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Transmission line

In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmission must be taken into account. This applies especially to radio-frequency engineering because the short wavelengths mean that wave phenomena arise over very short distances (this can be as short as millimetres depending on frequency). However, the theory of transmission lines was historically developed to explain phenomena on very long telegraph lines, especially submarine telegraph cables.

Transmission lines are used for purposes such as connecting radio transmitters and receivers with their antennas (they are then called feed lines or feeders), distributing cable television signals, trunklines routing calls between telephone switching centres, computer network connections and high speed computer data buses. RF engineers commonly use short pieces of transmission line, usually in the form of printed planar transmission lines, arranged in certain patterns to build circuits such as filters. These circuits, known as distributed-element circuits, are an alternative to traditional circuits using discrete capacitors and inductors.

Overview
Ordinary electrical cables suffice to carry low frequency alternating current (AC), such as mains power, which reverses direction 100 to 120 times per second, and audio signals. However, they are not generally used to carry currents in the radio frequency range, above about 30 kHz, because the energy tends to radiate off the cable as radio waves, causing power losses. Radio frequency currents also tend to reflect from discontinuities in the cable such as connectors and joints, and travel back down the cable toward the source. These reflections act as bottlenecks, preventing the signal power from reaching the destination. Transmission lines use specialized construction, and impedance matching, to carry electromagnetic signals with minimal reflections and power losses. The distinguishing feature of most transmission lines is that they have uniform cross sectional dimensions along their length, giving them a uniform impedance, called the characteristic impedance, to prevent reflections. Types of transmission line include parallel line (ladder line, twisted pair), coaxial cable, and planar transmission lines such as stripline and microstrip. The higher the frequency of electromagnetic waves moving through a given cable or medium, the shorter the wavelength of the waves. Transmission lines become necessary when the transmitted frequency's wavelength is sufficiently short that the length of the cable becomes a significant part of a wavelength.

At microwave frequencies and above, power losses in transmission lines become excessive, and waveguides are used instead, which function as "pipes" to confine and guide the electromagnetic waves. Some sources define waveguides as a type of transmission line; however, this article will not include them.

History
Mathematical analysis of the behaviour of electrical transmission lines grew out of the work of James Clerk Maxwell, Lord Kelvin, and Oliver Heaviside. In 1855, Lord Kelvin formulated a diffusion model of the current in a submarine cable. The model correctly predicted the poor performance of the 1858 trans-Atlantic submarine telegraph cable. In 1885, Heaviside published the first papers that described his analysis of propagation in cables and the modern form of the telegrapher's equations.

The four terminal model

For the purposes of analysis, an electrical transmission line can be modelled as a two-port network (also called a quadripole), as follows:

In the simplest case, the network is assumed to be linear (i.e. the complex voltage across either port is proportional to the complex current flowing into it when there are no reflections), and the two ports are assumed to be interchangeable. If the transmission line is uniform along its length, then its behaviour is largely described by a two parameters called characteristic impedance, symbol Z0 and propagation delay, symbol $$ \tau_p $$. Z0 is the ratio of the complex voltage of a given wave to the complex current of the same wave at any point on the line. Typical values of Z0 are 50 or 75 ohms for a coaxial cable, about 100 ohms for a twisted pair of wires, and about 300 ohms for a common type of untwisted pair used in radio transmission. Propagation delay is proportional to the length of the transmission line and is never less than the length divided by the speed of light. Typical delays for modern communication transmission lines vary from $3.33 ns/m$ to $5 ns/m$.

When sending power down a transmission line, it is usually desirable that as much power as possible will be absorbed by the load and as little as possible will be reflected back to the source. This can be ensured by making the load impedance equal to Z0, in which case the transmission line is said to be matched.

Some of the power that is fed into a transmission line is lost because of its resistance. This effect is called ohmic or resistive loss (see ohmic heating). At high frequencies, another effect called dielectric loss becomes significant, adding to the losses caused by resistance. Dielectric loss is caused when the insulating material inside the transmission line absorbs energy from the alternating electric field and converts it to heat (see dielectric heating). The transmission line is modelled with a resistance (R) and inductance (L) in series with a capacitance (C) and conductance (G) in parallel. The resistance and conductance contribute to the loss in a transmission line.

The total loss of power in a transmission line is often specified in decibels per metre (dB/m), and usually depends on the frequency of the signal. The manufacturer often supplies a chart showing the loss in dB/m at a range of frequencies. A loss of 3 dB corresponds approximately to a halving of the power.

Propagation delay is often specified in units of nanoseconds per metre. While propagation delay usually depends on the frequency of the signal, transmission lines are typically operated over frequency ranges where the propagation delay is approximately constant.

Telegrapher's equations
The telegrapher's equations (or just telegraph equations) are a pair of linear differential equations which describe the voltage ($$V$$) and current ($$I$$) on an electrical transmission line with distance and time. They were developed by Oliver Heaviside who created the transmission line model, and are based on Maxwell's equations.

The transmission line model is an example of the distributed-element model. It represents the transmission line as an infinite series of two-port elementary components, each representing an infinitesimally short segment of the transmission line:

 * The distributed resistance $$R$$ of the conductors is represented by a series resistor (expressed in ohms per unit length).
 * The distributed inductance $$L$$ (due to the magnetic field around the wires, self-inductance, etc.) is represented by a series inductor (in henries per unit length).
 * The capacitance $$C$$ between the two conductors is represented by a shunt capacitor (in farads per unit length).
 * The conductance $$G$$ of the dielectric material separating the two conductors is represented by a shunt resistor between the signal wire and the return wire (in siemens per unit length).

The model consists of an infinite series of the elements shown in the figure, and the values of the components are specified per unit length so the picture of the component can be misleading. $$R$$, $$L$$, $$C$$, and $$G$$ may also be functions of frequency. An alternative notation is to use $$R'$$, $$L'$$, $$C'$$ and $$G'$$ to emphasize that the values are derivatives with respect to length. These quantities can also be known as the primary line constants to distinguish from the secondary line constants derived from them, these being the propagation constant, attenuation constant and phase constant.

The line voltage $$V(x)$$ and the current $$I(x)$$ can be expressed in the frequency domain as

 * $$\frac{\partial V(x)}{\partial x} = -(R + j\,\omega\,L)\,I(x)$$

 * $$\frac{\partial I(x)}{\partial x} = -(G + j\,\omega\,C)\,V(x) ~\,.$$

 * (see differential equation, angular frequency ω and imaginary unit $j$)

Special case of a lossless line
When the elements $$R$$ and $$G$$ are negligibly small the transmission line is considered as a lossless structure. In this hypothetical case, the model depends only on the $$L$$ and $$C$$ elements which greatly simplifies the analysis. For a lossless transmission line, the second order steady-state Telegrapher's equations are:

 * $$\frac{\partial^2 V(x)}{\partial x^2}+ \omega^2 L\,C\,V(x) = 0$$

 * $$\frac{\partial^2 I(x)}{\partial x^2} + \omega^2 L\,C\,I(x) = 0 ~\,.$$

These are wave equations which have plane waves with equal propagation speed in the forward and reverse directions as solutions. The physical significance of this is that electromagnetic waves propagate down transmission lines and in general, there is a reflected component that interferes with the original signal. These equations are fundamental to transmission line theory.

General case of a line with losses
In the general case the loss terms, $$R$$ and $$G$$, are both included, and the full form of the Telegrapher's equations become:

 * $$\frac{\partial^2 V(x)}{\partial x^2} = \gamma^2 V(x)\,$$

 * $$\frac{\partial^2 I(x)}{\partial x^2} = \gamma^2 I(x)\,$$

where $$\gamma$$ is the (complex) propagation constant. These equations are fundamental to transmission line theory. They are also wave equations, and have solutions similar to the special case, but which are a mixture of sines and cosines with exponential decay factors. Solving for the propagation constant $$\gamma$$ in terms of the primary parameters $$R$$, $$L$$, $$G$$, and $$C$$ gives:

 * $$\gamma = \sqrt{(R + j\,\omega\,L)(G + j\,\omega\,C)\,}$$

and the characteristic impedance can be expressed as

 * $$Z_0 = \sqrt{\frac{R + j\,\omega\,L}{G + j\,\omega\,C}\,} ~\,.$$

The solutions for $$V(x)$$ and $$I(x)$$ are:

 * $$V(x) = V_{(+)} e^{-\gamma\,x} + V_{(-)} e^{+\gamma\,x} \,$$

 * $$I(x) = \frac{1}{Z_0}\,\left( V_{(+)} e^{-\gamma\,x} - V_{(-)} e^{+\gamma\,x} \right) ~\,. $$

The constants $$V_{(\pm)}$$ must be determined from boundary conditions. For a voltage pulse $$V_{\mathrm{in}}(t) \,$$, starting at $$x = 0$$ and moving in the positive $$x$$ direction, then the transmitted pulse $$V_{\mathrm{out}}(x,t) \,$$ at position $$x$$ can be obtained by computing the Fourier Transform, $$\tilde{V}(\omega)$$, of $$V_{\mathrm{in}}(t) \,$$, attenuating each frequency component by $$e^{-\operatorname{Re}(\gamma)\,x} \,$$, advancing its phase by $$-\operatorname{Im}(\gamma)\,x\,$$, and taking the inverse Fourier Transform. The real and imaginary parts of $$\gamma$$ can be computed as

 * $$\operatorname{Re}(\gamma) = \alpha = (a^2 + b^2)^{1/4} \cos(\psi ) \,$$

 * $$\operatorname{Im}(\gamma) = \beta = (a^2 + b^2)^{1/4} \sin(\psi) \,$$

with

 * $$a ~ \equiv ~ R\, G\, - \omega^2 L\,C\ ~ = ~ \omega^2 L\,C\,\left[ \left( \frac{R}{\omega L} \right) \left( \frac{G}{\omega C} \right) - 1 \right] $$

 * $$b ~ \equiv ~ \omega\,C\,R + \omega\,L\,G ~ = ~ \omega^2 L\,C\,\left( \frac{R}{\omega\,L} + \frac{G}{\omega\,C} \right) $$

the right-hand expressions holding when neither $$L$$, nor $$C$$, nor $$\omega$$ is zero, and with

 * $$\psi ~ \equiv ~ \tfrac{1}{2}\operatorname{atan2}(b,a)\,$$

where atan2 is the everywhere-defined form of two-parameter arctangent function, with arbitrary value zero when both arguments are zero.

