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| '''Thévenin's theorem''' holds, to illustrate in DC [[circuit theory]] terms, that (see image):
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| :*Any [[linearity|linear]] [[electrical network]] with [[voltage source|voltage]] and [[current source|current sources]] and only [[resistance (electricity)|resistance]]s can be replaced at terminals A-B by an equivalent voltage source V<sub>th</sub> in [[series and parallel circuit|series]] connection with an equivalent resistance R<sub>th</sub>.
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| :*This equivalent voltage V<sub>th</sub> is the voltage obtained at terminals A-B of the network with terminals A-B [[open circuit voltage|open circuit]]ed.
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| :*This equivalent resistance R<sub>th</sub> is the resistance obtained at terminals A-B of the network with all its independent current sources open circuited and all its independent voltage sources [[short circuit]]ed.
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| For AC systems, the theorem can be applied to [[Reactive power|reactive]] [[Electrical impedance|impedance]]s as well as resistances.
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| The theorem was independently derived in 1853 by the German scientist [[Hermann von Helmholtz]] and in 1883 by [[Léon Charles Thévenin]] (1857–1926), an [[electrical engineering|electrical engineer]] with France's national [[Postes, télégraphes et téléphones (France)|Postes et Télégraphes]] telecommunications organization.<ref>Helmholtz</ref><ref>Thévenin (1883a)</ref><ref>Thévenin (1883b)</ref><ref>Johnson (2003a)</ref><ref>Brittain</ref><ref name="Dorf">Dorf</ref>
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| Thévenin's theorem and its dual, [[Norton's theorem]], are widely used for circuit analysis simplification and to study circuit's initial-condition and steady-state response.<ref>Brenner</ref><ref>Elgerd</ref> Thévenin's theorem can be used to convert any circuit's sources and impedances to a '''Thévenin equivalent'''; use of the theorem may in some cases be more convenient than use of [[Kirchhoff's circuit laws]].<ref name="Dorf"/><ref>Dwight</ref>
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| [[Image:Thevenin equivalent.png|frame|right|Any [[Black box (systems)|black box]] containing resistances only and voltage and current sources can be replaced to a Thévenin [[equivalent circuit]] consisting of an equivalent voltage source in series connection with an equivalent resistance.]]
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| == Calculating the Thévenin equivalent ==
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| To calculate the equivalent circuit, the resistance and voltage are needed, so [[Simultaneous equations|two equations]] are required. These two equations are usually obtained by using the following steps, but any conditions placed on the terminals of the circuit should also work:
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| # Calculate the output voltage, ''V''<sub>AB</sub>, when in [[Open-circuit voltage|open circuit]] condition (no [[External electric load|load resistor]]—meaning infinite resistance). This is ''V''<sub>Th</sub>.
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| # Calculate the output current, ''I''<sub>AB</sub>, when the output terminals are [[short circuit]]ed (load resistance is 0). ''R''<sub>Th</sub> equals ''V''<sub>Th</sub> divided by this ''I''<sub>AB</sub>.
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| The equivalent circuit is a voltage source with voltage ''V''<sub>Th</sub> in series with a resistance ''R''<sub>Th</sub>.
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| Step 2 could also be thought of as:
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| :2a. Replace the independent voltage sources with short circuits, and independent current sources with open circuits.
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| :2b. Calculate the resistance between terminals A and B. This is ''R''<sub>Th</sub>.
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| The Thévenin-equivalent voltage is the voltage at the output terminals of the original circuit. When calculating a Thévenin-equivalent voltage, the [[voltage divider]] principle is often useful, by declaring one terminal to be ''V''<sub>out</sub> and the other terminal to be at the ground point.
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| The Thévenin-equivalent resistance is the resistance measured across points A and B "looking back" into the circuit. It is important to first replace all voltage- and current-sources with their internal resistances. For an ideal voltage source, this means replace the voltage source with a short circuit. For an ideal current source, this means replace the current source with an open circuit. Resistance can then be calculated across the terminals using the formulae for [[series and parallel circuits]]. This method is valid only for circuits with independent sources. If there are [[dependent source]]s in the circuit, another method must be used such as connecting a test source across A and B and calculating the voltage across or current through the test source.
