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In a nonlinear system, impasse points are singularities beyond which solutions are not continuable. In this article, we study two families of nonlinear electrical circuits, which can be represented by nonlinear Implicit Differential Equations. We set conditions that ensure the existence of impasse points in both families of circuits. In the literature, there exist general results to analyse the presence of such singularities in given differential equations of this type. However, the method proposed in this work allows detecting their existence in these electrical topologies in an extremely straightforward way, as illustrated by the examples of application.

In the literature, there are some results that ensure the existence of impasse points [

This paper is organised as follows. In Section

A first-order Implicit Differential Equation (IDE), defined on a dimension

A

For

The points

Let us consider a dimension

A curve

For a given IDE (

From now on, we shall consider (

For a given point

If there is a curve solution

In the analysis of the existence of impasse points in (

In fact, we can decompose the singular set as the disjoint union

The following theorem [

First, we recursively define

Let

Let the equation given in (

In this section, we find a necessary and sufficient condition for the existence of singular impasse points in two families of electrical circuits: the first one of parallel nonlinear circuits and the other one of series nonlinear circuits. In each case, this condition is obtained from the successive derivatives of a particular function that appears in the data of the corresponding circuit.

We consider the generic parallel circuit (see Figure

Parallel nonlinear circuit.

The corresponding currents are

From the Kirchhoff Current Law, we get

Equation (

By considering that

If

In order to use the results set in the previous section, we introduce

In the main result of this subsection, we get a necessary and sufficient condition for an essential singularity to be an impasse point of (

Let the equation given in (

The proof is given by induction on

For

Let the equation given in (

By Lemma

Any essential singularity of (

Let

We conclude that any essential singularity of (

We consider a particular case in the family of parallel circuits with nonlinear

Then, the set of essential singularities is

In Figure

Impasse solutions of Example

In this subsection, we consider the case dual to the one in Section

Series nonlinear circuit.

Let

Then, with

If

In order to use the results developed in Section

Analogous results, as those we proved for parallel nonlinear circuits in the previous subsection, are also valid in this family of series nonlinear circuits. Then, we can conclude again that all the essential singularities are impasse points of (

We consider the model of a diode-tunnel circuit [

Then, the set of essential singularities is

In Figure

Impasse solutions of Example

In addition, by simple calculations, we conclude that

This paper considered the analysis of impasse singular points in nonlinear electrical circuits, specifically in series and parallel connection. To this end, a theorem that provides a sufficient and necessary condition to determine the existence of impasse points in the aforementioned electrical systems is proposed and proved. Moreover, as a corollary, it has been established that all the essential singularities of those circuits are effectively impasse points.

Through the results obtained in this work, the presence of an impasse point can be analytically recognised by performing straightforward computations. This comes to be a useful instrument in the adequate modelling of nonlinear electrical circuits, given that the detection of impasse points suggests that the circuit model is defective and, consequently, a model refinement could be required.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This research was supported by the Universidad Nacional de La Plata, the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), and the Agencia Nacional de Promoción Científica y Tecnológica (ANPCyT), from Argentina.