Alternatively, the complex square root can be evaluated algebraically, to yield:

 * $$ \alpha = \frac{\pm b}{\sqrt{2 \left( - a + \sqrt{a^2 + b^2} \right)~}},$$

and
 * $$ \beta = \pm { \sqrt{\tfrac{1}{2}\left( - a + \sqrt{a^2 + b^2} \right)~} },$$

with the plus or minus signs chosen opposite to the direction of the wave's motion through the conducting medium. ($a$ is usually negative, since $$G$$ and $$R$$ are typically much smaller than $$\omega C$$ and $$\omega L$$, respectively, so  $−a$  is usually positive. $b$ is always positive.)

Special, low loss case
For small losses and high frequencies, the general equations can be simplified: If $$\tfrac{R}{\omega\,L} \ll 1 $$ and $$\tfrac{G}{\omega\,C} \ll 1$$ then

 * $$\operatorname{Re}(\gamma) = \alpha \approx \tfrac{1}{2}\sqrt{L\,C\,}\,\left( \frac{R}{L} + \frac{G}{C} \right) \,$$

 * $$\operatorname{Im}(\gamma) = \beta \approx \omega\,\sqrt{L\,C\,} ~.\,$$

Since an advance in phase by $$- \omega\,\delta$$ is equivalent to a time delay by $$\delta$$, $$V_{out}(t)$$ can be simply computed as

 * $$V_{\mathrm{out}}(x,t) \approx V_{\mathrm{in}}(t - \sqrt{L\,C\,}\,x)\,e^{- \tfrac{1}{2}\sqrt{L\,C\,}\,\left( \frac{R}{L} + \frac{G}{C} \right)\,x }. \,$$

Heaviside condition
The Heaviside condition is $$ \frac {G}{C} = \frac {R}{L} $$.

If R, G, L, and C are constants that are not frequency dependent and the Heaviside condition is met, then waves travel down the transmission line without dispersion distortion.

Input impedance of transmission line

The characteristic impedance $$Z_0$$ of a transmission line is the ratio of the amplitude of a single voltage wave to its current wave. Since most transmission lines also have a reflected wave, the characteristic impedance is generally not the impedance that is measured on the line.

The impedance measured at a given distance $$\ell$$ from the load impedance $$Z_\mathrm{L}$$ may be expressed as

 * $$Z_\mathrm{in}\left(\ell\right)=\frac{V(\ell)}{I(\ell)} = Z_0 \frac{1 + \mathit{\Gamma}_\mathrm{L} e^{-2 \gamma \ell}}{1 - \mathit{\Gamma}_\mathrm{L} e^{-2 \gamma \ell}}$$,

where $$\gamma$$ is the propagation constant and $$\mathit{\Gamma}_\mathrm{L} = \frac{\,Z_\mathrm{L} - Z_0\,}{Z_\mathrm{L} + Z_0}$$ is the voltage reflection coefficient measured at the load end of the transmission line. Alternatively, the above formula can be rearranged to express the input impedance in terms of the load impedance rather than the load voltage reflection coefficient:

 * $$Z_\mathrm{in}(\ell) = Z_0\,\frac{Z_\mathrm{L} + Z_0 \tanh\left(\gamma \ell\right)}{Z_0 + Z_\mathrm{L}\,\tanh\left(\gamma \ell \right)}$$.

Input impedance of lossless transmission line
For a lossless transmission line, the propagation constant is purely imaginary, $$\gamma = j\,\beta$$, so the above formulas can be rewritten as

Z_\mathrm{in}(\ell) = Z_0 \frac{Z_\mathrm{L} + j\,Z_0\,\tan(\beta \ell)}{Z_0 + j\,Z_\mathrm{L}\tan(\beta \ell)} $$

where $$\beta = \frac{\,2 \pi\,}{\lambda}$$ is the wavenumber.

In calculating $$\beta,$$ the wavelength is generally different inside the transmission line to what it would be in free-space. Consequently, the velocity factor of the material the transmission line is made of needs to be taken into account when doing such a calculation.

Half wave length
For the special case where $$\beta\,\ell= n\,\pi$$ where n is an integer (meaning that the length of the line is a multiple of half a wavelength), the expression reduces to the load impedance so that
 * $$Z_\mathrm{in} = Z_\mathrm{L} \,$$

for all $$n\,.$$ This includes the case when $$n=0$$, meaning that the length of the transmission line is negligibly small compared to the wavelength. The physical significance of this is that the transmission line can be ignored (i.e. treated as a wire) in either case.

Quarter wave length
For the case where the length of the line is one quarter wavelength long, or an odd multiple of a quarter wavelength long, the input impedance becomes

Z_\mathrm{in}=\frac{Z_0^2}{Z_\mathrm{L}} ~\,. $$

Matched load
Another special case is when the load impedance is equal to the characteristic impedance of the line (i.e. the line is matched), in which case the impedance reduces to the characteristic impedance of the line so that
 * $$Z_\mathrm{in}=Z_\mathrm{L}=Z_0 \,$$

for all $$\ell$$ and all $$\lambda$$.

Short

For the case of a shorted load (i.e. $$Z_\mathrm{L} = 0$$), the input impedance is purely imaginary and a periodic function of position and wavelength (frequency)

 * $$Z_\mathrm{in}(\ell) = j\,Z_0\,\tan(\beta \ell). \,$$

Open
For the case of an open load (i.e. $$Z_\mathrm{L} = \infty$$), the input impedance is once again imaginary and periodic

 * $$Z_\mathrm{in}(\ell) = -j\,Z_0 \cot(\beta \ell). \,$$

Matrix parameters
The simulation of transmission lines embedded into larger systems generally utilize admittance parameters (Y matrix), impedance parameters (Z matrix), and/or scattering parameters (S matrix) that embodies the full transmission line model needed to support the simulation.

Admittance parameters
Admittance (Y) parameters may be defined by applying a fixed voltage to one port (V1) of a transmission line with the other end shorted to ground and measuring the resulting current running into each port (I1, I2) and computing the admittance on each port as a ratio of I/V  The admittance parameter Y11 is I1/V1, and the admittance parameter Y12 is I2/V1. Since transmission lines are electrically passive and symmetric devices, Y12 = Y21, and Y11 = Y22.

For lossless and lossy transmission lines respectively, the Y parameter matrix is as follows:

$$Y_{\text{Lossless}}=\begin{bmatrix} \frac{-jcot(\beta l)}{Z_o} & \frac{jcsc(\beta l)}{Z_o} \\ \frac{jcsc(\beta l)}{Z_o} & \frac{-jcot(\beta l)}{Z_o} \end{bmatrix} \text{   } Y_{\text{Lossy}}=\begin{bmatrix} \frac{coth(\gamma l)}{Z_o} & \frac{-csch(\gamma l)}{Z_o} \\ \frac{-csch(\gamma l)}{Z_o} & \frac{coth(\gamma l)}{Z_o} \end{bmatrix}$$

Impedance parameters
Impedance (Z) parameter may defines by applying a fixed current into one port (I1) of a transmission line with the other port open and measuring the resulting voltage on each port (V1, V2) and computing the impedance parameter Z11 is V1/I1, and the impedance parameter Z12 is V2/I1. Since transmission lines are electrically passive and symmetric devices, V12 = V21, and V11 = V22.

In the Y and Z matrix definitions, $$Y = Z^{-1}$$ and $$Z = Y^{-1}$$. Unlike ideal lumped 2 port elements (resistors, capacitors, inductors, etc.) which do not have defined Z parameters, transmission lines have an internal path to ground, which permits the definition of Z parameters.

For lossless and lossy transmission lines respectively, the Z parameter matrix is as follows:

$$Z_{\text{Lossless}}=\begin{bmatrix} -jZ_ocot(\beta l) & -jZ_ocsc(\beta l) \\ -jZ_ocsc(\beta l) & -jZ_ocot(\beta l) \end{bmatrix} \text{    } Z_{\text{Lossy}}=\begin{bmatrix} Z_ocoth(\gamma l) & Z_ocsch(\gamma l) \\ Z_ocsch(\gamma l) & Z_ocoth(\gamma l) \end{bmatrix}$$

Scattering parameters
Scattering (S) matrix parameters model the electrical behavior of the transmission line with matched loads at each termination.