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| ===Example===
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| {|
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| |[[Image:Thevenin and norton step 1.png|framed|left|Step 0: The original circuit]] || [[Image:Thevenin step 2.png|framed|right|Step 1: Calculating the equivalent output voltage]] || [[Image:Thevenin and norton step 3.png|framed|right|Step 2: Calculating the equivalent resistance]] || [[Image:Thevenin step 4.png|framed|right|Step 3: The equivalent circuit]]
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| |}
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| In the example, calculating the equivalent voltage:
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| :<math>
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| V_\mathrm{Th}
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| = {R_2 + R_3 \over (R_2 + R_3) + R_4} \cdot V_\mathrm{1}
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| </math>
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| ::<math>
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| = {1\,\mathrm{k}\Omega + 1\,\mathrm{k}\Omega \over (1\,\mathrm{k}\Omega + 1\,\mathrm{k}\Omega) + 2\,\mathrm{k}\Omega} \cdot 15 \,\mathrm{V}
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| </math>
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| ::<math>
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| = {1 \over 2} \cdot 15 \, \mathrm{V} = 7.5 \,\mathrm{V}
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| </math>
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| (notice that ''R''<sub>1</sub> is not taken into consideration, as above calculations are done in an open circuit condition between A and B, therefore no current flows through this part, which means there is no current through R<sub>1</sub> and therefore no voltage drop along this part)
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| Calculating equivalent resistance:
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| : <math>
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| R_\mathrm{Th} = R_1 + \left[ \left( R_2 + R_3 \right) \| R_4 \right]
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| </math>
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| :: <math>
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| = 1\,\mathrm{k}\Omega + \left[ \left( 1\,\mathrm{k}\Omega + 1\,\mathrm{k}\Omega \right) \| 2\,\mathrm{k}\Omega \right]
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| </math>
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| :: <math>
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| = 1\,\mathrm{k}\Omega + \left({1 \over ( 1\,\mathrm{k}\Omega + 1\,\mathrm{k}\Omega )} + {1\over (2\,\mathrm{k}\Omega ) }\right)^{-1} = 2\,\mathrm{k}\Omega.
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| </math>
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| ==Conversion to a Norton equivalent==
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| {{main|Norton's theorem}}
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| [[Image:Thevenin to Norton2.PNG|right]]
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| A [[Norton's theorem|Norton equivalent circuit]] is related to the Thévenin equivalent by the following:
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| :<math>R_\mathrm{Th} = R_\mathrm{No} \!</math>
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| :<math>V_\mathrm{Th} = I_\mathrm{No} R_\mathrm{No} \!</math>
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| :<math>I_\mathrm{No} = V_\mathrm{Th} / R_\mathrm{Th}. \!</math>
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| ==Practical limitations==
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| * Many, if not most circuits are only linear over a certain range of values, thus the Thévenin equivalent is valid only within this linear range and may not be valid outside the range.
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| * The Thévenin equivalent has an equivalent I–V characteristic only from the point of view of the load.
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| * The power dissipation of the Thévenin equivalent is not necessarily identical to the power dissipation of the real system. However, the power dissipated by an external resistor between the two output terminals is the same regardless of how the internal circuit is represented.
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| ==A proof of the theorem==
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| The proof involves two steps. First use [[superposition theorem]] to construct a solution, and then use [[uniqueness theorem]] to show the solution is unique. The second step is usually implied. Firstly, using the superposition theorem, in general for any linear "black box" circuit which contains voltage sources and resistors, one can always write down its voltage as a linear function of the corresponding current as follows
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| :<math>V = V_\mathrm{Eq}-Z_\mathrm{Eq}I \!</math>
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| where the first term reflects the linear summation of contributions from each voltage source, while the second term measures the contribution from all the resistors. The above argument is due to the fact that the voltage of the black box for a given current <math>I</math> is identical to the linear superposition of the solutions of the following problems: (1) to leave the black box open circuited but activate individual voltage source one at a time and, (2) to short circuit all the voltage sources but feed the circuit with a certain ideal voltage source so that the resulting current exactly reads <math>I</math> (or an ideal current source of current <math>I</math>). Once the above expression is established, it is straightforward to show that <math> V_\mathrm{Eq} </math> and <math> Z_\mathrm{Eq} </math> are the single voltage source and the single series resistor in question.
