Thévenin's theorem
1
Thévenin's theorem
As originally stated in terms of DC resistive circuits only, the
Thévenin's theorem holds that:
• Any linear electrical network with voltage and current sources
and only resistances can be replaced at terminals A-B by an
equivalent voltage source Vth in series connection with an
equivalent resistance Rth.
• This equivalent voltage Vth is the voltage obtained at terminals
A-B of the network with terminals A-B open circuited.
Any 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.
• This equivalent resistance Rth 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 circuited.
In circuit theory terms, the theorem allows any one-port network to be reduced to a single voltage source and a single
impedance.
The theorem also applies to frequency domain AC circuits consisting of reactive and resistive impedances.
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 engineer with France's national Postes et Télégraphes
telecommunications organization.[1][2][3][4][5][6]
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.[7][8] 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.[9]
Calculating the Thévenin equivalent
To calculate the equivalent circuit, the resistance and voltage are needed, so 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:
1. Calculate the output voltage, VAB, when in open circuit condition (no load resistor—meaning infinite resistance).
This is VTh.
2. Calculate the output current, IAB, when the output terminals are short circuited (load resistance is 0). RTh equals
VTh divided by this IAB.
The equivalent circuit is a voltage source with voltage VTh in series with a resistance RTh.
Step 2 could also be thought of as:
2a. Replace the independent voltage sources with short circuits, and independent current sources with open
circuits.
2b. Calculate the resistance between terminals A and B. This is RTh.
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 Vout and
the other terminal to be at the ground point.
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
Thévenin's theorem
2
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
sources 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.
Example
Step 1: Calculating the equivalent output
voltage
Step 0: The original circuit
Step 3: The equivalent circuit
Step 2: Calculating the equivalent
resistance
In the example, calculating the equivalent voltage:
(notice that R1 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 R1 and therefore no
voltage drop along this part)
Calculating equivalent resistance:
Conversion to a Norton equivalent
Main article: Norton's theorem
A Norton equivalent circuit is related to the Thévenin equivalent by the
following:
Thévenin's theorem
3
Practical limitations
• 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.
• The Thévenin equivalent has an equivalent I–V characteristic only from the point of view of the load.
• 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.
A proof of the theorem
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
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 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
(or an ideal current source of current
and
). Once the above expression is established, it is straightforward to show that
are the single voltage source and the single series resistor in question.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
Helmholtz
Thévenin (1883a)
Thévenin (1883b)
Johnson (2003a)
Brittain
Dorf
Brenner
Elgerd
Dwight
Bibliography
• Brenner, Egon; Javid, Mansour (1959). "Chapter 12 - Network Functions" (http://books.google.ca/books/
about/Analysis_of_electric_circuits.html?id=V4FrAAAAMAAJ&redir_esc=y). Analysis of Electric Circuits.
McGraw-Hill. pp. 268–269.
• Brittain, J.E. (March 1990). "Thevenin's theorem" (http://ieeexplore.ieee.org/search/searchresult.
jsp?newsearch=true&queryText=James+E.+Brittain+Thevenin's+theorem&.x=41&.y=17). IEEE Spectrum
27 (3): 42. doi: 10.1109/6.48845 (http://dx.doi.org/10.1109/6.48845). Retrieved 1 February 2013.
• Dorf, Richard C.; Svoboda, James A. (2010). "Chapter 5 - Circuit Theorems" (http://ca.wiley.com/WileyCDA/
WileyTitle/productCd-EHEP000347.html). Introduction to Electric Circuits (8th ed.). Hoboken, NJ: John Wiley
& Sons. pp. 162–207. ISBN 978-0-470-52157-1.
• Dwight, Herbert B. (1949). "Sec. 2 - Electric and Magnetic Circuits". In Knowlton, A.E. Standard Handbook for
Electrical Engineers (8th ed.). McGraw-Hill. p. 26.
• Elgerd, Olle I. (2007). "Chapter 10, Energy System Transients - Surge Phenomena and Symmetrical Fault
Analysis" (http://books.google.ca/books/about/Electric_Energy_Systems_Theory.html?id=AKTi3UxfhlgC&
Thévenin's theorem
•
•
•
•
redir_esc=y). Electric Energy Systems Theory: An Introduction. Tata McGraw-Hill. pp. 402–429.
ISBN 978-0070192300.
Helmhotz, H. (1853). "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)" (http://gallica.bnf.fr/ark:/12148/
bpt6k151746.image.f225.langFR). Annalen der Physik und Chemie 89 (6): 211–233.
Johnson, D.H. (2003a). "Origins of the equivalent circuit concept: the voltage-source equivalent" (http://www.
ece.rice.edu/~dhj/paper1.pdf). Proceedings of the IEEE 91 (4): 636–640. doi: 10.1109/JPROC.2003.811716
(http://dx.doi.org/10.1109/JPROC.2003.811716).
Johnson, D.H. (2003b). "Origins of the equivalent circuit concept: the current-source equivalent" (http://www.
ece.rice.edu/~dhj/paper2.pdf). Proceedings of the IEEE 91 (5): 817–821. doi: 10.1109/JPROC.2003.811795
(http://dx.doi.org/10.1109/JPROC.2003.811795).
Thévenin, L. (1883a). "Extension de la loi d’Ohm aux circuits électromoteurs complexes (Extension of Ohm’s law
to complex electromotive circuits)" (http://books.google.com/?id=shUAAAAAMAAJ&pg=PA222&
lpg=PA222&dq=Extension+de+la+loi+d’Ohm+aux+circuits+électromoteurs+complexes#v=onepage&
q=Extension de la loi d’Ohm aux circuits électromoteurs complexes&f=false). Annales Télégraphiques. 3e
series 10: 222–224.
• Thévenin, L. (1883b). "Sur un nouveau théorème d'électricité dynamique (On a new theorem of dynamic
electricity)". Comptes rendus hebdomadaires des séances de l'Académie des Sciences 97: 159–161.
• Wenner, F. (1926). Sci. Paper S531, A principle governing the distribution of current in systems of linear
conductors. Washington, D.C.: Bureau of Standards.
External links
• Thevenin's theorem at allaboutcircuits.com (http://www.allaboutcircuits.com/vol_1/chpt_10/8.html)
• Filter-Order Filters: Shortcut via Thévenin Equivalent Source (http://www.tedpavlic.com/teaching/osu/
ece209/support/circuits_sys_review.pdf) — 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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