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As originally stated in terms of direct-current resistive circuits only, **Thévenin's theorem** states that *"For any linear electrical network containing only voltage sources, current sources and resistances can be replaced at terminals A–B by an equivalent combination of a voltage source V _{th} in a series connection with a resistance R_{th}."*

- Calculating the Thévenin equivalent
- Example
- Conversion to a Norton equivalent
- Practical limitations
- A proof of the theorem
- In three-phase circuits
- See also
- References
- Further reading
- External links

- The equivalent voltage
*V*_{th}is the voltage obtained at terminals A–B of the network with terminals A–B open circuited. - The equivalent resistance
*R*_{th}is the resistance that the circuit between terminals A and B would have if all ideal voltage sources in the circuit were replaced by a short circuit and all ideal current sources were replaced by an open circuit. - If terminals A and B are connected to one another, the current flowing from A to B will be
*V*_{th}/*R*_{th}. This means that*R*_{th}could alternatively be calculated as*V*_{th}divided by the short-circuit current between A and B when they are connected together.

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. It means the theorem applies for AC in an exactly same way to DC except that resistances are generalized to 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] }^{ [7] }

Thévenin's theorem and its dual, Norton's theorem, are widely used to make circuit analysis simpler and to study a circuit's initial-condition and steady-state response.^{ [8] }^{ [9] } 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.^{ [7] }^{ [10] }

The equivalent circuit is a voltage source with voltage *V*_{Th} in series with a resistance *R*_{Th}.

The Thévenin-equivalent voltage *V*_{Th} is the open-circuit 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*_{out} and the other terminal to be at the ground point.

The Thévenin-equivalent resistance *R*_{Th} is the resistance measured across points A and B "looking back" into the circuit. The resistance is measured after replacing all voltage- and current-sources with their internal resistances. That means an ideal voltage source is replaced with a short circuit, and an ideal current source is replaced 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.

As a mnemonic, the Thevenin replacements for voltage and current sources can be remembered as we turn the sources' values (meaning their voltage or current) to zero. A zero valued voltage source would create a potential difference of zero volts between its terminals, just like an ideal short circuit would do, with two leads touching; therefore we replace the source with a short circuit. Similarly, a zero valued current source and an *open* circuit both pass zero current.

In the example, calculating the equivalent voltage:

(Notice that *R*_{1} 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_{1} and therefore no voltage drop along this part.)

Calculating equivalent resistance ( is the total resistance of two parallel resistors):

A Norton equivalent circuit is related to the Thévenin equivalent by

- Many circuits are only linear over a certain range of values, thus the Thévenin equivalent is valid only within this linear 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 implemented.

The proof involves two steps. The first step is to use superposition theorem to construct a solution. Then, uniqueness theorem is employed to show that the obtained solution is unique. It is noted that the second step is usually implied in literature.

By using superposition of specific configurations, it can be shown that for any linear "black box" circuit which contains voltage sources and resistors, its voltage is a linear function of the corresponding current as follows

Here, the first term reflects the linear summation of contributions from each voltage source, while the second term measures the contributions from all the resistors. The above expression is obtained by using 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 (Alternatively, one can use an ideal current source of current ). Moreover, it is straightforward to show that and are the single voltage source and the single series resistor in question.

As a matter of fact, the above relation between and is established by superposition of some particular configurations. Now, the uniqueness theorem guarantees that the result is general. To be specific, there is one and only one value of once the value of is given. In other words, the above relation holds true independent of what the "black box" is plugged to.

In 1933, A. T. Starr published a generalization of Thévenin's theorem in an article of the magazine *Institute of Electrical Engineers Journal*, titled *A New Theorem for Active Networks*,^{ [11] } which states that any three-terminal active linear network can be substituted by three voltage sources with corresponding impedances, connected in wye or in delta.

An **electrical network** is an interconnection of electrical components or a model of such an interconnection, consisting of electrical elements. An **electrical circuit** is a network consisting of a closed loop, giving a return path for the current. Linear electrical networks, a special type consisting only of sources, linear lumped elements, and linear distributed elements, have the property that signals are linearly superimposable. They are thus more easily analyzed, using powerful frequency domain methods such as Laplace transforms, to determine DC response, AC response, and transient response.

In electrical engineering, **impedance** is the opposition to alternating current presented by the combined effect of resistance and reactance in a circuit.

