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2014-12-21T03:58:21Z

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2014-01-23T03:01:19Z

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The '''Amott test''' is one of the most widely used empirical [[wettability]] measurements for reservoir [[Core sample|cores]] in [[petroleum engineering]]. The method combines two spontaneous [[imbibition]] measurements and two forced displacement measurements. This test defines two different indices: the Amott water index (<math>I_w</math>) and the Amott oil index (<math>I_o</math>).

==Amott–Harvey index==
The two Amott indices are often combined to give the '''Amott–Harvey index'''. It is a number between -1 and 1 describing wettability of a rock in drainage processes. It is defined as:
:<math>AI=I_w-I_o</math>
[[Image:Amott-Harvey_Index_calculation.png|thumb|300px|right|Figure 1: Amott–Harvey index and USBM number calculation.]]
These two indices are obtained from [[special core analysis]] (SCAL) experiments ([[porous plate]] or [[centrifuge]]) by plotting the capillary pressure curve as a function of the water saturation as shown on figure 1:
:<math>I_w=\frac{S_{spw}-S_{cw}}{1-S_{cw}-S_{or}}</math>
with <math>S_{spw}</math> is the water saturation for a zero capillary pressure during the imbibition process, <math>S_{cw}</math> is the irreducible water saturation and <math>S_{or}</math> is the residual oil saturation after imbibition.
:<math>I_o=\frac{S_{spo}-S_{or}}{1-S_{cw}-S_{or}}</math>
with <math>S_{spo}</math> is the oil saturation for a zero capillary pressure during the [[secondary drainage]] process, <math>S_{cw}</math> is the irreducible water saturation and <math>S_{or}</math> is the residual non-wetting phase saturation after imbibition.

A rock is defined as:
* Water wet when the Amott–Harvey index is between 0.3 and 1,
* Weakly water wet when the Amott–Harvey index is between 0 and 0.3,
* Weakly oil wet when the Amott–Harvey index is between -0.3 and 0,
* Oil wet when the Amott–Harvey index is between -1 and -0.3.

==See also==
* [[USBM index]] - an alternative wettability index
* [[Relative permeability]]
* [[Multiphase flow]]
* [[Capillary pressure]]
* [[Leverett J-function]]
* [[Imbibition]]

==External links==
* http://www.jgmaas.com/scores/facts.html

==References==
* Dake, L.P., "Fundamentals of Reservoir Engineering", Elsevier Scientific Publishing Company, Amsterdam, 1977.
* Amott, E., "Observations relating to the wettability of porous rock", Trans. AIME 219, pp. 156–162, 1959.
[[Category:Fluid dynamics]]

{{fluiddynamics-stub}}

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2014-01-21T21:59:37Z

71.167.58.9: /* H(jω) vs H(ω) */ new section

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State space representation

2014-01-20T21:10:13Z

71.167.58.9: /* Proper transfer functions */ roman for labels

[[File:Iraqi voters in Baghdad2.jpg|right|thumb|300px|Voters lining up outside a [[Baghdad]] polling station during the [[Iraqi legislative election, January 2005|2005 Iraqi election]]. Voter turnout was considered high despite widespread concerns of violence.]]
{{voting}}
'''Voter turnout''' is the percentage of eligible [[voting|voters]] who cast a [[ballot]] in an [[election]]. (Who is eligible varies by country, and should not be confused with the total adult population. For example, some countries discriminate based on sex, race, and/or religion. Age and citizenship are usually among the criteria.) After increasing for many decades, there has been a trend of decreasing voter turnout in most established [[democracy|democracies]] since the 1960s.<ref name="Niemi and Weisberg p. 31">Niemi and Weisberg p. 31</ref> In general, low turnout may be due to [[disenchantment]], [[apathy|indifference]], or contentment. Low turnout is often considered to be undesirable, and there is much debate over the factors that affect turnout and how to increase it. In spite of significant study into the issue, scholars are divided on reasons for the decline. Its cause has been attributed to a wide array of [[economics|economic]], [[demographics|demographic]], cultural, [[technology|technological]], and institutional factors. There have been many efforts to increase turnout and encourage voting.

Different [[country|countries]] have very different voter turnouts. For example, in the [[United States presidential election, 2008|United States 2008 presidential election]] turnout was 64%.<ref name="turnout-report final">{{cite press release |url=http://timeswampland.files.wordpress.com/2008/12/2008turnout-report_final11.pdf |format=PDF |title=African-Americans, Anger, Fear and Youth Propel Turnout to Highest Level Since 1964 |accessdate=2008-12-18 |publisher=Center for the Study of the American Electorate, American University}}</ref><ref name=social>{{cite web|url=http://www.gallup.com/poll/112807/Blacks-Conservative-Republicans-Some-Moral-Issues.aspx |title=Gallup.com |publisher=Gallup.com |date= |accessdate=2011-01-20}}</ref> In [[Belgium]], which has [[compulsory voting]], and [[Malta]], which does not, participation reaches 95%. These differences are caused by a mix of cultural and institutional factors.

==Reasons for voting==
In any large election the chance of any one vote determining the outcome is low. Some studies show that a single vote in a voting scheme such as the [[Electoral College (United States)|Electoral College]] in the United States has an even lower chance of determining the outcome.<ref>Satoshi Kanazawa. "A Possible Solution to the Paradox of Voter Turnout." ''The Journal of Politics.'' p. 974</ref> Other studies claim that the Electoral College actually increases voting power.<ref>Gelman, Katz,and Teurlinckx. [http://www.stat.columbia.edu/~gelman/research/published/STS027.pdf "The Mathematics and Statistics of Voting Power."] 'Statistical Science' 2002, vol 17, no 4</ref> Studies using [[game theory]], which takes into account the ability of voters to [[interaction|interact]], have also found that the expected turnout for any large election should be zero.<ref name="Kanazawa">Kanazawa p. 975</ref>

The basic formula for determining whether someone will vote, on the questionable assumption that people act completely rationally, is<ref>The basic idea behind this formula was developed by [[Anthony Downs]] in ''An Economic Theory of Democracy.'' published in 1957. The formula itself was developed by [[William H. Riker]] and [[Peter Ordeshook]] and published in "A Theory of the Calculus of Voting." ''American Political Science Review.'' 1968. 62:25–42.</ref>

: <math>PB + D > C,</math>

where
* ''P'' is the [[probability]] that an individual's vote will affect the outcome of an election,
* ''B'' is the perceived benefit that would be received if that person's favored [[political party]] or candidate were elected,
* ''D'' originally stood for democracy or [[civic duty]], but today represents any social or personal [[Utility|gratification]] an individual gets from voting, and
* ''C'' is the time, effort, and financial cost involved in voting.

Since ''P'' is virtually zero in most elections, ''PB'' is also near zero, and ''D'' is thus the most important element in motivating people to vote. For a person to vote, these factors must outweigh ''C''.

Riker and Ordeshook developed the modern understanding of ''D''. They listed five major forms of gratification that people receive for voting: complying with the social obligation to vote; affirming one's allegiance to the political system; affirming a partisan preference (also known as expressive voting, or voting for a candidate to express support, not to achieve any outcome); affirming one's importance to the political system; and, for those who find politics interesting and entertaining, researching and making a decision.<ref>Riker and Ordeshook, 1968</ref> Other political scientists have since added other motivators and questioned some of Riker and Ordeshook's assumptions.{{Citation needed|date=August 2009}} All of these concepts are inherently imprecise, making it difficult to discover exactly why people choose to vote.

Recently, several scholars have considered the possibility that B includes not only a personal interest in the outcome, but also a concern for the welfare of others in the society (or at least other members of one's favorite group or party).<ref>Jankowski, Richard. 2002. "Buying a Lottery Ticket to Help the Poor: Altruism, Civic Duty, and Self-Interest in the Decision to Vote." Rationality and Society 14(1): 55–77.</ref><ref>[[Aaron Edlin|Edlin, Aaron]], [[Andrew Gelman]], and Noah Kaplan. 2007. "Voting as a Rational Choice: Why and How People Vote to Improve the Well-Being of Others." Rationality and Society.</ref> In particular, experiments in which subject [[altruism]] was measured using a [[dictator game]] showed that concern for the well-being of others is a major factor in predicting turnout<ref>[[James H. Fowler|Fowler, James H.]] "Altruism and Turnout," ''Journal of Politics'' 68 (3): 674–683 (August 2006)</ref> and political participation.<ref>[[James H. Fowler|Fowler, James H.]], Kam CD "Beyond the Self: Altruism, Social Identity, and Political Participation," ''Journal of Politics'' 69 (3): 811–825 (August 2007)</ref><ref>Loewen, PJ "Antipathy, Affinity, and Political Participation," ''Canadian Journal of Political Science'' (Forthcoming 2010)</ref> Note that this motivation is distinct from D, because voters must think others benefit from the ''outcome'' of the election, not their ''act'' of voting in and of itself.

==The significance of voter turnout==
High voter turnout is often considered to be desirable, though among political scientists and economists specialising in public choice, the issue is still debated.<ref>See Mark N. Franklin. "Electoral Engineering and Cross National Turnout Differences." ''British Journal of Political Science,'' who attempts to challenge some of this consensus</ref> A high turnout is generally seen as evidence of the [[legitimacy (political science)|legitimacy]] of the current system. [[Dictator]]s have often fabricated high turnouts in [[show election|showcase elections]] for this purpose. For instance, [[Saddam Hussein]]'s 2002 referendum was claimed to have had 100% participation.<ref>[http://archives.cnn.com/2002/WORLD/meast/10/16/iraq.vote/ CNN – Saddam gets perfect poll result]</ref> Opposition parties sometimes boycott votes they feel are unfair or illegitimate, or if the election is for a government that is considered illegitimate. For example, the [[Holy See]] instructed Italian Catholics to boycott national elections for several decades after the [[Italian unification|creation]] of the [[Italy|State of Italy]].<ref>Katz p. 242</ref> In some countries, there are threats of violence against those who vote, such as during the [[2005 Iraq election]]s, an example of [[voter suppression]]. However, some political scientists question the view that high turnout is an implicit endorsement of the system. Mark N. Franklin contends that in [[European Union elections]] opponents of the federation, and of its legitimacy, are just as likely to vote as proponents.<ref name="Franklin">Franklin. "Electoral Engineering"</ref>

Assuming that low turnout is a reflection of disenchantment or indifference, a poll with very low turnout may not be an accurate reflection of the [[popular sovereignty|will of the people]]. On the other hand, if low turnout is a reflection of contentment of voters about likely winners or parties, then low turnout is as legitimate as high turnout, as long as the right to vote exists. Still, low turnouts can lead to unequal representation among various parts of the population. In developed countries, non-voters tend to be concentrated in particular demographic and socioeconomic groups, especially the [[youth|young]] and the [[poverty|poor]]. However, in [[India]], which boasts an electorate of more than 670 million people, the opposite is true. The poor, who comprise the majority of the demographic, are more likely to vote than the rich and the middle classes{{Citation needed|date=October 2008}}, and turnout is higher in rural areas than urban areas.<ref>Gupta, D. (2004). [http://rspas.anu.edu.au/papers/asarc/india_forum/IndiaForum_Gupta-AnalysisIndianElections-Final.pdf An analysis of Indian elections], Appendix D. Australia South Asia Research Centre, Australian National University. Retrieved 2008-11-20.</ref> In low-turnout countries, these groups{{Clarify|date=July 2009}} are often significantly under-represented in elections.{{Citation needed|date=July 2009}} This has the potential to skew policy. For instance, a high voter turnout among [[old age|seniors]] coupled with a low turnout among the young may lead to more money for seniors' [[health care]], and less for youth employment schemes. Some nations thus have rules that render an election invalid if too few people vote, such as [[Serbia]], where three successive presidential elections were rendered invalid in 2003.{{Citation needed|date=May 2012}}

==Socio-economic factors==
{|align="right" class="wikitable" style="margin: 0 15px 0 15px;"
|+'''Socio-Economic Status and Voting Turnout in USA and India<ref>{{cite book
| last = Linz
| first = Juan
| coauthors = Alfred Stephan, Yogendra Yadav
| title = Democracy and Diversity
| publisher = Oxford University Press
| year = 2007
| location = New Delhi
| pages = 99
| isbn = 978-0-19-568368-4}}
</ref>
|-
!USA (1988)
!India (1988)
|-
!colspan="2"|Turnout
|-
|50.1 %<ref>Federal Election Commission via [http://www.infoplease.com/ipa/A0781453.html National Voter Turnout in Federal Elections: 1960–2008], infoplease.com</ref>
|62 %
|-
!colspan="2"|Income (Quinitile)
|-
|Lowest 20%: 36.4%
|57 %
|-
|52
|65
|-
|59
|73
|-
|67
|60
|-
|Highest 20%: 63.1
|47
|-
!colspan="2"|Education
|-
|No high school 38%
|Illiterate 57%
|-
|Some high school 43
|Up to middle 83
|-
|High school graduate 57
|College 57
|-
|Some college 66
|Post-graduate 41
|-
|College grad 79
|
|-
|Post-graduate 84
|
|-
!colspan="2"|Community (1996)
|-
|White 56
|Hindu 60
|-
|Black 50
|Hindu (OBC) 58
|-
|Latino 27
|SC 75
|-
|
|ST 59
|-
|
|Muslim 70
|-
|
|Sikh 89
|}

In each country, some parts of society are more likely to vote than others. In high-turnout countries, these differences tend to be limited. As turnout approaches 90%, it becomes difficult to find significant differences between voters and nonvoters, but in low turnout nations the differences between voters and non-voters can be quite marked.<ref name="Franklin"/> These differences appear to persist over time; in fact, the strongest predictor of individual turnout is whether or not one voted in the previous election.<ref>[[James H. Fowler|Fowler, James H.]] "Habitual Voting and Behavioral Turnout," ''Journal of Politics'' 68 (2): 335–344 (May 2006)</ref> As a result, many scholars think of turnout as habitual behavior that can be learned or unlearned, especially among young adults.<ref>Plutzer, E. "Becoming a Habitual Voter: Inertia, Resources, and Growth in Young Adulthood." ''American Political Science Review'' 96, no. 1 (2002): 41–56.</ref>

Socioeconomic factors significantly affect whether or not individuals develop the habit of voting. The most important socioeconomic factor affecting voter turnout is [[education]]. The more educated a person is, the more likely he or she is to vote, even controlling for other factors that are closely associated with education level, such as [[income]] and [[Social class|class]]. Income has some effect independently: wealthier people are more likely to vote, regardless of their educational background. There is some debate over the effects of [[ethnicity]], [[Race (classification of human beings)|race]], and [[gender]]. In the past, these factors unquestionably influenced turnout in many nations, but nowadays the consensus among political scientists is that these factors have little effect in Western democracies when education and income differences are taken into account.<ref name="Sigelman">Sigelman, L., Roeder, P. W., Jewell, M. E., & Baer, M. A. (1985). Voting and nonvoting: A multi-election perspective. American Journal of Political Science, 29(4), 749–765.</ref> However, since different ethnic groups typically have different levels of education and income, there are important differences in turnout between such groups in many societies. Other demographic factors have an important influence: young people are far less likely to vote than the elderly; and single people are less likely to vote than those who are married.{{Citation needed|date=December 2009}} Occupation has little effect on turnout, with the notable exception of higher voting rates among government employees in many countries.<ref name="Sigelman"/>

There can also be regional differences in voter turnout. One issue that arises in continent-spanning nations, such as Australia, [[Canada]], the United States and [[Russia]], is that of [[time zone]]s. Canada banned the broadcasting of election results in any region where the polls have not yet closed; this ban was upheld by the [[Supreme Court of Canada]]. In several recent Australian national elections, the citizens of Western Australia knew which party would form the new government up to an hour before the polling booths in their State closed.

==Hereditary factors==
While socioeconomic factors undoubtedly play a role in determining voter turnout, new evidence suggests that genetic factors may also be important. Scholars recently used [[twin studies]] of validated turnout in Los Angeles and self-reported turnout in the [[National Longitudinal Study of Adolescent Health]] to establish that the decision to vote in the United States has very strong [[heritability]].<ref>{{cite journal|title=Genetic Variation in Political Participation |last=Fowler |first=James H. |coauthors=Laura A. Baker, Christopher T. Dawes |journal=American Political Science Review |url=http://jhfowler.ucsd.edu/genetic_basis_of_political_cooperation.pdf | issue=2 |volume=102 |pages=233–248 |date=May 2008 | doi=10.1017/S0003055408080209|format=PDF}}</ref> If so, it could help to explain why parental turnout is such a strong predictor of voting in young people,<ref>Plutzer "Becoming a Habitual Vote"</ref> as people inherit genes as well as behaviors from their parents. It might also help to explain why voting appears to be habitual.<ref>Fowler, "Habitual Voting and Behavioral Turnout"</ref> If there is an innate predisposition to vote or abstain, this would explain why past voting behavior is such a good predictor of future voter reaction.

