One-Ports Composed of Power-Law Resistors

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 51, NO. 9, SEPTEMBER 2004

One-Ports Composed of Power-Law Resistors Emanuel Gluskin

Abstract—Some resistive (more generally, algebraic, e.g., dc magnetic or dielectric) 1-ports, composed of elements having a singledegree characteristic ( ( ) ) are considered and some general relations relevant to such 1-ports of an arbitrary structure are proved. For series-parallel (s-p) 1-ports and also fractal 1-ports, this specific characteristic permits constructive (formulas) circuit solution. A generalization of the concept of conductivity is thus obtained. Index Terms—Circuit analysis, power-law characteristics, resistive (algebraic) one-ports.

I. INTRODUCTION ESISTIVE circuits composed of identical elements are relevant to spatial filtering effects, modeling of partial differential equations, and are met in some other fields [1]–[6].1 One of the interesting problems is that of the input conductivity (the 1-port problem) of some nonlinear resistive grids [3], which is a generalization of the classical linear problem [7]. In terms of the input characteristic, the 1-port problem is illustrated by Fig. 1. Algebraic characteristic may also be relevant to the elements of a magnetic or ferroelectric dc circuit, but we shall speak only about resistive realizations. Having a certain topology of the 1-port and the elements’ current characteristic , we are interested voltage , in determining the input conductivity function , defined by i.e., we study the function to function map that is the focus the circuit topology. For the single-degree below, the map possesses some general scaling features that are independent of the 1-port structure. The terminal (see Fig. 1) will always be the “common ground.” The nodal voltages (potentials), all measured with respect to , will be denoted as , and the branches’ voltages (voltage drops) as , with the integer subscripts. The following single-degree model:

R

(1) includes some finite positive constants and . This model is a flexible theoretical tool, both in the sense that it is possible to develop a consistent straightforward theory, and in view of an Manuscript received October 16, 2002; revised August 1, 2003 and November 6, 2003. This paper was recommended by Associate Editor J. Vandewalle. The author is with The Academic Institute of Technology, Holon 58102, Israel and also with the Electrical Engineering Department, Ben-Gurion University, Beer-Sheva 84105, Israel (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TCSII.2004.834532 1Note: In [1], equation fonts in the article in the Proceedings were damaged. However, the file 1474.pdf in the conference’s CD-ROM is correct and it may be found also in http://www.it.dtu.dk/ecs/tutor-01.htm#T3. A tutorial under the same title was also given at ECCTD’01, and the “handout” (that does not appear at this http site), distributed at the conference, includes a very detailed list of references on the theory and applications of resistive grids and on some close topics, with this author’s essential comments.

Fig. 1. The 1-port (the “f ! F circuit”) of a given topology, composed of identical conductors f ( 1 ). In some developments we allow the conductors to have different characteristics in the sense of different constant factors before the same function that defines the dependence on the argument.

application [1],[5], [6] for 1-ports with polynomial characteristics of the elements. We shall prove some general statements associated with such a model and to give some rules for the relevant calculations. , for an integer odd , (1) simply beSince , including the linear case of . comes The power consumptions of the conductors defined by (1) are

In terms of [8], the elements are strictly passive. According to (1), and tend to infinity simultaneously. This ensures the existence of the resistive circuit solution [8] and [9], while the uniqueness is provided by the monotonicity of (1). The general solvability is explicitly seen below via some specific circuit parameters defined in Section II, which we calculate. Following [1] and [5], we simplify (1) to (1a) The physical requirement of each of the to be real valued, generally excludes in (1a) negative (i.e., negative ), and thus, (1a) is, generally, defined only for positive . Using only the simplified characteristic (1a), we shall not specifically limit , does not alternate and but always assume that the polarity of that the orientation (if needed for actual nonsymmetric characteristics) is done (see also, Section V, for this point). The analytically very simple model (1a) of the single-degree characteristic with the parameter having the physical dimension (units) A/V is sufficient for our targets here. Another form (i.e., of the single-degree characteristic is ), including two given positive constants and , which have, respectively, the dimensions of the current and voltage. This more detailed form appears to be necessary, for some applications [6]. We allow the factor to be different for different conductors, i.e., for the internal branches of the circuit

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(1b)

