Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 743189, 7 pages

http://dx.doi.org/10.1155/2015/743189

## Two Mathematical Comments on the Thevenin Theorem: An “Algebraic Ideal” and the “Affine Nonlinearity”

Kinneret College on the Sea of Galilee, 15132 Jordan Valley, Israel

Received 10 March 2015; Accepted 26 May 2015

Academic Editor: Tito Busani

Copyright © 2015 Emanuel Gluskin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We discuss the most important and simple concept of basic circuit theory—the concept of the unideal source—or the Thevenin circuit. It is explained firstly how the Thevenin circuit can be interpreted in the algebraic sense. Then, we critically consider the common opinion that it is a linear circuit, showing that linearity (or nonlinearity) depends on the use of the port. The difference between the cases of a source being an input or an internal element (as it is in Thevenin’s circuit) is important here. The distinction in the definition of linear operator in algebra (here in system theory) and in geometry is also important for the subject, and we suggest the wide use of the concept of “affine nonlinearity.” This kind of nonlinearity should be relevant for the development of complicated circuitry (perhaps in a biological modeling context) with nonprescribed definition of subsystems, when the interpretation of a port as input or output can become dependent on the local intensity of a process.

#### 1. Introduction

For the first time, the equivalent circuit was suggested by Herman von Helmholtz in 1853. Later, it was rediscovered (and proved for complicated linear 1-ports) by Léon Charles Thevenin (in 1883) and then by Edward Lawry Norton and Hans Ferdinand Mayer (both in 1926) [1]. Since passing from the series circuit of Thevenin, including a voltage source, to the parallel Norton version, including a current source, is an immediate application of the equivalent generator theorem, we will always speak about “Thevenin theorem,” or just “theorem,” meaning the series circuit.

Though the rigor proof (in [2, 3] using the basic substitution theorem) of the theorem is nontrivial, the very result is not surprising. Indeed, when having some voltage at the port of a circuit, one can naturally try to use the circuit as a voltage source. Since, furthermore, it is difficult to create a good voltage source, the idea of an equivalent circuit with an internal resistor (or an internal impedance , or , but we will use the simplest classical model with a usual resistor) providing the dependence of the output voltage on the load, that is, the nonideality of the source, naturally appears.

What is really surprising is not the technical but the logical side, namely, the fact that Helmholtz, who was able to formulate in 1847 the law of conservation of energy for both mechanical and electrical systems, suggested in 1853 the theorem only for electrical systems, while it is not difficult to replace the voltage source by a source of a mechanical force and find mechanical equivalent for the 1-port circuit. Thus, Helmholtz (who, of course, knew the circuit equations suggested by Kirchhoff in 1847 and could expect intensive development of circuit theory) saw in electrical engineering something relevant to the theorem, which is not found in mechanics. It is clear today that this “something” is associated with the concept of port (input, output), involved in the formulation of the theorem, because this concept is flexible not in mechanics, but in electrical engineering that easily creates very complicated structures that can be seen as composed of some subsystems relevant to the theorem. That is, the point is technology, but we will stress the theoretical side. See also [4].

The closely associated question to be asked is that of whether a subsystem is an active one, having a load, or by itself is a load of a stronger circuit. The first case is more common in the applications of the Thevenin theorem, but the “affine nonlinearity,” with which we will be concerned, is better seen in the second case, and it is methodologically important that the question about linearity or nonlinearity of a subsystem (or of the whole system) can be influenced by the thus-seen degree of activity of the circuit. This circumstance, perhaps unexpected for many, is, in fact, not surprising, because we always consider linearity (or nonlinearity) of a certain system, and any correct definition of a system must include definition of its ports. (See also [4] and references there, for a development of this outlook, showing, in particular, that nonlinearity does not just mean a curviness of a characteristic.)

The classical theorem, related to the most basic circuit-theory concept of nonideal source, is taught at the very beginning of the standard electrical engineering education, but revising just the most basic concepts is most useful, and this relates also to the Thevenin theorem. In the relatively recent works [5, 6] a criterion is suggested, not appearing in the classical theory, defining the conditions when a 1-port, including a dependent source, is, as a whole, an ideal, or a nonideal source, and here we make some new steps.

Regarding the use here of the “unpopular” concepts of affine nonlinearity and algebraic ideal, it should be noted that in [4] a state of the practically very important fluorescent lamp circuit is classified using the concept of affine nonlinearity. Without this interpretation, the (actually very strong) nonlinearity of the lamp circuit is not obvious. The fact that the circuit specialists should pay more attention to the concept of affine nonlinearity is not only because of the Thevenin circuit.

The present completion of the conceptual frame can attract mathematical students to circuit theory and, on the contrary, the electrical engineering students to some additional mathematics, and the vision of a complicated system, suggested in Section 4.2, may be relevant for a biological modeling. Hopefully, our discussion will finally motivate some applications that would become some “tools in hand” for an ordinary electrical engineer.

#### 2. The Circuit

Consider the well-known Thevenin circuit to which many linear 1-port circuits are reduced [1–3]. It is an equivalent circuit, which means that it influences the external circuit just as the original circuit does.

For simplicity, we will consider the Thevenin circuit as it is usually introduced, that is, for the simple resistive circuits. See Figure 1.