For lossless and lossy transmission lines respectively, the S parameter matrix is as follows, using standard hyperbolic to circular complex translations.

$$S_{\text{Lossless}}=\begin{bmatrix} \frac{(Z_o^2-Z_p^2)sin(\beta l)}{(Z_o^2+Z_p^2)sin(\beta l)-j2Z_oZ_pcos(\beta l)} & \frac{2Z_oZ_p}{j(Z_o^2+Z_p^2)sin(\beta l)+2Z_oZ_pcos(\beta l)} \\ \frac{2Z_oZ_p}{j(Z_o^2+Z_p^2)sin(\beta l)+2Z_oZ_pcos(\beta l)} & \frac{(Z_o^2-Z_p^2)sin(\beta l)}{(Z_o^2+Z_p^2)sin(\beta l)-j2Z_oZ_pcos(\beta l)} \end{bmatrix} \text{   } S_{\text{Lossy}}=\begin{bmatrix} \frac{(Z_o^2-Z_p^2)sinh(\gamma l)}{(Z_o^2+Z_p^2)sinh(\gamma l)+2Z_oZ_pcosh(\gamma l)} & \frac{2Z_oZ_p}{(Z_o^2+Z_p^2)sinh(\gamma l)+2Z_oZ_pcosh(\gamma l)} \\ \frac{2Z_oZ_p}{(Z_o^2+Z_p^2)sinh(\gamma l)+2Z_oZ_pcosh(\gamma l)} & \frac{(Z_o^2-Z_p^2)sinh(\gamma l)}{(Z_o^2+Z_p^2)sinh(\gamma l)+2Z_oZ_pcosh(\gamma l)} \end{bmatrix}$$

Variable definitions
In all matrix parameters above, the following variable definitions apply:

$$Z_o$$ = characteristic impedance

Zp = port impedance, or termination impedance

$$\gamma = \alpha + j\beta$$ = the propagation constant per unit length

$$\alpha$$ = attenuation constant in nepers per unit length

$$\beta = \frac{2\pi}{\lambda} = \frac{\omega}{V}$$ = wave number or phase constant radians per unit length

$$\omega$$ = frequency radians / second

$$V=\frac{1}{\sqrt{LC}} = \frac{V_C}{\sqrt{E_{re}}}$$ = Speed of propagation

$$\lambda$$ = wave length in unit length

L = inductance per unit length

C = capacitance per unit length

$$E_{re}$$ = effective dielectric constant

$$V_C$$ = 299,792,458 meters / second = Speed of light in a vacuum

Coaxial cable
Coaxial lines confine virtually all of the electromagnetic wave to the area inside the cable. Coaxial lines can therefore be bent and twisted (subject to limits) without negative effects, and they can be strapped to conductive supports without inducing unwanted currents in them. In radio-frequency applications up to a few gigahertz, the wave propagates in the transverse electric and magnetic mode (TEM) only, which means that the electric and magnetic fields are both perpendicular to the direction of propagation (the electric field is radial, and the magnetic field is circumferential). However, at frequencies for which the wavelength (in the dielectric) is significantly shorter than the circumference of the cable other transverse modes can propagate. These modes are classified into two groups, transverse electric (TE) and transverse magnetic (TM) waveguide modes. When more than one mode can exist, bends and other irregularities in the cable geometry can cause power to be transferred from one mode to another.

The most common use for coaxial cables is for television and other signals with bandwidth of multiple megahertz. In the middle 20th century they carried long distance telephone connections.

Planar lines
Planar transmission lines are transmission lines with conductors, or in some cases dielectric strips, that are flat, ribbon-shaped lines. They are used to interconnect components on printed circuits and integrated circuits working at microwave frequencies because the planar type fits in well with the manufacturing methods for these components. Several forms of planar transmission lines exist.

Microstrip

A microstrip circuit uses a thin flat conductor which is parallel to a ground plane. Microstrip can be made by having a strip of copper on one side of a printed circuit board (PCB) or ceramic substrate while the other side is a continuous ground plane. The width of the strip, the thickness of the insulating layer (PCB or ceramic) and the dielectric constant of the insulating layer determine the characteristic impedance. Microstrip is an open structure whereas coaxial cable is a closed structure.

Stripline
A stripline circuit uses a flat strip of metal which is sandwiched between two parallel ground planes. The insulating material of the substrate forms a dielectric. The width of the strip, the thickness of the substrate and the relative permittivity of the substrate determine the characteristic impedance of the strip which is a transmission line.

Coplanar waveguide
A coplanar waveguide consists of a center strip and two adjacent outer conductors, all three of them flat structures that are deposited onto the same insulating substrate and thus are located in the same plane ("coplanar"). The width of the center conductor, the distance between inner and outer conductors, and the relative permittivity of the substrate determine the characteristic impedance of the coplanar transmission line.

Balanced lines
A balanced line is a transmission line consisting of two conductors of the same type, and equal impedance to ground and other circuits. There are many formats of balanced lines, amongst the most common are twisted pair, star quad and twin-lead.

Twisted pair
Twisted pairs are commonly used for terrestrial telephone communications. In such cables, many pairs are grouped together in a single cable, from two to several thousand. The format is also used for data network distribution inside buildings, but the cable is more expensive because the transmission line parameters are tightly controlled.

Star quad
Star quad is a four-conductor cable in which all four conductors are twisted together around the cable axis. It is sometimes used for two circuits, such as 4-wire telephony and other telecommunications applications. In this configuration each pair uses two non-adjacent conductors. Other times it is used for a single, balanced line, such as audio applications and 2-wire telephony. In this configuration two non-adjacent conductors are terminated together at both ends of the cable, and the other two conductors are also terminated together.

When used for two circuits, crosstalk is reduced relative to cables with two separate twisted pairs.

When used for a single, balanced line, magnetic interference picked up by the cable arrives as a virtually perfect common mode signal, which is easily removed by coupling transformers.

The combined benefits of twisting, balanced signalling, and quadrupole pattern give outstanding noise immunity, especially advantageous for low signal level applications such as microphone cables, even when installed very close to a power cable. The disadvantage is that star quad, in combining two conductors, typically has double the capacitance of similar two-conductor twisted and shielded audio cable. High capacitance causes increasing distortion and greater loss of high frequencies as distance increases.

Twin-lead
Twin-lead consists of a pair of conductors held apart by a continuous insulator. By holding the conductors a known distance apart, the geometry is fixed and the line characteristics are reliably consistent. It is lower loss than coaxial cable because the characteristic impedance of twin-lead is generally higher than coaxial cable, leading to lower resistive losses due to the reduced current. However, it is more susceptible to interference.

Lecher lines
Lecher lines are a form of parallel conductor that can be used at UHF for creating resonant circuits. They are a convenient practical format that fills the gap between lumped components (used at HF/VHF) and resonant cavities (used at UHF/SHF).

Single-wire line
Unbalanced lines were formerly much used for telegraph transmission, but this form of communication has now fallen into disuse. Cables are similar to twisted pair in that many cores are bundled into the same cable but only one conductor is provided per circuit and there is no twisting. All the circuits on the same route use a common path for the return current (earth return). There is a power transmission version of single-wire earth return in use in many locations.

Signal transfer
Electrical transmission lines are very widely used to transmit high frequency signals over long or short distances with minimum power loss. One familiar example is the down lead from a TV or radio aerial to the receiver.

Transmission line circuits
A large variety of circuits can also be constructed with transmission lines including impedance matching circuits, filters, power dividers and directional couplers.

Stepped transmission line
A stepped transmission line is used for broad range impedance matching. It can be considered as multiple transmission line segments connected in series, with the characteristic impedance of each individual element to be $$Z_\mathrm{0,i}$$. The input impedance can be obtained from the successive application of the chain relation

 * $$Z_\mathrm{i+1} = Z_\mathrm{0,i}\,\frac{\,Z_\mathrm{i} + j\,Z_\mathrm{0,i}\,\tan(\beta_\mathrm{i} \ell_\mathrm{i})\,}{Z_\mathrm{0,i} + j\,Z_\mathrm{i}\,\tan(\beta_\mathrm{i} \ell_\mathrm{i})}\,$$

where $$\beta_\mathrm{i}$$ is the wave number of the $$\mathrm{i}$$-th transmission line segment and $$\ell_\mathrm{i}$$ is the length of this segment, and $$Z_\mathrm{i}$$ is the front-end impedance that loads the $$\mathrm{i}$$-th segment.

Because the characteristic impedance of each transmission line segment $$Z_\mathrm{0,i}$$ is often different from the impedance $$Z_0$$ of the fourth, input cable (only shown as an arrow marked $$Z_0$$ on the left side of the diagram above), the impedance transformation circle is off-centred along the $$x$$ axis of the Smith Chart whose impedance representation is usually normalized against $$Z_0$$.

Approximating lumped elements
At higher frequencies, the reactive parasitic effects of real world lumped elements, including inductors and capacitors, limits their usefulness. Therefore, it is sometimes useful to approximate the electrical characteristics of inductors and capacitors with transmission lines at the higher frequencies using Richards' Transformations and then substitute the transmission lines for the lumped elements.

More accurate forms of multimode high frequency inductor modeling with transmission lines exist for advanced designers.

Stub filters
If a short-circuited or open-circuited transmission line is wired in parallel with a line used to transfer signals from point A to point B, then it will function as a filter. The method for making stubs is similar to the method for using Lecher lines for crude frequency measurement, but it is 'working backwards'. One method recommended in the RSGB's radiocommunication handbook is to take an open-circuited length of transmission line wired in parallel with the feeder delivering signals from an aerial. By cutting the free end of the transmission line, a minimum in the strength of the signal observed at a receiver can be found. At this stage the stub filter will reject this frequency and the odd harmonics, but if the free end of the stub is shorted then the stub will become a filter rejecting the even harmonics.

Wideband filters can be achieved using multiple stubs. However, this is a somewhat dated technique. Much more compact filters can be made with other methods such as parallel-line resonators.

Pulse generation
Transmission lines are used as pulse generators. By charging the transmission line and then discharging it into a resistive load, a rectangular pulse equal in length to twice the electrical length of the line can be obtained, although with half the voltage. A Blumlein transmission line is a related pulse forming device that overcomes this limitation. These are sometimes used as the pulsed power sources for radar transmitters and other devices.

Sound
The theory of sound wave propagation is very similar mathematically to that of electromagnetic waves, so techniques from transmission line theory are also used to build structures to conduct acoustic waves; and these are called acoustic transmission lines.

title: Transmission line


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Transmission medium

A transmission medium is a system or substance that can mediate the propagation of signals for the purposes of telecommunication. Signals are typically imposed on a wave of some kind suitable for the chosen medium. For example, data can modulate sound, and a transmission medium for sounds may be air, but solids and liquids may also act as the transmission medium. Vacuum or air constitutes a good transmission medium for electromagnetic waves such as light and radio waves. While a material substance is not required for electromagnetic waves to propagate, such waves are usually affected by the transmission media they pass through, for instance, by absorption or reflection or refraction at the interfaces between media. Technical devices can therefore be employed to transmit or guide waves. Thus, an optical fiber or a copper cable is used as transmission media.

