#### 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 [15]. 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 [915], 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.

The corresponding currents are , , and , respectively, and is the common voltage. Nonlinear relations are assumed between the capacitor charge and and the inductor flow and . and are real analytic nonlinear functions defined as and , with and . In the branch of the NME, and are related by means of the equation , with being an analytic real function that is nonconstant and nonlinear.

From the Kirchhoff Current Law, we get . In addition, from electromagnetic relations, and . Calling , , , , and , the QLIDE that models the circuit iswithAs , then is singular of rank , for all

Equation (6) is equivalent to the following system defined for all [19]:withas , and then

By considering that and are both positive functions, the singular set is

If , then, in particular, (8) has no singular impasse points. So, from now on, we shall assume that Moreover, as is not identically null, Then, we are interested in solutions on achieving singular impasse points (backward or forward) (i.e., impasse solutions).

In order to use the results set in the previous section, we introducewhere by simple calculations we get The set of essential singularities defined by is not empty.

In the main result of this subsection, we get a necessary and sufficient condition for an essential singularity to be an impasse point of (8). Previously, we set a lemma and finally, as a corollary, we conclude that any essential singularity of (8) is an impasse point.

Lemma 2. Let the equation given in (8) be valid. By considering and defined as in (11) and (12), respectively, and the functions defined recursively as in (5), then, for each , there are functions such thatwith

Proof. The proof is given by induction on .
For , by simple calculations, it is easy to prove that withFor , assuming the inductive hypothesis for , we calculate the expression of :Finally, for , it can be easily shown that the coefficients are defined recursively as

Theorem 3. Let the equation given in (8) be valid and an essential singularity of (8). Then, is an impasse point of (8) iff there exists , , such that .

Proof. By Lemma 2, the general formula for isUsing the expression set in Lemma 2 for the coefficients , the following statement can be proved by induction on , : By applying this statement and the characterisation of an impasse point given in Theorem 1, we can conclude the proof.

Corollary 4. Any essential singularity of (8) is an impasse point.

Proof. Let be an essential singularity of (8). If is not an impasse point of (8), then, by Theorem 3, , for all Moreover, as is an essential singularity, it holds that Finally the analyticity of allows us to ensure that , for all ; that is, is a constant function on , which contradicts one of the assumptions set for the circuit at the beginning of this section.
We conclude that any essential singularity of (8) is an impasse point.

Example 5. We consider a particular case in the family of parallel circuits with nonlinear [9] and, for the sake of simplicity, constants and . By calling , , , and , we obtain and , for all
Then, the set of essential singularities isBy Corollary 4, we conclude that all the essential singularities are impasse points.
In Figure 2, we show the graphic of and some impasse solutions. In addition, by simple calculations, we conclude that and are nonessential singularities and , , and are regular equilibrium points.

##### 3.2. Impasse Points in a Series Nonlinear Circuit

In this subsection, we consider the case dual to the one in Section 3.1, that is, a generic series circuit (see Figure 3) with a nonlinear capacitor , a nonlinear inductor , and a nonlinear memoryless two-terminal element. Similar to the parallel case, if any of the circuit components is linear, it can be treated as a particular case of a broader nonlinear family.

Let and be real analytic functions, in general nonlinear ones, defined as and , where depends on the capacitor voltage and depends on the inductor current . The voltage of the NME is , with the real analytic nonlinear nonconstant function , representing the voltage-current relation of the NME. The inductor voltage is .

Then, with , the QLIDE that models the circuit iswhere By considering the voltage-current relation, the original model of circuit (23) is equivalent to an order system depending on the variables and For all , the reduced model iswith The singular set is

If , there are no singular impasse points for (25). So, in order to analyse the existence of impasse solutions on , we shall assume that

In order to use the results developed in Section 2, we calculateThe set of essential singularities is

Remark 6. 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 (25).

Example 7. We consider the model of a diode-tunnel circuit [12] as a particular case in the family of series nonlinear circuits with the voltage-current relation given by , , and
Then, the set of essential singularities iswhich are all impasse points of the circuit.
In Figure 4, we show the graphic of and some impasse solutions.

In addition, by simple calculations, we conclude that and are nonessential singularities and is a regular equilibrium.

#### 4. Conclusions

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.

#### Conflicts of Interest

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

#### Acknowledgments

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.