formulasearchengine - User contributions [en]

Cyclonic separation

2014-11-19T10:13:19Z

173.174.242.125: typo

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Synchrotron light source

2013-12-07T12:37:13Z

173.174.52.213: /* Compact synchrotron light sources */

{{About||the green vehicle|Triac (car)|triiodothyroacetic acid|Tiratricol}}

[[Image:triac.svg|thumb|TRIAC schematic symbol]]

'''TRIAC''', from '''Triode for Alternating Current''', is a [[genericized tradename]] for an [[electronic component]] that can conduct [[electric current|current]] in either direction when it is triggered (turned on), and is formally called a '''bidirectional triode thyristor''' or '''bilateral triode thyristor'''.

TRIACs are part of the [[thyristor]] family and are closely related to [[silicon-controlled rectifier]]s (SCR). However, unlike SCRs, which are unidirectional devices (that is, they can conduct current only in one direction), TRIACs are bidirectional and so current can flow in either direction. Another difference from SCRs is that TRIAC current flow can be enabled by either a positive or negative current applied to its ''gate'' electrode, whereas SCRs can be triggered only by current going into the gate. To create a triggering current, a positive or negative voltage has to be applied to the gate with respect to the MT1 terminal (otherwise known as A1).

Once triggered, the device continues to conduct until the current drops below a certain threshold called the [[Silicon controlled rectifier#Modes of operation|holding current]].

The bidirectionality makes TRIACs very convenient switches for [[alternating current]] circuits, also allowing them to control very large power flows with [[Ampere|milliampere]]-scale gate currents. In addition, applying a trigger pulse at a controlled phase angle in an A.C. cycle allows control of the percentage of current that flows through the TRIAC to the load ([[phase control]]), which is commonly used, for example, in controlling the speed of low-power [[induction motors]], in dimming lamps, and in controlling A.C. heating resistors.

==Physics of the device==
{{anchor|figure1}}{{Gallery
|height=265
|lines=2
|align=right
|Image:TRIAC Equivalent Circuit.png|alt1=Figure 1: Triggering modes.|Figure 1: Triggering modes. Quadrants II, I, III, IV, top left, top right, bottom left, bottom right respectively
|File:Triac structure.svg|alt2=Figure 2: TRIAC semiconductor construction.|Figure 2: TRIAC semiconductor construction.
}}
To explain how TRIACs work, one has to individually analyze the triggering in each one of the four quadrants. The four quadrants are illustrated in Figure 1, according to the voltage on the gate and the MT2 terminals with respect to the MT1 terminal. The MT1 and MT2 terminals are also commonly referred to as A1 and A2, respectively.<ref name="ThyristorTheory"/>

The relative sensitivity depends on the physical structure of a particular triac, but as a rule, quadrant I is the most sensitive (least gate current required) and quadrant IV is the least sensitive (most gate current required).{{Clarify|post-text=Why is Q-IV the least sensitive? See [[Talk:TRIAC#Request of explanation: why Q-IV is the least sensitive quadrant?|discussion]]| date=October 2011}}

In quadrants 1 and 2, MT2 is positive, and current flows from MT2 to MT1 through P, N, P and N layers. The N region attached to MT2 does not participate significantly. In quadrants 3 and 4, MT2 is negative, and current flows from MT1 to MT2, also through P, N, P and N layers. The N region attached to MT2 is active, but the N region attached to MT1 only participates in the initial triggering, not the bulk current flow.

In most applications, the gate current comes from MT2, so quadrants 1 and 3 are the only operating modes.
{{-}}

===Triggering in Quadrant I===
{{Gallery
|width=220
|height=323
|lines=2
|align=right
|Image:Triac Quad I.svg|alt1=Figure 3: Operation in Quadrant I|Figure 3: Operation in Quadrant I
|Image:Triac Quad I like SCR.JPG|alt2=Figure 4: Equivalent electric circuit for a triac in Q-I operation mode.|Figure 4: Equivalent electric circuit for a triac in Q-I operation mode.
}}
Quadrant I operation occurs when the gate and MT2 are positive with respect to MT1. [[#figure1|Figure 1]]

The precise mechanism is illustrated in Figure 3. The gate current makes an equivalent NPN transistor switch on, which in turn draws current from the base of an equivalent PNP transitor, turning it on also. Part of the gate current (dotted line) is lost through the ohmic path across the p-silicon, flowing directly into MT1 without passing through the NPN transistor base. In this case, the injection of holes in the p-silicon makes the stacked n, p and n layers beneath MT1 behave like a NPN transistor, which turns on due to the presence of a current in its base. This, in turn, causes the p, n and p layers over MT2 to behave like a PNP transistor, which turns on because its n-type base becomes forward-biased with respect to its emitter, (MT2). Thus, the triggering scheme is the same as an SCR. The equivalent circuit is depicted in Figure 4.