Electromagnetic radiation can be transmitted through an optical medium, such as optical fiber, or through twisted pair wires, coaxial cable, or dielectric-slab waveguides. It may also pass through any physical material that is transparent to the specific wavelength, such as water, air, glass, or concrete. Sound is, by definition, the vibration of matter, so it requires a physical medium for transmission, as do other kinds of mechanical waves and heat energy. Historically, science incorporated various aether theories to explain the transmission medium. However, it is now known that electromagnetic waves do not require a physical transmission medium, and so can travel through the vacuum of free space. Regions of the insulative vacuum can become conductive for electrical conduction through the presence of free electrons, holes, or ions.

Telecommunications
A physical medium in data communications is the transmission path over which a signal propagates. Many different types of transmission media are used as communications channel.

In many cases, communication is in the form of electromagnetic waves. With guided transmission media, the waves are guided along a physical path; examples of guided media include phone lines, twisted pair cables, coaxial cables, and optical fibers. Unguided transmission media are methods that allow the transmission of data without the use of physical means to define the path it takes. Examples of this include microwave, radio or infrared. Unguided media provide a means for transmitting electromagnetic waves but do not guide them; examples are propagation through air, vacuum and seawater.

The term direct link is used to refer to the transmission path between two devices in which signals propagate directly from transmitters to receivers with no intermediate devices, other than amplifiers or repeaters used to increase signal strength. This term can apply to both guided and unguided media.

Simplex versus duplex
A signal transmission may be simplex, half-duplex, or full-duplex.

In simplex transmission, signals are transmitted in only one direction; one station is a transmitter and the other is the receiver. In the half-duplex operation, both stations may transmit, but only one at a time. In full-duplex operation, both stations may transmit simultaneously. In the latter case, the medium is carrying signals in both directions at the same time.

Types
In general, a transmission medium can be classified as
 * linear, if different waves at any particular point in the medium can be superposed;
 * bounded, if it is finite in extent, otherwise unbounded;
 * uniform or homogeneous, if its physical properties are unchanged at different points;
 * isotropic, if its physical properties are the same in different directions.

There are two main types of transmission media:
 * guided media—waves are guided along a solid medium such as a transmission line;
 * unguided media—transmission and reception are achieved by means of an antenna.

One of the most common physical media used in networking is copper wire. Copper wire to carry signals to long distances using relatively low amounts of power. The unshielded twisted pair (UTP) is eight strands of copper wire, organized into four pairs.

Twisted pair
Twisted pair cabling is a type of wiring in which two conductors of a single circuit are twisted together for the purposes of improving electromagnetic compatibility. Compared to a single conductor or an untwisted balanced pair, a twisted pair reduces electromagnetic radiation from the pair and crosstalk between neighboring pairs and improves rejection of external electromagnetic interference. It was invented by Alexander Graham Bell.

Coaxial cable
Coaxial cable, or coax (pronounced ) is a type of electrical cable that has an inner conductor surrounded by a tubular insulating layer, surrounded by a tubular conducting shield. Many coaxial cables also have an insulating outer sheath or jacket. The term coaxial comes from the inner conductor and the outer shield sharing a geometric axis. Coaxial cable was invented by English physicist, engineer, and mathematician Oliver Heaviside, who patented the design in 1880.

Coaxial cable is a type of transmission line, used to carry high frequency electrical signals with low losses. It is used in such applications as telephone trunk lines, broadband internet networking cables, high-speed computer data busses, carrying cable television signals, and connecting radio transmitters and receivers to their antennas. It differs from other shielded cables because the dimensions of the cable and connectors are controlled to give a precise, constant conductor spacing, which is needed for it to function efficiently as a transmission line.

Optical fiber

Optical fiber, which has emerged as the most commonly used transmission medium for long-distance communications, is a thin strand of glass that guides light along its length. Four major factors favor optical fiber over copper: data rates, distance, installation, and costs. Optical fiber can carry huge amounts of data compared to copper. It can be run for hundreds of miles without the need for signal repeaters, in turn, reducing maintenance costs and improving the reliability of the communication system because repeaters are a common source of network failures. Glass is lighter than copper allowing for less need for specialized heavy-lifting equipment when installing long-distance optical fiber. Optical fiber for indoor applications cost approximately a dollar a foot, the same as copper.

Multimode and single mode are two types of commonly used optical fiber. Multimode fiber uses LEDs as the light source and can carry signals over shorter distances, about 2 kilometers. Single mode can carry signals over distances of tens of miles.

An optical fiber is a flexible, transparent fiber made by drawing glass (silica) or plastic to a diameter slightly thicker than that of a human hair. Optical fibers are used most often as a means to transmit light between the two ends of the fiber and find wide usage in fiber-optic communications, where they permit transmission over longer distances and at higher bandwidths (data rates) than electrical cables. Fibers are used instead of metal wires because signals travel along them with less loss; in addition, fibers are immune to electromagnetic interference, a problem from which metal wires suffer excessively. Fibers are also used for illumination and imaging, and are often wrapped in bundles so they may be used to carry light into, or images out of confined spaces, as in the case of a fiberscope. Specially designed fibers are also used for a variety of other applications, some of them being fiber optic sensors and fiber lasers.

Optical fibers typically include a core surrounded by a transparent cladding material with a lower index of refraction. Light is kept in the core by the phenomenon of total internal reflection which causes the fiber to act as a waveguide. Fibers that support many propagation paths or transverse modes are called multi-mode fibers, while those that support a single mode are called single-mode fibers (SMF). Multi-mode fibers generally have a wider core diameter and are used for short-distance communication links and for applications where high power must be transmitted. Single-mode fibers are used for most communication links longer than 1000 m.

Being able to join optical fibers with low loss is important in fiber optic communication. This is more complex than joining electrical wire or cable and involves careful cleaving of the fibers, precise alignment of the fiber cores, and the coupling of these aligned cores. For applications that demand a permanent connection a fusion splice is common. In this technique, an electric arc is used to melt the ends of the fibers together. Another common technique is a mechanical splice, where the ends of the fibers are held in contact by mechanical force. Temporary or semi-permanent connections are made by means of specialized optical fiber connectors.

The field of applied science and engineering concerned with the design and application of optical fibers is known as fiber optics. The term was coined by Indian physicist Narinder Singh Kapany, who is widely acknowledged as the father of fiber optics.

Radio
Radio propagation is the behavior of radio waves as they travel, or are propagated, from one point to another, or into various parts of the atmosphere. As a form of electromagnetic radiation, like light waves, radio waves are affected by the phenomena of reflection, refraction, diffraction, absorption, polarization, and scattering. Understanding the effects of varying conditions on radio propagation has many practical applications, from choosing frequencies for international shortwave broadcasters, to designing reliable mobile telephone systems, to radio navigation, to operation of radar systems.

Different types of propagation are used in practical radio transmission systems. Line-of-sight propagation means radio waves that travel in a straight line from the transmitting antenna to the receiving antenna. Line of sight transmission is used to medium-range radio transmission such as cell phones, cordless phones, walkie-talkies, wireless networks, FM radio and television broadcasting and radar, and satellite communication, such as satellite television. Line-of-sight transmission on the surface of the Earth is limited to the distance to the visual horizon, which depends on the height of transmitting and receiving antennas. It is the only propagation method possible at microwave frequencies and above. At microwave frequencies, moisture in the atmosphere (rain fade) can degrade transmission.

At lower frequencies in the MF, LF, and VLF bands, due to diffraction radio waves can bend over obstacles like hills, and travel beyond the horizon as surface waves which follow the contour of the Earth. These are called ground waves. AM broadcasting stations use ground waves to cover their listening areas. As the frequency gets lower, the attenuation with distance decreases, so very low frequency (VLF) and extremely low frequency (ELF) ground waves can be used to communicate worldwide. VLF and ELF waves can penetrate significant distances through water and earth, and these frequencies are used for mine communication and military communication with submerged submarines.

At medium wave and shortwave frequencies (MF and HF bands) radio waves can refract from a layer of charged particles (ions) high in the atmosphere, called the ionosphere. This means that radio waves transmitted at an angle into the sky can be reflected back to Earth beyond the horizon, at great distances, even transcontinental distances. This is called skywave propagation. It is used by amateur radio operators to talk to other countries and shortwave broadcasting stations that broadcast internationally. Skywave communication is variable, dependent on conditions in the upper atmosphere; it is most reliable at night and in the winter. Due to its unreliability, since the advent of communication satellites in the 1960s, many long-range communication that previously used skywaves now use satellites.

In addition, there are several less common radio propagation mechanisms, such as tropospheric scattering (troposcatter) and near vertical incidence skywave (NVIS) which are used in specialized communication systems.

Digital encoding
Transmission and reception of data is typically performed in four steps:
 * 1) At the transmitting end, the data is encoded to a binary representation.
 * 2) A carrier signal is modulated as specified by the binary representation.
 * 3) At the receiving end, the carrier signal is demodulated into a binary representation.
 * 4) The data is decoded from the binary representation.

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

title: Transmission medium


text:
Transmit-after-receive time delay

In telecommunication, transmit-after-receive time delay is the time interval from removal of RF energy at the local receiver input until the local transmitter is automatically keyed on and the transmitted rf signal amplitude has increased to 90% of its steady-state value. An Exception: High-frequency (HF) transceiver equipment is normally not designed with an interlock between receiver squelch and transmitter on-off key. The transmitter can be keyed on at any time, independent of whether or not a signal is being received at the receiver input.

title: Transmit-after-receive time delay


text:
Transponder

In telecommunications, a transponder is a device that, upon receiving a signal, emits a different signal in response. The term is a blend of transmitter and responder.

