A theory of nonlinear networks. I

This report describes a new approach to nonlinear RLC-networks which is based on the fact that the system of differential equations for such networks has the special form T/-x di dP(i, v) ., dv dP(i, v) L(i) dt ~ ~di~ • C(-V)dt dv The function, P(i, v), called the mixed potential function, can be used to construct Liapounov-type functions to prove stability under certain conditions. Several theorems on the stability of circuits are derived and examples are given to illustrate the results. A procedure is given to construct the mixed potential function directly from the circuit. The concepts of a complete set of mixed variables and a complete circuit are defined. Introduction. A. In the extensive theory of electrical circuits many impressive advances have led to a powerful tool for the engineer and the designer. For a wide class of problems one is able to construct a circuit with required properties using a rather complete theory which is available in several textbooks (see, e.g., [1], [2]). Most of these theories are based on the linear differential equations of electrical circuits. However, in recent times many engineering problems have led to the study of nonlinear networks which cannot appropriately be approximated by linear equations. Typical examples in this direction are the so-called flip-flop circuits which have several equilibrium states. Since a linear circuit obviously admits only one equilibrium, a flip-flop circuit can only be described by nonlinear differential equations. The main difference between such circuits and linear ones lies in the nonmonotone character of the voltage-current relations for the resistors. It will be a main point in the following to admit such "negative resistors". B. The electrical circuits considered in this paper are general RLC-circuits in which any or all of the elements may be nonlinear. One of the purposes of this paper is to show that the differential equations of such electrical circuits have a special form which has its ultimate basis in the conservation laws of Kirchhoff. It will be derived that under very general assumptions the differential equations have the form di„ dP L> dt di,' (p C'dtt=-%I' ('-r+l,-,r + .). (1) *Received May 29, 1963. The results reported in this paper were obtained in the course of research jointly sponsored by IBM and the Air Force Office of Scientific Research, Contract AF49(638)-1139. 2 R. K. BRAYTON AND J. K. MOSER [Vol. XXII, No. 1 where the i„ represent the currents in the inductors and v, the voltages across the capacitors. The function P(i, v) describes the physical properties of the resistive part of the circuit. Since it has the dimension of voltage times current, it will be called a potential function. This function can be formed additively from potential functions of the single elements similar to the way that the Hamiltonian is formed in particle dynamics from the potential energy and the kinetic energy of the different particles. However, it should be observed that equations (1) do not represent a Hamiltonian system since the latter describes nondissipative motion while in equations (1) the potential P contains dissipative terms. Also, the transformation properties of the above equations are different from Hamiltonian equations in that equations (1) preserve their form under coordinate transformations which leave the indefinite metric -Zi«2+ E ca(dv«)2 (2) p = 1 <r = l + r invariant. C. A geometrical interpretation of the special form of equations (1) is the following. We consider a box containing an electrical circuit with only resistive elements. There are n pairs of terminals on the box which are connected internally to the electrical circuit. To measure the external electrical properties of this box we connect each terminal to either a current source of prescribed current i„ (p = 1, • ■ • , r) or a voltage source of prescribed voltage v„ (a= r + 1, • • • , r + s = n). Under natural compatibility assumptions for the arrangement of these sources, an equilibrium state (t'„, vv) (v = 1, • • • , n) will be attained, i.e., the missing quantities v? (p = 1, • •, r) and i„ (<r = r + 1, • • •, r + s) will be determined. In other words the 2n voltages and currents satisfy n relations which define an n-dimensional surface in 2n-dirnensional space. We call this surface the characteristic surface 2 of the box. In fact, if n = 1, 2 is a curve usually called the voltagecurrent characteristic for an element or a circuit. The result that the equations have the form (1) can be expressed compactly by the identity, n E di, A dvv = 0, (3) y = ] i.e., this two-dimensional differential form in the sense of Cartan [3] vanishes identically on the surface 2. This fact will be explained and proved in section 13, part II. D. It is also the purpose of this study to draw some conclusions concerning the solutions of the differential equations (1) from their special form. To show that such implications can be expected, consider, for