However, the structure is different from SCRs. In particular, TRIAC always has a small current flowing directly from the gate to MT1 through the p-silicon without passing through the p-n junction between the base and the emitter of the equivalent NPN transistor. This current is indicated in Figure 3 by a dotted red line and it is the reason why a TRIAC needs more gate current to turn on than a comparably rated SCR.<ref name="PowerElec"/>

Generally, this quadrant is the most sensitive of the four. This is because it is the only quadrant where gate current is injected directly into the base of one of the main device transistors.{{Clarify|post-text=Why is Q-I the most sensitive? See [[Talk:TRIAC#Request of explanation: why Q-IV is the least sensitive quadrant?|discussion]]| date=October 2011}}
{{-}}

===Triggering in Quadrant II===
{{Gallery
|width=220
|height=323
|lines=2
|align=right
|Image:Triac Quad II.svg|alt1=Figure 5: Operation in Quadrant II|Figure 5: Operation in Quadrant II.
}}
Quadrant II operation occurs when the gate is negative and MT2 is positive with respect to MT1. [[#figure1|Figure 1]]

Figure 5 gives a graphical explanation of the triggering process. The turn-on of the device is three-fold and starts when the current from MT1 flows into the gate through the p-n junction under the gate. This switches on a structure composed by an NPN transistor and a PNP transistor, which has the gate as cathode (the turn-on of this structure is indicated by "1" in the figure). As current into the gate increases, the potential of the left side of the p-silicon under the gate rises towards MT1, since the difference in potential between the gate and MT2 tends to lower: this establishes a current between the left side and the right side of the p-silicon (indicated by "2" in the figure), which in turn switches on the NPN transistor under the MT1 terminal and as a consequence also the pnp transistor between MT2 and the right side of the upper p-silicon. So, in the end, the structure which is crossed by the major portion of the current is the same as Quadrant I operation ("3" in Figure 5).<ref name="PowerElec"/>
{{-}}

===Triggering in Quadrant III===
{{Gallery
|width=220
|height=323
|lines=2
|align=right
|Image:Triac_Quad_III.svg|alt1=Figure 6: Operation in Quadrant III|Figure 6: Operation in Quadrant III.
}}
Quadrant III operation occurs when the gate and MT2 are negative with respect to MT1. [[#figure1|Figure 1]]

The whole process is outlined in Figure 6. The process happens in different steps here too. In the first phase, the pn junction between the MT1 terminal and the gate becomes forward-biased (step 1). As forward-biasing implies the injection of minority carriers in the two layers joining the junction, electrons are injected in the p-layer under the gate. Some of these electrons do not recombine and escape to the underlying n-region (step 2). This in turn lowers the potential of the n-region, acting as the base of a pnp transistor which switches on (turning the transistor on without directly lowering the base potential is called '''remote gate control'''). The lower p-layer works as the collector of this PNP transistor and has its voltage heightened: actually, this p-layer also acts as the base of an NPN transistor made up by the last three layers just over the MT2 terminal, which, in turn, gets activated. Therefore, the red arrow labeled with a "3" in Figure 6 shows the final conduction path of the current.<ref name="PowerElec"/>
{{-}}

===Triggering in Quadrant IV===
{{Gallery
|width=220
|height=323
|lines=2
|align=right
|Image:Triac_Quad_IV.svg|alt1=Figure 7: Operation in Quadrant IV|Figure 7: Operation in Quadrant IV.
}}
Quadrant IV operation occurs when the gate is positive and MT2 is negative with respect to MT1. [[#figure1|Figure 1]]