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| == See also ==
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| *[[Millman's theorem]]
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| *[[Source transformation]]
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| *[[Superposition theorem]]
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| *[[Norton's theorem]]
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| *[[Maximum power transfer theorem]]
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| *[[Extra element theorem]]
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| == References ==
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| {{Reflist}}
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| ==Bibliography==
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| *{{cite conference|coauthors=Javid, Mansour|last=Brenner|first=Egon|title=Chapter 12 - Network Functions|booktitle=Analysis of Electric Circuits|year=1959|publisher=McGraw-Hill|pages=268–269|url=http://books.google.ca/books/about/Analysis_of_electric_circuits.html?id=V4FrAAAAMAAJ&redir_esc=y}}
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| *{{cite journal|last=Brittain|first=J.E.|title=Thevenin's theorem|journal=IEEE Spectrum|date=March 1990|volume=27|issue=3|pages=42|doi=10.1109/6.48845|url=http://ieeexplore.ieee.org/search/searchresult.jsp?newsearch=true&queryText=James+E.+Brittain+Thevenin%27s+theorem&.x=41&.y=17|accessdate=1 February 2013}}
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| *{{cite conference|last=Dorf|first=Richard C.|coauthors=Svoboda, James A.|title=Chapter 5 - Circuit Theorems|booktitle=Introduction to Electric Circuits|year=2010|publisher=John Wiley & Sons|location=Hoboken, NJ|isbn=978-0-470-52157-1|url=http://ca.wiley.com/WileyCDA/WileyTitle/productCd-EHEP000347.html|edition=8th|pages=162–207}}
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| *{{cite conference|booktitle=Standard Handbook for Electrical Engineers |editor-last=Knowlton|editor-first=A.E.|edition=8th|year=1949|title=Sec. 2 - Electric and Magnetic Circuits|first=Herbert B.|last=Dwight|publisher=McGraw-Hill|page=26}}
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| *{{cite conference|last=Elgerd|first=Olle I.|title=Chapter 10, Energy System Transients - Surge Phenomena and Symmetrical Fault Analysis|booktitle=Electric Energy Systems Theory: An Introduction|year=2007|publisher=Tata McGraw-Hill|pages=402–429|url=http://books.google.ca/books/about/Electric_Energy_Systems_Theory.html?id=AKTi3UxfhlgC&redir_esc=y|isbn=978-0070192300}}
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| *{{cite journal|last=Helmhotz|first=H.
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| |title=Über einige Gesetze der Vertheilung elektrischer Ströme in körperlichen Leitern mit Anwendung auf die thierisch-elektrischen Versuche (Some laws concerning the distribution of electrical currents in conductors with applications to experiments on animal electricity)|journal=Annalen der Physik und Chemie|volume=89|issue=6|pages=211–233|url=http://gallica.bnf.fr/ark:/12148/bpt6k151746.image.f225.langFR|year=1853}}
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| *{{cite journal|last=Johnson|first=D.H.|title=Origins of the equivalent circuit concept: the voltage-source equivalent|journal=Proceedings of the IEEE|year=2003a|volume=91|issue=4|pages=636–640|doi=10.1109/JPROC.2003.811716|url=http://www.ece.rice.edu/~dhj/paper1.pdf}}
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| *{{cite journal|last=Johnson|first=D.H.|title=Origins of the equivalent circuit concept: the current-source equivalent|journal=Proceedings of the IEEE|year=2003b|volume=91|issue=5|pages=817–821|doi=10.1109/JPROC.2003.811795|url=http://www.ece.rice.edu/~dhj/paper2.pdf}}
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| *{{cite journal|last=Thévenin|first=L.|title=Extension de la loi d’Ohm aux circuits électromoteurs complexes (Extension of Ohm’s law to complex electromotive circuits)|journal=Annales Télégraphiques|year=1883a|volume=10|series=3<sup>e</sup> series|pages=222–224|url=http://books.google.com/?id=shUAAAAAMAAJ&pg=PA222&lpg=PA222&dq=Extension+de+la+loi+d%E2%80%99Ohm+aux+circuits+%C3%A9lectromoteurs+complexes#v=onepage&q=Extension%20de%20la%20loi%20d%E2%80%99Ohm%20aux%20circuits%20%C3%A9lectromoteurs%20complexes&f=false}}
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| *{{cite journal|last=Thévenin|first=L.|title=Sur un nouveau théorème d'électricité dynamique (On a new theorem of dynamic electricity)|journal=[[Comptes rendus hebdomadaires des séances de l'Académie des Sciences]]|volume=97|pages=159–161|year=1883b}}
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| *{{cite journal|last=Wenner|first=F.|title=Sci. Paper S531, A principle governing the distribution of current in systems of linear conductors|year=1926|publisher=Bureau of Standards|location=Washington, D.C.}}
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| == External links ==
| |
| * [http://www.allaboutcircuits.com/vol_1/chpt_10/8.html Thevenin's theorem at allaboutcircuits.com]
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| * [http://www.tedpavlic.com/teaching/osu/ece209/support/circuits_sys_review.pdf Filter-Order Filters: Shortcut via Thévenin Equivalent Source] — showing on p. 4 complex circuit's Thévenin's theorem simplication to first-order [[low-pass filter]] and associated [[voltage divider]], [[time constant]] and [[gain]].
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| {{DEFAULTSORT:Thevenin's Theorem}}
| |
| [[Category:Circuit theorems]]
| |
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