In addition to the [[twin study]] method, scholars have used gene association studies to analyze voter turnout. Two genes that influence social behavior have been directly associated with voter turnout, specifically those regulating the [[serotonin]] system in the brain via the production of [[monoamine oxidase]] and 5HTT.<ref>{{cite journal|title=Two Genes Predict Voter Turnout |last=Fowler |first=James H. |coauthors=Christopher T. Dawes |journal=Journal of Politics |url=http://jhfowler.ucsd.edu/two_genes_predict_voter_turnout.pdf | issue=3 |volume=70 |pages=579–594 |date=July 2008 |format=PDF |doi=10.1017/S0022381608080638}}</ref> This study was recently reanalyzed and the findings suggested to be the result of several significant errors. Once these errors were corrected, there was no longer any statistically significant association between common variants of these two genes and voter turnout.<ref>{{cite journal|title=Candidate Genes and Political Behavior |last=Charney |first=Evan |coauthors=William English |journal=American Political Science Review |url=http://dl.dropbox.com/u/14943700/APSR%2C%20Candidate%20Genes%20and%20Political%20Behavior-1.pdf | issue=1 |volume=106 |pages=1–34 |date=February 2012 |format=PDF |doi=10.1017/S0003055411000554}}</ref>

==Differences between elections==
Within countries there can be important differences in turnout between individual elections
.<ref>{{cite journal|title=Timing of vote decision in first and second order Dutch elections 1978–1995. Evidence from artificial neural networks|last1=Eisinga|first1=R.|last2=Franses|first2=Ph.-H.|author2-link=Philip Hans Franses|last3=Van Dijk|first3=D. |journal=Political Analysis |issue=1|volume=7|pages=117–142| year=1998| doi=10.1093/pan/7.1.117}}</ref> Elections where control of the national [[executive (government)|executive]] is not at stake generally have much lower turnouts—often half that for general elections.{{Citation needed|date=June 2009}} Municipal and provincial elections, and by-elections to fill casual vacancies, typically have lower turnouts, as do elections for the parliament of the supranational [[European Union]], which is separate from the executive branch of the EU's government. In the United States, [[off-year elections|midterm congressional elections]] attract far lower turnouts than Congressional elections held concurrently with Presidential ones.<ref>Lijphart. p. 12</ref> [[Runoff election]]s also tend to attract lower turnouts.

In theory, one of the factors that is most likely to increase turnout is a close race. With an intensely polarized electorate and all polls showing a close finish between [[President of the United States|President]] [[George W. Bush]] and [[Democratic Party (United States)|Democratic]] challenger [[John F. Kerry]], the turnout in the [[United States presidential election, 2004|2004 U.S. presidential election]], was close to 60%, resulting in a record number of popular votes for both candidates; despite losing the election, Kerry even surpassed [[Ronald Reagan]]'s 1984 record in terms of the number of popular votes received. However, this race also demonstrates the influence that contentious social issues can have on voter turnout; for example, the voter turnout rate in 1860 wherein anti-[[slavery]] candidate Abraham Lincoln won the election was the second-highest on record (81.2 percent, second only to 1876, with 81.8 percent). Nonetheless, there is evidence to support the argument that predictable election results—where one vote is not seen to be able to make a difference—have resulted in lower turnouts, such as [[United States presidential election, 1996|Bill Clinton's 1996 re-election]] (which featured the lowest voter turnout in the United States since 1924), the [[United Kingdom general election, 2001|United Kingdom general election of 2001]], and the 2005 [[Spanish referendum on the European Constitution]]; all of these elections produced decisive results on a low turnout.

Bad weather can reduce turnouts,<ref name="Kanazawa"/><ref>{{cite journal|title= Weather conditions and voter turnout in Dutch national parliament elections, 1971–2010 |last1=Eisinga|first1=R.|last2=Te Grotenhuis|first2=M. |last3=Pelzer|first3=B. |journal=International Journal of Biometeorology |issue=4|volume=56|pages=783–786| year=2012| doi=10.1007/s00484-011-0477-7}}</ref><ref>{{cite journal|title= Weather conditions and political party vote share in Dutch national parliament elections, 1971–2010 |last1=Eisinga|first1=R.|last2=Te Grotenhuis|first2=M. |last3=Pelzer|first3=B. |journal=International Journal of Biometeorology |issue=6|volume=56|pages=1161–1165| year=2012| doi=10.1007/s00484-011-0504-8}}</ref> as can the season and the day of the week (although many nations hold all their elections on the same weekday). Weekend and summer elections find more of the population on holiday or uninterested in politics, and have lower turnouts. When nations set fixed election dates, these are usually midweek during the spring or autumn to maximize turnout. Variations in turnout between elections tend to be insignificant. It is extremely rare for factors such as competitiveness, weather, and time of year to cause an increase or decrease in turnout of more than five percentage points, far smaller than the differences between groups within society, and far smaller than turnout differentials between nations.<ref>G. Bingham Powell "Voter Turnout in Thirty Democracies." in ''Electoral Participation.''</ref>

==International differences==
{|align="right" class="wikitable sortable" style="margin: 0 15px 0 15px;" width="400"
|+'''Turnout in national [[lower house]] elections, 1960–1995
|-
!Country
!Compulsory
!№
!Turnout
|-
|{{MLT}}
|N
|6
|94%
|-
|{{CHI}}
|Y
|2
|93%†
|-
|{{AUT}}
|N
|9
|92%
|-
|{{BEL}}
|Y
|12
|91%
|-
|{{ITA}}
|N****
|9
|90%
|-
|{{LUX}}
|Y
|7
|90%
|-
|{{ISL}}
|N
|10
|89%
|-
|{{NZL}}
|N
|12
|88%
|-
|{{DEN}}
|N
|14
|87%
|-
|{{GER}}
|N
|9
|86%
|-
|{{SWE}}
|N
|14
|86%
|-
|{{GRE}}
|Y (not enforced)
|10
|86%
|-
|{{VEN}}
|N*
|7
|85%
|-
|{{CZE}} and {{SVK}}
|N
|6
|85%
|-
|{{ARG}}
|Y
|12
|83%
|-
|{{BRA}}
|Y
|9
|83%
|-
|{{NLD}}
|N**
|7
|83%
|-
|{{AUS}}
|Y
|19
|81%
|-
|{{CRC}}
|N
|8
|81%
|-
|{{NOR}}
|N
|9
|81%
|-
|{{ROM}}
|N
|2
|81%
|-
|{{BUL}}
|N
|2
|80%
|-
|{{ISR}}
|N
|9
|80%
|-
|{{POR}}
|N
|9
|79%
|-
|{{FIN}}
|N
|10
|78%
|-
|{{FRA}}
|N
|9
|76%
|-
|{{UK}}
|N
|9
|76%
|-
|{{KOR}}
|N
|11
|75%
|-
|{{IRL}}
|N
|11
|74%
|-
|{{CAN}}
|N
|12
|74%
|-
|{{ESP}}
|N
|6
|73%
|-
|{{JPN}}
|N
|12
|71%
|-
|{{POL}}
|N
|7
|71%
|-
|{{EST}}
|N
|2
|69%
|-
|{{HUN}}
|N
|6
|66%
|-
|{{RUS}}
|N
|2
|61%
|-
|{{PAK}}
|N
|6
|51%
|-
|{{IND}}
|N
|6
|58%
|-
|{{SUI}}
|N
|8
|54%
|-
|{{USA}}
|N
|18
|48%***
|-class="sortbottom"
|colspan="4"|*Compulsory voting until 1998
|-class="sortbottom"
|colspan="4"|**Excludes pre-1968 elections, when voting was compulsory. 
|-class="sortbottom"
|colspan="4"|***Turnout rates during the period ranged from 55% for general election years, to 40% to off-year elections (those for which the presidency was not on the ballot).
|-class="sortbottom"
|colspan="4"|****In Italy, voting used to be compulsory but only with "innocuous sanctions" (i.e., not enforced) up to 1992.<ref>[http://www.guardian.co.uk/politics/2005/jul/04/voterapathy.uk The Guardian]</ref><ref>[http://i.unu.edu/media/publication/000/002/224/full_participation_web.pdf United Nations University], p.5</ref>
|-class="sortbottom"
|colspan="4" |Statistics from Mark N. Franklin's "Electoral Participation", found in ''Controversies in Voting Behavior'' (2001). Includes only "free" elections.
|-class="sortbottom"
|colspan="4" |†Voting is no longer compulsory in Chile, but the turnout figures reflect a time when not voting was legally punished. 
|}
Voter turnout varies considerably between countries. It tends to be lower in the United States, Asia and Latin America than most of Europe, Canada{{Citation needed|date=August 2009}} and Oceania. Western Europe averages a 77% turnout, and South and Central America around 54% since 1945.<ref>[http://www.idea.int/vt/survey/voter_turnout3.cfm IDEA – Regional differences]</ref> The differences between nations tend to be greater than those between classes, ethnic groups, or regions within nations. Confusingly, some of the factors that cause internal differences do not seem to apply on a global level. For instance, nations with better-educated populaces do not have higher turnouts.
There are two main causes of these international differences—culture and institutions—although there is much debate over the relative impact of the various factors.

===Cultural factors===
Wealth and literacy have some effect on turnout, but are not reliable measures. Countries such as [[Angola]] and [[Ethiopia]] have long had high turnouts, but so have the wealthy states of Europe. The [[United Nations]] [[Human Development Index]] shows some correlation between higher standards of living and higher turnout. The age of a democracy is also an important factor. Elections require considerable involvement by the population, and it takes some time to develop the cultural habit of voting, and the associated understanding of and confidence in the electoral process. This factor may explain the lower turnouts in the newer democracies of Eastern Europe and Latin America. Much of the impetus to vote comes from a sense of civic duty, which takes time and certain social conditions to develop. that can take decades to develop:
*trust in government;
*degree of partisanship among the population;
*interest in politics, and
*belief in the efficacy of voting.<ref>G. Bingham Powell. "American Voter Turnout in Comparative Perspective." ''The American Political Science Review.'' 1986 p. 19.</ref>

Demographics also have an effect. Older people tend to vote more than youths, so societies where the average age is somewhat higher, such as Europe; have higher turnouts than somewhat younger countries such as the United States. Populations that are more mobile and those that have lower marriage rates tend to have lower turnout. In countries that are highly multicultural and multilingual, it can be difficult for national election campaigns to engage all sectors of the population.

The nature of elections also varies between nations. In the United States, [[negative campaigning]] and character attacks are more common than elsewhere, potentially suppressing turnouts. The focus placed on [[get out the vote]] efforts and mass-marketing can have important effects on turnout. Partisanship is an important impetus to turnout, with the highly partisan more likely to vote. Turnout tends to be higher in nations where political allegiance is closely linked to class, ethnic, linguistic, or religious loyalties.<ref>Powell "Thirty Democracies." p. 14</ref> Countries where [[multiparty]] systems have developed also tend to have higher turnouts. Nations with a party specifically geared towards the [[working class]] will tend to have higher turnouts among that class than in countries where voters have only [[big tent]] parties, which try to appeal to all the voters, to choose from.<ref>Powell. p. 13</ref>

===Institutional factors===
Institutional factors have a significant impact on voter turnout. Rules and laws are also generally easier to change than attitudes, so much of the work done on how to improve voter turnout looks at these factors. Making [[compulsory voting|voting compulsory]] has a direct and dramatic effect on turnout. Simply making it easier for candidates to stand through easier [[nomination rules]] is believed to increase voting. Conversely, adding barriers, such as a separate [[voter registration|registration]] process, can suppress turnout. The salience of an election, the effect that a vote will have on policy, and its proportionality, how closely the result reflects the will of the people, are two structural factors that also likely have important effects on turnout.

====Voter registration====
The modalities of how electoral registration is conducted can also affect turnout. For example until "rolling registration" was introduced in the United Kingdom, there was no possibility of the electoral register being updated during its currency, or even amending genuine mistakes after a certain cut off date. The register was compiled in October, and would come into force the next February, and would remain valid until the next January. The electoral register would become progressively more out of date during its period of validity, as electors moved or died (also people studying or working away from home often had difficulty voting). This meant that elections taking place later in the year tended to have lower turnouts than those earlier in the year. The introduction of rolling registration where the register is updated monthly has reduced but not entirely eliminated this issue since the process of amending the register is not automatic, and some individuals do not join the electoral register until the annual October compilation process.

Another country with a highly efficient registration process is France. At the age of eighteen, all youth are automatically registered. Only new residents and citizens who have moved are responsible for bearing the costs and inconvenience of updating their registration. Similarly, in [[Nordic countries]], all citizens and residents are included in the official population register, which is simultaneously a tax list, voter registration, and membership in the universal health system. Residents are required by law to report any change of address to register within a short time after moving. This is also the system in [[Germany]] (but without the membership in the health system).

The elimination of registration as a separate bureaucratic step can result in higher voter turnout. This is reflected in statistics from the United States Bureau of Census, 1982–1983. States that have same day registration, or no registration requirements, have a higher voter turnout than the national average. At the time of that report, the four states that allowed election day registration were Minnesota, Wisconsin, Maine, and Oregon. Since then, Idaho and Maine have changed to allow same day registration. North Dakota is the only state that requires no registration.<ref>U.S. Bureau of the Census, Statistical Abstract of the United States, 1982–83, Table no.804, p.492</ref>

====Compulsory voting====
One of the strongest factors affecting voter turnout is whether voting is [[compulsory voting|compulsory]]. In [[Australia]], voter registration and attendance at a polling booth have been mandatory since the 1920s (Australia has around 10% of eligible voters who are not registered to vote and around 6% of invalid votes, which are included in the 95% figure. Actual voter turnouts in Australia are closer to 81%.<ref>{{cite web|last=Killesteyn|first=Ed|title=Electoral Commissioner|url=http://www.aec.gov.au/about_aec/Publications/speeches/new-debate.htm|publisher=Australian Electoral Commission|accessdate=7 November 2012}}</ref>). Several other countries have similar laws, generally with somewhat reduced levels of enforcement. If a [[Bolivia]]n voter fails to participate in an election, the citizen may be denied withdrawal of their salary from the bank for three months.<ref>[http://politics.guardian.co.uk/apathy/story/0,,1521096,00.html The Guardian ''Compulsory voting around the world'' ]</ref> In [[Mexico]] and [[Brazil]], existing sanctions for non-voting are minimal or are rarely enforced. When enforced, compulsion has a dramatic effect on turnout. In [[Venezuela]] and the [[Netherlands]] compulsory voting has been rescinded, resulting in substantial decreases in turnout. In [[Greece]] voting is compulsory, however there are practically no sanctions for those who do not vote. In [[Belgium]] voting is compulsory, too, but not strongly enforced.

Sanctions for non-voting behaviour were foreseen sometimes even in absence of a formal requirement
to vote. In [[Italy]] the Constitution describes voting as a duty (art. 48), though electoral participation is not obligatory. From 1946 to 1992, thus, the Italian electoral law included light sanctions for non-voters (lists of non-voters were posted at polling stations).<ref>[http://i.unu.edu/media/publication/000/002/224/full_participation_web.pdf Sarah Birch], ''Full Participation. A comparative study of compulsory voting'', United Nations University, p.5</ref> Turnout rates have not declined substantially since 1992 in Italy, though, pointing to other factors than compulsory voting to explain high electoral participation.

====Salience====
Mark N. Franklin argues that salience, the perceived effect that an individual vote will have on how the country is run, has a significant effect on turnout. He presents [[Switzerland]] as an example of a nation with low salience. The nation's administration is highly decentralized, so that the federal government has limited powers. The government invariably consists of a coalition of parties, and the power wielded by a party is far more closely linked to its position relative to the coalition than to the number of votes it received. Important decisions are placed before the population in a [[referendum]]. Individual votes for the federal legislature are thus unlikely to have a significant effect on the nation, which probably explains the low average turnouts in that country. By contrast [[Malta]], with one of the world's highest voter turnouts, has a single legislature that holds a near monopoly on political power. Malta has a [[two-party system]] in which a small swing in votes can completely alter the executive.<ref>Mark N. Franklin. "Electoral Participation." in ''Controversies in Voting Behavior'' p. 87</ref> On the other hand, countries with a two party system can experience low turnout if large numbers of potential voters perceive little real difference between the main parties. Voters' perceptions of fairness also have an important effect on salience. If voters feel that the result of an election is more likely to be determined by fraud and corruption than by the will of the people, fewer people will vote.<ref>Richard S. Katz. Democracy and Elections. New York: Oxford University Press, 1997.</ref>

====Proportionality====
Another institutional factor that may have an important effect is proportionality, i.e., how closely the legislature reflects the views of the populace. A pure [[proportional representation]] system is fully proportional to the votes of the populace and a voter can be sure that he will be represented in parliament even if it is only the opposition bench; the only exception to this rule is for voters of parties that get less than a certain required percentage as a precondition to make it into parliament. Some countries have such [[electoral threshold]]s in place, e. g. 5% in Germany. By contrast, a [[Plurality electoral system|plurality system]] will almost always see districts in which one party is so dominant that there is little reason for voters of other parties to vote because votes for "losing" parties are in a sense lost.

