Mathematical Problems in Engineering

Volume 2018, Article ID 2613890, 7 pages

https://doi.org/10.1155/2018/2613890

## A Simple Method for Impasse Points Detection in Nonlinear Electrical Circuits

^{1}Departamento de Ciencias Básicas, Facultad de Ingeniería, Universidad Nacional de La Plata, Buenos Aires, Argentina^{2}Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Buenos Aires, Argentina^{3}LEICI, Facultad de Ingeniería, Universidad Nacional de La Plata and CONICET, Buenos Aires, Argentina

Correspondence should be addressed to María del Rosario Etchechoury; moc.liamg@etam.aliram

Received 27 October 2017; Accepted 30 January 2018; Published 5 March 2018

Academic Editor: Marco Spadini

Copyright © 2018 Diana Leonor Kleiman et al. 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

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.

#### 1. Introduction

*Implicit Differential Equations* appear frequently while modelling different physical systems in many areas. Many works about them have been inspired by applications in* circuit theory* [1–5]. In this context, as in many others, these equations are known as* Differential Algebraic Equations* [6, 7]. Certain initial value problems concerning the existence and extension of solutions for some singular points are of interest [8]. In particular, we are interested in those solutions for* impasse singular points,* that is, singular points where solutions collapse (sometimes with infinite velocity) and cannot continue; we call these curves* impasse solutions*. The existence of impasse points in a circuit indicates that the model is defective, so it must be remodelled by adding some parasitic capacitors and/or inductors. Moreover, by attempting to solve the system through numerical methods, solutions could oscillate near the impasse points. Then, it seems relevant to develop analytic methods in order to detect such points.

In the literature, there are some results that ensure the existence of impasse points [9–15], some of them concerned with electrical circuits. In this article, we obtained a new method to straightforwardly detect the existence of impasse points in two families of nonlinear electrical circuits. In fact, we set a characterisation of an impasse point in terms of a specific function which appears in the data of the corresponding circuit.

This paper is organised as follows. In Section 2, known definitions and general results that we need along all the article are presented. The new results, where we get conditions that ensure the existence of impasse points in both families of circuits, are introduced in Section 3; moreover, we develop some concrete examples to illustrate these new conditions. Conclusions are drawn in Section 4.

#### 2. Preliminaries

A first-order Implicit Differential Equation (IDE), defined on a dimension manifold , can be described aswhere is a function defined on the tangent bundle and is a dimension linear space.

A* solution curve* of (1) is a function , defined on an open real interval , such that is differentiable for all and

For sufficiently smooth, if and , for all in a neighbourhood of , it is possible to find a locally equivalent explicit ODE , with [16].

The points such that are called* singular points* of the IDE.

##### 2.1. Quasilinear Implicit Differential Equation

Let us consider a dimension manifold , a dimension linear space , a smooth application such that is linear in , and a given smooth map A* Quasilinear Implicit Differential Equation* (QLIDE) is represented by

##### 2.2. Solution Curve of a QLIDE

A curve , with being a real interval, , , , or , is a solution of (2) if is a continuous function in , differentiable in the interior of , such that satisfies (2), for all in the interior of Moreover, if , then satisfies (2), or if then satisfies (2), where and .

##### 2.3. Singular Set

For a given IDE (1), the* singular set* is and its elements are called* singular points.* In particular, for a given QLIDE (2), its singular set is We call the* regular set* of (2).

##### 2.4. Hypothesis

From now on, we shall consider (2) with the following restrictions: is a real analytic connected manifold with , and are real analytic maps, and finally is not identically null on . In this situation, the regular set is not empty.

##### 2.5. Impasse Point

For a given point , we say that a solution curve of (2) , with and , for all , has a* forward impasse point* (resp.,* backward impasse point*) in if when (resp., ) and (resp., ) does not exist.

If there is a curve solution with an impasse point (backward or forward) in , we call this curve an* impasse solution* of (2) in

##### 2.6. Essential and Nonessential Singularities

In the analysis of the existence of impasse points in (2), we shall use the classification of singularities given in [17]. If and , the classification corresponds to the analysis of the vector field and is related to the existence of a continuous extension of such vector field.

In fact, we can decompose the singular set as the disjoint union , where is the set of* essential singularities* and is the set of* nonessential singularities*.

The following theorem [11] sets a necessary and sufficient condition for an essential singularity to be an impasse point of (2).

First, we recursively define as follows:The family of maps generates a chain of ideals , with

Let be the set of zeros of and we consider Then, . Since the ring of analytic functions at one point is Noetherian [18], then it is possible to determine whether or not, in a finite number of steps.

Theorem 1. *Let the equation given in (2) be valid and let a point Let the chain , , be defined as in (5) with corresponding zero set of , Then is an impasse point of (2) iff *

#### 3. Impasse Points in Nonlinear Electrical Circuits

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.

##### 3.1. Impasse Points in a Parallel Nonlinear Circuit

We consider the generic parallel circuit (see Figure 1) with a nonlinear capacitor , a nonlinear inductor , and a nonlinear memoryless two-terminal element (NME) [14], such as a nonlinear resistor, a diode, and a tunnel diode. Note that if any of the components of the parallel circuit is linear, it can be treated as a particular case of a comprehensive nonlinear family.