Triggering in this quadrant is similar to triggering in Quadrant III. The process uses a remote gate control and is illustrated in Figure 7. As current flows from the p-layer under the gate into the n-layer under MT1, minority carriers in the form of free electrons are injected into the p-region and some of them are collected by the underlying np-junction and pass into the adjoining n-region without recombining. As in the case of a triggering in Quadrant III, this lowers the potential of the n-layer and turns on the PNP transistor formed by the n-layer and the two p-layers next to it. The lower p-layer works as the collector of this PNP transistor and has its voltage heightened: actually, this p-layer also acts as the base of an NPN transistor made up by the last three layers just over the MT2 terminal, which, in turn, gets activated. Therefore, the red arrow labeled with a "3" in Figure 6 shows the final conduction path of the current.<ref name="PowerElec"/>

Generally, this quadrant is the least sensitive of the four{{Clarify|post-text=Why is Q-IV the least sensitive? See [[Talk:TRIAC#Request of explanation: why Q-IV is the least sensitive quadrant?|discussion]]| date=October 2011}} In addition, some models of TRIACs cannot be triggered in this quadrant but only in the other three.

== Typical issues ==

There are some drawbacks one should know when using a TRIAC in a circuit. In this section, a few are summarized.

===Gate threshold current, latching current and holding current===

A TRIAC starts conducting when a current flowing into or out of its gate is sufficient to turn on the relevant junctions in the quadrant of operation. The minimum current able to do this is called '''gate threshold current''' and is generally indicated by IGT. In a typical TRIAC, the gate threshold current is generally a few milliampères, but one has to take into account also that:

* IGT depends on the temperature: The higher the temperature, the higher the reverse currents in the blocked junctions. This implies the presence of more free carriers in the gate region, which lowers the gate current needed.
* IGT depends on the quadrant of operation, because a different quadrant implies a different way of triggering ([[#Physics_of_the_device|see here]]). As a rule, the first quadrant is the most sensitive (i.e. requires the least current to turn on), whereas the fourth quadrant is the least sensitive.
* When turning on from an off-state, IGT depends on the voltage applied on the two main terminals MT1 and MT2. Higher voltage between MT1 and MT2 cause greater reverse currents in the blocked junctions requiring less gate current similar to high temperature operation. Generally, in datasheets, IGT is given for a specified voltage between MT1 and MT2.

When the gate current is discontinued, if the current flowing between the two main terminals is more than what is called the '''latching current,''' the device keeps conducting, otherwise the device might turn off. Latching current is the minimum that can make up for the missing gate current in order to keep the device internal structure latched. The value of this parameter varies with:

* gate current pulse (amplitude, shape and width)
* temperature
* control circuit (resistors or capacitors between the gate and MT1 increase the latching current because they steal some current from the gate before it can help the complete turn-on of the device)
* quadrant of operation

In particular, if the pulse width of the gate current is sufficiently large (generally some tens of microseconds), the TRIAC has completed the triggering process when the gate signal is discontinued and the latching current reaches a minimum level called '''holding current'''. Holding current is the minimum required current flowing between the two main terminals that keeps the device on after it has achieved commutation in every part of its internal structure.

In datasheets, the latching current is indicated as IL, while the holding current is indicated as IH. They are typically in the order of some milliampères.

=== Static dv/dt ===
A high d''v''/d''t'' between MT2 and MT1 may turn on the TRIAC when it is off. Typical values of critical static d''v''/d''t'' are in the terms of volts per microsecond.

The turn-on is due to a parasitic capacitive coupling of the gate terminal with the MT2 terminal, which lets currents flow into the gate in response to a large rate of voltage change at MT2. One way to cope with this limitation is to design a suitable RC or RCL [[snubber]] network. in many cases this is sufficient to lower the impedance of the gate towards MT1. By putting a resistor or a small capacitor (or both in parallel) between these two terminals, the capacitive current generated during the transient, flows out of the device without activating it. A careful reading of the application notes provided by the manufacturer and testing of the particular device model to design the correct network is in order. Typical values for capacitors and resistors between the gate and MT1 may be up to 100nF and 10Ω up to 1kΩ.<ref name="AN3008"/> Normal TRIACs, except for low-power types marketed as ''sensitive gate'',<ref>{{cite web |url=http://www.onsemi.com/pub/Collateral/2N6071-D.PDF |title=2N6071A/B Series Sensitive Gate Triacs |publisher=Semiconductor Components Industries, LLC |accessdate=June 28, 2012}}</ref> already have such a resistor built-in to safeguard against spurious dv/dt triggering. It should be noted that this will mask the gate's supposed diode-type behaviour when testing a TRIAC with a [[multimeter]].

