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.

Universidad Nacional de La Plata Consejo Nacional de Investigaciones Científicas y Técnicas Agencia Nacional de Promoción Científica y Tecnológica
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 . 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 . 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 , 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 m manifold M, can be described as(1)Φx,x˙=0,where Φ:TMF is a function defined on the tangent bundle TM and F is a dimension m linear space.

A solution curve of (1) is a function x:IM, defined on an open real interval I, such that x(t) is differentiable for all tI and x˙(t)=dx(t)/dt.

For Φ sufficiently smooth, if x0M and rgDvΦ(x,v)=m, for all x in a neighbourhood U of x0, it is possible to find a locally equivalent explicit ODE x˙=Φ1(x), with Φ1:UF .

The points xM such that rgDvΦ(x,v)<m are called singular points of the IDE.

2.1. Quasilinear Implicit Differential Equation

Let us consider a dimension m manifold M, a dimension m linear space F, a smooth application a:TMF such that a(x,x˙)a(x)x˙ is linear in x˙, and a given smooth map f:MF. A Quasilinear Implicit Differential Equation (QLIDE) is represented by(2)axx˙=fx.

2.2. Solution Curve of a QLIDE

A curve x:IM, with I being a real interval, I=(t0,t1), I=(t0,t1], I=[t0,t1), or I=[t0,t1], is a solution of (2) if x(t) is a continuous function in I, differentiable in the interior of I, such that (x(t),x˙(t)) satisfies (2), for all t in the interior of I. Moreover, if t0I, then (x(t0),x˙(t0+)) satisfies (2), or if t1I then (x(t1),x˙(t1-)) satisfies (2), where x˙(t0+)=limtt0+(xt-xt0/t-t0) and x˙(t1-)=limtt1-(xt-xt1/t-t1).

2.3. Singular Set

For a given IDE (1), the singular set is (3)Ms=xM:rgDvΦx,v<mand its elements are called singular points. In particular, for a given QLIDE (2), its singular set is (4)Ms=xM:rgax<m.We call Mr=M-Ms the regular set of (2).

2.4. Hypothesis

From now on, we shall consider (2) with the following restrictions: M is a real analytic connected manifold with dimM=m, a:MRm×Rm and f:MRm are real analytic maps, and finally deta(x) is not identically null on M. In this situation, the regular set Mr=M-Ms is not empty.

2.5. Impasse Point

For a given point x0Ms, we say that a solution curve of (2) x(t), with t(t0,t1) and x(t)Mr, for all t, has a forward impasse point (resp., backward impasse point) in x0 if x(t)x0 when tt1- (resp., tt0+) and x˙(t1-) (resp., x˙(t0+)) does not exist.

If there is a curve solution x(t) with an impasse point (backward or forward) in x0, we call this curve an impasse solution of (2) in x0.

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 . If h(x)=deta(x) and g(x)=(adja(x))Tf(x), the classification corresponds to the analysis of the vector field g(x)/h(x) 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 Ms=MseMsne, where Mse={xMs:g(x)0} is the set of essential singularities and Msne=Ms-Mse is the set of nonessential singularities.

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

First, we recursively define φ0,φ1,,φk, as follows:(5)φ0x=hx;φkx=φk-1x·gx,k1,kN.The family of maps φkCω(M) generates a chain of ideals Jk=φ0,,φk, with J0J1Jk.

Let Zk be the set of zeros of Jk and we consider Z=kZk. Then, Ms=Z0Z1Zk. Since the ring of analytic functions at one point is Noetherian , then it is possible to determine whether xZ or not, in a finite number of steps.

Theorem 1.

Let the equation given in (2) be valid and let a point xMse. Let the chain Jk, k=0,1,, be defined as in (5) with Zk corresponding zero set of Jk, k=0,1,. Then x is an impasse point of (2) iff xZ=kZk.

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 C, a nonlinear inductor L, and a nonlinear memoryless two-terminal element (NME) , 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.

Parallel nonlinear circuit.

