As originally stated in terms of direct-current resistive circuits only, **Thévenin's theorem** states that *"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}."*

- A proof of the theorem
- Helmholtz's proof
- Calculating the Thévenin equivalent
- Example
- Conversion to a Norton equivalent
- Practical limitations
- 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 (i.e., sources are set to zero). - If terminals A and B are connected to one another, the current flowing from A and B will be 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 (inductive and capacitive) 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] }

Various proofs have been given of Thévenin's theorem. Perhaps the simplest of these was the proof in Thévenin's original paper.^{ [3] } Not only is that proof elegant and easy to understand, but a consensus exists^{ [4] } that Thévenin's proof is both correct and general in its applicability. The proof goes as follows:

Consider an active network containing impedances, (constant-) voltage sources and (constant-) current sources. The configuration of the network can be anything. Access to the network is provided by a pair of terminals. Designate the voltage measured between the terminals as *V*_{θ}, as shown in the box on the left side of Figure 2.

Suppose that the voltage sources within the box are replaced by short circuits, and the current sources by open circuits. If this is done, no voltage appears across the terminals, and it is possible to measure the impedance between the terminals. Call this impedance *Z*_{θ}.

Now suppose that one attaches some linear network to the terminals of the box, having impedance *Z*_{e}, as in Figure 2a. We wish to find the current *I* through *Z*_{e}. The answer is not obvious, since the terminal voltage will not be *V*_{θ} after *Z*_{e} is connected.

Instead, we imagine that we attach, in series with impedance *Z*_{e}, a source with electromotive force *E* equal to *V*_{θ} but directed to oppose *V*_{θ}, as shown in Figure 2b. No current will then flow through *Z*_{e} since *E* balances *V*_{θ}.

Next, we insert another source of electromotive force, *E*_{1}, in series with *Z*_{e}, where *E*_{1} has the same magnitude as *E* but is opposed in direction (see Figure 2c). The current, *I*_{1}, can be determined as follows: it is the current that would result from *E*_{1} acting alone, with all other sources (within the active network and the external network) set to zero. This current is, therefore,

because *Z*_{e} is the impedance external to the box and *Z*_{θ} looking into the box when its sources are zero.

Finally, we note that *E* and *E*_{1} can be removed together without changing the current, and when they are removed, we are back to Figure 2a. Therefore *I*_{1} is the current, *I*, that we are seeking, i.e.

thus completing the proof. Figure 2d shows the Thévenin equivalent circuit.

As noted, Thévenin's theorem was first discovered and published by the German scientist Hermann von Helmholtz in 1853,^{ [1] } four years before Thévenin's birth. Thévenin's 1883 proof, described above, is nearer in spirit to modern methods of electrical engineering, and this may explain why his name is more commonly associated with the theorem.^{ [11] } Helmholtz's earlier formulation of the problem reflects a more general approach that is closer to physics.

In his 1853 paper, Helmholtz was concerned with the electromotive properties of "physically extensive conductors", in particular, with animal tissue. He noted that earlier work by physiologist Emil du Bois-Reymond had shown that "every smallest part of a muscle that can be stimulated is capable of producing electrical currents." At this time, experiments were carried out by attaching a galvanometer at two points to a sample of animal tissue and measuring current flow through the external circuit. Since the goal of this work was to understand something about the internal properties of the tissue, Helmholtz wanted to find a way to relate those internal properties to the currents measured externally.

Helmholtz's starting point was a result published by Gustav Kirchhoff in 1848. ^{ [12] } Like Helmholtz, Kirchhoff was concerned with three-dimensional, electrically conducting systems. Kirchhoff considered a system consisting of two parts, which he labelled parts A and B. Part A (which played the part of the "active network" in Fig. 2 above) consisted of a collection of conducting bodies connected end to end, each body characterized by an electromotive force and a resistance. Part B was assumed to be connected to the endpoints of A via two wires. Kirchhoff then showed (p. 195) that "without changing the flow at any point in B, one can substitute for A a conductor in which an electromotive force is located which is equal to the sum of the voltage differences in A, and which has a resistance equal to the summed resistances of the elements of A".

In his 1853 paper, Helmholtz acknowledged Kirchhoff's result, but noted that it was only valid in the case that, "as in hydroelectric batteries", there are no closed current curves in A, but rather that all such curves pass through B. He therefore set out to generalize Kirchhoff's result to the case of an arbitrary, three-dimensional distribution of currents and voltage sources within system A.

Helmholtz began by providing a more general formulation than had previously been published of the superposition principle, which he expressed (p. 212-213) as follows:

If any system of conductors contains electromotive forces at various locations, the electrical voltage at every point in the system through which the current flows is equal to the algebraic sum of those voltages which each of the electromotive forces would produce independently of the others. And similarly, the components of the current intensity that are parallel to three perpendicular axes are equal to the sum of the corresponding components that belong to the individual forces.

Using this theorem, as well as Ohm's law, Helmholtz proved the following three theorems about the relation between the internal voltages and currents of "physical" system A, and the current flowing through the "linear" system B, which was assumed to be attached to A at two points on its surface:

- For every conductor A, within whose interior electromotive forces are arbitrarily distributed, a certain distribution of electromotive forces can be specified on its surface, which would produce the same currents as the internal forces of A in every applied conductor B.
- The voltages and current components inside the conductor A when an external circuit is attached are equal to the sum of the voltages and current components that occur in it in the absence of the attached circuit and those of the surface.
- Different ways of distributing electromotive forces on the surface of the conductor A, which should give the same derived currents as its internal forces, can only differ by a difference that has the same constant value at all points on the surface.