In electrical engineering, the **maximum power transfer theorem** states that, to obtain *maximum* external power from a power source with internal resistance, the resistance of the load must equal the resistance of the source as viewed from its output terminals. Moritz von Jacobi published the maximum power (transfer) theorem around 1840; it is also referred to as "**Jacobi's law**".

**Johnson–Nyquist noise** is the electronic noise generated by the thermal agitation of the charge carriers inside an electrical conductor at equilibrium, which happens regardless of any applied voltage. Thermal noise is present in all electrical circuits, and in sensitive electronic equipment can drown out weak signals, and can be the limiting factor on sensitivity of electrical measuring instruments. Thermal noise increases with temperature. Some sensitive electronic equipment such as radio telescope receivers are cooled to cryogenic temperatures to reduce thermal noise in their circuits. The generic, statistical physical derivation of this noise is called the fluctuation-dissipation theorem, where generalized impedance or generalized susceptibility is used to characterize the medium.

In direct-current circuit theory, **Norton's theorem**, also called the **Mayer–Norton theorem**, is a simplification that can be applied to networks made of linear time-invariant resistances, voltage sources, and current sources. At a pair of terminals of the network, it can be replaced by a current source and a single resistor in parallel.

In electronics, **negative resistance** (**NR**) is a property of some electrical circuits and devices in which an increase in voltage across the device's terminals results in a decrease in electric current through it.

In electronics, a **voltage divider ** is a passive linear circuit that produces an output voltage (*V*_{out}) that is a fraction of its input voltage (*V*_{in}). **Voltage division** is the result of distributing the input voltage among the components of the divider. A simple example of a voltage divider is two resistors connected in series, with the input voltage applied across the resistor pair and the output voltage emerging from the connection between them.

A **gyrator** is a passive, linear, lossless, two-port electrical network element proposed in 1948 by Bernard D. H. Tellegen as a hypothetical fifth linear element after the resistor, capacitor, inductor and ideal transformer. Unlike the four conventional elements, the gyrator is non-reciprocal. Gyrators permit network realizations of two-(or-more)-port devices which cannot be realized with just the conventional four elements. In particular, gyrators make possible network realizations of isolators and circulators. Gyrators do not however change the range of one-port devices that can be realized. Although the gyrator was conceived as a fifth linear element, its adoption makes both the ideal transformer and either the capacitor or inductor redundant. Thus the number of necessary linear elements is in fact reduced to three. Circuits that function as gyrators can be built with transistors and op-amps using feedback.

The **output impedance** of an electrical network is the measure of the opposition to current flow (impedance), both static (resistance) and dynamic (reactance), into the load network being connected that is *internal* to the electrical source. The output impedance is a measure of the source's propensity to drop in voltage when the load draws current, the source network being the portion of the network that transmits and the load network being the portion of the network that consumes.

A network, in the context of electrical engineering and electronics, is a collection of interconnected components. **Network analysis** is the process of finding the voltages across, and the currents through, all network components. There are many techniques for calculating these values. However, for the most part, the techniques assume linear components. Except where stated, the methods described in this article are applicable only to *linear* network analysis.

A **current source** is an electronic circuit that delivers or absorbs an electric current which is independent of the voltage across it.

A practical electrical power source which is a linear electric circuit may, according to Thévenin's theorem, be represented as an ideal voltage source in series with an impedance. This impedance is termed the **internal resistance** of the source. When the power source delivers current, the measured voltage output is lower than the no-load voltage; the difference is the voltage drop caused by the internal resistance. The concept of internal resistance applies to all kinds of electrical sources and is useful for analyzing many types of electrical circuits.

A **voltage source** is a two-terminal device which can maintain a fixed voltage. An ideal voltage source can maintain the fixed voltage independent of the load resistance or the output current. However, a real-world voltage source cannot supply unlimited current.

In electronics, a **current divider ** is a simple linear circuit that produces an output current (*I*_{X}) that is a fraction of its input current (*I*_{T}). **Current division** refers to the splitting of current between the branches of the divider. The currents in the various branches of such a circuit will always divide in such a way as to minimize the total energy expended.