In datasheets, the static d''v''/d''t'' is usually indicated as <math> \left (\frac{\operatorname{d}v}{\operatorname{d}t}\right )_s </math> and, as mentioned before, is in relation to the tendency of a TRIAC to turn on '''from the off state''' after a large voltage rate of rise even without applying any current in the gate.

===Critical di/dt===
A high rate of rise of the current flowing between MT1 and MT2 (in either direction) '''when the device is turning on''' can damage or destroy the TRIAC even if the pulse duration is very short. The reason is that during the commutation, the power dissipation is not uniformly distributed across the device. When switching on, the device starts to conduct current before the conduction finishes to spread across the entire junction. The device typically starts to conduct the current imposed by the external circuitry after some nanoseconds or microseconds but the complete switch on of the whole junction takes a much longer time, so too swift a current rise may cause local hot spots that can permanently damage the TRIAC.

In datasheets, this parameter is usually indicated as <math>\frac{\operatorname{d}i}{\operatorname{d}t}</math> and is typically in the order of the tens of ampère per microsecond.<ref name="ThyristorTheory"/>

===Commutating dv/dt and di/dt===
The commutating d''v''/d''t'' rating applies when a TRIAC has been conducting and attempts to turn off with a partially reactive load, such as an inductor. The current and voltage are out of phase, so when the current decreases below the holding value, the triac attempts to turn off, but because of the phase shift between current and voltage, a sudden voltage step takes place between the two main terminals, which turns the device on again.

In datasheets, this parameter is usually indicated as <math> \left ( \frac{\operatorname{d}v}{\operatorname{d}t} \right ) _c </math> and is generally in the order of up to some volts per microsecond.

The reason why '''commutating d''v''/d''t'' is less than static d''v''/d''t''''' is that, shortly before the device tries to turn off, there is still some excess minority charge in its internal layers as a result of the previous conduction. When the TRIAC starts to turn off, these charges alter the internal potential of the region near the gate and MT1, so it is easier for the capacitive current due to d''v''/d''t'' to turn on the device again.

Another important factor during a commutation from on-state to off-state is the d''i''/d''t'' of the current from MT1 to MT2. This is similar to the recovery in standard diodes: the higher the d''i''/d''t'', the greater the reverse current. Because in the TRIAC there are parasitic resistances, a high reverse current in the p-n junctions inside it can provoke a voltage drop between the gate region and the MT1 region which may make the TRIAC stay turned on.

In a datasheet, the commutating d''i''/d''t'' is usually indicated as <math> \left ( \frac{\operatorname{d}i}{\operatorname{d}t} \right ) _c </math> and is generally in the order of some ampères per microsecond.

The commutating d''v''/d''t'' is very important when the TRIAC is used to drive a load with a phase shift between current and voltage, such as an inductive load. Suppose one wants to turn the inductor off: when the current goes to zero, if the gate is not fed, the TRIAC attempts to turn off, but this causes a step in the voltage across it due to the afore-mentioned phase shift. If the commutating d''v''/d''t'' rating is exceeded, the device will not turn off.

==Snubber circuits==
When used with [[inductor|inductive]] loads such as electric fans, care must be taken to assure that the TRIAC will turn off correctly at the end of each half-cycle of the AC power. Indeed, TRIACs can be very sensitive to high values of dv/dt between MT1 and MT2, so a phase shift between current and voltage (as in the case of an inductive load) leads to sudden voltage step that can make the device turn on in an unwanted manner.<ref name="PowerElec"/>

Unwanted turn-ons can be avoided by using a snubber circuit (usually of the RC or RCL type) between MT1 and MT2. Snubber circuits are also used to prevent premature triggering, caused for example by voltage spikes in the mains supply.

Because turn-ons are caused by internal capacitive currents flowing into the gate as a consequence of a high voltage d''v''/d''t'', a gate resistor or capacitor (or both in parallel) may be connected between the gate and MT1 to provide a low-impedance path to MT1 and further prevent false triggering. This, however, increases the required trigger current or adds latency due to capacitor charging. On the other hand, a resistor between the gate and MT1 helps draw leakage currents out of the device, thus improving the performance of the TRIAC at high temperature, where the maximum allowed d''v''/d''t'' is lower. Values of resistors less than 1kΩ and capacitors of 100nF are generally suitable for this purpose, although the fine-tuning should be done on the particular device model.<ref name="AN3008"/>

For higher-powered, more-demanding loads, two [[Silicon-controlled rectifier|SCRs]] in inverse parallel may be used instead of one TRIAC. Because each SCR will have an entire half-cycle of reverse polarity voltage applied to it, turn-off of the SCRs is assured, no matter what the character of the load. However, due to the separate gates, proper triggering of the SCRs is more complex than triggering a TRIAC.