GLUSKIN: ONE-PORTS COMPOSED OF POWER-LAW RESISTORS

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but must be the same for all the elements. Taking a different does not introduce any essential difficulty. In calcuvalue of lation, sub-1-ports with different may naturally arise as composing the whole 1-port. We shall use vectors and related to the branches, i.e., , and . The input current is a component (and in ), the index possesses one more of ; i.e., in value than in (1b), including also the branch with the source. are different, we can introduce, for the internal branches If . Then, the [i.e., relevant to (1b)], the vectory notation 1-port problem is written as . It is also convenient to use the following definition. Definition: A 1-port with the elements described by (1a) will be named the “ -circuit.” II. SCALING PROPERTIES FOR INPUT CURRENT AND NODAL VOLTAGES Statement 1: For any -circuit (2) independent of , but, generally, depenwith some factor dent on . This feature of the input current means some analytical similarity of the whole 1-port and its elements (1a). Formally, a proof of this statement uses the existence of the solution for every . At the same time, it is seen from the proof that it would be sufficient to know that a solution exists for a certain . Proof: Denoting, for brevity, as , we first take , and denote the associated circuit state as . By the . circuit definition, Then, we take any arbitrary and note that since and , respectively, satisfy KVL and KCL, the sets of the variables (vectors) and , which differ from and only by the constant factors, also satisfy these equations. This includes, of course, the KCL equation at the input node (i.e., the input current is involved), and the KVL equation at the input loop. We check then that for the internal circuit currents associated with the voltage drops on the elements, the variables and satisfy (1b). Using that , we have for the new state that , in agreement with (1b). Thus, is the circuit state for the input . Applying to the branch with the input equals the input current, we obtain (2). (In particular, .) current for Corollary 1: Since for any structure of the 1-port , we can realize, for any given , the same by means of different structures, for an infinite range of . Thus, for instance, an infinite grid 1-port may be approximated, as regards , by means of an s-p structure, or by means of a fractal structure (see below) without any limitation on . In order to formulate the next corollary, we consider that a basic feature of any 1-port is the possibility to create fractal circuits of any desired order using that the branches of the given circuit are also 1-ports. That is, for any 1-port composed of some

Fig. 2. The -circuit with the elements f ( 1 ) = ( 1 ) (D calculation of a D () and a d() in Section III.

= 1

8

s) for

identical elements , we can, disregarding what is, recursively create “fractal” circuits, realizing the given map “ ” , in each recursive step, as where of is the number of steps, by replacing the branches by the already obtained circuit as many times as one wishes, obtaining a fractal circuit of a degree , which can be written as . (Consider this, e.g., for Fig. 2.) The point now is that, in the context of this recursive opera. Indeed, both functions are . tion, (2) means Corollary 2: If the circuit is solvable, then, the circuit is also solvable, by similar recursive steps. This . is obvious from the relation are different, i.e., we Remark 2.1: If in the initial circuit, are given the vector-to-scalar map , then, the recursive are simpler than , because they involve maps only scalars. Statement 2: For the nodal voltages (potentials), we have (3) with some nondimensional factors that are independent of , but generally depend on . (In particular, ). Proof: This follows from the equality i.e., , used in the proof of (2), and the obvious fact that each of the nodal voltages (potentials) is a sum of some of the voltage drops (possibly including as a separate term). Remark 2.2: The dependence of on may be much stronger than that of . Indeed, the KCL equations at the upper by means of terms of the type input node present , with the parameters (labeled by ) related to the branches that are directly connected to the upper node, , with or, alternatively, by means of terms of the type the parameters (labeled by ) related to the branches that are directly connected to the lower (the grounded) node. That is, is defined by the terms of type , or . For or weakly dependent or may be strongly dependent on . on , quickly tend (so that Remark 2.3: For is already a “large” value) to certain limits that are solely defined by circuit topology [1], [5], [6], and one easily finds are completely independent of . cases [6] when It is easy to see that determination of allows one to find all of the nodal voltages and branch currents, and thus a complete circuit solution. III. EXAMPLE OF Consider the circuit of Fig. 2.

AND

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 51, NO. 9, SEPTEMBER 2004

The nodal voltage is unknown. Together with the input KCL equation, the nodal equations are (4) (5)

that

from which

From (5)

(10)

yielding (6) Thus

Using (9) and (10), one easily obtains (8). (or ) that exThe following easily found formula for presses the “voltage division” in the series connection

(7) Substituting (6) into (4), we obtain

that is, (8)

is also useful. For instance, the condition for a Wheatstone bridge, composed of such elements, to be balanced (i.e., with no current in the central branch) is immediately found to be similar to the well-known condition for a linear bridge, with , replacing the linear conductivities (resistances) in the latter condition. , then, (10) becomes . For If identical conductors

It is directly seen from (4) and (5), that if we add factor in , this factor appears also in , i.e., in . Observe, in view of Remark 2.3, from (6) that (i.e., ) as , which is because the elements become identical voltage hardlimiters, and the two series hardlimiters on the right are “blocked” by their (vertical) neighbor, connected in parallel, etc. See also [1], [5], [6], especially [5], for this point. and may be explicitly systematWe now show how ically found for any s-p circuit.

which directly follows from the fact that the voltage on each conductor is , and thus, . If (two parallel conductors connected in series with one such), then, from (10)

IV. PARALLEL AND SERIES CONNECTIONS OF NONLINEAR CONDUCTORS

Since the input function of any s-p 1-port may be calculated by sequential calculations of series and parallel connections, (9) and (10) necessarily lead to a constructive form of (2):

For the parallel connection of any kind of conductors, the , is the sum of the individual total conductivity, denoted as conductivities. In particular, for two conductivities and , with the same and for (9) For

identical elements

For the series connection of two conductors characterized by and we obtain, using the equations and

(11)

(12) [or (2)] with some factor independent of , but, generally, dependent on . It also follows from the above formulas that when starting from elements with the same , we obtain (13) with some nondimensional (unit less) function (for the first . Function time introduced in [5]), i.e., is most important in the theory of these circuits [1], [5], [6]. Using (9) and (10), one can solve any s-p circuit similarly to what is done with linear s-p circuits in basic circuit analysis. Namely, calculating s-p connections using (9) and (10), one