Proportional systems tend to produce multiparty governments ([[coalition government]]s). This may reduce salience, since the voters have little influence over which parties are included in the coalition.<ref>Robert W. Jackman and Ross A. Miller. "Voter Turnout in the Industrial Democracies During the 1980s." in ''Elections and Voting Behaviour: New Challenges, New Perspectives.'' p. 308</ref> For instance, after the [[2005 German election]], the creation of the executive not only expressed the will of the voters of the majority party but also was the result of political deal-making. Although there is no guarantee, this is lessened as the parties usually state with whom they will favour a coalition after the elections.{{Citation needed|date=August 2009}}

Political scientists are divided on whether proportional representation increases voter turnout, though in countries with proportional representation voter turnout is higher.<ref>Katz p. 240</ref><ref>"Unequal Participation: Democracy's Unresolved Dilemma," in American Political Science Review (March 1997).</ref><ref>AU: ANDRÉ BLAIS
AU: R. K. CARTY
TI: Does proportional representation foster voter turnout?
SO: European Journal of Political Research
VL: 18
NO: 2
PG: 167–181
YR: 1990
ON: 1475-6765
PN: 0304-4130
AD: Université de Montréal, Canada;; University of British Columbia, Canada {{doi|10.1111/j.1475-6765.1990.tb00227.x}}</ref> There are other systems that attempt to preserve both salience and proportionality, for example, the [[Mixed member proportional representation]] system in [[New Zealand]] (in operation since 1996), in Germany and several other countries. However, these tend to be complex electoral systems, and in some cases complexity appears to suppress voter turnout.<ref>Powell "Thirty Democracies." p. 12</ref> The dual system in Germany, though, seems to have had no negative impact on voter turnout.

====Ease of voting====
Ease of voting is a factor in rates of turnout. In the United States and most Latin American nations, voters must go through separate [[voter registration]] procedures before they are allowed to vote. This two-step process quite clearly decreases turnout. U.S. states with no, or easier, registration requirements have larger turnouts.<ref>Richard G. Niemi and Herbert F. Weisberg. ''Controversies in Voting Behavior'' p. 31</ref> Other methods of improving turnout include making voting easier through more available [[absentee voting|absentee polling]] and improved access to polls, such as increasing the number of possible voting locations, lowering the average time voters have to spend waiting in line, or requiring companies to give workers some time off on voting day{{which| which nations enact these measures|date=March 2011}}. In some areas, generally those where some polling centres are relatively inaccessible, such as [[India]], elections often take several days. Some countries have considered [[internet voting]] as a possible solution. In other countries, like [[France]], voting is held on the weekend, when most voters are away from work. Therefore, the need for time off from work as a factor in voter turnout is greatly reduced.

Many countries have looked into internet voting as a possible solution for low voter turnout. Some countries like France and Switzerland use internet voting. However, it has only been used sparingly by a few states in the US. This is due largely to security concerns, although the US Department of Defense has been looking into making internet voting secure. The idea would be that voter turnout would increase because people could cast their vote from the comfort of their own homes, although the few experiments with internet voting have produced mixed results.<ref>{{cite web|title=Voting Drops 83 Percent In All-Digital Election|url=http://www.kitv.com/Voting-Drops-83-Percent-In-All-Digital-Election/-/8906042/5389636/-/8fij32z/-/index.html|work=26 May 2009|publisher=KITV News|accessdate=2 September 2013}}</ref>

====Voter fatigue====
{{Main|Voter fatigue}}
Voter fatigue can lower turnout. If there are many elections in close succession, voter turnout will decrease as the public tires of participating. In low-turnout Switzerland, the average voter is invited to go to the polls an average of seven times a year; the United States has frequent elections, with two votes per year on average, if one includes all levels of government as well as [[primary election|primaries]].<ref>Franklin "Electoral Participation." p. 98</ref> Holding multiple elections at the same time can increase turnout; however, presenting voters with massive multipage ballots, as occurs in some parts of the United States, can reduce turnouts.<ref>Arend Lijphart. "[http://www.cuhk.edu.hk/gpa/wang_files/Dem15.pdf Unequal Participation: Democracy's Unresolved Dilemma]." ''American Political Science Review.''</ref>

====Measuring turnout====
Differing methods of measuring voter turnout can contribute to reported differences between nations. There are difficulties in measuring both the numerator, the number of voters who cast votes, and the denominator, the number of voters eligible to vote.

For the numerator, it is often assumed that the number of voters who went to the polls should equal the number of ballots cast, which in turn should equal the number of votes counted, but this is not the case. Not all voters who arrive at the polls necessarily cast ballots. Some may be turned away because they are ineligible, some may be turned away improperly, and some who sign the voting register may not actually cast ballots. Furthermore, voters who do cast ballots may abstain, deliberately voting for nobody, or they may [[spoiled ballot|spoil]] their votes, either accidentally or as an act of protest.

In the United Kingdom, the [[Electoral Commission (United Kingdom)|Electoral Commission]] distinguishes between "valid vote turnout", which excludes spoilt ballots, and "ballot box turnout", which does not.

In the United States, it has been common to report turnout as the sum of votes for the top race on the ballot, because not all jurisdictions report the actual number of people who went to the polls nor the number of undervotes or overvotes.<ref name="eac.gov">Kimball W. Brace, [http://www.eac.gov/News/meetings/050504/ploneexfile.2006-04-18.4617096900/attachment_download/file Overview of Voting Equipment Usage in United States, Direct Recording Electronic (DRE) Voting], statement to the [[Election Assistance Commission]], May 5, 2004.</ref> Overvote rates of around 0.3 percent are typical of well-run elections, but in Gadsden County Florida, the overvote rate was 11 percent in November 2000.<ref name="Human Factors in Voting Technology">[[Douglas W. Jones]], [http://www.cs.uiowa.edu/~jones/voting/cogel/ Human Factors in Voting Technology], presentation to the Council on Governmental Ethics Laws September 29, 2002, Ottawa Canada.</ref>

For the denominator, it is often assumed that the number of eligible voters was well defined, but again, this is not the case. In the United States, for example, there is no accurate registry of exactly who is eligible to vote, since only about 70–75% of people choose to register themselves.<ref>Katz p. 239</ref> Thus, turnout has to be calculated based on population estimates. Some political scientists have argued that these measures do not properly account for the large number of [[Alien (law)|illegal aliens]], disenfranchised [[felony|felon]]s and persons who are considered 'mentally incompetent' in the United States, and that American voter turnout is higher than is normally reported.<ref>Niemi and Weisberg "Introduction." ''Controversies in Voting Behavior.'' p. 25</ref> Professor Michael P. McDonald constructed an estimation of the turnout against the '[[voting eligible population]]' (VEP), instead of the '[[voting age population]]' (VAP). For the American presidential elections of 2004, turnout could then be expressed as 60.32% of VEP, rather than 55.27% of VAP.{{Dead link|date=August 2009}}<ref>{{Dead link|date=August 2009}}McDonald "2004 Voting-Age and Voting-Eligible Population Estimates and Voter Turnout" http://elections.gmu.edu/Voter_Turnout_2004.htm</ref>

In New Zealand, registration is supposed to be universal. This does not eliminate uncertainty in the eligible population because this system has been shown to be unreliable, with a large number of eligible but unregistered citizens, creating inflated turnout figures.<ref>Katz p. 334</ref>

A second problem with turnout measurements lies in the way turnout is computed. One can count the number of voters, or one can count the number of ballots, and in a vote-for-one race, one can sum the number of votes for each candidate. These are not necessarily identical because not all voters who sign in at the polls necessarily cast ballots, although they ought to, and because voters may cast [[spoiled ballot|spoil]] their votes.

==Trends of decreasing turnout==
[[File:Turnout.png|right|frame|Change in voter turnout over time for five selected countries]]
Over the last 40 years, voter turnout has been steadily declining in the established democracies.<ref name="Niemi and Weisberg p. 31"/> This trend has been significant in the United States, Western Europe, Japan and Latin America. It has been a matter of concern and controversy among political scientists for several decades. During this same period, other forms of political participation have also declined, such as voluntary participation in political parties and the attendance of observers at town meetings. The decline in voting has also accompanied a general decline in civic participation, such as church attendance, membership in professional, fraternal, and student societies, youth groups, and parent-teacher associations.<ref>Robert D. Putnam "Tuning In, Tuning Out: The Strange Disappearance of Social Capital in America." in ''Controversies in Voting Behavior'' p. 40</ref> At the same time, some forms of participation have increased. People have become far more likely to participate in [[boycott]]s, [[protest|demonstration]]s, and to donate to political campaigns.<ref>Niemi and Weisberg. p. 30</ref>

Before the late 20th century, [[suffrage]] — the right to vote — was so limited in most nations that turnout figures have little relevance to today. One exception was the United States, which had near universal white male suffrage by 1840. The U.S. saw a steady rise in voter turnout during the century, reaching its peak in the years after the [[American Civil War|Civil War]]. Turnout declined from the 1890s until the 1930s, then increased again until 1960 before beginning its current long decline.<ref>Walter Dean Burnham. "The Appearance and Disappearance of the American Voter."</ref> In Europe, voter turnouts steadily increased from the introduction of universal suffrage before peaking in the mid-to-late 1960s, with modest declines since then. These declines have been smaller than those in the United States, and in some European countries turnout have remained stable and even slightly increased. Globally, voter turnout has decreased by about five percentage points over the last four decades.<ref>Lijphart p. 6</ref>

===Reasons for decline===
Many causes have been proposed for this decline; a combination of factors is most likely. When asked why they do not vote, many people report that they have too little free time. However, over the last several decades, studies have consistently shown that the amount of [[Free time|leisure time]] has not decreased.{{Citation needed|date=July 2009}} The perception that one is busier is common, and might be just as important as a real decrease in leisure time. Geographic mobility has increased over the last few decades. There are often barriers to voting in a district where one is a recent arrival, and a new arrival is likely to know little about the local candidate and local issues. [[Francis Fukuyama]] has blamed the [[welfare state]], arguing that the decrease in turnout has come shortly after the government became far more involved in people's lives. He argues in ''Trust: The Social Virtues and The Creation of Prosperity'' that the [[social capital]] essential to high voter turnouts is easily dissipated by government actions. However, on an international level those states with the most extensive social programs tend to be the ones with the highest turnouts. Richard Sclove argues, in ''Democracy and Technology,'' that technological developments in society such as "automobilization," suburban living, and "an explosive proliferation of home entertainment devices" have contributed to a loss of community, which in turn has weakened participation in civic life.<ref>Sclove p. 241</ref>{{Nonspecific|date=September 2009}}

'''Trust''' in government and in politicians has decreased in many nations. However, the first signs of decreasing voter turnout occurred in the early 1960s, which was before the major upheavals of the late 1960s and 1970s. [[Robert D. Putnam]] argues that the collapse in civil engagement is due to the introduction of television. In the 1950s and 1960s, television quickly became the main leisure activity in developed nations. It replaced earlier more social entertainments such as bridge clubs, church groups, and bowling leagues. Putnam argues that as people retreated within their homes and general social participation declined so too did voting.<ref>Putnam p. 61</ref> Rosenstone and Hansen contend that the decline in turnout is the product of a change in campaigning strategies as a result of the so-called new media. Before the introduction of television, almost all of a party's resources would be directed towards intensive local campaigning and [[get out the vote]] initiatives. In the modern era, these resources have been redirected to expensive media campaigns in which the potential voter is a passive participant.<ref>Steven J. Rosenstone and John Mark Hansen. "Solving the Puzzle of Participation in Electoral Politics." p. 73</ref> During the same period, [[negative campaigning]] has become ubiquitous in the United States and elsewhere and has been shown to impact voter turnout.<ref>Yanna Krupnikov. ""[http://onlinelibrary.wiley.com/doi/10.1111/j.1540-5907.2011.00522.x/full When Does Negativity Demobilize? Tracing the Conditional Effect of Negative Campaigning on Voter Turnout]." American Journal of Political Science, Volume 55, Issue 4, pages 797–813, October 2011.</ref> [[Attack ad]]s and smear campaigns give voters a negative impression of the entire political process. The evidence for this is mixed: elections involving highly unpopular incumbents generally have high turnout; some studies have found that mudslinging and character attacks reduce turnout, but that substantive attacks on a party's record can increase it.<ref>Niemi and Weisberg p. 30.</ref>

The decline in voter turnout is almost wholly concentrated among non-seniors.{{Citation needed|date=August 2009}} Those who began voting prior to 1960 maintain the same high turnout rates of that era. For each subsequent generation, starting with the one that came of age in the 1960s, turnout has steadily declined. Recent programs to increase the rates of voting among young people—such as MTV's "[[Rock the Vote]]" and the "[[Vote or Die]]" initiatives in the United States—may have marginally increased turnouts of those between the ages of 18 and 25 to vote.{{Failed verification|date=August 2009}}<ref>{{Failed verification|date=August 2009}}Eisner, Jane. "[http://www.centredaily.com/mld/centredaily/news/opinion/11872741.htm Rock the Vote, now 15, eager to help drive policy.]" ''Philadelphia Inquirer'' 12 June 2005. 12 July 2005</ref> A number of governments and [[Election management body|electoral commissions]] have also launched efforts to boost turnout. For instance [[Elections Canada]] has launched mass media campaigns to encourage voting prior to elections, as have bodies in Taiwan and the United Kingdom.

===Ineligibility===
Much of the above analysis is predicated on voter turnout as measured as a percentage of the voting-age population. In a 2001 article in the [[American Political Science Review]], Michael McDonald and Samuel Popkin argued, that at least in the United States, voter turnout since 1972 has not actually declined when calculated for those eligible to vote, what they term the voting-eligible population.<ref>Michael McDonald and Samual Popkin. "The Myth of the Vanishing Voter" in ''American Political Science Review.''</ref> In 1972, noncitizens and ineligible felons (depending on state law) constituted about 2% of the voting-age population. By 2004, ineligible voters constituted nearly 10%. Ineligible voters are not evenly distributed across the country – 20% of California's voting-age population is ineligible to vote – which confounds comparisons of states. Furthermore, they argue that an examination of the Census Bureau's Current Population Survey shows that turnout is low but not declining among the youth, when the high youth turnout of 1972 (the first year 18–20 year olds were eligible to vote in most states) is removed from the trendline.

==Notes==
{{reflist|colwidth=30em}}

==References==
*Franklin, Mark N. "Electoral Engineering and Cross National Turnout Differences." ''British Journal of Political Science.'' 1999
*Kanazawa, Satoshi. "A Possible Solution to the Paradox of Voter Turnout." ''The Journal of Politics.''
*Lijphart, Arend. "Unequal Participation: Democracy's Unresolved Dilemma." ''American Political Science Review.'' vol. 91 (March 1997): 1–14. p. 12
*McDonald, Michael and Samuel Popkin. "The Myth of the Vanishing Voter." ''American Political Science Review.'' 2001.
*Niemi, Richard G. and Herbert F. Weisberg. eds. ''Controversies in Voting Behavior.'' Washington, D.C: CQ Press, 2001.
*Norris, Pippa. ''Elections and Voting Behaviour: New Challenges, New Perspectives.'' Aldershot: Ashgate, Dartmouth, 1998.
*Rose, Richard, ed. ''Electoral Participation: A Comparative Analysis.'' Beverly Hills: Sage Publications, 1980.
*Wolfinger, Raymond E. and Steven J. Rosenstone. 1980. ''Who Votes?'' New Haven, CT: Yale University Press.
*Wolfinger, R., Glass, D., Squire, P.(1990). Predictors of electoral turnout:an international comparison. Policy Studies Review, 9(3), p551–574, 24p
*Highton, B. (1997, May). Easy registration and voter turnout. The Journal of Politics, 59(2), pp. 565–575.