In air navigation or radio frequency identification, a flight transponder is an automated transceiver in an aircraft that emits a coded identifying signal in response to an interrogating received signal.

In a communications satellite, a satellite transponder receives signals over a range of uplink frequencies, usually from a satellite ground station; the transponder amplifies them, and re-transmits them on a different set of downlink frequencies to receivers on Earth, often without changing the content of the received signal or signals.

Satellite/broadcast communications
A communications satellite’s channels are called transponders because each is a separate transceiver or repeater. With digital video data compression and multiplexing, several video and audio channels may travel through a single transponder on a single wideband carrier. Original analog video only has one channel per transponder, with subcarriers for audio and automatic transmission identification service (ATIS). Non-multiplexed radio stations can also travel in single channel per carrier (SCPC) mode, with multiple carriers (analog or digital) per transponder. This allows each station to transmit directly to the satellite, rather than paying for a whole transponder, or using landlines to send it to an earth station for multiplexing with other stations.

Optical communications
In fiber-optic communications, a transponder is the element that sends and receives the optical signal from a fiber. A transponder is typically characterized by its data rate and the maximum distance the signal can travel.

The term "transponder" can apply to different items with important functional differences, mentioned across academic and commercial literature:
 * according to one description, a transponder and transceiver are both functionally similar devices that convert a full-duplex electrical signal into a full-duplex optical signal. The difference between the two is that transceivers interface electrically with the host system using a serial interface, whereas transponders use a parallel interface to do so. In this view, transponders provide easier-to-handle lower-rate parallel signals, but are bulkier and consume more power than transceivers.
 * according to another description, transceivers are limited to providing an electrical-optical function only (not differentiating between serial or parallel electrical interfaces), whereas transponders convert an optical signal at one wavelength to an optical signal at another wavelength (typically ITU standardized for DWDM communication). As such, transponders can be considered as two transceivers placed back-to-back. This view also seems to be held by, for example, Fujitsu.

As a result, differences in transponder functionality also might influence the functional description of related optical modules like transceivers and muxponders.

Aviation

Another type of transponder occurs in identification friend or foe (IFF) systems in military aviation and in air traffic control secondary surveillance radar (beacon radar) systems for general aviation and commercial aviation.

Primary radar works best with large all-metal aircraft, but not so well on small, composite aircraft. Its range is also limited by terrain and rain or snow and also detects unwanted objects such as automobiles, hills and trees. Furthermore, it cannot always estimate the altitude of an aircraft. Secondary radar overcomes these limitations but it depends on a transponder in the aircraft to respond to interrogations from the ground station to make the plane more visible.

Depending on the type of interrogation, the transponder sends back a transponder code (or "squawk code", Mode A) or altitude information (Mode C) to help air traffic controllers to identify the aircraft and to maintain separation between planes. Another mode called Mode S (Mode Select) is designed to help avoiding over-interrogation of the transponder (having many radars in busy areas) and to allow automatic collision avoidance. Mode S transponders are backward compatible with Modes A and C. Mode S is mandatory in controlled airspace in many countries. Some countries have also required, or are moving toward requiring, that all aircraft be equipped with Mode S, even in uncontrolled airspace. However, in the field of general aviation there have been objections to these moves, because of the cost, size, limited benefit to the users in uncontrolled airspace, and, in the case of balloons and gliders, the power requirements during long flights.

Transponders are used on some military aircraft to ensure ground personnel can verify the functionality of a missile’s flight termination system prior to launch. Such radar-enhancing transponders are needed as the enclosed weapon bays on modern aircraft interfere with prelaunch, flight termination system verification performed by range safety personnel during training test launches. The transponders re-radiate the signals allowing for much longer communication distances.

Marine
The International Maritime Organization's International Convention for the Safety of Life at Sea (SOLAS) requires the Automatic Identification System (AIS) to be fitted aboard international voyaging ships with, and all passenger ships regardless of size. AIS transmitters/receivers are generally called transponders, but they generally transmit autonomously, although coast stations can interrogate class B transponders on smaller vessels for additional information. In addition, navigational aids often have transponders called RACON (radar beacons) designed to make them stand out on a ship's radar screen.

Sonar transponders operate under water and are used to measure distance and form the basis of underwater location marking, position tracking and navigation.

Electronic toll collection
Electronic toll collection systems such as E-ZPass in the eastern United States use RFID transponders to identify vehicles.

Lap timing
Transponders are used in races for lap timing. A cable loop is dug into the race circuit near to the start/finish line. Each individual runner or car has an active transponder with a unique ID code. When the individual passes the start/finish line, the lap time and the racing position is shown on the score board.

Passive and active RFID systems are used in motor sports, and off-road events such as Enduro and Hare and Hounds racing, the riders have a transponder on their person, normally on their arm. When they complete a lap they swipe or touch the receiver which is connected to a computer and log their lap time.

NASCAR uses transponders and cable loops placed at numerous points around the track to determine the lineup during a caution period. This system replaced a dangerous race back to the start-finish line.

Car keys
Many modern automobiles have keys with transponders hidden inside the plastic head of the key. The user of the car may not even be aware that the transponder is there, because there are no buttons to press. When a key is inserted into the ignition lock cylinder and turned, the car's computer sends a signal to the transponder. Unless the transponder replies with a valid code, the computer will not allow the engine to be started. Transponder keys have no battery; they are energized by the signal itself.

Gated communities
Transponders may also be used by residents to enter their gated communities. However, having more than one transponder causes problems. If a resident's car with simple transponder is parked in the vicinity, any vehicle can come up to the automated gate, triggering the gate interrogation signal, which may get an acceptable response from the resident's car. Such units properly installed might involve beamforming, unique transponders for each vehicle, or simply obliging vehicles to be stored away from the gate.

title: Transponder


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Transposition

Transposition may refer to:

Logic and mathematics

 * Transposition (mathematics), a permutation which exchanges two elements and keeps all others fixed
 * Transposition, producing the transpose of a matrix AT, which is computed by swapping columns for rows in the matrix A
 * Transpose of a linear map
 * Transposition (logic), a rule of replacement in philosophical logic
 * Transpose relation, another name for converse relation

Games

 * Transposition (chess), different moves or a different move order leading to the same position, especially during the openings
 * Transposition table, used in computer games to speed up the search of the game tree

Biology

 * Transposition (birth defect), a group of congenital defects involving an abnormal spatial arrangement of tissue or organ
 * Transposition of the great vessels, cardiac transposition, a congenital heart defect with malformation of any of the major vessels
 * Transposition of teeth
 * Penoscrotal transposition
 * Transposition (horizontal gene transfer), the transfer of genetic material between organisms other than by vertical gene transfer
 * Transposons, or genetic transposition, a mutation in which a chromosomal segment is transferred to a new position on the same or another chromosome

Other uses

 * Transposition (law), the incorporation of the provisions of a European Union directive into a Member State's domestic law
 * Transposition (music), moving a note or collection of notes up or down in pitch by a constant number of semitones
 * Transposition (transmission lines), periodic swapping of positions of the conductors of a transmission line
 * Transposition cipher, an elementary cryptographic operation
 * Transposition, docking, and extraction an orbital maneuver performed on the Apollo lunar missions
 * Transposition, sleight of hand (magic), a performer appears to make two different objects ([usually] coins or cards) switch places with one another faster than physically possible.
 * Transpose, a database of academic journal preprinting policies maintained by ASAPbio

title: Transposition


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Transverse redundancy check

In telecommunications, a transverse redundancy check (TRC) or vertical redundancy check is a redundancy check for synchronized parallel bits applied once per bit time, across the bit streams. This requires additional parallel channels for the check bit or bits.

The term usually applies to a single parity bit, although it could also be used to refer to a larger Hamming code.

The adjective "transverse" is most often used when it is used in combination with additional error control coding, such as a longitudinal redundancy check. Although parity alone can only detect and not correct errors, it can be part of a system for correcting errors.

An example of a TRC is the parity written to the 9th track of a 9-track tape.

title: Transverse redundancy check


text:
Tree structure

A tree structure, tree diagram, or tree model is a way of representing the hierarchical nature of a structure in a graphical form. It is named a "tree structure" because the classic representation resembles a tree, although the chart is generally upside down compared to a biological tree, with the "stem" at the top and the "leaves" at the bottom.

A tree structure is conceptual, and appears in several forms. For a discussion of tree structures in specific fields, see Tree (data structure) for computer science; insofar as it relates to graph theory, see tree (graph theory) or tree (set theory). Other related articles are listed below.

Terminology and properties
The tree elements are called "nodes". The lines connecting elements are called "branches". Nodes without children are called leaf nodes, "end-nodes", or "leaves".

Every finite tree structure has a member that has no superior. This member is called the "root" or root node. The root is the starting node. But the converse is not true: infinite tree structures may or may not have a root node.

The names of relationships between nodes model the kinship terminology of family relations. The gender-neutral names "parent" and "child" have largely displaced the older "father" and "son" terminology. The term "uncle" is still widely used for other nodes at the same level as the parent, although it is sometimes replaced with gender-neutral terms like "ommer".

 * A node's "parent" is a node one step higher in the hierarchy (i.e. closer to the root node) and lying on the same branch.
 * "Sibling" ("brother" or "sister") nodes share the same parent node.
 * A node's "uncles" (sometimes "ommers") are siblings of that node's parent.
 * A node that is connected to all lower-level nodes is called an "ancestor". The connected lower-level nodes are "descendants" of the ancestor node.

In the example, "encyclopedia" is the parent of "science" and "culture", its children. "Art" and "craft" are siblings, and children of "culture", which is their parent and thus one of their ancestors. Also, "encyclopedia", as the root of the tree, is the ancestor of "science", "culture", "art" and "craft". Finally, "science", "art" and "craft", as leaves, are ancestors of no other node.