The corresponding currents are Ic, Il, and Inme, respectively, and V is the common voltage. Nonlinear relations are assumed between the capacitor charge and VQ=γc(V) and the inductor flow and Ilϕ=γlIl. C and L are real analytic nonlinear functions defined as C(V)=dγc/dV and LIl=dγl/dIl, with C:R(0,) and L:R(0,). In the branch of the NME, Inme and V are related by means of the equation V=θ(Inme), with θ:RR being an analytic real function that is nonconstant and nonlinear.

From the Kirchhoff Current Law, we get Inme+Ic+Il=0. In addition, from electromagnetic relations, Ic=C(V)V˙ and V=L(Il)I˙l. Calling x1=Inme, x2=Ic, x3=Il, x4=V, and X=(x1,x2,x3,x4), the QLIDE that models the circuit is(6)AXX˙=FXwith(7)AX=0000000000Lx30000Cx4,FX=x1+x2+x3x4-θx1x4x2.As L(x3),C(x4)>0, then A(X) is singular of rank 2, for all XR4.

Equation (6) is equivalent to the following system defined for all x=(x1,x2)M=R2 :(8)axx˙=fxwith(9)ax=-L-x1-x2-L-x1-x2Cθx1dθx1dx10,fx=θx1x2,as x1+x2+x3=0, and then L(x3)=L(-x1-x2).

By considering that L and C are both positive functions, the singular set is(10)Ms=x1,x2R2:dθx1dx1=0.

If Ms=, then, in particular, (8) has no singular impasse points. So, from now on, we shall assume that Ms. Moreover, as dθ(x1)/dx1 is not identically null, Mr. Then, we are interested in solutions on Mr achieving singular impasse points (backward or forward) (i.e., impasse solutions).

In order to use the results set in the previous section, we introduce(11)hx1,x2=detax1,x2,(12)gx1,x2=adjax1,x2Tfx1,x2,where by simple calculations we get (13)hx1,x2=Cθx1dθx1dx1L-x1-x2,gx1,x2=g1x1,x2g2x1,x2=x2L-x1-x2-Cθx1dθx1dx1θx1+x2L-x1-x2.The set of essential singularities defined by (14)Mse=x1,x2R2:dθx1dx1=0,x20is 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 h and g defined as in (11) and (12), respectively, and the functions φn defined recursively as in (5), then, for each n1, there are functions κn,1(x),κn,2(x),,κn,n+1(x) such that(15)φnx=dθx1dx12κn,1x+d2θx1dx12κn,2x++dnθx1dx1nκn,nx+dn+1θx1dx1n+1κn,n+1x,with x=(x1,x2).

Proof.

The proof is given by induction on n.

For n=1, by simple calculations, it is easy to prove that (16)φ1x=φ0x·gx=dθx1dx12κ1,1x+d2θx1dx12κ1,2x,with(17)κ1,1x=dCtdtt=θx1L-x1-x2·g1x-C2θx1L-x1-x2x2,κ1,2x=Cθx1L-x1-x2·g1x.For n>1, assuming the inductive hypothesis for φn-1(x), we calculate the expression of φn(x):(18)φnx=φn-1xx·gx=φn-1xx1·g1x+φn-1xx2·g2x=dθx1dx12κn-1,1xx1·g1x+κn-1,1xx2·g2x+d2θx1dx122dθx1dx1κn-1,1x+κn-1,2xx1·g1x+κn-1,2xx2·g2x+d3θx1dx13κn-1,2x·g1x+κn-1,3xx1·g1x+κn-1,3xx2·g2x++dnθx1dx1nκn-1,n-1x·g1x+κn-1,nxx1·g1x+κn-1,nxx2·g2x+dn+1θx1dx1n+1·κn-1,nx·g1x=dθx1dx12·κn,1x+d2θx1dx12·κn,2x+d3θx1dx13·κn,3x++dnθx1dx1n·κn,nx+dn+1θx1dx1n+1·κn,n+1x.Finally, for n>1, it can be easily shown that the coefficients κn,i(x) are defined recursively as (19)κn,ix=κn-1,1x·gx,if  i=12dθx1dx1·κn-1,1x·g1x+κn-1,2x·gx,if  i=2κn-1,i-1x·g1x+κn-1,ix·gx,if  3inκn-1,nx·g1x,if  i=n+1.

Theorem 3.