==Further reading==
''alphabetical by title and work''
*{{cite news | author=Charles Q. Choi | title=The Genetics of Politics | url= | format=Print | work=[[Scientific American]]
| publisher=Scientific American, Inc. | pages=18, 21 | date=November 2007 | accessdate=2008-06-26
| quote= ...the desire to vote or abstain from politics might largely be hardwired into our biology }} 
*{{cite web | url=http://elections.lib.tufts.edu/aas_portal/index.xq
| title=A New Nation Votes: American Elections Returns 1787–1825 | accessdate=2008-06-24 | work=Digital Collections and Archives
| date=2008-05-29 | publisher=[[Tufts University]] | author=Philip Lampi
| quote=A New Nation Votes is a searchable collection of election returns from the earliest years of American democracy. }}
*{{cite web | url=http://makeitanissue.org.uk/devlog/2007/01/the_power_commission_was_estab.php | title=The Power Report | accessdate=2008-06-24 | work=makeitanissue.org.uk | date=2007-01-19 | publisher=The Power Inquiry
| quote=The Power Commission was established to discover what is happening to our democracy. It sought to establish why people were disengaging from formal democratic politics in Britain and how these trends could be reversed. }}
*{{cite web | url=http://www.electionguide.org/voter-turnout.php | title=Voter Turnout | accessdate=2008-06-24
| work=ElectionGuide | publisher=[[International Foundation for Electoral Systems]]
| quote=...ElectionGuide is the most comprehensive and timely source of verified election information and results available online. }}
*{{cite web | url=http://www.fairvote.org/?page=262 | title=Voter Turnout | accessdate=2008-06-24 | work=[[FairVote]]
| publisher=Voting and Democracy Research Center
| quote=Voter Turnout is a fundamental quality of fair elections and is generally considered to be a necessary factor for a healthy democracy.}}
*{{cite web | url=http://www.idea.int/vt | title=Voter Turnout | accessdate=2008-06-23 | work=International IDEA website
| date=2008-06-16 | publisher=[[International Institute for Democracy and Electoral Assistance]]
| quote=The International IDEA Voter Turnout Website contains the most comprehensive global collection of political participation statistics available. }}
*{{cite web | url=http://elections.gmu.edu/voter_turnout.htm | title=Voter Turnout | author=Michael McDonald
| accessdate=2008-06-24 | work=United States Elections Project | date=2008-04-01
| quote=Statistics on voter turnout presented here show that the much-lamented decline in voter participation is an artifact of the way in which it is measured. }}
*{{cite web | url=http://www.mapleleafweb.com/features/voter-turnout-canada | title=Voter Turnout in Canada | accessdate=2008-06-23
| work=Maple Leaf Web | date=2007-03-01 | author=Rhonda Parkinson
| quote=Since the 1980s, voter turnout in federal elections has fallen sharply. }}

{{DEFAULTSORT:Voter Turnout}}
[[Category:Elections]]

{{Link GA|zh}}

Color wheel graphs of complex functions

2014-01-19T18:18:15Z

71.167.61.80:

The '''random walker algorithm''' is an algorithm for [[image segmentation]]. In the first description of the algorithm,<ref name="grady2006random">L. Grady: [http://www.cns.bu.edu/~lgrady/grady2006random.pdf Random Walks for Image Segmentation], IEEE Trans. on Pattern Analysis and Machine Intelligence, Vol. 28, No. 11, pp. 1768–1783, Nov., 2006.</ref> a user interactively labels a small number of pixels with known labels (called seeds), e.g., "object" and "background". The unlabeled pixels are each imagined to release a random walker, and the probability is computed that each pixel's random walker first arrives at a seed bearing each label, i.e., if a user places K seeds, each with a different label, then it is necessary to compute, for each pixel, the probability that a random walker leaving the pixel will first arrive at each seed. This computation may be determined analytically by solving a system of linear equations. After computing these probabilities for each pixel, the pixel is assigned to the label for which it is most likely to send a random walker. The image is modeled as a [[Graph (mathematics)|graph]], in which each pixel corresponds to a node which is connected to neighboring pixels by edges, and the edges are weighted to reflect the similarity between the pixels. Therefore, the random walk occurs on the weighted graph (see Doyle and Snell for an introduction to random walks on graphs<ref>P. Doyle, J. L. Snell: Random Walks and Electric Networks, Mathematical Association of America, 1984</ref>).

Although the initial algorithm was formulated as an interactive method for image segmentation, it has been extended to be a fully automatic algorithm, given a data fidelity term (e.g., an intensity prior).<ref name="grady2005multilabel">Leo Grady: Multilabel Random Walker Image Segmentation Using Prior Models, Proc. of CVPR, Vol. 1, pp. 763–770, 2005. [http://www.cns.bu.edu/%7Elgrady/grady2005multilabel.pdf]</ref> It has also been extended to other applications, such as Image Matching (R. Shen, I. Cheng, X.li and A. Basu), ICPR 2008, and Image Fusion, (R. Shen, I. Cheng, J.Shi and A. Basu), IEEE Trans. on Image Processing, 2011, and other applications.

The algorithm was initially published as a conference paper<ref>Leo Grady, Gareth Funka-Lea: Multi-Label Image Segmentation for Medical Applications Based on Graph-Theoretic Electrical Potentials, Proc. of the 8th ECCV Workshop on Computer Vision Approaches to Medical Image Analysis and Mathematical Methods in Biomedical Image Analysis, pp. 230–245, 2004.</ref> and later as a journal paper.<ref name="grady2006random" />

==Mathematics==

Although the algorithm was described in terms of random walks, the probability that each node sends a random walker to the seeds may be calculated analytically by solving a sparse, positive-definite system of linear equations with the graph [[Laplacian matrix of a graph|Laplacian matrix]], which we may represent with the variable <math>L</math>. The algorithm was shown to apply to an arbitrary number of labels (objects), but the exposition here is in terms of two labels (for simplicity of exposition).

Assume that the image is represented by a [[Graph (mathematics)|graph]], with each node <math>v_i</math> associated with a pixel and each edge <math>e_{ij}</math> connecting neighboring pixels <math>v_i</math> and <math>v_j</math>. The edge weights are used to encode node similarity, which may be derived from differences in image intensity, color, texture or any other meaningful features. For example, using image intensity <math>g_i</math> at node <math>v_i</math>, it is common to use the edge weighting function

:<math>w_{ij} = \exp{\left(-\beta (g_i - g_j)^2\right)}.
</math>

The nodes, edges and weights can then be used to construct the graph [[Laplacian matrix of a graph|Laplacian matrix]].

The random walker algorithm optimizes the energy
:<math>Q(x) = x^T L x = \sum_{e_{ij}} w_{ij} \left(x_i - x_j\right)^2</math>
where <math>x_i</math> represents a real-valued variable associated with each node in the graph and the optimization is constrained by <math>x_i = 1</math> for <math>v_i \in F</math> and <math>x_i = 0</math> for <math>v_i \in B</math>, where <math>F</math> and <math>B</math> represent the sets of foreground and background seeds, respectively. If we let <math>S</math> represent the set of nodes which are seeded (i.e., <math>S = F \cup B</math>) and <math>\overline{S}</math> represent the set of unseeded nodes (i.e., <math>S \cup \overline{S} = V</math> where <math>V</math> is the set of all nodes), then the optimum of the energy minimization problem is given by the solution to
:<math>
L_{\overline{S},\overline{S}} x_{\overline{S}} = - L_{\overline{S},S} x_{S},
</math>
where the subscripts are used to indicate the portion of the graph Laplacian matrix <math>L</math> indexed by the respective sets.

To incorporate likelihood (unary) terms into the algorithm, it was shown in <ref name="grady2005multilabel" /> that one may optimize the energy
:<math>Q(x) = x^T L x + \gamma \left((1-x)^T F (1-x) + x^T B x\right) = \sum_{e_{ij}} w_{ij} \left(x_i - x_j\right)^2 + \gamma \left(\sum_{v_i} f_i (1-x_i)^2 + \sum_{v_i} b_i x_i^2 \right),</math>
for positive, diagonal matrices <math>F</math> and <math>B</math>. Optimizing this energy leads to the system of linear equations
:<math>
\left(L_{\overline{S},\overline{S}} + \gamma F_{\overline{S},\overline{S}} + \gamma B_{\overline{S},\overline{S}}\right) x_{\overline{S}} = - L_{\overline{S},S} x_{S} - \gamma F_{\overline{S},\overline{S}}.
</math>
The set of seeded nodes, <math>S</math>, may be empty in this case (i.e., <math>\overline{S}=V</math>), but the presence of the positive diagonal matrices allows for a unique solution to this linear system.

For example, if the likelihood/unary terms are used to incorporate a color model of the object, then <math>f_i</math> would represent the confidence that the color at node <math>v_i</math> would belong to object (i.e., a larger value of <math>f_i</math> indicates greater confidence that <math>v_i</math> belonged to the object label) and <math>b_i</math> would represent the confidence that the color at node <math>v_i</math> belongs to the background.

==Algorithm interpretations==

The random walker algorithm was initially motivated by labeling a pixel as object/background based on the probability that a random walker dropped at the pixel would first reach an object (foreground) seed or a background seed. However, there are several other interpretations of this same algorithm which have appeared in.<ref name="grady2006random" />

===Circuit theory interpretations===

There are well-known connections between [[electrical circuit]] theory and random walks on graphs.<ref>P. G. Doyle, J. L. Snell: Random Walks and Electrical Networks, Carus Mathematical Monographs, 1984</ref> Consequently, the random walker algorithm has two different interpretations in terms of an electric circuit. In both cases, the graph is viewed as an electric circuit in which each edge is replaced by a passive linear [[resistor]]. The resistance, <math>r_{ij}</math>, associated with edge <math>e_{ij}</math> is set equal to <math>r_{ij} = \frac{1}{w_{ij}}</math> (i.e., the edge weight equals [[electrical conductance]]).

In the first interpretation, each node associated with a background seed, <math>v_i \in B</math>, is tied directly to [[Ground (electricity)|ground]] while each node associated with an object/foreground seed, <math>v_i \in F</math> is attached to a unit [[direct current]] ideal [[voltage source]] tied to ground (i.e., to establish a unit potential at each <math>v_i \in F</math>). The steady-state electrical circuit potentials established at each node by this circuit configuration will exactly equal the random walker probabilities. Specifically, the electrical potential, <math>x_i</math> at node <math>v_i</math> will equal the probability that a random walker dropped at node <math>v_i</math> will reach an object/foreground node before reaching a background node.

In the second interpretation, labeling a node as object or background by thresholding the random walker probability at 0.5 is equivalent to labeling a node as object or background based on the relative effective conductance between the node and the object or background seeds. Specifically, if a node has a higher effective conductance (lower effective resistance) to the object seeds than to the background seeds, then node is labeled as object. If a node has a higher effective conductance (lower effective resistance) to the background seeds than to the object seeds, then node is labeled as background.

==Extensions==

The traditional random walker algorithm described above has been extended in several ways:

* Random walks with restart<ref>T. H. Kim, K. M. Lee, S. U. Lee: Generative Image Segmentation Using Random Walks with Restart, Proc. of ECCV 2008, pp. 264–275</ref>
* Alpha matting<ref>J. Wang, M. Agrawala, M. F. Cohen: Soft scissors: an interactive tool for realtime high quality matting, Proc. of SIGGRAPH 2007</ref>
* Threshold selection<ref>S. Rysavy, A. Flores, R. Enciso, K. Okada: Classifiability Criteria for Refining of Random Walks Segmentation, Proc. of ICPR 2008</ref>
* Soft inputs<ref>W. Yang, J. Cai, J. Zheng, J. Luo: User-friendly Interactive Image Segmentation through Unified Combinatorial User Inputs, IEEE Trans. on Image Proc., 2010</ref>
* Run on a presegmented image<ref>C. Chefd'hotel, A. Sebbane: Random walk and front propagation on watershed adjacency graphs for multilabel image segmentation, Proc. of ICV 2007</ref>
* Scale space random walk<ref>R. Rzeszutek, T. El-Maraghi, D. Androutsos: Image segmentation using scale-space random walks, Proc. of the 16th international conference on Digital Signal Processing, pp. 458–461, 2009</ref>
* Fast random walker using offline [[precomputation]] <ref>L. Grady, A.K. Sinop: Fast approximate random walker segmentation using eigenvector
precomputation. In IEEE Conf. CVPR, pp. 1–8, 2008</ref><ref>S. Andrews, G. Hamarneh, A. Saad. Fast random walker with priors using precomputation for interactive medical image segmentation, Proc. of MICCAI 2010</ref>

==Applications==

Beyond image segmentation, the random walker algorithm has been additionally applied to several problems in computer vision and graphics:

* Image Colorization<ref>X. Liu, J. Liu, Z. Feng: Colorization Using Segmentation with Random Walk, Computer Analysis of Images and Patterns, pp. 468–475, 2009</ref>
* Interactive rotoscoping<ref>R. Rzeszutek, T. El-Maraghi, D. Androutsos: Interactive rotoscoping through scale-space random walks, Proc. of the 2009 IEEE international conference on Multimedia and Expo</ref>
* Medical image segmentation<ref>S. P. Dakua, J. S. Sahambi: LV Contour Extraction from Cardiac MR
Images Using Random Walks Approach, Int. Journal of Recent Trends in Engineering, Vol 1, No. 3, May 2009</ref><ref>F. Maier, A. Wimmer, G. Soza, J. N. Kaftan, D. Fritz, R. Dillmann: Automatic Liver Segmentation Using the Random Walker Algorithm, Bildverarbeitung für die Medizin 2008</ref><ref>P. Wighton, M. Sadeghi, T. K. Lee, M. S. Atkins: A Fully Automatic Random Walker Segmentation for Skin Lesions in a Supervised Setting, Proc. of MICCAI 2009</ref>
* Merging multiple segmentations<ref>P. Wattuya, K. Rothaus, J. S. Prassni, X. Jiang: A random walker based approach to combining multiple segmentations, Proc. of ICPR 2008</ref>
* Mesh segmentation<ref>Y.-K. Lai, S.-M. Hu, R. R. Martin, P. L. Rosin: Fast mesh segmentation using random walks, Proc. of the 2008 ACM symposium on Solid and physical modeling</ref><ref>J. Zhang, J. Zheng, J. Cai: Interactive Mesh Cutting Using Constrained Random Walks, IEEE Trans. on Visualization and Computer Graphics, 2010.</ref>
* Mesh denoising<ref>X. Sun, P. L. Rosin, R. R. Martin, F. C. Langbein: Random walks for feature-preserving mesh denoising, Computer Aided Geometric Design, Vol. 25, No. 7, Oct. 2008, pp. 437–456</ref>
* Segmentation editing<ref>L. Grady, G. Funka-Lea: An Energy Minimization Approach to the Data Driven Editing of Presegmented Images/Volumes, Proc. of MICCAI, Vol. 2, 2006, pp. 888–895</ref>
* Shadow elimination<ref>G. Li, L. Qingsheng, Q. Xiaoxu: Moving Vehicle Shadow Elimination Based on Random Walk and Edge Features, Proc. of IITA 2008</ref>
* Image matching<ref>R. Shen, I. Cheng, X. Li, A. Basu: Stereo matching using random walks, Proc. of ICPR 2008</ref>
* Image Fusion<ref>R. Shen, I. Cheng, J. Shi, A. Basu: Generalized Random Walks for Fusion of Multi-exposure Images, IEEE Trans. on Image Processing, 2011.</ref>

==References==

{{Reflist}}


==External links==
*[http://www.cns.bu.edu/~lgrady/random_walker_matlab_code.zip Matlab code implementing the original random walker algorithm]
*[http://fastrw.cs.sfu.ca/ Matlab code implementing the random walker algorithm with precomputation]
*[http://scikits-image.org/docs/dev/auto_examples/plot_random_walker_segmentation.html Python implementation of the original random walker algorithm] in the image processing toolbox [http://scikits-image.org/ scikits-image]

[[Category:Image segmentation]]

Electronic filter topology

2014-01-17T22:23:21Z

71.167.61.77:

'''Tarski–Grothendieck set theory''' ('''TG''', named after mathematicians [[Alfred Tarski]] and [[Alexander Grothendieck]]) is an [[axiomatic set theory]] that was introduced as part of the [[Mizar system]] for formal verification of proofs.

Tarski–Grothendieck set theory is a [[non-conservative extension]] of [[Zermelo–Fraenkel set theory]] (ZFC) and is distinguished from other axiomatic set theories by the inclusion of '''Tarski's axiom''' which states that for each set there is a [[Grothendieck universe]] it belongs to (see below). Tarski's axiom implies the existence of [[inaccessible cardinal]]s, providing a richer [[ontology]] than that of conventional set theories such as ZFC.

==Axioms==

While the [[axiom]]s and [[definition]]s defining Mizar's basic objects and processes are fully [[Formal system|formal]], they are described informally below.

* Given any set <math>A</math>, the singleton <math>\{A\}</math> exists.
* Given any two sets, their unordered and ordered pairs exist.
* Given any family of sets, its union exists.

'''TG''' includes the following axioms, which are conventional because also part of [[ZFC]]:
* Set axiom: Quantified variables range over sets alone; everything is a set (the same [[ontology]] as [[ZFC]]).
* [[Extensionality]] axiom: Two sets are identical if they have the same members.
* [[Axiom of regularity]]: No set is a member of itself, and circular chains of membership are impossible.
* [[Axiom schema of replacement]]: Let the [[domain (mathematics)|domain]] of the [[function (mathematics)|function]] <math>F</math> be the set <math>A</math>. Then the [[range (mathematics)|range]] of <math>F</math> (the values of <math>F(x)</math> for all members <math>x</math> of <math>A</math>) is also a set.

It is Tarski's axiom that distinguishes '''TG''' from other axiomatic set theories. Tarski's axiom also implies the axioms of [[axiom of infinity|infinity]], [[axiom of choice|choice]],<ref>Tarski (1938)</ref><ref>http://mmlquery.mizar.org/mml/current/wellord2.html#T26</ref> and [[axiom of power set|power set]].<ref>Robert Solovay, [http://www.cs.nyu.edu/pipermail/fom/2008-March/012783.html Re: AC and strongly inaccessible cardinals].</ref><ref>[http://us.metamath.org/mpegif/grothpw.html Metamath '''grothpw'''.]</ref> It also implies the existence of [[inaccessible cardinal]]s, thanks to which the [[ontology]] of '''TG''' is much richer than that of conventional set theories such as [[ZFC]].