From these, Helmholtz derived his final result (p. 222):

If a physical conductor with constant electromotive forces in two specific points on its surface is connected to any linear conductor, then in its place one can always substitute a linear conductor with a certain electromotive force and a certain resistance, which in all applied linear conductors would excite exactly the same currents as the physical one. ... The resistance of the linear conductor to be substituted is equal to that of the body when a current is passed through it from the two entry points of the linear conductor.

He then noted that his result, derived for a general "physical system", also applied to "linear" (in a geometric sense) circuits like those considered by Kirchhoff:

What applies to every physical conductor also applies to the special case of a branched linear current system. Even if two specific points of such a system are connected to any other linear conductors, it behaves compared to them like a linear conductor of certain resistance, the magnitude of which can be found according to the well-known rules for branched lines, and of certain electromotive force, which is given by the voltage difference of the derived points as it existed before the added circuit.

This formulation of the theorem is essentially the same as Thévenin's, published 30 years later.

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 the sources' values (meaning their voltage or current) are set 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 the source is replaced 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 (*R _{x}* ||

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.

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*,^{ [13] } 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. Thus all circuits are networks, but not all networks are circuits. 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.

An **inductor**, also called a **coil**, **choke**, or **reactor**, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a coil.

The **characteristic impedance** or **surge impedance** (usually written Z_{0}) of a uniform transmission line is the ratio of the amplitudes of voltage and current of a wave travelling in one direction along the line in the absence of reflections in the other direction. Equivalently, it can be defined as the input impedance of a transmission line when its length is infinite. Characteristic impedance is determined by the geometry and materials of the transmission line and, for a uniform line, is not dependent on its length. The SI unit of characteristic impedance is the ohm.

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

The **electrical resistance** of an object is a measure of its opposition to the flow of electric current. Its reciprocal quantity is **electrical conductance**, measuring the ease with which an electric current passes. Electrical resistance shares some conceptual parallels with mechanical friction. The SI unit of electrical resistance is the ohm, while electrical conductance is measured in siemens (S).

In electromagnetism and electronics, **electromotive force** is an energy transfer to an electric circuit per unit of electric charge, measured in volts. Devices called electrical *transducers* provide an emf by converting other forms of energy into electrical energy. Other types of electrical equipment also produce an emf, such as batteries, which convert chemical energy, and generators, which convert mechanical energy. This energy conversion is achieved by physical forces applying physical work on electric charges. However, electromotive force itself is not a physical force, and ISO/IEC standards have deprecated the term in favor of **source voltage** or **source tension** instead.

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**".

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.

**Kirchhoff's circuit laws** are two equalities that deal with the current and potential difference in the lumped element model of electrical circuits. They were first described in 1845 by German physicist Gustav Kirchhoff. This generalized the work of Georg Ohm and preceded the work of James Clerk Maxwell. Widely used in electrical engineering, they are also called **Kirchhoff's rules** or simply **Kirchhoff's laws**. These laws can be applied in time and frequency domains and form the basis for network analysis.

An **LC circuit**, also called a **resonant circuit**, **tank circuit**, or **tuned circuit**, is an electric circuit consisting of an inductor, represented by the letter L, and a capacitor, represented by the letter C, connected together. The circuit can act as an electrical resonator, an electrical analogue of a tuning fork, storing energy oscillating at the circuit's resonant frequency.

In electrical engineering, 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.

In electrical engineering and electronics, a *network* 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.

In electrical engineering, a practical electric power source which is a linear 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 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 (

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 amplifier's input and output terminals, and 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.

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.

The **impedance analogy** is a method of representing a mechanical system by an analogous electrical system. The advantage of doing this is that there is a large body of theory and analysis techniques concerning complex electrical systems, especially in the field of filters. By converting to an electrical representation, these tools in the electrical domain can be directly applied to a mechanical system without modification. A further advantage occurs in electromechanical systems: Converting the mechanical part of such a system into the electrical domain allows the entire system to be analysed as a unified whole.

The **mobility analogy, **also called **admittance analogy** or **Firestone analogy**, is a method of representing a mechanical system by an analogous electrical system. The advantage of doing this is that there is a large body of theory and analysis techniques concerning complex electrical systems, especially in the field of filters. By converting to an electrical representation, these tools in the electrical domain can be directly applied to a mechanical system without modification. A further advantage occurs in electromechanical systems: Converting the mechanical part of such a system into the electrical domain allows the entire system to be analysed as a unified whole.

- 1 2 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. - 1 2 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. - 1 2 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 [in German] (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. - ↑ Maloberti, Franco; Davies, Anthony C. (2016).
*A Short History of Circuits and Systems*. Delft: River Publishers. p. 37. ISBN 978-87-93379-71-8. - ↑ Kirchhoff, Gustav (1848). "Ueber die Anwendbarkeit der Formeln für die Intensitäten der galvanischen Ströme in einem Systeme linearer Leiter auf Systeme, die zum Theil aus nicht linearen Leitern bestehen" [On the applicability of the formulas for the intensities of the galvanic currents in a system of linear conductors to systems that partly consist of non-linear conductors].
*Annalen der Physik und Chemie*(in German).**75**: 189–205. doi:10.1002/andp.18481511003. - ↑ 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*.**39**(1). Washington, D.C.: Bureau of Standards: 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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