Let the equation given in (8) be valid and x=(x1,x2) an essential singularity of (8). Then, x is an impasse point of (8) iff there exists N, n2, such that (dnθ(x1)/dx1n)x1=x10.

Proof.

By Lemma 2, the general formula for φn(x1,x2) is(20)φnx1,x2=dθx1dx12κn,1x1,x2+d2θx1dx12κn,2x1,x2++dnθx1dy1nκn,nx1,x2+dn+1θx1dx1n+1κn,n+1x1,x2.Using the expression set in Lemma 2 for the coefficients κn,i, the following statement can be proved by induction on n, n1: (21)φkx=0,1kn,iffdkθx1x1kx1=x1=0,2kn+1.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 x=(x1,x2) be an essential singularity of (8). If x is not an impasse point of (8), then, by Theorem 3, (dnθ(x1)/dx1n)x1=x1=0, for all n2. Moreover, as x is an essential singularity, it holds that (dθx1/dx1)x1=x1=0. Finally the analyticity of θ(x1) allows us to ensure that θ(x1)=θ(x1), for all x1R; that is, θ(x1) is a constant function on R, 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 V=θ(Inme)=Inme3-Inme  and, for the sake of simplicity, constants C(V)=1/4 and L(I3)=4. By calling x1=Inme, x2=Ic, x3=Il, and x4=V, we obtain x4=θ(x1)=x13-x1 and dθ(x1)/dx1=3x12-1, for all x1R.

Then, the set of essential singularities is(22)Mse=x1,x2R2:dθx1dx1=0,x20=33,x2R2:x20-33,x2R2:x20.By Corollary 4, we conclude that all the essential singularities are impasse points.

In Figure 2, we show the graphic of Mse and some impasse solutions. In addition, by simple calculations, we conclude that (-3/3,0) and (3/3,0) are nonessential singularities and (-1,0), (0,0), and (1,0) are regular equilibrium points.

Impasse solutions of Example 5.

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 C, a nonlinear inductor L, 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.

Series nonlinear circuit.

Let C:R(0,) and L:R(0,) be real analytic functions, in general nonlinear ones, defined as C(Vc)=dγc/dVc and LI=dγl/dIl, where C depends on the capacitor voltage x1=Vc and L depends on the inductor current x2=I. The voltage of the NME is x3=Vnme, with the real analytic nonlinear nonconstant function ψ:RR, x2=ψ(x3) representing the voltage-current relation of the NME. The inductor voltage is x4=Vl.

Then, with X=(x1,x2,x3,x4), the QLIDE that models the circuit is(23)AXX˙=FX,where (24)AX=Cx10000Lx20000000000,FX=x2x4x2-ψx3x1+x3+x4.By considering the voltage-current relation, the original model of circuit (23) is equivalent to an order 2 system depending on the variables x1 and x3. For all x=(x1,x3)M=R2, the reduced model is(25)axx˙=fxwith (26)ax=Cx100Lψx3·dψx3dx3,fx=ψx3-x1-x3.The singular set is (27)Ms=x1,x3R2:dψx3dx3=0.

If Ms=, there are no singular impasse points for (25). So, in order to analyse the existence of impasse solutions on R2, we shall assume that Ms.

In order to use the results developed in Section 2, we calculate(28)hx=detax1,x3=Cx1Lψx3dψx3dx3,gx=g1x1,x3g2x1,x3=adjax1,x3Tfx1,x3=Lψx3·dψx3dx3·ψx3Cx1·-x1-x3.The set of essential singularities is (29)Mse=x1,x3R2:dψx3dx3=0,x3-x1.

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  as a particular case in the family of series nonlinear circuits with the voltage-current relation given by x2=ψx3=x33-9x32+24x3, C(x1)=(0.2x12+0.3)/(x12+1), and L(x2)=(3.8x22+4.2)/(x22+1).

Then, the set of essential singularities is(30)Mse=x1,2R2:x1-2x1,4R2:x1-4,which are all impasse points of the circuit.

In Figure 4, we show the graphic of Mse and some impasse solutions.

Impasse solutions of Example 7.

In addition, by simple calculations, we conclude that (-2,2) and (-4,4) are nonessential singularities and (0,0) 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.

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