In electronics, the **Miller effect** accounts for the increase in the equivalent input capacitance of an inverting voltage amplifier due to amplification of the effect of capacitance between the input and output terminals. The virtually increased input capacitance due to the Miller effect is given by

**Source transformation** is the process of simplifying a circuit solution, especially with mixed sources, by transforming voltage sources into current sources, and vice versa, using Thévenin's theorem and Norton's theorem respectively.

**Ripple** in electronics is the residual periodic variation of the DC voltage within a power supply which has been derived from an alternating current (AC) source. This ripple is due to incomplete suppression of the alternating waveform after rectification. Ripple voltage originates as the output of a rectifier or from generation and commutation of DC power.

The **superposition theorem** is a derived result of the superposition principle suited to the network analysis of electrical circuits. The superposition theorem states that for a linear system the response in any branch of a bilateral linear circuit having more than one independent source equals the algebraic sum of the responses caused by each independent source acting alone, where all the other independent sources are replaced by their internal impedances.

An **equivalent impedance** is an equivalent circuit of an electrical network of impedance elements which presents the same impedance between all pairs of terminals as did the given network. This article describes mathematical transformations between some passive, linear impedance networks commonly found in electronic circuits.

An **RLC circuit** is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components of this circuit, where the sequence of the components may vary from RLC.

- ↑ von Helmholtz, Hermann (1853). "Ueber 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].
*Annalen der Physik und Chemie*(in German).**89**(6): 211–233. Bibcode:1853AnP...165..211H. doi:10.1002/andp.18531650603. - ↑ Thévenin, Léon Charles (1883). "Extension de la loi d'Ohm aux circuits électromoteurs complexes" [Extension of Ohm's law to complex electromotive circuits].
*Annales Télégraphiques*. 3^{e}series (in French).**10**: 222–224. - ↑ Thévenin, Léon Charles (1883). "Sur un nouveau théorème d'électricité dynamique" [On a new theorem of dynamic electricity].
*Comptes rendus hebdomadaires des séances de l'Académie des Sciences*(in French).**97**: 159–161. - ↑ Johnson, Don H. (2003). "Origins of the equivalent circuit concept: the voltage-source equivalent" (PDF).
*Proceedings of the IEEE*.**91**(4): 636–640. doi:10.1109/JPROC.2003.811716. hdl: 1911/19968 . - ↑ Johnson, Don H. (2003). "Origins of the equivalent circuit concept: the current-source equivalent" (PDF).
*Proceedings of the IEEE*.**91**(5): 817–821. doi:10.1109/JPROC.2003.811795. - ↑ Brittain, James E. (March 1990). "Thevenin's theorem".
*IEEE Spectrum*.**27**(3): 42. doi:10.1109/6.48845. S2CID 2279777 . Retrieved 2013-02-01. - 1 2 Dorf, Richard C.; Svoboda, James A. (2010). "Chapter 5: Circuit Theorems".
*Introduction to Electric Circuits*(8th ed.). Hoboken, NJ, USA: John Wiley & Sons. pp. 162–207. ISBN 978-0-470-52157-1. - ↑ Brenner, Egon; Javid, Mansour (1959). "Chapter 12: Network Functions".
*Analysis of Electric Circuits*. McGraw-Hill. pp. 268–269. - ↑ Elgerd, Olle Ingemar (2007). "Chapter 10: Energy System Transients - Surge Phenomena and Symmetrical Fault Analysis".
*Electric Energy Systems Theory: An Introduction*. Tata McGraw-Hill. pp. 402–429. ISBN 978-0-07019230-0. - ↑ Dwight, Herbert Bristol (1949). "Section 2: Electric and Magnetic Circuits". In Knowlton, Archer E. (ed.).
*Standard Handbook for Electrical Engineers*(8th ed.). McGraw-Hill. p. 26. - ↑ Starr, A. T. (1933). "A new theorem for active networks".
*Journal of the Institution of Electrical Engineers*.**73**(441): 303–308. doi:10.1049/jiee-1.1933.0129.

- Wenner, Frank (1926). "A principle governing the distribution of current in systems of linear conductors".
*Proceedings of the Physical Society*. Washington, D.C.: Bureau of Standards.**39**(1): 124–144. Bibcode:1926PPS....39..124W. doi:10.1088/0959-5309/39/1/311. hdl: 2027/mdp.39015086551663 . Scientific Paper S531. - First-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.

- Media related to Thévenin's theorem at Wikimedia Commons

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