* '''Tarski's axiom''' (adapted from Tarski 1939<ref>Tarski (1939)</ref>). For every set <math>x</math>, there exists a set <math>y</math> whose members include:

- <math>x</math> itself;

- every subset of every member of <math>y</math>;

- the power set of every member of <math>y</math>;

- every subset of <math>y</math> of [[cardinality]] less than that of <math>y</math>.

More formally:

:<math>\exists y [x\in y \wedge \forall z\in y(\mathcal P(z)\subseteq y\wedge\mathcal P(z)\in y) \wedge \forall z\in\mathcal P(y)(\neg z\approx y\to z\in y)]</math>

where "<math>\mathcal P(x)</math>" denotes the power class of ''x'' and "<math>\approx</math>" denotes [[equinumerosity]]. What Tarski's axiom states (in the vernacular) for each set <math>x</math> there is a [[Grothendieck universe]] it belongs to.

==Implementation in the Mizar system==

The Mizar language, underlying the implementation of '''TG''' and providing its logical syntax, is typed and the types are assumed to be non-empty. Hence, the theory is implicitly taken to be [[Axiom of empty set|non-empty]]. The existence axioms, e.g. the existence of the unordered pair, is also implemented indirectly by the definition of term constructors.

The system includes equality, the membership predicate and the following standard definitions:
* [[Singleton (mathematics)|Singleton]]: A set with one member;
* [[Unordered pair]]: A set with two distinct members. <math>\{a,b\} = \{b,a\}</math>;
* [[Ordered pair]]: The set <math>\{\{a,b\},\{a\}\} = (a,b) \neq (b,a)</math>;
* [[Subset]]: A set all of whose members are members of another given set;
* The [[union (set theory)|union]] of a family of sets <math>Y</math>: The set of all members of every member of <math>Y</math>.

==See also==
*[[Mizar system]]
*[[Grothendieck universe]]
*[[Axiom of limitation of size]]

==Notes==
<references/>

==References==
*Andreas Blass, I.M. Dimitriou, and [[Benedikt Löwe]] (2007) "[http://dare.uva.nl/document/25381 Inaccessible Cardinals without the Axiom of Choice,]" ''Fundamenta Mathematicae'' 194: 179-89.
* {{cite conference
| first = Nicolas
| last = Bourbaki
| authorlink = Nicolas Bourbaki
| year = 1972
| title = Univers
| booktitle = Séminaire de Géométrie Algébrique du Bois Marie – 1963-64 – Théorie des topos et cohomologie étale des schémas – (SGA 4) – vol. 1 (Lecture notes in mathematics '''269''')
| editor = [[Michael Artin]], [[Alexandre Grothendieck]], [[Jean-Louis Verdier]], eds.
| publisher = [[Springer Science+Business Media|Springer-Verlag]]
| location = Berlin; New York
| language = French
| pages = 185–217
| url = http://modular.fas.harvard.edu/sga/sga/4-1/4-1t_185.html
}}
* [[Patrick Suppes]] (1960) ''Axiomatic Set Theory''. Van Nostrand. Dover reprint, 1972.
* {{cite journal
| last = Tarski
| first = Alfred
| authorlink = Alfred Tarski
| year = 1938
| title = Über unerreichbare Kardinalzahlen
| journal = Fundamenta Mathematicae
| volume = 30
| pages = 68–89
| url = http://matwbn.icm.edu.pl/ksiazki/fm/fm30/fm30113.pdf
}}
* {{cite journal
| last = Tarski
| first = Alfred
| authorlink = Alfred Tarski
| year = 1939
| title = On the well-ordered subsets of any set
| journal = Fundamenta Mathematicae
| volume = 32
| pages = 176–183
| url = http://matwbn.icm.edu.pl/ksiazki/fm/fm32/fm32115.pdf
}}

==External links==
*Trybulec, Andrzej, 1989, "[http://mizar.uwb.edu.pl/JFM/Axiomatics/tarski.html Tarski–Grothendieck Set Theory]", ''Journal of Formalized Mathematics''.
*[[Metamath]]: "[http://us.metamath.org/mpegif/mmset.html Proof Explorer Home Page.]" Scroll down to "Grothendieck's Axiom."
* [[PlanetMath]]: "[http://planetmath.org/encyclopedia/TarskisAxiom.html Tarski's Axiom]"

{{DEFAULTSORT:Tarski-Grothendieck set theory}}
[[Category:Systems of set theory]]

Sallen–Key topology

2014-01-17T15:46:32Z

71.167.61.77: roman labels, subscript inside superscript

[[Image:Hot metalwork.jpg|250px|thumb|right|Thermal radiation in visible light can be seen on this hot metalwork. Thermal energy would ideally be the amount of heat required to warm the metal to its temperature, but this quantity is not well-defined, as there are many ways to obtain a given body at a given temperature, and each of them may require a different amount of total [[heat]] input. Thermal energy, unlike [[internal energy]], is therefore not a state function.]]
'''Thermal energy''' is the part of the total potential [[energy]] and kinetic energy of an [[Physical body|object]] or sample of matter that results in the system [[temperature]].<ref name=eb>[http://www.britannica.com/eb/article-9072068/thermal-energy Thermal energy entry in Britannica Online]</ref> It is represented by the variable Q, and can be measured in [[Joule]]s. This quantity may be difficult to determine or even meaningless unless the system has attained its temperature only through warming (heating), and not been subjected to work input or output, or any other energy-changing processes. Because the total amount of [[heat]] that enters an object is not a conserved quantity like mass or energy, and may be destroyed or created by many processes, the idea of an object's thermal energy or "heat content," something that remains a measureable and objective part of the [[internal energy]] of a body, cannot be strictly upheld. The idea of a thermal (part) of object internal energy is therefore useful only as an ideal model, in special cases where the total integrated energy of heat added or removed from a system happens to stay approximately constant as heat is conducted through the system.

The [[internal energy]] of a system, also often called the thermodynamic energy, includes other forms of energy in a thermodynamic system in addition to thermal energy, namely forms of [[potential energy]] that do not influence temperature and do not absorb heat, such as the [[chemical energy]] stored in its molecular structure and electronic configuration, and the nuclear [[binding energy]] that binds the sub-atomic particles of matter.

Microscopically, the thermal energy may include both the [[kinetic energy]] and [[potential energy]] of a system's constituent particles, which may be atoms, molecules, electrons, or particles in plasmas. It originates from the individually random, or disordered, motion of particles in a large ensemble, as consequence of absorbing [[heat]]. In ideal monatomic gases, thermal energy is entirely kinetic energy. In other substances, in cases where some of thermal energy is stored in atomic vibration, this vibrational part of the thermal energy is stored equally partitioned between potential energy of atomic vibration, and kinetic energy of atomic vibration. Thermal energy is thus [[equipartition theorem|equally partitioned]] between all available quadratic [[degrees of freedom (physics and chemistry)|degrees of freedom]] of the particles. As noted, these degrees of freedom may include pure translational motion in gases, in rotational states, and as potential and kinetic energy in [[normal mode]]s of vibrations in intermolecular or crystal [[lattice vibration]]s. In general, due to quantum mechanical reasons, the availability of any such degrees of freedom is a function of the energy in the system, and therefore depends on the temperature (see [[heat capacity]] for discussion of this phenomenon).

Macroscopically, the thermal energy of a system at a given temperature is related proportionally to its [[heat capacity]]. However, since the heat capacity differs according to whether or not constant volume or constant pressure is specified, or phase changes permitted, the heat capacity cannot be used define thermal energy unless it is done in such a way as to insure that only heat gain or loss (not work) makes any changes in the internal energy of the system. Usually, this means specifying the "constant volume heat capacity" of the system so that no work is done. Also the heat capacity of a system for such purposes must not include heat absorbed by any chemical reaction or process.

As noted, thermal energy is not a state function, or a property of a system, since the total thermal energy needed to warm a system to a given temperature depends on the path taken to attain the temperature, unless all forms of work and chemical potential change in the system are zero or negligible (in which case thermal energy is a subset of the internal energy). Thus, thermal energy is process-dependent except in systems in which processes to change internal energy other than heating, can be neglected. Nevertheless, when this is true, thermal energy and heat capacity may be a useful concept in the study of heat transfer in solids and liquids, in engineering and other disciplines.

== Differentiation from heat ==
[[Heat]], in the strict use in physics, is characteristic only of a process, i.e. it is absorbed or produced as an energy ''exchange'', always as a result of a temperature difference. Heat is thermal energy in the process of transfer or conversion across a boundary of one region of matter to another, as a result of a temperature difference.<ref name=speyer>{{cite book
|title=Thermal Analysis of Materials
|author=Robert F. Speyer
|publisher=Marcel Dekker, Inc.
|year=2012
|isbn=0-8247-8963-6
|series=Materials Engineering
|page=2
}}</ref> In engineering, the terms "heat" and "heat transfer" are thus used nearly interchangeably, since heat is always understood to be in the process of transfer. The energy transferred by heat is called by other terms (such as thermal energy or latent energy) when this energy is no longer in net transfer, and has become static.<ref name="Incrop">{{cite book
| author = Frank P. Incropera
| authorlink = Frank P. Incropera
| coauthors = David P. De Witt and D. P. Dewitt
| title = Fundamentals of Heat and Mass Transfer
| edition = 3rd
| publisher = [[John Wiley & Sons]]
| year = 1990
| page = 2
| isbn = 0-471-51729-1}} See box definition: "Heat transfer (or heat) is energy in transit due to a temperature difference." See page 14 for the definition of the thermal component of the thermodynamic [[internal energy]].</ref> Thus, heat is not a static property of matter. Matter does not contain heat, but rather thermal energy, and even the thermal energy is subject to transformations into and out of other types of energy, and so can be considered to be "conserved" only when these processes are small. The heat transfer rate or heating rate is the amount of energy per unit time being transferred as heat, or the heat [[power (physics)|power]].

When two thermodynamic systems with different temperatures are brought into diathermic contact, they spontaneously exchange energy as heat, the exchange being transfer of thermal energy from the system of higher temperature to the colder system. Heat may cause work to be performed on a system, for example, in form of volume or pressure changes. This work may be used in heat engines to convert thermal energy into other forms of energy. When two systems have reached a [[thermodynamic equilibrium]], they have attained the same exact temperature and the net exchange of thermal energy vanishes, and heat flow ceases.

==Definitions==
Thermal energy is the portion of the thermodynamic or internal energy of a system that is responsible for the temperature of the system.<ref name=eb /><ref name=speyer /> The thermal energy of a system scales with its size and is therefore an [[extensive property]]. It is not a [[state function]] of the system unless the system has been constructed so that all changes in internal energy are due to changes in thermal energy, as a result of heat transfer (not work). Otherwise thermal energy is dependent on the way or method by which the system attained its temperature.

From a macroscopic thermodynamic description, the thermal energy of a system is given by its constant volume specific [[heat capacity]] ''C(T)'', a temperature coefficient also called thermal capacity, at any given [[absolute temperature]] (''T''):
:<math>U_{thermal} = C(T) \cdot T.</math>
The heat capacity is a function of temperature itself, and is typically measured and specified for certain standard conditions and a specific [[amount of substance]] (molar heat capacity) or [[mass]] units (specific heat capacity). At constant volume (''V''), ''C''V it is the temperature coefficient of energy.<ref name=fuchs>{{cite book
|title=The Dynamics of Heat: A Unified Approach to Thermodynamics and Heat Transfer
|author=Hans U. Fuchs
|edition=2
|publisher=Springer
|year=2010
|isbn=978-1-4419-7603-1
|page=211
}}</ref> In practice, given a narrow temperature range, for example the operational range of a heat engine, the heat capacity of a system is often constant, and thus thermal energy changes are conveniently measured as temperature fluctuations in the system.

In the microscopical description of [[statistical physics]], the thermal energy is identified with the mechanical kinetic energy of the constituent particles or other forms of kinetic energy associated with quantum-mechanical [[Microstate (statistical mechanics)|microstates]].

The distinguishing difference between the terms ''kinetic energy'' and ''thermal energy'' is that thermal energy is the ''mean'' energy of disordered, i.e. random, motion of the particles or the oscillations in the system. The conversion of energy of ordered motion to thermal energy results from collisions.<ref name=blundell>{{cite book
|author=S. Blundell, K. Blundell
|title=Concepts in Thermal Physics
|year=2006
|publisher=Oxford University Press
|isbn=0-19-856769-3
|page=366
}}</ref>

All kinetic energy is partitioned into the degrees of freedom of the system. The average energy of a single particle with ''f'' quadratic degrees of freedom in a thermal bath of temperature ''T'' is a statistical mean energy given by the [[equipartition theorem]] as
:<math>E_{thermal} = f \cdot \tfrac 1 2 kT \,\!</math>
where ''k'' is the [[Boltzmann constant]]. The total thermal energy of a sample of matter or a thermodynamic system is consequently the average sum of the kinetic energies of all particles in the system. Thus, for a system of ''N'' particles its thermal energy is<ref name=schroeder>{{cite book
|title=An Introduction to Thermal Physics
|author=D.V. Schroeder
|publisher=Addison-Wesley
|year=1999
|isbn=0-201-38027-7
|page=15
}}</ref>
:<math>U_{thermal} = N \cdot f \cdot \tfrac{1}{2} kT.</math>
For gaseous systems, the factor ''f'', the number of degrees of freedom, commonly has the value 3 in the case of the monatomic gas, 5 for many diatomic gases, and 7 for larger molecules at ambient temperatures. In general however, it is a function of the temperature of the system as internal modes of motion, vibration, or rotation become available in higher energy regimes.

''Uthermal'' is not the total energy of a system. Physical systems also contains static [[potential energy]] (such as [[chemical energy]]) that arises from interactions between particles, [[Nuclear potential energy|nuclear energy]] associated with atomic nuclei of particles, and even the [[rest mass energy]] due to the equivalence of energy and mass.

==Thermal energy of the ideal gas==
Thermal energy is most easily defined in the context of the [[ideal gas]], which is well approximated by a [[monatomic]] gas at low pressure. The ideal gas is a gas of particles considered as point objects of perfect spherical symmetry that interact only by elastic collisions and fill a volume such that their mean free path between collisions is much larger than their diameter.

The mechanical kinetic energy of a single particle is
:<math>E_{kinetic} = \tfrac 1 2 m v^2 \,\!</math>
where ''m'' is the particle's mass and ''v'' is its velocity. The thermal energy of the gas sample consisting of ''N'' atoms is given by the sum of these energies, assuming no losses to the container or the environment:
:<math>U_{thermal} = \tfrac 1 2 N m \overline{v^2} = \tfrac{3}{2} N k T,</math>
where the line over the velocity term indicates that the average value is calculated over the entire ensemble. The total thermal energy of the sample is proportional to the macroscopic temperature by a constant factor accounting for the three translational degrees of freedom of each particle and the Boltzmann constant. The Boltzmann constant converts units between the microscopic model and the macroscopic temperature. This formalism is the basic assumption that directly yields the [[ideal gas law]] and it shows that for the ideal gas, the internal energy ''U'' consists only of its thermal energy:
:<math> U = U_{thermal}.\;</math>

==Historical context==
In an 1847 lecture entitled ''On Matter, Living Force, and Heat'', [[James Prescott Joule]] characterized various terms that are closely related to thermal energy and heat.
He identified the terms [[latent heat]] and [[sensible heat]] as forms of heat each effecting distinct physical phenomena, namely the potential and kinetic energy of particles, respectively.<ref>{{citation
|author=J. P. Joule
|title=Matter, Living Force, and Heat
|work=The Scientific Papers of James Prescott Joule
|year=1884
|publisher=The Physical Society of London
|page=274
|quote=I am inclined to believe that both of these hypotheses will be found to hold good,—that in some instances, particularly in the case of ''sensible'' heat, or such as is indicated by the thermometer, heat will be found to consist in the living force of the particles of the bodies in which it is induced; whilst in others, particularly in the case of ''latent'' heat, the phenomena are produced by the separation of particle from particle, so as to cause them to attract one another through a greater space.
|url=http://www.archive.org/details/scientificpapers01joul
|accessdate=2 January 2013}}
</ref>
He describes latent energy as the energy of interaction in a given configuration of particles, i.e. a form of [[potential energy]], and the sensible heat as an energy affecting temperature measured by the thermometer due to the thermal energy, which he called the ''living force''.

