Threshold," from the Scott Pilgrim vs. the World soundtrack, 2010
 * "The Threshold", by Connie Smith from Connie Smith, 1965

Other uses

 * Threshold (video game), a 1981 video game
 * Thresholds, a nonprofit organization serving persons with mental illnesses and substance abuse problems in Illinois
 * Threshold Audio, a manufacturer of high end audio amplifiers
 * Displaced threshold, an aviation term
 * Election threshold, a concept in voting systems
 * Poverty thresholds (United States Census Bureau)
 * In government procurement, the expenditure value which determines which rules and procedures are to be applied to a proposed purchase

title: Threshold


text:
Time-assignment speech interpolation

In telecommunication, a time-assignment speech interpolation (TASI) was an analog technique used on certain long transmission links to increase voice transmission capacity.

TASI was invented by Bell Labs in the early 1960s to increase the capacity of transatlantic telephone cables. It was one of their first applications requiring electronic switching of voice circuits.

Later digital circuit multiplication equipment included TASI as a feature, not as distinct hardware.

Operation
TASI takes advantage of the fact that in typical person-person conversation, speech in a single direction occurs for approximately 40% of the time, the remaining time being occupied with pauses and/or silence. Statistical analysis demonstrated that for an average voice channel usage of 40%, over 74 speech conversations could be handled using 37 full Duplex speech circuits thereby doubling potential revenue for a small capital outlay relative to a highly expensive cable. e.g. £12.5 million (£263 million as of 2014) cost of the TAT-1 cable on which TASI was implemented.

TASI worked by switching additional users onto any voice channel temporarily idled because an original user has stopped speaking. When the original user resumes speaking, that user would, in turn, be switched to any channel that happened to be idle. The speech detector function is called voice activity detection. Clipping or loss of speech would occur for all conversations that needed to be assigned to an available idle channel and in practice lasted at least 17 ms whilst information required to re-connect both parties was signalled by the TASI control circuits. An additional freezeout period lasting between 0 and 500 ms would depend on the instantaneous loading of voice circuits. In actual use, these delays presented few problems in typical conversations.

One of the issues with using this type of technology was that the users listening on an idled channel can sometimes hear the conversation that has been switched onto it. Generally the sound heard was of very low volume and individual words are not distinguishable. See also crosstalk for a similar phenomenon in telecommunications. Another potential issue was ensuring that non-voice type circuits (e.g. Music or radio type circuits where pauses would occur infrequently) were not routed via TASI speech channels since these could seriously degrade the level of service where callers would encounter frequent clipped speech and breaks in the conversation.

title: Time-assignment speech interpolation