Tree structures can depict all kinds of taxonomic knowledge, such as family trees, the biological evolutionary tree, the evolutionary tree of a language family, the grammatical structure of a language (a key example being S → NP VP, meaning a sentence is a noun phrase and a verb phrase, with each in turn having other components which have other components), the way web pages are logically ordered in a web site, mathematical trees of integer sets, et cetera.

The Oxford English Dictionary records use of both the terms "tree structure" and "tree-diagram" from 1965 in Noam Chomsky's Aspects of the Theory of Syntax.

In a tree structure there is one and only one path from any point to any other point.

Computer science uses tree structures extensively (see Tree (data structure) and telecommunications.)

For a formal definition see set theory, and for a generalization in which children are not necessarily successors, see prefix order.

Examples of tree structures

 * Internet:
 * usenet hierarchy
 * Vacuum tubes
 * Document Object Model's logical structure, Yahoo! subject index, Curlie
 * Operating system: directory structure
 * Information management: Dewey Decimal System, PSH, this hierarchical bulleted list
 * Management: hierarchical organizational structures
 * Computer science:
 * binary search tree
 * red–black tree
 * AVL tree
 * R-tree
 * doubly logarithmic tree
 * Biology: evolutionary tree
 * Business: pyramid selling scheme
 * Project management: work breakdown structure
 * Linguistics:
 * (Syntax) Phrase structure trees
 * (Historical Linguistics) Tree model of language change
 * Sports: business chess, playoffs brackets
 * Mathematics: Von Neumann universe
 * Group theory: descendant trees

Representing trees
There are many ways of visually representing tree structures. Almost always, these boil down to variations, or combinations, of a few basic styles:

Classical node-link diagrams
Classical node-link diagrams, that connect nodes together with line segments:

Nested sets
Nested sets that use enclosure or containment to show parenthood; examples include TreeMaps, fractal maps, and Euler diagrams:

Layered "icicle" diagrams
Layered "icicle" diagrams that use alignment/adjacency.

Outlines and tree views
Lists or diagrams that use indentation, sometimes called "outlines" or "tree views".

An outline:
 * encyclopedia
 * culture
 * art
 * craft
 * science

A tree view:

 * encyclopedia
 * culture
 * art
 * craft
 * science

Nested parentheses
A correspondence to nested parentheses was first noticed by Sir Arthur Cayley:  ((art,craft)culture,science)encyclopedia

or

encyclopedia(culture(art,craft),science)

Radial trees
Trees can also be represented radially:

title: Tree structure


text:
Troposphere

The troposphere is the lowest layer of the atmosphere of Earth. It contains 80% of the total mass of the planetary atmosphere and 99% of the total mass of water vapor and aerosols, and is where most weather phenomena occur. From the planetary surface of the Earth, the average height of the troposphere is 18 km in the tropics; 17 km in the middle latitudes; and 6 km in the high latitudes of the polar regions in winter; thus the average height of the troposphere is 13 km.

The term troposphere derives from the Greek words tropos  (rotating) and sphaira (sphere) indicating that rotational turbulence mixes the layers of air and so determines the structure and the phenomena of the troposphere. The rotational friction of the troposphere against the planetary surface affects the flow of the air, and so forms the planetary boundary layer (PBL) that varies in height from hundreds of meters up to 2 km. The measures of the PBL vary according to the latitude, the landform, and the time of day when the meteorological measurement is realized. Atop the troposphere is the tropopause, which is the functional atmospheric border that demarcates the troposphere from the stratosphere. As such, because the tropopause is an inversion layer in which air-temperature increases with altitude, the temperature of the tropopause remains constant. The layer has the largest concentration of nitrogen.

Composition
The Earth's planetary atmosphere contains, besides other gases, water vapour and carbon dioxide, which produce carbonic acid in rain water, which therefore has an approximate natural pH of 5.0 to 5.5 (slightly acidic). (Water other than atmospheric water vapour fallen as fresh rain, such as fresh/sweet/potable/river water, will usually be affected by the physical environment and may not be in this pH range.) Atmospheric water vapour holds suspended gasses in it (not by mass),78.08% nitrogen as N2, 20.95% oxygen as O2, 0.93% argon, trace gases, and variable amounts of condensing water (from saturated water vapor). Any carbon dioxide released into the atmosphere from a pressurised source combines with the carbonic acid water vapour and momentarily reduces the atmospheric pH by negligible amounts. Respiration from animals releases out of equilibrium carbonic acid and low levels of other ions. Combustion of hydrocarbons which is not a chemical reaction releases to atmosphere carbonic acid water as; saturates, condensates, vapour or gas (invisible steam). Combustion can releases particulates (carbon/soot and ash) as well as molecules forming nitrites and sulphites which will reduce the atmospheric pH of the water slightly or harmfully in highly industrialised areas where this is classed as air pollution and can create the phenomena of acid rain, a pH lower than the natural pH5.56. The negative effects of the by-products of combustion released into the atmospheric vapour can be removed by the use of scrubber towers and other physical means, the captured pollutants can be processed into a valuable by-product. The sources of atmospheric water vapor are the bodies of water (oceans, seas, lakes, rivers, swamps), and vegetation on the planetary surface, which humidify the troposphere through the processes of evaporation and transpiration respectively, and which influences the occurrence of weather phenomena; the greatest proportion of water vapor is in the atmosphere nearest the surface of the Earth. The temperature of the troposphere decreases at high altitude by way of the inversion layers that occur in the tropopause, which is the atmospheric boundary that demarcates the troposphere from the stratosphere. At higher altitudes, the low air-temperature consequently decreases the saturation vapor pressure, the amount of atmospheric water vapor in the upper troposphere.

Pressure
The maximum air pressure (weight of the atmosphere) is at sea level and decreases at high altitude because the atmosphere is in hydrostatic equilibrium, wherein the air pressure is equal to the weight of the air above a given point on the planetary surface. The relation between decreased air pressure and high altitude can be equated to the density of a fluid, by way of the following hydrostatic equation:

 * $$ \frac{dP}{dz} = -\rho g_n = - \frac {mPg_n}{RT}$$

where:

 * gn is the standard gravity
 * ρ is the density

 * z is the altitude
 * P is the pressure
 * R is the gas constant
 * T is the thermodynamic (absolute) temperature
 * m is the molar mass

Temperature
The planetary surface of the Earth heats the troposphere by means of latent heat, thermal radiation, and sensible heat. The gas layers of the troposphere are less dense at the geographic poles and denser at the equator, where the average height of the tropical troposphere is 13 km, approximately 7.0 km greater than the 6.0 km average height of the polar troposphere at the geographic poles; therefore, surplus heating and vertical expansion of the troposphere occur in the tropical latitudes. At the middle latitudes, tropospheric temperatures decrease from an average temperature of 15 °C at sea level to approximately -55 °C at the tropopause. At the equator, the tropospheric temperatures decrease from an average temperature of 20 °C at sea level to approximately -70 to -75 C at the tropopause. At the geographical poles, the Arctic and the Antarctic regions, the tropospheric temperature decreases from an average temperature of 0 °C at sea level to approximately -45 °C at the tropopause.

Altitude
The temperature of the troposphere decreases with increased altitude, and the rate of decrease in air temperature is measured with the Environmental Lapse Rate ($$-dT/dz$$) which is the numeric difference between the temperature of the planetary surface and the temperature of the tropopause divided by the altitude. Functionally, the ELR equation assumes that the planetary atmosphere is static, that there is no mixing of the layers of air, either by vertical atmospheric convection or winds that could create turbulence.

The difference in temperature derives from the planetary surface absorbing most of the energy from the sun, which then radiates outwards and heats the troposphere (the first layer of the atmosphere of Earth) while the radiation of surface heat to the upper atmosphere results in the cooling of that layer of the atmosphere. The ELR equation also assumes that the atmosphere is static, but heated air becomes buoyant, expands, and rises. The dry adiabatic lapse rate (DALR) accounts for the effect of the expansion of dry air as it rises in the atmosphere, and the wet adiabatic lapse rate (WALR) includes the effect of the condensation-rate of water vapor upon the environmental lapse rate.

Compression and expansion
A parcel of air rises and expands because of the lower atmospheric pressure at high altitudes. The expansion of the air parcel pushes outwards against the surrounding air, and transfers energy (as work) from the parcel of air to the atmosphere. Transferring energy to a parcel of air by way of heat is a slow and inefficient exchange of energy with the environment, which is an adiabatic process (no energy transfer by way of heat). As the rising parcel of air loses energy while it acts upon the surrounding atmosphere, no heat energy is transferred from the atmosphere to the air parcel to compensate for the heat loss. The parcel of air loses energy as it reaches greater altitude, which is manifested as a decrease in the temperature of the air mass. Analogously, the reverse process occurs within a cold parcel of air that is being compressed and is sinking to the planetary surface.

The compression and the expansion of an air parcel are reversible phenomena in which energy is not transferred into or out of the air parcel; atmospheric compression and expansion are measured as an isentropic process ($$dS = 0$$) wherein there occurs no change in entropy as the air parcel rises or falls within the atmosphere. Because the heat exchanged ($$dQ = 0$$) is related to the change in entropy ($$dS$$ by $$dQ = T dS$$) the equation governing the air temperature as a function of altitude for a mixed atmosphere is: $$ \frac{\, dS\,}{dz} = 0 $$ where $S$ is the entropy. The isentropic equation states that atmospheric entropy does not change with altitude; the adiabatic lapse rate measures the rate at which temperature decreases with altitude under such conditions.