In addition to commutation, a TRIAC may also not turn on reliably with non-resistive loads if the [[Phase (waves)#Phase shift|phase shift]] of the current prevents achieving [[Silicon controlled rectifier#Modes of operation|holding current]] at trigger time. To overcome that, [[pulse wave|pulse train]]s may be used to repeatedly try to trigger the TRIAC until it finally turns on. The advantage is that the gate current does not need to be maintained throughout the entire [[phase control|conduction angle]], which can be beneficial when there is only limited drive capability available.

==Application==
[[File:Triac funktion.gif|thumb|right|Typical use as a dimmer]]
Low power TRIACs are used in many applications such as [[electric light|light]] [[dimmer]]s, speed controls for [[electric fan]]s and other [[electric motor]]s, and in the modern computerized control circuits of many household [[Small appliance|small]] and [[major appliance]]s.

== Example data ==
{| class="wikitable" style=text-align:center
|+ Some typical TRIAC specifications<ref name="nxp138">{{cite web |title=Philips Semiconductors Product specification Triacs BT138 series |url=http://www.nxp.com/acrobat_download/datasheets/BT138_SERIES_2.pdf}} 090119 nxp.com</ref><ref name="st3035/50">{{cite web |title=STMicroelectronics T3035H, T3050H Snubberless high temperature 30 A Triacs |url=http://www.st.com/internet/com/TECHNICAL_RESOURCES/TECHNICAL_LITERATURE/DATASHEET/CD00263568.pdf}} st.com 100922</ref>
|-
! Variable name !! Parameter !! Typical value !! {{nowrap| Unit }}
|-
| <math>V_{gt}</math>
|align=left| Gate threshold voltage
|align=right| 1.5
| [[Volt|V]]
|-
| <math>I_{gt}</math>
|align=left| Gate threshold current
|align=right| {{nowrap|10–50}}
| [[Ampere|mA]]
|-
| <math>V_{drm}</math>
|align=left| Repetitive peak off-state forward voltage
|align=right| {{nowrap|600–800}}
| [[Volt|V]]
|-
| <math>V_{rrm}</math>
|align=left| Repetitive peak off-state reverse voltage
|align=right| {{nowrap|600–800}}
| [[Volt|V]]
|-
| <math>I_{t}</math>
|align=left| [[Root mean square|RMS]] on-state current
|align=right| {{nowrap|4–40}}
| [[Ampere|A]]
|-
| <math>I_{tsm}</math>
|align=left| On-state current, non-repetitive peak
|align=right| {{nowrap|100–270}}
| [[Ampere|A]]
|-
| <math>V_{t}</math>
|align=left| On-state forward voltage
|align=right| {{nowrap|1.5}}
| [[Volt|V]]
|}

==Three-quadrant TRIAC==
A TRIAC which can only operate in quadrants I through III, and cannot be triggered in quadrant IV, has improved turn-off (commutation) characteristics.

These devices are made specifically for improved commutation when controlling a highly-inductive load, such as a motor or [[solenoid]], an application where normal TRIACs have problems due to high voltage/current angles; as soon as they turn off due to the current falling to zero, they experience a voltage spike which can turn them back on again. Most TRIACs' commutation with inductive loads can be improved by use of a [[snubber]] network, but these components are often designed to be able to dispense with need for such a circuit. This improvement is achieved at the expense of the ability to trigger the device in the 4th quadrant (negative voltage and positive gate current). However, this is usually no problem, because this trigger mode is seldom used since even normal TRIACs are least sensitive there.

The first were marketed by Thomson Semiconductors (now [[ST Microelectronics]]) under the name '''Alternistor''', and now sells additional models under the trademark "SNUBBERLESS".

[[Littelfuse]] also uses the name "Alternistor". [[NXP Semiconductors]] calls them "High commutation" ('''Hi-Com''') TRIACs.