==Distinction of thermal energy and heat==

In thermodynamics, [[heat]] must always be defined as energy in exchange between two systems, or a single system and its surroundings.<ref name=leland>{{citation
|title=Basic Principles of Classical and Statistical Thermodynamics
|author=Thomas W. Leland, Jr.
|editor=G. A. Mansoori
|url=http://www.uic.edu/labs/trl/1.OnlineMaterials/BasicPrinciplesByTWLeland.pdf
}}</ref> According to the [[zeroth law of thermodynamics]], heat is exchanged ''between'' thermodynamic systems in thermal contact only if their temperatures are different, as this is the condition when the net exchange of thermal energy is non-zero. For the purpose of distinction, a system is defined to be enclosed by a well-characterized boundary. If heat traverses the boundary in direction ''into'' the system, the internal energy change is considered to be a positive quantity, while ''exiting'' the system, it is negative. As a process variable, heat is never a property of the system, nor is it ''contained'' within the boundary of the system.<ref name=speyer/>

In contrast to heat, thermal energy exists on both sides of a boundary. It is the statistical mean of the microscopic fluctuations of the kinetic energy of the systems' particles, and it is the source and the effect of the transfer of heat across a system boundary. Statistically, thermal energy is always exchanged between systems, even when the temperatures on both sides is the same, i.e. the systems are in thermal equilibrium. However, at equilibrium, the ''net'' exchange of thermal energy is zero, and therefore there is no heat.

Thermal energy may be increased in a system by other means than heat, for example when mechanical or electrical work is performed on the system. No qualitative difference exists between the thermal energy added by other means. Thermal energy is not a state function, although it may be closely related to the [[internal energy]] of some systems, which is a state function. There is also no need in classical thermodynamics to characterize the thermal energy in terms of atomic or molecular behavior. A change in thermal energy induced in a system is the product of the change in entropy and the temperature of the system.

Heat exchanged across a boundary may cause changes other than a change in temperature. For example, it may cause phase transitions, such as melting or evaporation, which are changes in the configuration of a material. Since such an energy exchange is not observable by a change in temperature, it is called a [[latent heat]] and represents a change in the potential energy of the system.

Rather than being itself the thermal energy involved in a transfer, heat is sometimes also understood as the process of that transfer, i.e. ''heat'' functions as a verb.

Today's narrow definition of ''heat'' in physics contrasts with its use in common language, in some engineering disciplines, and in the historical scientific development of thermodynamics in the [[caloric theory]] of heat. The phenomenon of ''heat'' in these instances is today properly identified as the [[entropy]].<ref name=fuchs/>

== The origin of heat energy on Earth ==



[[Earth|Earth's]] proximity to the [[Sun]] is the reason that almost everything near Earth's surface is warm with a temperature substantially above absolute zero.<ref>The deepest ocean depths (3 to 10 km) are no colder than about 274.7–275.7 K (1.5–2.5 °C). Even the world-record cold surface temperature established on July 21, 1983 at [[Vostok Station]], Antarctica is 184 K (a reported value of −89.2 °C). The residual heat of gravitational contraction left over from earth's formation, tidal friction, and the decay of radioisotopes in earth's core provide insufficient heat to maintain earth's surface, oceans, and atmosphere "substantially above" absolute zero in this context. Also, the qualification of "most-everything" provides for the exclusion of lava flows, which derive their temperature from these deep-earth sources of heat.</ref> [[Solar radiation]] constantly replenishes heat energy that Earth loses into space and a relatively stable state of near equilibrium is achieved. Because of the wide variety of heat diffusion mechanisms (one of which is black-body radiation which occurs at the speed of light), objects on Earth rarely vary too far from the global mean surface and air temperature of 287 to 288 K (14 to 15 °C). The more an object's or system's temperature varies from this average, the more rapidly it tends to come back into equilibrium with the ambient environment.

==Thermal energy of individual particles==
The term ''thermal energy'' is also often used as a property of single particles to designate the kinetic energy of the particles. An example is the description of [[thermal neutron]]s having a certain thermal energy, which means that the kinetic energy of the particle is equivalent to the temperature of its surroundings.

==See also==
*[[Heat transfer]]
*[[Ocean thermal energy conversion]]
*[[Thermal science]]

==References==
{{reflist}}

==External links==
*[http://solar.calfinder.com/library/thermal Example of incorrect use of ''heat'' and ''thermal energy'']

[[Category:Thermodynamics]]
[[Category:Forms of energy]]

Total harmonic distortion

2013-12-02T22:12:24Z

71.167.68.30: Undid revision 583916328 by Kvng (talk) I proposed the merge and I agree with those who say they should be separate

{{About|the radio-frequency transmission line|the power transmission line|electric power transmission}}
[[File:Transmission line animation.gif|right|thumb|300px|Schematic showing how a wave flows down a lossless transmission line. Red color indicates high [[voltage]], and blue indicates low voltage. Black dots represent [[electron]]s. The line is terminated at an [[impedance matching|impedance-matched]] load resistor (box on right), which fully absorbs the wave.]]
[[Image:F-Stecker und Kabel.jpg|thumb|One of the most common types of transmission line, [[coaxial cable]]. ]]

In [[Telecommunications engineering|communications]] and [[electronic engineering]], a '''transmission line''' is a specialized cable or other structure designed to carry [[alternating current]] of [[radio frequency]], that is, currents with a [[frequency]] high enough that their [[wave]] nature must be taken into account. Transmission lines are used for purposes such as connecting [[Transmitter|radio transmitters]] and [[Radio receiver|receivers]] with their [[antenna (radio)|antennas]], distributing [[cable television]] signals, [[trunking|trunklines]] routing calls between telephone switching centers, computer network connections, and high speed computer [[data bus]]es.

This article covers two-conductor transmission line such as parallel line ([[ladder line]]), [[coaxial cable]], [[stripline]], and [[microstrip]]. Some sources also refer to [[waveguide]], [[dielectric waveguide]], and even [[optical fiber]] as transmission line, however these lines require different analytical techniques and so are not covered by this article; see [[Waveguide (electromagnetism)]].

== 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 signal]]s. However, they cannot be used to carry currents in the [[radio frequency]] range or higher,<ref name="Jackman">{{cite book
| last = Jackman
| first = Shawn M.
| coauthors = Matt Swartz, Marcus Burton, Thomas W. Head
| title = CWDP Certified Wireless Design Professional Official Study Guide: Exam PW0-250
| publisher = John Wiley & Sons
| year = 2011
| location =
| pages = Ch. 7
| url = http://books.google.com/books?id=AQ8WJGshLBEC&pg=PT300&lpg=PT300&dq=%22what+is+a+transmission+line?%22+%22A+cable's+nature
| doi =
| id =
| isbn = 1118041615}}</ref> which reverse direction millions to billions of times per second, because the energy tends to radiate off the cable as [[radio wave]]s, causing power losses. Radio frequency currents also tend to reflect from discontinuities in the cable such as [[electrical connector|connectors]] and joints, and travel back down the cable toward the source.<ref name="Jackman" /><ref name="Oklobdzija">{{cite book
| last = Oklobdzija
| first = Vojin G.
| coauthors = Ram K. Krishnamurthy
| title = High-Performance Energy-Efficient Microprocessor Design
| publisher = Springer
| year = 2006
| location =
| pages = 297
| url = http://books.google.com/books?id=LmfHof1p3qUC&pg=PA297&dq=%22transmission+line%22+%22uniform
| doi =
| id =
| isbn = 0387340475}}</ref> 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 ''[[Electrical impedance|impedance]]'', called the [[characteristic impedance]],<ref name="Oklobdzija" /><ref name="Guru">{{cite book
| last = Guru
| first = Bhag Singh
| coauthors = Hüseyin R. Hızıroğlu
| title = Electromagnetic Field Theory Fundamentals, 2nd Ed.
| publisher = Cambridge Univ. Press
| year = 2004
| location =
| pages = 422–423
| url = http://books.google.com/books?id=qzNdDtZUPXMC&pg=PA422&dq=%22transmission+line%22+uniform
| doi =
| id =
| isbn = 1139451928}}</ref><ref name="Schmitt">{{cite book
| last = Schmitt
| first = Ron Schmitt
| title = Electromagnetics Explained: A Handbook for Wireless/ RF, EMC, and High-Speed Electronics
| publisher = Newnes
| year = 2002
| location =
| pages = 153
| url = http://books.google.com/books?id=7gJ4RocvEskC&pg=PA153&dq=%22transmission+line%22+uniform
| doi =
| id =
| isbn = 0080505236}}</ref> to prevent reflections. Types of transmission line include parallel line ([[ladder line]], [[twisted pair]]), [[coaxial cable]], [[stripline]], and [[microstrip]].<ref name="Carr">{{cite book
| last = Carr
| first = Joseph J.
| title = Microwave & Wireless Communications Technology
| publisher = Newnes
| year = 1997
| location = USA
| pages = 46–47
| url = http://books.google.com/books?id=1j1E541LKVoC&pg=PA46&dq=%22parallel+line%22+%22coaxial+cable%22+stripline+waveguide
| doi =
| id =
| isbn = 0750697075}}</ref><ref name="Raisanen">{{cite book
| last = Raisanen
| first = Antti V.
| coauthors = Arto Lehto
| title = Radio Engineering for Wireless Communication and Sensor Applications
| publisher = Artech House
| year = 2003
| location =
| pages = 35–37
| url = http://books.google.com/books?id=m8Dgkvf84xoC&pg=PA35
| doi =
| id =
| isbn = 1580536697}}</ref> 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 length of the cable is longer than a significant fraction of the transmitted frequency's wavelength.

At [[microwave]] frequencies and above, power losses in transmission lines become excessive, and [[waveguide]]s are used instead,<ref name="Jackman" /> which function as "pipes" to confine and guide the electromagnetic waves.<ref name="Raisanen" /> Some sources define waveguides as a type of transmission line;<ref name="Raisanen" /> however, this article will not include them. At even higher frequencies, in the [[terahertz]], [[infrared]] and [[light]] range, waveguides in turn become lossy, and [[optics|optical]] methods, (such as lenses and mirrors), are used to guide electromagnetic waves.<ref name="Raisanen" />

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 line]]s.

==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 communications cable|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]].<ref>Ernst Weber and Frederik Nebeker, ''The Evolution of Electrical Engineering'', IEEE Press, Piscataway, New Jersey USA, 1994 ISBN 0-7803-1066-7</ref>

==Applicability==

In many [[electric circuit]]s, the length of the wires connecting the components can for the most part be ignored. That is, the voltage on the wire at a given time can be assumed to be the same at all points. However, when the voltage changes in a time interval comparable to the time it takes for the signal to travel down the wire, the length becomes important and the wire must be treated as a transmission line. Stated another way, the length of the wire is important when the signal includes [[Harmonic analysis|frequency components]] with corresponding [[wavelength]]s comparable to or less than the length of the wire.

A common rule of thumb is that the cable or wire should be treated as a transmission line if the length is greater than 1/10 of the wavelength. At this length the phase delay and the interference of any reflections on the line become important and can lead to unpredictable behavior in systems which have not been carefully designed using transmission line theory.

==The four terminal model==

[[Image:Transmission line symbols.svg|thumb|Variations on the [[electronic schematic|schematic]] [[electronic symbol]] for a transmission line.]]

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

[[Image:Transmission line 4 port.svg]]

In the simplest case, the network is assumed to be linear (i.e. the [[complex number|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 single parameter called the ''[[characteristic impedance]]'', symbol Z0. This 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 [[Ohm (unit)|ohm]]s 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.

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 ''[[impedance matching|matched]]''.

[[File:TransmissionLineDefinitions.svg|thumb|310px|A transmission line is drawn as two black wires. At a distance ''x'' into the line, there is current ''I(x)'' traveling through each wire, and there is a voltage difference ''V(x)'' between the wires. If the current and voltage come from a single wave (with no reflection), then ''V''(''x'') / ''I''(''x'') = ''Z''0, where ''Z''0 is the ''[[characteristic impedance]]'' of the line.]]

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 modeled 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.

High-frequency transmission lines can be defined as those designed to carry electromagnetic waves whose [[wavelength]]s are shorter than or comparable to the length of the line. Under these conditions, the approximations useful for calculations at lower frequencies are no longer accurate. This often occurs with [[radio]], [[microwave]] and [[light|optical]] signals, [[metal mesh optical filters]], and with the signals found in high-speed [[digital circuit]]s.

==Telegrapher's equations==
{{Main|Telegrapher's equations}}
{{See also|Reflections on copper lines}}

The '''telegrapher's equations''' (or just '''telegraph equations''') are a pair of linear differential equations which describe the [[voltage]] and [[Electric current|current]] 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]].

[[Image:Transmission line element.svg|thumb|right|250px|Schematic representation of the elementary component of a transmission line.]]
The transmission line model 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 <math>R</math> of the conductors is represented by a series resistor (expressed in ohms per unit length).
* The distributed inductance <math>L</math> (due to the [[magnetic field]] around the wires, [[self-inductance]], etc.) is represented by a series [[inductor]] ([[henry (unit)|henries]] per unit length).
* The capacitance <math>C</math> between the two conductors is represented by a [[Shunt (electrical)|shunt]] [[capacitor]] C ([[farad]]s per unit length).
* The [[Electric conductance|conductance]] <math>G</math> of the dielectric material separating the two conductors is represented by a shunt resistor between the signal wire and the return wire ([[Siemens (unit)|siemens]] per unit length).

The model consists of an ''infinite series'' of the elements shown in the figure, and that the values of the components are specified ''per unit length'' so the picture of the component can be misleading. <math>R</math>, <math>L</math>, <math>C</math>, and <math>G</math> may also be functions of frequency. An alternative notation is to use <math>R'</math>, <math>L'</math>, <math>C'</math> and <math>G'</math> 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 <math>V(x)</math> and the current <math>I(x)</math> can be expressed in the frequency domain as

:<math>\frac{\partial V(x)}{\partial x} = -(R + j \omega L)I(x)</math>

:<math>\frac{\partial I(x)}{\partial x} = -(G + j \omega C)V(x).</math>

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

:<math>\frac{\partial^2V(x)}{\partial x^2}+ \omega^2 LC\cdot V(x)=0</math>

:<math>\frac{\partial^2I(x)}{\partial x^2} + \omega^2 LC\cdot I(x)=0.</math>

These are [[wave equation]]s which have [[plane wave]]s 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.

If <math>R</math> and <math>G</math> are not neglected, the Telegrapher's equations become:

:<math>\frac{\partial^2V(x)}{\partial x^2} = \gamma^2 V(x)</math>

:<math>\frac{\partial^2I(x)}{\partial x^2} = \gamma^2 I(x)</math>

where

:<math>\gamma = \sqrt{(R + j \omega L)(G + j \omega C)}</math>

and the characteristic impedance is:

:<math>Z_0 = \sqrt{\frac{R + j \omega L}{G + j \omega C}}.</math>

The solutions for <math>V(x)</math> and <math>I(x)</math> are:

:<math>V(x) = V^+ e^{-\gamma x} + V^- e^{\gamma x} \,</math>

:<math>I(x) = \frac{1}{Z_0}(V^+ e^{-\gamma x} - V^- e^{\gamma x}). \,</math>

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

:<math>\mathrm{Re}(\gamma) = (a^2 + b^2)^{1/4} \cos(\mathrm{atan2}(b,a)/2) \,</math>

:<math>\mathrm{Im}(\gamma) = (a^2 + b^2)^{1/4} \sin(\mathrm{atan2}(b,a)/2) \,</math>

where [[atan2]] is the two-parameter arctangent, and

:<math>a \equiv \omega^2 LC \left[ \left( \frac{R}{\omega L} \right) \left( \frac{G}{\omega C} \right) - 1 \right] </math>

:<math>b \equiv \omega^2 LC \left( \frac{R}{\omega L} + \frac{G}{\omega C} \right). </math>

For small losses and high frequencies, to first order in <math>R / \omega L</math> and <math>G / \omega C</math> one obtains

:<math>\mathrm{Re}(\gamma) \approx \frac{\sqrt{LC}}{2} \left( \frac{R}{L} + \frac{G}{C} \right) \,</math>

:<math>\mathrm{Im}(\gamma) \approx \omega \sqrt{LC}. \, </math>

Noting that an advance in phase by <math>- \omega \delta</math> is equivalent to a time delay by <math>\delta</math>, <math>V_{out}(t)</math> can be simply computed as

:<math>V_{\mathrm{out}}(x,t) \approx V_{\mathrm{in}}(t - \sqrt{LC}x) e^{- \frac{\sqrt{LC}}{2} \left( \frac{R}{L} + \frac{G}{C} \right) x }. \,</math>

==Input impedance of transmission line==
[[File:SmithChartLineLength.svg|thumb|right|350px|Looking towards a load through a length {{math|''l''}} of lossless transmission line, the impedance changes as {{math|''l''}} increases, following the blue circle on this [[Smith chart|impedance Smith chart]]. (This impedance is characterized by its [[reflection coefficient]] {{math|''Vreflected'' / ''Vincident''}}.) The blue circle, centered within the chart, is sometimes called an ''SWR circle'' (short for ''constant [[standing wave ratio]]'').]]