instance, an RC-circuit (i.e., a circuit without inductors or r = 0 in (1)). In this case, the quadratic form (2) is positive definite and can be used as a metric ids)2 = t: c,(dv,)\ (T*=l One verifies immediately that in this case P(i, v) decreases along solutions of (1) since dP = ydPdK = _(d^2 dt dv„ dt \dt. which is negative except at the equilibrium points. This implies that all solutions of an RC-circuit approach equilibrium states for t —* <» even if the resistors are negative in 1964] THEORY OF NONLINEAR NETWORKS 3 some regions. Of course, some natural assumptions have to be added and these will be found in section 8. Especially in case a circuit contains negative resistors is it of interest to find criteria which guarantee that the solutions approach the equilibria as time increases and therefore do not oscillate. We saw that this is generally the case for RC-circuits and similarly for RL-circuits. On the other hand, RLC-circuits certainly will admit oscillations in general even in the linear case. But one would expect a nonoscillatory behavior of circuits in which the inductance—or a quantity of the dimension L/R2C—is sufficiently small. Such criteria for nonoscillation will be derived in section 8. The main idea is to associate with the differential equation another metric which is positive and so find a function P* which decreases along the solutions. Such criteria are especially valuable for large circuits which contain many loops. It is usually hard to judge intuitively whether the presence of many loops may lead to oscillatory behavior. In section 9 we discuss an example of an arbitrarily large ladder network containing nonlinear elements, which demonstrates that our criteria are the best possible in general. In section 20, part II, similar methods are used to establish the existence of periodic solutions for periodically excited nonlinear circuits. This result can be considered as an extension of a theorem of R. Duffin [4], This paper is divided into two parts. The more important part is the first which leads to the main results rather directly without containing all the detailed proofs and refinements. The second part contains several additional results as well as detailed proofs complementing part I. Originally, this work started with the study of some nonlinear circuits proposed by Goto and others [5]. Some preliminary investigations in this direction have been published earlier (see [6, 7]). In this paper, we present these ideas in a more systematic fashion in the hope that it will be useful to the theoretically inclined electrical engineers as well as mathematicians. 1. Complete sets of variables for a network. A network is an idealized concept in circuit theory which can be defined as a set of points, called nodes, and a set of connecting lines, called branches. It is irrelevant whether nodes and branches lie in a plane or whether the branches can be realized by straight lines. It is essential, however, that every branch connect exactly two nodes. Such a network is frequently called a graph. Actually, for applications other natural restrictions—like connectedness of the graph— could be imposed which, however, we will not need. In each branch labeled by m = 1, • • • , b we specify a direction arbitrarily, indicating it by an arrow ("directed" graph). Accordingly, we distinguish the two connected nodes as initial and end nodes. The current flow in such a network is completely described by giving the amount of current i„ flowing in the direction of the arrow; that means is negative if the flow is against the specified direction and positive otherwise. Similarly, we associate with each branch a voltage with a specified sign by taking the voltage level at the end node minus the voltage level at the initial node of the branch. The 2b variables i„ , v„ (n = I, • • • , b) are restricted by the well-known Kirchhoff laws. The node law expresses that the currents arriving at any node (taken with proper sign) add up to zero which we write symbolically in the form yi ± = o. (i .i) 4 R. K. BRAYTON AND J. K. MOSER [Vol. XXII, No. 1 Kirchhoff's loop law expresses that the voltage drop over any loop (closed chain of branches) is zero, or Z±t>„ = 0. (1.2) loop Another way of describing this loop law is that to every node one can assign a voltage level such that v„ equals the difference of the defined voltage levels between the end node and initial node.* In the investigation of the circuit dynamics it will be of first importance to know how restrictive Kirchhoff's laws are. They form a set of linear equations, and we study first which of the currents and voltages can be chosen independently. More precisely, we call a set of variables i\ , • • • , ir , vr+1 , • • ■ , vr+** "complete" if they can be chosen independently without leading to a violation of Kirchhoff's laws and if they determine in each branch at least one of the two variables, the current or the voltage. The problem is to describe a complete set of variable