== See also ==
* [[Thyristor]]
* [[DIAC|Diode for alternating current (DIAC)]]
* [[Silicon-controlled rectifier|Silicon-controlled rectifier (SCR)]]
* [[Quadrac]]

== References ==
{{Reflist|refs=
<ref name="ThyristorTheory">"Thyristor Theory and Design Considerations", ON Semiconductor, available at [http://www.onsemi.com/pub/Collateral/HBD855-D.PDF www.onsemi.com/pub/Collateral/HBD855-D.PDF]</ref>
<ref name="PowerElec">M.D. Singh, K.B. Khanchandani, Power Electronics, Second Edition, Tata McGraw-Hill, New Delhi, 2007, pages 148-152</ref>
<ref name="AN3008">Application Note AN-3008, ''RC Snubber Networks for Thyristor Power Control and Transient Suppression'', Fairchild Semiconductor, available at http://www.fairchildsemi.com/an/AN/AN-3008.pdf, pages 1-5</ref>
}}

==Further reading==
* ''Thyristor Theory and Design Considerations''; ON Semiconductor; 240 pages; 2006; HBD855/D. [http://www.onsemi.com/pub_link/Collateral/HBD855-D.PDF (Free PDF download)]

== External links ==
{{Commons category|Triacs}}
*[http://www.du.edu/~etuttle/electron/elect5.htm A site about thyristors]

{{Electronic component}}

{{DEFAULTSORT:Triac}}
[[Category:Solid state switches]]
[[Category:Power electronics]]

SL2

2013-11-20T01:41:39Z

173.174.157.72:

In differential geometry, the '''integration along fibers''' of a [[differential form|''k''-form]] yields a <math>(k-m)</math>-form where ''m'' is the dimension of the fiber, via "integration". More precisely, let <math>\pi: E \to B</math> be a fiber bundle over a manifold with compact oriented fibers. If <math>\alpha</math> is a ''k''-form on ''E'', then let:

: <math>(\pi_* \alpha)_b(w_1, \dots, w_{k-m}) = \int_{\pi^{-1}(b)} \beta</math>

where <math>\beta</math> is the induced top-form on the fiber <math>\pi^{-1}(b)</math>; i.e., an <math>m</math>-form given by

:<math>\beta(v_1, \dots, v_m) = \alpha(\widetilde{w_1}, \dots, \widetilde{w_{k-m}}, v_1, \dots, v_m), \quad \widetilde{w_i}\text{ the lifts of } w_i.</math>

(To see <math>b \mapsto (\pi_* \alpha)_b</math> is smooth, work it out in coordinates; cf. an example below.)

<math>\pi_*</math> is then a linear map <math>\Omega^k(E) \to \Omega^{k-m}(B)</math>, which is in fact surjective. By Stokes' formula, if the fibers have no boundaries, the map descends to [[de Rham cohomology]]:

:<math>\pi_*: \operatorname{H}^k(E) \to \operatorname{H}^{k-m}(B).</math>

This is also called the fiber integration. Now, suppose <math>\pi</math> is a sphere bundle; i.e., the typical fiber is a sphere. Then there is an exact sequence <math>0 \to K \to \Omega^*(E) \overset{\pi_*}\to \Omega^*(B) \to 0</math>, ''K'' the kernel,
which leads to a long exact sequence, using <math>\operatorname{H}^k(B) \simeq \operatorname{H}^{k+m}(K)</math>:
:<math>\dots \rightarrow \operatorname{H}^k(B) \overset{\delta}\to \operatorname{H}^{k+m+1}(B) \overset{\pi^*} \rightarrow \operatorname{H}^{k+m+1}(E) \overset{\pi_*} \rightarrow \operatorname{H}^{k+1}(B) \rightarrow \dots</math>,
called the [[Gysin sequence]].

== Example ==
Let <math>\pi: M \times [0, 1] \to M</math> be an obvious projection. For simplicity, assume <math>M = \mathbb{R}^n</math> with coordinates <math>x_j</math> and consider a ''k''-form:

:<math>\alpha = f \, dx_{i_1} \wedge \dots \wedge dx_{i_k} + g \, dt \wedge dx_{j_1} \wedge \dots \wedge dx_{j_{k-1}}.</math>

Then, at each point in ''M'',

:<math>\pi_*(\alpha) = \pi_*(g \, dt \wedge dx_{j_1} \wedge \dots \wedge dx_{j_{k-1}}) = \left( \int_0^1 g(\cdot, t) \, dt \right) \, {dx_{j_1} \wedge \dots \wedge dx_{j_{k-1}}}.</math>

From this the next formula follows easily: if <math>\alpha</math> is any ''k''-form on <math>M \times I,</math>