The [[characteristic impedance]] {{math|''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, {{math|''l''}}, from the load impedance {{math|''ZL''}} may be expressed as,

:<math>Z_{in}\left(l\right)=\frac{V(l)}{I(l)}=Z_0 \frac{1 + \Gamma_L e^{-2 \gamma l}}{1 - \Gamma_L e^{-2 \gamma l}}</math>,

where {{math|''γ''}} is the propagation constant and <math>\Gamma_L=\left(Z_L - Z_0\right)/\left(Z_L + Z_0\right)</math> is the voltage [[reflection coefficient]] 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:

:<math>Z_{in}\left(l\right)=Z_0 \frac{Z_L + Z_0 \tanh\left(\gamma l\right)}{Z_0 + Z_L\tanh\left(\gamma l\right)}</math>.

===Input impedance of lossless transmission line===
For a lossless transmission line, the propagation constant is purely imaginary, {{math|''γ''{{=}}''jβ''}}, so the above formulas can be rewritten as,

:<math>
Z_\mathrm{in} (l)=Z_0 \frac{Z_L + jZ_0\tan(\beta l)}{Z_0 + jZ_L\tan(\beta l)}
</math>

where <math>\beta=\frac{2\pi}{\lambda}</math> is the [[wavenumber]].

In calculating {{math|''β''}}, the wavelength is generally different inside the transmission line to what it would be in free-space and the velocity constant of the material the transmission line is made of needs to be taken into account when doing such a calculation.

===Special cases of lossless transmission lines===

====Half wave length====
For the special case where <math>\beta l= n\pi</math> 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
:
<math>Z_\mathrm{in}=Z_L \,</math>

for all {{math|''n''}}. This includes the case when {{math|''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====
{{Main|quarter wave impedance transformer}}
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
:<math>
Z_\mathrm{in}=\frac{{Z_0}^2}{Z_L}. \,
</math>

====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
:<math>
Z_\mathrm{in}=Z_L=Z_0 \,
</math>
for all <math>l</math> and all <math>\lambda</math>.

====Short====
[[File:Transmission line animation open short.gif|thumb|right|300px|Standing waves on a transmission line with an open-circuit load (top), and a short-circuit load (bottom). Colors represent voltages, and black dots represent electrons.]]
{{main|stub (electronics)#Short circuited stub|l1=stub}}
For the case of a shorted load (i.e. <math>Z_L=0</math>), the input impedance is purely imaginary and a periodic function of position and wavelength (frequency)

:<math>
Z_\mathrm{in} (l)=j Z_0 \tan(\beta l). \,
</math>

====Open====
{{main|stub (electronics)#Open_circuited_stub|l1=stub}}
For the case of an open load (i.e. <math>Z_L=\infty</math>), the input impedance is once again imaginary and periodic

:<math>
Z_\mathrm{in} (l)=-j Z_0 \cot(\beta l). \,
</math>

===Stepped transmission line===
[[Image:Segments.jpg|thumb|left|A simple example of stepped transmission line consisting of three segments.]]A stepped transmission line<ref>{{cite web|url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WJX-4W2122T-1&_user=5755111&_rdoc=1&_fmt=&_orig=search&_sort=d&_docanchor=&view=c&_acct=C000000150&_version=1&_urlVersion=0&_userid=5755111&md5=fe79f204b33cf7eb6d03cb89ff250c91 |title=Journal of Magnetic Resonance - Impedance matching with an adjustable segmented transmission line |publisher=ScienceDirect.com |date= |accessdate=2013-06-15}}</ref> 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 Z0,i. The input impedance can be obtained from the successive application of the chain relation

:<math>
Z_\mathrm{i+1}=Z_\mathrm{0,i} \frac{Z_i + jZ_\mathrm{0,i}\tan(\beta_i l_i)}{Z_\mathrm{0,i} + jZ_i\tan(\beta_i l_i)}
</math>

where <math>\beta_i</math> is the wave number of the ''i''th transmission line segment and li is the length of this segment, and Zi is the front-end impedance that loads the ''i''th segment. [[Image:PolarSmith.jpg|thumb|right|The impedance transformation circle along a transmission line whose characteristic impedance Z0,i is smaller than that of the input cable Z0. And as a result, the impedance curve is off-centered towards the -x axis. Conversely, if Z0,i > Z0, the impedance curve should be off-centered towards the +x axis.]]Because the characteristic impedance of each transmission line segment Z0,i is often different from that of the input cable Z0, the impedance transformation circle is off centered along the x axis of the [[Smith Chart]] whose impedance representation is usually normalized against Z0.

==Practical types==

===Coaxial cable===

{{Main|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, transverse electric (TE) and transverse magnetic (TM) [[waveguide]] modes can also propagate. 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.

[[Image:Solec Kujawski longwave antenna feeder.jpg|thumb|right|A type of transmission line called a ''cage line'', used for high power, low frequency applications. It functions similarly to a large coaxial cable. This example is the antenna [[feedline]] for a [[longwave]] radio transmitter in [[Poland]], which operates at a frequency of 225 kHz and a power of 1200 kW.]]

===Microstrip===

{{Main|microstrip}}

A microstrip circuit uses a thin flat conductor which is [[Parallel (geometry)|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===

:''Main article : [[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.

===Balanced lines===
{{Main|Balanced line}}
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====
{{Main|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.<ref>Syed V. Ahamed, Victor B. Lawrence, ''Design and engineering of intelligent communication systems'', pp.130-131, Springer, 1997 ISBN 0-7923-9870-X.</ref> 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 circuit, 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.

Interference picked up by the cable arrives as a virtually perfect common mode signal, which is easily removed by coupling transformers. Because the conductors are always the same distance from each other, cross talk is reduced relative to cables with two separate twisted pairs.

The combined benefits of twisting, differential signalling, and quadrupole pattern give outstanding noise immunity, especially advantageous for low signal level applications such as long 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.<ref>{{cite book|last=Lampen|first=Stephen H.|title=Audio/Video Cable Installer's Pocket Guide|year=2002|publisher=McGraw-Hill|isbn=0071386211|pages=32, 110, 112}}</ref><ref>{{cite book|last=Rayburn|first=Ray|title=Eargle's The Microphone Book: From Mono to Stereo to Surround - A Guide to Microphone Design and Application|edition=3|year=2011|publisher=Focal Press|isbn=0240820754|pages=164–166}}</ref>

==== Twin-lead ====
{{Main|Twin-lead}}
Twin-lead consists of a pair of conductors held apart by a continuous insulator.

====Lecher lines====
{{Main|Lecher lines}}
Lecher lines are a form of parallel conductor that can be used at [[Ultra high frequency|UHF]] for creating resonant circuits. They are a convenient practical format that fills the gap between [[Lumped element model|lumped]] components (used at [[High frequency|HF]]/[[VHF]]) and [[Resonant cavity|resonant cavities]] (used at [[Ultra high frequency|UHF]]/[[Super high frequency|SHF]]).

===Single-wire line===
[[Unbalanced line]]s 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.

==General applications==

===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 [[Antenna (radio)|aerial]] to the receiver.

===Pulse generation===

Transmission lines are also 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.

===Stub filters===
{{see also|Distributed element filter#Stub band-pass 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 [[Radio Society of Great Britain|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.

==Acoustic transmission lines==
{{Main|Acoustic transmission line}}

An acoustic transmission line is the [[Acoustics|acoustic]] [[analogy|analog]] of the electrical transmission line, typically thought of as a rigid-walled tube that is long and thin relative to the wavelength of sound present in it.

==Solutions of the Telegrapher's Equations as Circuit Components==
{{cleanup|section|reason=Poor style|date=June 2012}}

[[Image:Unbalanced Transmission Line Equivalent Sub Circuit.jpg|right|thumb|300px|Equivalent circuit of a transmission line described by the Telegrapher's equations.]]

[[Image:Balanced Transmission Line Equivalent Circuit.jpg|right|thumb|300px|Solutions of the Telegrapher's Equations as Components in the Equivalent Circuit of a Balanced Transmission Line Two-Port Implementation.]]

The solutions of the telegrapher's equations can be inserted directly into a circuit as components. The circuit in the top figure implements the solutions of the telegrapher's equations.<ref>{{Citation | last = McCammon | first = Roy | title = SPICE Simulation of Transmission Lines by the Telegrapher's Method | url=http://i.cmpnet.com/rfdesignline/2010/06/C0580Pt1edited.pdf | accessdate = 22 Oct 2010 }}</ref>

The bottom circuit is derived from the top circuit by source transformations.<ref>{{cite book
|author=William H. Hayt|title=Engineering Circuit Analysis
|edition=second
|publisher=McGraw-Hill
|location=New York, NY|year=1971|isbn=070273820}}, pp. 73-77</ref> It also implements the solutions of the telegrapher's equations.

The solution of the telegrapher's equations can be expressed as an ABCD type '' [[Two-port network]]'' with the following defining equations<ref>
{{cite book
|author=John J. Karakash|title=Transmission Lines and Filter Networks
|edition=First
|publisher=Macmillan
|location=New York, NY|year=1950}}, p. 44
</ref>

:<math>V_1 = V_2 \cosh ( \gamma x) + I_2 Z \sinh (\gamma x) \,</math>
:<math>I_1 = V_2 \frac{1}{Z} \sinh (\gamma x) + I_2 \cosh(\gamma x). \,</math>

:The symbols: <math>E_s, E_L, I_s, I_L, l \,</math> in the source book have been replaced by the symbols :<math> V_1, V_2, I_1, I_2, x \,</math> in the preceding two equations.

The ABCD type two-port gives <math>V_1 \, </math> and <math>I_1 \, </math> as functions of <math>V_2 \, </math> and <math>I_2 \, </math>. Both of the circuits above, when solved for <math>V_1 \, </math> and <math>I_1 \, </math> as functions of <math>V_2 \, </math> and <math>I_2 \, </math> yield exactly the same equations.

In the bottom circuit, all voltages except the port voltages are with respect to ground and the differential amplifiers have unshown connections to ground. An example of a transmission line modeled by this circuit would be a balanced transmission line such as a telephone line. The impedances Z(s), the voltage dependent current sources (VDCSs) and the difference amplifiers (the triangle with the number "1") account for the interaction of the transmission line with the external circuit. The T(s) blocks account for delay, attenuation, dispersion and whatever happens to the signal in transit. One of the T(s) blocks carries the ''forward wave'' and the other carries the ''backward wave''. The circuit, as depicted, is fully symmetric, although it is not drawn that way. The circuit depicted is equivalent to a transmission line connected from <math>V_1 \, </math> to <math>V_2 \, </math> in the sense that <math>V_1 \, </math>, <math>V_2 \, </math>, <math>I_1 \, </math> and <math>I_2 \, </math> would be same whether this circuit or an actual transmission line was connected between <math>V_1 \, </math> and <math>V_2 \, </math>. There is no implication that there are actually amplifiers inside the transmission line.

Every two-wire or balanced transmission line has an implicit (or in some cases explicit) third wire which may be called shield, sheath, common, Earth or ground. So every two-wire balanced transmission line has two modes which are nominally called the differential and common modes. The circuit shown on the bottom only models the differential mode.

In the top circuit, the voltage doublers, the difference amplifiers and impedances Z(s) account for the interaction of the transmission line with the external circuit. This circuit, as depicted, is also fully symmetric, and also not drawn that way. This circuit is a useful equivalent for an unbalanced transmission line like a coaxial cable or a micro strip line.

These are not the only possible equivalent circuits.

==See also==
{{multicol}}
* [[Distributed element model]]
* [[Electric power transmission]]
* [[Heaviside condition]]
* [[Longitudinal wave|Longitudinal electromagnetic wave]]
* [[Lumped components]]
* [[Propagation velocity]]
{{multicol-break}}
* [[Radio frequency power transmission]]
* [[Smith chart]], a graphical method to solve transmission line equations
* [[Standing wave]]
* [[Time domain reflectometer]]
* [[Transverse wave|Transverse electromagnetic wave]]
{{multicol-end}}

==References==
''Part of this article was derived from [[Federal Standard 1037C]].''
{{Reflist}}

*{{Citation
|last= Steinmetz
|first= Charles Proteus
|authorlink= Charles Proteus Steinmetz
|title= The Natural Period of a Transmission Line and the Frequency of lightning Discharge Therefrom
|journal=The Electrical World
|date= August 27, 1898
|pages= 203–205
|issn=
|doi=}}
*{{Citation
|title= Electromagnetism
|edition= 2nd
|last=Grant
|first= I. S.
|last2= Phillips
|first2= W. R.
|publisher= John Wiley
|isbn= 0-471-92712-0
|doi=}}
*{{Citation
|title=Fundamentals of Applied Electromagnetics
|edition= 2004 media
|last= Ulaby
|first= F. T.
|publisher= Prentice Hall
|isbn= 0-13-185089-X
|doi= }}
*{{Citation
|title=Radio communication handbook
|year= 1982
|page= 20
|chapter= Chapter 17
|publisher= [[Radio Society of Great Britain]]
|isbn= 0-900612-58-4
|doi= }}
*{{Citation
|last= Naredo
|first= J. L.
|first2= A. C.
|last2= Soudack
|first3= J. R.
|last3= Marti
|title= Simulation of transients on transmission lines with corona via the method of characteristics
|journal= IEE Proceedings. Generation, Transmission and Distribution.
|volume= 142
|issue= 1
|publisher= Institution of Electrical Engineers
|location= Morelos 
|date= Jan 1995
|issn= 1350-2360
|doi=}}

==Further reading==
{{Commons category|Transmission lines}}
* [http://earlyradiohistory.us/1902wt.htm Annual Dinner of the Institute at the Waldorf-Astoria]. [[Transactions of the American Institute of Electrical Engineers]], New York, January 13, 1902. (Honoring of [[Guglielmo Marconi]], January 13, 1902)
* Avant! software, [http://web.archive.org/web/20050925041320/http://www.ece.cmu.edu/~ee762/hspice-docs/html/hspice_and_qrg/hspice_2001_2-124.html Using Transmission Line Equations and Parameters]. Star-Hspice Manual, June 2001.
* Cornille, P, [http://www.iop.org/EJ/abstract/0022-3727/23/2/001 On the propagation of inhomogeneous waves]. J. Phys. D: Appl. Phys. 23, February 14, 1990. (Concept of inhomogeneous waves propagation — Show the importance of the telegrapher's equation with Heaviside's condition.)
*Farlow, S.J., ''Partial differential equations for scientists and engineers''. J. Wiley and Sons, 1982, p. 126. ISBN 0-471-08639-8.
* Kupershmidt, Boris A., [http://arxiv.org/abs/math-ph/9810020 Remarks on random evolutions in Hamiltonian representation]. Math-ph/9810020. J. Nonlinear Math. Phys. 5 (1998), no. 4, 383-395.
* [[Mihajlo Pupin|Pupin, M.]], {{US patent|1541845}}, ''Electrical wave transmission''.
* [http://cktse.eie.polyu.edu.hk/eie403/ Transmission line matching]. EIE403: High Frequency Circuit Design. Department of Electronic and Information Engineering, Hong Kong Polytechnic University. ([[Portable Document Format|PDF]] format)
* Wilson, B. (2005, October 19). ''[http://cnx.rice.edu/content/m1044/latest/ Telegrapher's Equations]''. Connexions.
* John Greaton Wöhlbier, "''[http://www.wildwestwohlbiers.org/john/files/ms_thesis.pdf "Fundamental Equation''" and "''Transforming the Telegrapher's Equations"]''. Modeling and Analysis of a Traveling Wave Under Multitone Excitation.
* Agilent Technologies. Educational Resources. ''Wave Propagation along a Transmission Line''. [http://education.tm.agilent.com/index.cgi?CONTENT_ID=6 Edutactional Java Applet].
* Qian, C., [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WJX-4W2122T-1&_user=5755111&_rdoc=1&_fmt=&_orig=search&_sort=d&_docanchor=&view=c&_acct=C000000150&_version=1&_urlVersion=0&_userid=5755111&md5=fe79f204b33cf7eb6d03cb89ff250c91 Impedance matching with adjustable segmented transmission line]. J. Mag. Reson. 199 (2009), 104-110.

==External links==
* [http://www.cvel.clemson.edu/emc/calculators/TL_Calculator/index.html Transmission Line Parameter Calculator]
* [http://www.amanogawa.com/archive/transmissionB.html Interactive applets on transmission lines]
* [http://www.eetimes.com/design/microwave-rf-design/4200760/SPICE-Simulation-of-Transmission-Lines-by-the-Telegrapher-s-Method-Part-1-of-3-?Ecosystem=microwave-rf-design SPICE Simulation of Transmission Lines]

{{Telecommunications}}

{{DEFAULTSORT:Transmission Line}}
[[Category:Cables]]
[[Category:Telecommunications engineering]]
[[Category:Distributed element circuits]]

Proper transfer function

2013-11-15T04:47:01Z

71.167.62.76:

The '''steering law''' in [[human–computer interaction]] and [[ergonomics]] is a predictive [[model (abstract)|model]] of human movement that describes the time required to navigate, or ''steer'', through a 2-dimensional tunnel. The tunnel can be thought of as a path or trajectory on a plane that has an associated thickness or width, where the width can vary along the tunnel. The goal of a steering task is to navigate from one end of the tunnel to the other as quickly as possible, without touching the boundaries of the tunnel. A real world example that approximates this task is driving a car down a road that may have twists and turns, where the car must navigate the road as quickly as possible without touching the sides of the road. The steering law predicts both the instantaneous speed at which we may navigate the tunnel, and the total time required to navigate the entire tunnel.