GLUSKIN: ONE-PORTS COMPOSED OF POWER-LAW RESISTORS

“rolls up” the circuit close to the input, and then “unrolls” it, sequentially finding the potentials of the “appearing” nodes, all the time dealing with some local s-p 1-ports with known input characteristics, connected into the larger 1-port. In such a sequential calculation, each branch current is the input current of some internal 1-port, and each branch voltage is the input voltage of some such 1-port. Thus, for any finite s-p 1-port, composed of elements (1a), and any input , the branch currents and the nodal voltages can be found as explicit functions of , and , and the same is true for fractal s-p one-ports.

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we have to take care to have a similar orientation of the eleon a chosen digraph. ments with respect to the actual (Consider here, in particular, that PSpice does not always realize such a prescribed odd characteristic as (1) well enough, creating , which nonsymmetric characteristics.) The polarities of the define the necessary orientation of the elements, may be determined, for any topology of the resistive 1-port, and for any certain polarity of , using the linear version of the circuit that always may be easily calculated. In many cases (e.g., the circuit in Fig. 2), the positive directions of the currents are obvious. For the general theory of resistive circuits, see also, [9]–[12].

V. CONCLUSION AND FINAL REMARKS As a part of an extended research related to resistive (algebraic) 1-ports composed of identical 1-port elements [1], [4]–[6], this paper has focused on the case of the elements , (or ), showing characterized by some unique scaling relations. , and for It is proved that for the nodal voltages the input conductivity , i.e., . The latter relation means that our 1-port realizes some analogy of linear conductance. The problem of finding the 1-port’s input characteristic naturally leads to the creation of fractal-type circuits. If the basic , is solvable, then, the 1-port 1-port, (also symbolically written as , where is the number of the recursion building steps) is also solvable (at each step in the same way, and see Remark 2.1). Complicated constructively solvable nonlinear circuits can be thus easily created. Together with the analysis of the s-p circuits, this gives some impression regarding the variety of the topologies of such solvable 1-ports, named -circuits. The -circuits have an important application which had been synoptically introduced in [1] (some approximate analytical sufor polynomial) and will be systemperposition of atically presented in [6]. Last but not the least, despite the specificity of the elements (1a), they are quite well modeled by PSpice and MatLab, and thus, the topic of such 1-ports is found to be appropriate for students’ projects, in a course on nonlinear circuits. The modeling requires some remarks as follows. , for the negative When actually using elements whose , are not defined symmetrically with respect to the positive,

ACKNOWLEDGMENT The author would like to thank the unknown reviewers for their useful comments, especially Reviewer 1 who helped to formulate Statement 1 for arbitrary topology of the -circuit. REFERENCES [1] E. Gluskin, “An introduction to the nonlinear conductive grid,” in Proc. Eur. Conf. Circuit Theory and Design (ECCTD’01), vol. III, Espoo, Finland, Aug. 27–31, 2001, pp. 261–264. [2] B. E. Shi and L. O. Chua, “Resistive grid image filtering: Input/output analysis via the CNN framework,” IEEE Trans. Circuits Syst. I, vol. 39, pp. 531–547, July 1992. [3] V. A. Chang and A. H. Zemanian, “Operating points in n-dimensional nonlinear nonuniform infinite and finite grounded grids,” IEEE Trans. Circuits Syst. I, vol. 47, pp. 144–149, Feb. 2000. [4] E. Gluskin, “An approximation for the input conductivity function of the nonlinear resistive grid,” Int. J. Circuit Theory Applicat., vol. 29, no. 5, pp. 517–526, 2001. , “Calculations and hypotheses relating to the input conductivity [5] function of an infinite grid of similar nonlinear resistors,” Int. J. Syst. Sci., vol. 32, no. 6, pp. 753–766, 2001. [6] , -test: A test for algebraic 1-ports. unpublished. [7] A. H. Zemanian, “A classical puzzle: The driving point resistance of infinite grids,” IEEE Circuits Syst. Mag., pp. 7–9, Mar. 1984. [8] L. O. Chua, Ch. A. Desoer, and E. Kuh, “General resistive circuits,” in Linear and Nonlinear Circuits. New York: McGraw Hill, 1987, pp. 248–249. [9] L. V. Danilov, P. N. Matchanov, and E. S. Philippov, Teorija Nelinejnich Electricheskich Zepej . Leningrad, Russia: Energoatomizdat, 1990. [10] M. Hasler and J. Neiryneck, Nonlinear Circuits. Norwood, MA: Artech House, 1986. [11] J. Vandewalle and L. O. Chua, “The colored branch theorem and its applications in circuit theory,” IEEE Trans. Circuits Syst., vol. CAS-27, pp. 816–825, Sept. 1980. [12] L. O. Chua and N. N. Wang, “On the application of the degree theory to the analysis of resistive nonlinear networks,” Int. J. Circuit Theory Applicat., vol. 5, pp. 35–68, 1977.