Thévenin's theorem

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 Vth in a series connection with a resistance Rth."

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.

Designate the voltage measured between the terminals as Vθ, as shown in the box on the left side of Figure 2.

Now suppose that one attaches some linear network to the terminals of the box, having impedance Ze, as in Figure 2a.

The answer is not obvious, since the terminal voltage will not be Vθ after Ze is connected.

Instead, we imagine that we attach, in series with impedance Ze, a source with electromotive force E equal to Vθ but directed to oppose Vθ, as shown in Figure 2b.

Next, we insert another source of electromotive force, E1, in series with Ze, where E1 has the same magnitude as E but is opposed in direction (see Figure 2c).

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 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.

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: 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.

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.

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.

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.

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.