The steering law has been independently discovered and studied three times (Rashevsky, 1959; Drury, 1971; Accot and Zhai, 1997). Its most recent discovery has been within the [[human–computer interaction]] community, which has resulted in the most general mathematical formulation of the law.

==The steering law in human–computer interaction==

Within human–computer interaction, the law was rediscovered by Johnny Accot and Shumin Zhai, who mathematically derived it in a novel way from [[Fitts's law]] using [[integral calculus]], experimentally verified it for a class of tasks, and developed the most general mathematical statement of it. Some researchers within this community have sometimes referred to the law as the '''Accot–Zhai steering law''' or '''Accot's law''' (Accot is pronounced ''ah-cot'' in [[English language|English]] and ''ah-koh'' in [[French language|French]]). In this context, the steering law is a predictive [[model (abstract)|model]] of [[human musculoskeletal system|human movement]], concerning the speed and total time with which a user may steer a [[pointing device]] (such as a [[computer mouse|mouse]] or [[stylus]]) through a 2D tunnel presented on a screen (i.e. with a bird's eye view of the tunnel), where the user must travel from one end of the path to the other as quickly as possible, while staying within the confines of the path. One potential practical application of this law is in modelling a user's performance in navigating a hierarchical cascading [[menu (computing)|menu]].

Many researchers in [[human–computer interaction]], including Accot himself, find it surprising or even amazing that the steering law model predicts performance as well as it does, given the almost purely mathematical way in which it was derived. Some consider this a testament to the robustness of [[Fitts's law]].

In its general form, the steering law can be expressed as

:<math>T=a + b \int_{C} \frac{ds}{W(s)}</math>

where ''T'' is the average time to navigate through the path, ''C'' is the path parameterized by ''s'', ''W(s)'' is the width of the path at ''s'', and ''a'' and ''b'' are experimentally fitted constants. In general, the path may have a complicated curvilinear shape (such as a spiral) with variable thickness ''W(s)''.

Simpler paths allow for mathematical simplifications of the general form of the law. For example, if the path is a straight tunnel of constant width ''W'', the equation reduces to

:<math>T=a + b \frac{A}{W}</math>

where ''A'' is the length of the path. We see, especially in this simplified form, a ''speed–accuracy'' tradeoff, somewhat similar to that in [[Fitts's law]].

We can also differentiate both sides of the integral equation with respect to ''s'' to obtain the local, or instantaneous, form of the law:

:<math>\frac{ds}{dT} = \frac{W(s)}{b}</math>

which says that the instantaneous speed of the user is proportional to the width of the tunnel. This makes intuitive sense if we consider the analogous task of driving a car down a road: the wider the road, the faster we can drive and still stay on the road, even if there are curves in the road.

==Derivation of the model from Fitts's law==

This derivation is only meant as a high level sketch. It lacks the illustrations of, and may differ in detail from, the derivation given by Accot and Zhai (1997).

Assume that the time required for goal passing (i.e. passing a pointer through a goal at distance ''A'' and of width ''W'',
oriented perpendicular to the axis of motion) can be modeled with this form of [[Fitts's law]]:

:<math>T_\text{goal} = b \log_2 \left( \frac{A}{W} + 1 \right)</math>

Then, a straight tunnel of length ''A'' and constant width ''W'' can be approximated as a sequence of ''N'' evenly spaced goals, each separated from its neighbours by a distance of ''A/N''. We can let ''N'' grow arbitrarily large, making the distance between successive goals become infinitesimal. The total time to navigative through all the goals, and thus through the tunnel, is

{|
|-
|''T''straight tunnel
|<math>= \lim_{N \to \infty} \sum_{i=1}^N b \log_2 \left( \frac{A/N}{W} + 1 \right)</math>
|-
|
|<math>= \lim_{N \to \infty} N b \log_2 \left( \frac{A}{N W} + 1 \right)</math>
|-
|
|<math>= b \lim_{N \to \infty} \frac{\log_2 \left( \frac{A}{N W} + 1 \right)}{1/N}</math>
|(applying [[L'Hôpital's rule]] ...)
|-
|
|<math>= b \lim_{N \to \infty} \frac{\frac{1}{\left( \frac{A}{N W} + 1 \right)}\frac{A}{W}(-1/N^2)}{-1/N^2}</math>
|-
|
|<math>= b \frac{A}{W} \lim_{N \to \infty} \frac{1}{\left( \frac{A}{N W} + 1 \right)}</math>
|-
|
|<math>= b \frac{A}{W}</math>
|}

Next, consider a curved tunnel of total length ''A'', parameterized by ''s'' varying from 0 to ''A''. Let ''W(s)'' be the variable width of the tunnel. The tunnel can be approximated as a sequence of ''N'' straight tunnels, numbered 1 through ''N'', each located at ''si'' where ''i'' = 1 to ''N'', and each of length ''s''''i''+1 − ''s''''i'' and of width ''W''(''s''''i''). We can let ''N'' grow arbitrarily large, making the length of successive straight tunnels become infinitesimal. The total time to navigative through the curved tunnel is

{|
|-
|''T''curved tunnel
|<math>= \lim_{N \to \infty} \sum_{i=1}^N b \frac{s_{i+1} - s_i}{W(s_i)}</math>
|-
|
|<math>= b \int_0^A \frac{ds}{W(s)}</math>
|(... by the definition of the definite [[integral]])
|}

yielding the general form of the steering law.

==Modeling steering in layers ==
Steering law has been extended to predict movement time for steering in layers of thickness ''t''. The relation is given{{cn|date=February 2013}} by

: <math>T = a+b\sqrt{(A/W)^2+(A/t)^2}.</math>

==References==
* Drury, C. G. (1971). Movements with lateral constraint. Ergonomics, 14, 293–305. http://www.ncbi.nlm.nih.gov/pubmed/5093722
* Johnny Accot and Shumin Zhai (1997). Beyond Fitts' law: models for trajectory-based HCI tasks. Proceedings of [[Association for Computing Machinery|ACM]] CHI 1997 Conference on Human Factors in Computing Systems, pp. 295–302. http://doi.acm.org/10.1145/258549.258760 http://www.almaden.ibm.com/u/zhai/papers/steering/chi97.pdf
* Johnny Accot and Shumin Zhai (1999). Performance evaluation of input devices in trajectory-based tasks: An application of the steering law. In Proceedings of [[Association for Computing Machinery|ACM]] CHI 1999 Conference on Human Factors in Computing Systems, pages 466–472. http://www.almaden.ibm.com/u/zhai/papers/steering/chi97.pdf
* Johnny Accot and Shumin Zhai (2001). Scale effects in steering law tasks. In Proceedings of ACM CHI 2001 Conference on Human Factors in Computing Systems, pages 1–8. http://doi.acm.org/10.1145/365024.365027 http://www.almaden.ibm.com/u/zhai/papers/EASEChinese/Scale.pdf
* Kattinakere, Raghavendra S., Grossman, Tovi and Subramanian, Sriram (2007): Modeling steering within above-the-surface interaction layers. In Proceedings of ACM CHI 2007 Conference on Human Factors in Computing Systems 2007. pp. 317–326. http://doi.acm.org/10.1145/1240624.1240678 http://www.dgp.toronto.edu/~tovi/papers/chi%202007%20steering.pdf
* Rashevsky, N. (1959). Mathematical biophysics of automobile driving. Bulletin of Mathematical Biophysics, 21, 375–385. http://www.springerlink.com/content/e21715050741p065/
* Shumin Zhai and Johnny Accot and Rogier Woltjer (2004). Human Action Laws in Electronic Virtual Worlds: An Empirical Study of Path Steering Performance in VR. Presence, Vol. 13, No. 2, April 2004, 113–127. http://www.almaden.ibm.com/u/zhai/papers/LawsOfActionManuscript.pdf
** Contains references to, and discusses differences with, earlier work on the "steering law" by Rashevsky and by Drury.

==See also==

* [[Crossing-based interface]] — any graphical user interface that uses ''goal crossing tasks'' as the basic interaction paradigm

==External links==
* http://www.almaden.ibm.com/u/zhai/topics/LawsOfAction.htm

[[Category:Human–computer interaction]]

Decarburization

2013-05-14T10:15:41Z

71.167.173.50: /* As a secondary effect */

The '''Steinhart–Hart equation''' is a model of the [[Electrical resistance|resistance]] of a [[semiconductor]] at different [[temperature]]s. The equation is:

:<math>{1 \over T} = A + B \ln(R) + C (\ln(R))^3 \,</math>

where:
* <math>T</math> is the temperature (in kelvins)
* ''R'' is the resistance at ''T'' (in ohms)
* <math>A</math>, <math>B</math>, and <math>C</math> are the '''Steinhart–Hart coefficients''' which vary depending on the type and model of [[thermistor]] and the temperature range of interest. (The most general form of the applied equation contains a <math>(\ln(R))^2</math> term, but this is frequently neglected because it is typically much smaller than the other coefficients, and is therefore not shown above.)

==Uses of the equation==
The equation is often used to derive a precise temperature of a thermistor since it provides a closer approximation to actual temperature than simpler equations, and is useful over the entire working temperature range of the sensor. Steinhart–Hart coefficients are usually published by thermistor manufacturers.

Where Steinhart–Hart coefficients are not available, they can be derived. Three accurate measures of resistance are made at precise temperatures, then the coefficients are derived by solving three [[simultaneous equations]].

==Inverse of the equation==
To find the resistance of a semiconductor given the temperature the inverse of the Steinhart–Hart equation must be used. See the [http://www.cornerstonesensors.com/reports/ABC%20Coefficients%20for%20Steinhart-Hart%20Equation.pdf Application Note], "A, B, C Coefficients for Steinhart–Hart Equation".
:<math>R = \exp\left(\sqrt[3]{x - y} - \sqrt[3]{x + y}\right),</math>
where
:<math>y = {A - {1 \over T} \over 2C},</math>
:<math>x = \sqrt{\left({B \over 3C}\right)^3 + y^2}.</math>

==Steinhart–Hart coefficients==
To find the coefficients of Steinhart–Hart, we need to know at-least three operating points. For this, we use three values of resistance data for three known temperatures.
:<math>\begin{cases} A + \left(\ln R_1 \right) B + \left(\ln R_1 \right)^3 C=\frac{1}{T_1} \\ A + \left(\ln R_2 \right) B + \left(\ln R_2 \right)^3 C = \frac{1}{T_2} \\ A + \left(\ln R_3 \right) B + \left(\ln R_3 \right)^3 C = \frac{1}{T_3} \end{cases}</math>

With <math>R_1</math>, <math>R_2</math> and <math>R_3</math> values of resistance at the temperatures <math>T_1</math>, <math>T_2</math> and <math>T_3</math>, one can express <math>A</math>, <math>B</math> and <math>C</math> (all calculations):

<math>L_1 = \ln\left(R_1\right)</math>,   <math>L_2=\ln\left(R_2\right)</math>   and   <math>L_3=\ln\left(R_3\right)</math>,

<math>Y_1=\frac{1}{T_1}</math>,  <math>Y_2=\frac{1}{T_2}</math>   and   <math>Y_3=\frac{1}{T_3}</math>,

<math>\gamma_2=\frac{Y_2-Y_1}{L_2-L_1}</math>, <math>\gamma_3=\frac{Y_3-Y_1}{L_3-L_1}</math>

:<math>\Rightarrow C=\left( \frac{ \gamma_3 - \gamma_2 }{ L_3 - L_2} \right) \left(L_1 + L_2 + L_3\right)^{-1}</math>

:<math>\Rightarrow B=\gamma_2 - C \left(L_1^2+L_1 L_2+L_2^2\right)</math>

:<math>\Rightarrow A=Y_1 - \left(B+L_1^2 C\right) L_1</math>

==Developers of the equation==
The equation is named after [[John S. Steinhart]] and [[Stanley R. Hart]] who first published the relationship in 1968.<ref>John S. Steinhart, Stanley R. Hart, Calibration curves for thermistors, Deep Sea Research and Oceanographic Abstracts, Volume 15, Issue 4, August 1968, Pages 497-503, ISSN 0011-7471, {{doi|10.1016/0011-7471(68)90057-0}}.</ref> Professor Steinhart (1929–2003), a fellow of the [[American Geophysical Union]] and of the [[American Association for the Advancement of Science]], was a member of the faculty of [[University of Wisconsin–Madison]] from 1969 to 1991.[http://www.secfac.wisc.edu/senate/2004/0405/1775(mem_res).pdf] Dr. Hart, a Senior Scientist at [[Woods Hole Oceanographic Institution]] since 1989 and fellow of the [[Geological Society of America]], the American Geophysical Union, the [[Geochemical Society]] and the [[European Association of Geochemistry]], [http://www.whoi.edu/science/GG/people/shart/cv.htm] was associated with Professor Steinhart at the [[Carnegie Institution of Washington]] when the equation was developed.

==References==
<references/>

{{DEFAULTSORT:Steinhart-Hart equation}}
[[Category:Condensed matter physics]]

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'''Underactuation''' is a technical term used in [[robotics]] and [[control theory]] to describe mechanical systems that cannot be commanded to follow arbitrary trajectories in [[configuration space]]. This condition can occur for a number of reasons, the simplest of which is when the system has a lower number of [[actuator]]s than [[degrees of freedom (engineering)|degrees of freedom]]. In this case, the system is said to be ''trivially underactuated''.

The class of underactuated mechanical systems is very rich and includes such diverse members as [[automobile]]s, [[airplanes]], and even [[animal]]s.

==Definition==
To understand the mathematical conditions which lead to underactuation, one must examine the dynamics that govern the systems in question. [[Newton's laws of motion]] dictate that the dynamics of mechanical systems are inherently second order. In general, these dynamics can be described by a second order [[differential equation]]:

 
<math> \ddot{q} = f(q,\dot{q},u,t) </math>

Where:

<math>q \in \mathbb{R}^n</math> is the position state vector <math> u \in \mathbb{R}^m </math> is the vector of control inputs <math> t </math> is time.

Furthermore, the dynamics for these systems can be rewritten to be affine in the control inputs:

 
<math> \ddot{q} = f_1(q,\dot{q},t) + f_2(q, \dot{q}, t)u </math>

When expressed in this form, the system is said to be underactuated if:<ref>{{cite book
| last = Tedrake
| first = Russ
| last2 =
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| authorlink =
| title = Underactuated Robotics: Learning, Planning, and Control for Efficient and Agile Machines
| publisher =
| series =
| volume =
| edition =
| year = 2008
| location =
| pages =
| language =
| url = http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-832-underactuated-robotics-spring-2009/readings/MIT6_832s09_read_ch01.pdf
| doi =
| id =
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| jfm = }}</ref>

 
<math>rank[{f_2(q, \dot{q}, t)}] < dim[q]</math>

When this condition is met, there are acceleration directions that can not be produced no matter what the control vector is.

Note that <math>f_2(q, \dot{q}, t)</math> does not explicitly represent the number of actuators present in the system. Indeed, there may be more actuators than degrees of freedom and the system may still be underactuated. Also worth noting is the dependence of <math>f_2(q, \dot{q}, t)</math> on the state <math>q, \dot{q}</math>. That is, there may exist states in which an otherwise fully actuated system becomes underactuated.

==Examples==

The classic [[inverted pendulum]] is an example of a trivially underactuated system: it has two degrees of freedom (one for its support's motion in the horizontal plane, and one for the angular motion of the pendulum), but only one of them (the cart position) is actuated, and the other is only indirectly controlled. Although naturally extremely unstable, this underactuated system is still controllable.

A standard automobile is underactuated due to the nonholonomic constraints imposed by the wheels. That is, a car cannot accelerate in a direction perpendicular to the direction the wheels are facing. A similar argument can be made for boats, planes and most other vehicles.

==See also==
* [[Passive dynamics]]

==References==
{{reflist}}

==Further reading==
* M. Saliba, and C.W. de Silva, "An Innovative Robotic Gripper for Grasping and Handling Research," ''IEEE Journal of Robotics and Automation'', pp. 975–979, 1991.
* N. Dechev, W.L. Cleghorn, and S. Naumann, “Multiple Finger, Passive Adaptive Grasp Prosthetic Hand,” ''Journal of Mechanism and Machine Theory'', Vol. 36, No. 10, pp. 1157–1173, 2001.

==External links==
* Canudas-de-Wit, C. [http://www.lag.ensieg.inpg.fr/canudas/publications/Oscillations/Virtual_constraints_Anual_review_04.pdf On the concept of virtual constraints as a tool for walking robot control and balancing] Annual Reviews in Control, 28 (2004), pp. 157–166. (Elsevier)
*[http://www.mne.ksu.edu/static/nlc/tiki-index.php?page=NL+Systems&highlight=underactuated Nonlinear Systems] College of Mechanical and Nuclear Engineering, Kansas State University

[[Category:Robot control]]
[[Category:Control theory]]