Humidity
If the air contains water vapor, then cooling of the air can cause the water to condense, and the air no longer functions as an ideal gas. If the air is at the saturation vapor pressure, then the rate at which temperature decreases with altitude is called the saturated adiabatic lapse rate. The actual rate at which the temperature decreases with altitude is the environmental lapse rate. In the troposphere, the average environmental lapse rate is a decrease of about 6.5 °C for every 1.0 km (1,000m) of increased altitude. For dry air, an approximately ideal gas, the adiabatic equation is: $$ p(z) \Bigl[T(z)\Bigr]^{-\frac{\gamma}{\,\gamma\,-\,1\,}} = \text{constant}$$ wherein $$\gamma$$ is the heat capacity ratio ($$\gamma \approx \,$$$7/5$) for air. The combination of the equation for the air pressure yields the dry adiabatic lapse rate:$$\frac{\,dT\,}{dz} = - \frac{\; m g \;}{R} \frac{\;\gamma\,-\,1\;}{\gamma} = -9.8^\circ\mathrm{C/km}$$.

Environment
The environmental lapse rate ($$dT/dz$$), at which temperature decreases with altitude, usually is unequal to the adiabatic lapse rate ($$dS/dz \ne 0$$). If the upper air is warmer than predicted by the adiabatic lapse rate ($$dS/dz > 0$$), then a rising and expanding parcel of air will arrive at the new altitude at a lower temperature than the surrounding air. In which case, the air parcel is denser than the surrounding air, and so falls back to its original altitude as an air mass that is stable against being lifted. If the upper air is cooler than predicted by the adiabatic lapse rate, then, when the air parcel rises to a new altitude, the air mass will have a higher temperature and a lower density than the surrounding air and will continue to accelerate and rise.

Tropopause
The tropopause is the atmospheric boundary layer between the troposphere and the stratosphere, and is located by measuring the changes in temperature relative to increased altitude in the troposphere and in the stratosphere. In the troposphere, the temperature of the air decreases at high altitude, however, in the stratosphere the air temperature initially is constant, and then increases with altitude. The increase of air temperature at stratospheric altitudes results from the ozone layer's absorption and retention of the ultraviolet (UV) radiation that Earth receives from the Sun. The coldest layer of the atmosphere, where the temperature lapse rate changes from a positive rate (in the troposphere) to a negative rate (in the stratosphere) locates and identifies the tropopause as an inversion layer in which limited mixing of air layers occurs between the troposphere and the stratosphere.

Atmospheric flow
The general flow of the atmosphere is from west to east, which, however, can be interrupted by polar flows, either north-to-south flow or a south-to-north flow, which meteorology describes as a zonal flow and as a meridional flow. The terms are used to describe localized areas of the atmosphere at a synoptic scale; the three-cell model more fully explains the zonal and meridional flows of the planetary atmosphere of the Earth.

Three-cell model

The three-cell model of the atmosphere of the Earth describes the actual flow of the atmosphere with the tropical-latitude Hadley cell, the mid-latitude Ferrel cell, and the polar cell to describe the flow of energy and the circulation of the planetary atmosphere. Balance is the fundamental principle of the model — that the solar energy absorbed by the Earth in a year is equal to the energy radiated (lost) into outer space. The Earth's energy balance does not equally apply to each latitude because of the varying strength of the sunlight that strikes each of the three atmospheric cells, consequent to the inclination of the axis of planet Earth within its orbit of the Sun. The resultant atmospheric circulation transports warm tropical air to the geographic poles and cold polar air to the tropics. The effect of the three cells is the tendency to the equilibrium of heat and moisture in the planetary atmosphere of Earth.

Zonal flow
A zonal flow regime is the meteorological term meaning that the general flow pattern is west to east along the Earth's latitude lines, with weak shortwaves embedded in the flow. The use of the word "zone" refers to the flow being along the Earth's latitudinal "zones". This pattern can buckle and thus become a meridional flow.

Meridional flow
When the zonal flow buckles, the atmosphere can flow in a more longitudinal (or meridional) direction, and thus the term "meridional flow" arises. Meridional flow patterns feature strong, amplified troughs of low pressure and ridges of high pressure, with more north–south flow in the general pattern than west-to-east flow. '''

title: Troposphere


text:
Tropospheric wave

In telecommunication, a tropospheric wave is a radio wave that travels via reflection in the troposphere. Trophospheric waves are propagated from a place of abrupt change in the dielectric constant, or its gradient. In some cases, a ground wave may be so altered that new components appear to arise from reflection in regions of rapidly changing dielectric constant. When these components are distinguishable from the other components, they are called "tropospheric waves."

title: Tropospheric wave


text:
Trunk

Trunk may refer to:

Biology

 * Trunk (anatomy), synonym for torso
 * Trunk (botany), a tree's central superstructure
 * Trunk of corpus callosum, in neuroanatomy
 * Elephant trunk, the proboscis of an elephant

Computing

 * Trunk (software), in revision control
 * Trunk line, a system of shared network access
 * VLAN, which uses a trunk port

Entertainment and media

 * Trunk (album), 2013 album by Ulf Lundell
 * The Trunk, a 1961 British film
 * "The Trunk" (The Twilight Zone), a television episode
 * Trunk Records, a record label

Physical containers

 * Trunk (car), a large storage compartment
 * Trunk (luggage)
 * Trunk (motorcycle), a storage compartment

Other uses

 * Trunk (surname), a German-language surname
 * Trunk carrier, a type of airline regulated by the Civil Aeronautics Board of the United States up until 1978
 * Trunk road, a major road
 * Trunk shot, a cinematic shot from within a car trunk
 * Trunk show, a merchandising event

title: Trunk


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Trusted computing base

The trusted computing base (TCB) of a computer system is the set of all hardware, firmware, and/or software components that are critical to its security, in the sense that bugs or vulnerabilities occurring inside the TCB might jeopardize the security properties of the entire system. By contrast, parts of a computer system that lie outside the TCB must not be able to misbehave in a way that would leak any more privileges than are granted to them in accordance to the system's security policy.

The careful design and implementation of a system's trusted computing base is paramount to its overall security. Modern operating systems strive to reduce the size of the TCB so that an exhaustive examination of its code base (by means of manual or computer-assisted software audit or program verification) becomes feasible.

Definition and characterization
The term trusted computing base goes back to John Rushby, who defined it as the combination of operating system kernel and trusted processes. The latter refers to processes which are allowed to violate the system's access-control rules. In the classic paper Authentication in Distributed Systems: Theory and Practice Lampson et al. define the TCB of a computer system as simply
 * a small amount of software and hardware that security depends on and that we distinguish from a much larger amount that can misbehave without affecting security.

Both definitions, while clear and convenient, are neither theoretically exact nor intended to be, as e.g. a network server process under a UNIX-like operating system might fall victim to a security breach and compromise an important part of the system's security, yet is not part of the operating system's TCB. The Orange Book, another classic computer security literature reference, therefore provides a more formal definition of the TCB of a computer system, as

 * the totality of protection mechanisms within it, including hardware, firmware, and software, the combination of which is responsible for enforcing a computer security policy.

In other words, trusted computing base (TCB) is a combination of hardware, software, and controls that work together to form a trusted base to enforce your security policy.

The Orange Book further explains that

 *  [t] he ability of a trusted computing base to enforce correctly a unified security policy depends on the correctness of the mechanisms within the trusted computing base, the protection of those mechanisms to ensure their correctness, and the correct input of parameters related to the security policy.

In other words, a given piece of hardware or software is a part of the TCB if and only if it has been designed to be a part of the mechanism that provides its security to the computer system. In operating systems, this typically consists of the kernel (or microkernel) and a select set of system utilities (for example, setuid programs and daemons in UNIX systems). In programming languages designed with built-in security features, such as Java and E, the TCB is formed of the language runtime and standard library.

Predicated upon the security policy
As a consequence of the above Orange Book definition, the boundaries of the TCB depend closely upon the specifics of how the security policy is fleshed out. In the network server example above, even though, say, a Web server that serves a multi-user application is not part of the operating system's TCB, it has the responsibility of performing access control so that the users cannot usurp the identity and privileges of each other. In this sense, it definitely is part of the TCB of the larger computer system that comprises the UNIX server, the user's browsers and the Web application; in other words, breaching into the Web server through e.g. a buffer overflow may not be regarded as a compromise of the operating system proper, but it certainly constitutes a damaging exploit on the Web application.

This fundamental relativity of the boundary of the TCB is exemplified by the concept of the 'target of evaluation' ('TOE') in the Common Criteria security process: in the course of a Common Criteria security evaluation, one of the first decisions that must be made is the boundary of the audit in terms of the list of system components that will come under scrutiny.

A prerequisite to security
Systems that don't have a trusted computing base as part of their design do not provide security of their own: they are only secure insofar as security is provided to them by external means (e.g. a computer sitting in a locked room without a network connection may be considered secure depending on the policy, regardless of the software it runs). This is because, as David J. Farber et al. put it,  [i]n a computer system, the integrity of lower layers is typically treated as axiomatic by higher layers. As far as computer security is concerned, reasoning about the security properties of a computer system requires being able to make sound assumptions about what it can, and more importantly, cannot do; however, barring any reason to believe otherwise, a computer is able to do everything that a general Von Neumann machine can. This obviously includes operations that would be deemed contrary to all but the simplest security policies, such as divulging an email or password that should be kept secret; however, barring special provisions in the architecture of the system, there is no denying that the computer could be programmed to perform these undesirable tasks.

These special provisions that aim at preventing certain kinds of actions from being executed, in essence, constitute the trusted computing base. For this reason, the Orange Book (still a reference on the design of secure operating systems ) characterizes the various security assurance levels that it defines mainly in terms of the structure and security features of the TCB.

Software parts of the TCB need to protect themselves
As outlined by the aforementioned Orange Book, software portions of the trusted computing base need to protect themselves against tampering to be of any effect. This is due to the von Neumann architecture implemented by virtually all modern computers: since machine code can be processed as just another kind of data, it can be read and overwritten by any program. This can be prevented by special memory management provisions that subsequently have to be treated as part of the TCB. Specifically, the trusted computing base must at least prevent its own software from being written to.