:<math>\pi_*(d \alpha) = \alpha_1 - \alpha_0 - d \pi_*(\alpha)</math>

where <math>\alpha_i</math> is the restriction of <math>\alpha</math> to <math>M \times \{i\}</math>. This formula is a special case of Stokes' formula. As an application of this, let <math>f: M \times [0, 1] \to N</math> be a smooth map (thought of as a homotopy). Then the composition <math>h = \pi_* \circ f^*</math> is a [[homotopy operator]]:

:<math>d \circ h + h \circ d = f_1^* - f_0^*: \Omega^k(N) \to \Omega^k(M),</math>

which implies <math>f_1, f_0</math> induces the same map on cohomology. For example, let ''U'' be an open ball with center at the origin and let <math>f_t: U \to U, x \mapsto tx</math>. Then <math>\operatorname{H}^k(U) = \operatorname{H}^k(pt)</math>, the fact known as the [[Poincaré lemma]].


== See also ==
*[[Gysin homomorphism]]
*[[Transgression map]]

== References ==
*[[Michele Audin]], Torus actions on symplectic manifolds, Birkhauser, 2004

[[Category:Differential geometry]]

Cotangent complex

2013-05-24T20:04:37Z

173.174.110.162: Grammar: In the first sentence I changed "... is a roughly a universal..." to "... is roughly a universal...".

{{Orphan|date=December 2011}}

The '''noisy channel model''' is a framework used in [[spell checker]]s,
[[question answering]], [[speech recognition]], and [[machine translation]].
In this model, the goal is to find the intended word given a word where the
letters have been scrambled in some manner.

== Definition ==

Given an alphabet <math>\Sigma</math>, let <math>\Sigma^*</math> be the set
of all finite strings over <math>\Sigma</math>. Let the dictionary
<math>D</math> of valid words be some subset of <math>\Sigma^*</math>, i.e.,
<math>D\subseteq\Sigma^*</math>.

The '''noisy channel''' is the matrix

:<math>\Gamma_{ws} = \Pr(s|w)</math>,

where <math>w\in D</math> is the intended word and <math>s\in\Sigma^*</math>
is the scrambled word that was actually received.

== Example ==

Consider the English alphabet
<math>\Sigma = \{a, b, c, ..., y, z, A, B, ..., Z, ...\}</math>. Some subset
<math>D\subseteq\Sigma^*</math> makes up the dictionary of valid English
words.

There are several mistakes that may occur while typing, including:

# Missing letters, e.g., ''leter'' instead of ''letter''
# Accidental letter additions, e.g., ''misstake'' instead of ''mistake''
# Swapping letters, e.g., ''recieved'' instead of ''received''
# Replacing letters, e.g., ''fimite'' instead of ''finite''

To construct the noisy channel matrix <math>\Gamma</math>, we must consider
the probability of each mistake, given the intended word
(<math>\Pr(s|w)</math> for all <math>w\in D</math> and
<math>s\in\Sigma^*</math>). These probabilities may be gathered, for
example, by considering the [[Levenshtein distance]] between <math>s</math>
and <math>w</math> or by comparing the draft of an essay with one that has
been manually edited for spelling.

== Error-correction ==

The goal of the noisy channel model is to find the intended word given the
scrambled word that was received. The '''decision function'''
<math>\sigma : \Sigma^* \to D</math> is a function that, given a scrambled word, returns
the intended word.

Methods of constructing a decision function include the
[[maximum likelihood|maximum likelihood rule]], the
[[maximum a posteriori|maximum a posteriori rule]], and the
[[Minimum_distance#Coding_Theory|minimum distance rule]].

In some cases, it may be better to accept the scrambled word as the intended
word rather than attempt to find an intended word in the dictionary. For
example, the word ''[[schönfinkeling]]'' may not be in the dictionary, but might
in fact be the intended word.

== See also ==

* [[Coding theory]]

== References ==
{{Refbegin}}
* {{cite journal
| last1 = Brill | first1 = Eric | last2 = Moore | first2 = Robert C.
| title = An Improved Error Model for Noisy Channel Spelling Correction
|date=Jan 2000
| journal = Proceedings of ACL 2000
| url = https://research.microsoft.com/apps/pubs/default.aspx?id=66833
}}
{{Refend}}

[[Category:Automatic identification and data capture]]
[[Category:Computational linguistics]]
[[Category:Statistical natural language processing]]