In many modern CPUs, the protection of the memory that hosts the TCB is achieved by adding in a specialized piece of hardware called the memory management unit (MMU), which is programmable by the operating system to allow and deny a running program's access to specific ranges of the system memory. Of course, the operating system is also able to disallow such programming to the other programs. This technique is called supervisor mode; compared to more crude approaches (such as storing the TCB in ROM, or equivalently, using the Harvard architecture), it has the advantage of allowing security-critical software to be upgraded in the field, although allowing secure upgrades of the trusted computing base poses bootstrap problems of its own.

Trusted vs. trustworthy
As stated above, trust in the trusted computing base is required to make any progress in ascertaining the security of the computer system. In other words, the trusted computing base is “trusted” first and foremost in the sense that it has to be trusted, and not necessarily that it is trustworthy. Real-world operating systems routinely have security-critical bugs discovered in them, which attests to the practical limits of such trust.

The alternative is formal software verification, which uses mathematical proof techniques to show the absence of bugs. Researchers at NICTA and its spinout Open Kernel Labs have recently performed such a formal verification of seL4, a member of the L4 microkernel family, proving functional correctness of the C implementation of the kernel. This makes seL4 the first operating-system kernel which closes the gap between trust and trustworthiness, assuming the mathematical proof is free from error.

TCB size
Due to the aforementioned need to apply costly techniques such as formal verification or manual review, the size of the TCB has immediate consequences on the economics of the TCB assurance process, and the trustworthiness of the resulting product (in terms of the mathematical expectation of the number of bugs not found during the verification or review). In order to reduce costs and security risks, the TCB should therefore be kept as small as possible. This is a key argument in the debate preferring microkernels to monolithic kernels.

Examples
AIX materializes the trusted computing base as an optional component in its install-time package management system.

title: Trusted computing base


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Turnkey

A turnkey, a turnkey project, or a turnkey operation (also spelled turn-key) is a type of project that is constructed so that it can be sold to any buyer as a completed product. This is contrasted with build to order, where the constructor builds an item to the buyer's exact specifications, or when an incomplete product is sold with the assumption that the buyer would complete it.

A turnkey project or contract as described by Duncan Wallace (1984) is

A turnkey contract is typically a construction contract under which a contractor is employed to plan, design and build a project or an infrastructure and do any other necessary development to make it functional or ‘ready to use’ at an agreed price and by a fixed date.

In turnkey contracts, most of the time the employer provides the primary design. The contractor must follow the primary design provided by the employer.

A turnkey computer system is a complete computer including hardware, operating system and application(s) designed and sold to satisfy specific business requirements.

Common usage
Turnkey refers to something that is ready for immediate use, generally used in the sale or supply of goods or services. The word is a reference to the fact that the customer, upon receiving the product, just needs to turn the ignition key to make it operational, or that the key just needs to be turned over to the customer. Turnkey is commonly used in the construction industry, for instance, in which it refers to bundling of materials and labour by the home builder or general contractor to complete the home without owner involvement. The word is often used to describe a home built on the developer's land with the developer's financing ready for the customer to move in. If a contractor builds a "turnkey home" it frames the structure and finish the interior; everything is completed down to the cabinets and carpet. Turnkey is also commonly used in motorsports to describe a car being sold with powertrain (engine, transmission, etc.) to contrast with a vehicle sold without one so that other components may be re-used.

Similarly, this term may be used to advertise the sale of an established business, including all the equipment necessary to run it, or by a business-to-business supplier providing complete packages for business start-up. An example would be the creation of a "turnkey hospital" which would be building a complete medical.

In manufacturing, the turnkey manufacturing contractor (the business that takes on the turnkey project) normally provide help during the initial design process, machining and tooling, quality assurance, to production, packaging and delivery. Turnkey manufacturing have advantages in saving production time, single point of contact, cost savings and price certainty and quality assurance.

Specific usage
The term turnkey is also often used in the technology industry, most commonly to describe pre-built computer "packages" in which everything needed to perform a certain type of task (e.g. audio editing) is put together by the supplier and sold as a bundle. This often includes a computer with pre-installed software, various types of hardware, and accessories. Such packages are commonly called appliances. A website with a ready-made solutions and some configurations is called a turnkey website.

In real estate, turnkey is defined as a home or property that is ready for occupation for its intended purpose, i.e., a home that is fully functional, needs no upgrading or repairs (move-in ready). In commercial use, a building set up to do auto repairs would be defined as turnkey if it came fully stocked with all needed machinery and tools for that particular trade. The turnkey process includes all of the steps involved to open a location including the site selection, negotiations, space planning, construction coordination and complete installation. "Turnkey real estate" also refers to a type of investment. This process includes the purchase, construction or rehab (of an existing site), the leasing out to tenants, and then the sale of the property to a buyer. The buyer is purchasing an investment property which is producing a stream of income.

In drilling, the term indicates an arrangement where a contractor must fully complete a well up to some milestone to receive any payment (in exchange for greater compensation upon completion).

title: Turnkey


text:
Two-out-of-five code

A two-out-of-five code is a constant-weight code that provides exactly ten possible combinations of two bits, and is thus used for representing the decimal digits using five bits. Each bit is assigned a weight, such that the set bits sum to the desired value, with an exception for zero.

According to Federal Standard 1037C:
 * each decimal digit is represented by a binary numeral consisting of five bits of which two are of one kind, called ones, and three are of the other kind, called zeros, and
 * the usual weights assigned to the bit positions are 0-1-2-3-6. However, in this scheme, zero is encoded as binary 01100; strictly speaking the 0-1-2-3-6 previously claimed is just a mnemonic device.

The weights give a unique encoding for most digits, but allow two encodings for 3: 0+3 or 10010 and 1+2 or 01100. The former is used to encode the digit 3, and the latter is used to represent the otherwise unrepresentable zero.

The IBM 7070, IBM 7072, and IBM 7074 computers used this code to represent each of the ten decimal digits in a machine word, although they numbered the bit positions 0-1-2-3-4, rather than with weights. Each word also had a sign flag, encoded using a two-out-of-three code, that could be A Alphanumeric, − Minus, or + Plus. When copied to a digit, the three bits were placed in bit positions 0-3-4. (Thus producing the numeric values 3, 6 and 9, respectively.)

A variant is the United States Postal Service POSTNET barcode, used to represent the ZIP Code for automated mail sorting and routing equipment. This uses two tall bars as ones and three short bars as zeros. Here, the weights assigned to the bit positions are 7-4-2-1-0. Again, zero is encoded specially, using the 7+4 combination (binary 11000) that would naturally encode 11. This method was also used in North American telephone multi-frequency and crossbar switching systems.

The USPS Postal Alpha Numeric Encoding Technique (PLANET) uses the same weights, but with the opposite bar-height convention.

The Code 39 barcode uses weights 1-2-4-7-0 (i.e. LSB first, Parity bit last) for the widths of its bars, but it also encodes two bits of extra information in the spacing between bars. The || ||| spacing is used for digits.

The following table represents decimal digits from 0 to 9 in various two-out-of-five code systems:

The requirement that exactly two bits be set is strictly stronger than a parity check; like all constant-weight codes, a two-out-of-five code can detect not only any single-bit error, but any unidirectional error -- cases in which all the individual bit errors are of a single type (all 0→1 or all 1→0).

title: Two-out-of-five code


text:
NSA product types

The U.S. National Security Agency (NSA) used to rank cryptographic products or algorithms by a certification called product types. Product types were defined in the National Information Assurance Glossary (CNSSI No. 4009, 2010) which used to define Type 1, 2, 3, and 4 products. The definitions of numeric type products have been removed from the government lexicon and are no longer used in government procurement efforts.

Type 1 product
A Type 1 product was a device or system certified by NSA for use in cryptographically securing classified U.S. Government information. A Type 1 product was defined as:

"Cryptographic equipment, assembly or component classified or certified by NSA for encrypting and decrypting classified and sensitive national security information when appropriately keyed. Developed using established NSA business processes and containing NSA approved algorithms. Used to protect systems requiring the most stringent protection mechanisms."

They were available to U.S. Government users, their contractors, and federally sponsored non-U.S. Government activities subject to export restrictions in accordance with International Traffic in Arms Regulations.

Type 1 certification was a rigorous process that included testing and formal analysis of (among other things) cryptographic security, functional security, tamper resistance, emissions security (EMSEC/TEMPEST), and security of the product manufacturing and distribution process.

Type 2 product
A Type 2 product was unclassified cryptographic equipment, assemblies, or components, endorsed by the NSA, for use in telecommunications and automated information systems for the protection of national security information, as defined as:

"Cryptographic equipment, assembly, or component certified by NSA for encrypting or decrypting sensitive national security information when appropriately keyed. Developed using established NSA business processes and containing NSA approved algorithms. Used to protect systems requiring protection mechanisms exceeding best commercial practices including systems used for the protection of unclassified national security information."

Type 3 product
A Type 3 product was a device for use with Sensitive, But Unclassified (SBU) information on non-national security systems, defined as:

"Unclassified cryptographic equipment, assembly, or component used, when appropriately keyed, for encrypting or decrypting unclassified sensitive U.S. Government or commercial information, and to protect systems requiring protection mechanisms consistent with standard commercial practices. Developed using established commercial standards and containing NIST approved cryptographic algorithms/modules or successfully evaluated by the National Information Assurance Partnership (NIAP)."

Approved encryption algorithms included three-key Triple DES, and AES (although AES can also be used in NSA-certified Type 1 products). Approvals for DES, two-key Triple DES and Skipjack have been withdrawn as of 2015.

Type 4 product
A Type 4 product was an encryption algorithm that was registered with NIST but is not a Federal Information Processing Standard (FIPS), defined as:

"Unevaluated commercial cryptographic equipment, assemblies, or components that neither NSA nor NIST certify for any Government usage. These products are typically delivered as part of commercial offerings and are commensurate with the vendor’s commercial practices. These products may contain either vendor proprietary algorithms, algorithms registered by NIST, or algorithms registered by NIST and published in a FIPS."

title: NSA product types