ON THE RESISTIVE FUNCTION MEASURED BETWEEN TWO POINTS ON A GRID OR A LATTICE OF SIMILAR NON-LINEAR RESISTORS

Transcription

1 INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS LETTER TO THE EDITOR ON THE RESISTIVE FUNCTION MEASURED BETWEEN TWO POINTS ON A GRID OR A LATTICE OF SIMILAR NON-LINEAR RESISTORS EMANUEL GLUSKIN* Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel SUMMARY The known problem of the resistance measured between two closest nodes on a grid of similar resistors, often considered in a linear version in physics and electrical engineering textbooks, is considered in a non-linear version. An estimation, based on a simple polynomial model, shows that with increase in the number of the dimensions of the grid (or a lattice), the non-linearity of the characteristic of a single resistor is expressed more and more weakly in the input resistive function, and that the input resistive function has the same (in the sense of the degrees involved and polarities of the coefficients) polynomial structure as the individual characteristic of the single resistor John Wiley & Sons, Ltd. 1. INTRODUCTION In References 1 5 the well-known problem of the input resistance of an infinite square grid of similar linear resistors is considered. The input resistance is measured between a pair of close nodes which are denoted as P and Q. The input resistance is found in References 1 and 2 (see also References 3 6 for some generalizations of the linear problem) by means of an ingenious use of a superposition procedure, in which in each step the input current is supplied to only one of the points, P or Q, (and a point at infinity is involved), and thus the local electrical current distribution is symmetrical in each step, so that the computational problem of the infinity of the grid is overcome. The superposition procedure was possible because of the linearity of the system presented by the grid. Our intention here is to consider this problem in the context of a non-linear resistive individual characteristic, which is both mathematically and methodologically interesting, and may be helpful in, e.g. using the relatively easily measurable non-linear input resistive function of the grid for obtaining information about the distribution of the currents in the grid. Not seeking a complete solution, which, in a constructive form, would be very cumbersome in the non-linear case, we shall see in a simple approximation associated with a cut grid, how a non-linear term added to the voltage current (v i) characteristic of the resistors from which the grid is composed qualitatively influences the input voltage current characteristic of the grid. Some estimations and conclusions will be obtained also for the infinite non-linear grid. * Correspondence to: E. Gluskin, Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Beer-Sheva, Israel. A more detailed work by the author on this topic is currently in preparation. CCC /98/ $17.50 Received 20 May John Wiley & Sons, Ltd. Revised 19 February 1997

2 208 LETTER TO THE EDITOR It will be somewhat more suitable to consider at the start not the resistivity v i, but the conductivity i v characteristic of the single resistor: i"f (v) (v and i relate to a single resistor) (1) Because of the non-linearity of the resistors, the input (between P and Q) conductivity characteristic of the grid is also a non-linear function of the input voltage i"ψ(v) (v and i relate to the input of the grid) We shall see that there is an analytical self-similarity of the grid which means that for a polynomial model of f (v), ψ(v) includes the same powers of v asf(v). The coefficients before the powers are generally different to the corresponding ones in f (v), but are of the same polarity. As a simple model we shall consider the individual characteristic f (v)"gv#εv (1a) If ε is small, the inverse resistivity characteristic v"f (i) will be approximately v+ri!εri with R"g. Introducing the notation α"!εr, we write this as v"ri#αi. Apart from the inversion of f (v), and apart from the fact that approximations of the type (1a) are usually physically good only for a small ε, there is no formal reason in what follows for ε to be small. Dealing with cut grids/lattices below, we shall obtain in our calculations expressions for the input characteristic in the form ψ(v)"a(gv#bεv) (2) with precisely calculated constants A and B which are, generally, different for different cuttings of the grid. The form (2) is correct also for the infinite grid, and we can find A precisely for the infinite grid, by noting that for any certain (here infinite) grid, setting ε"0 leads to a linear grid which can be easily computed. As for B, we shall be able to give for the infinite case, only an estimation for this parameter, based on a half-empirical assumption. Our main focus here is B, as associated with the non-linearity. We therefore shall concentrate our attention first of all on the relation ψ(v)&(gv#bεv) finding in some cases B to be much smaller than the value B"1 in the individual characteristic f (v)"gv#εv, which means a relative linearity of ψ(v), compared to f (v), which is an important point for any application Calculation for a cut grid 2. AN ESTIMATION OF ψ(v) FOR A NON-LINEAR 2D-GRID For an estimation, we shall reduce (see Figure 1) the infinite 2D-grid to the resistor, to which the terminals of the voltage source are connected, and the two meshes which are closest to this resistor. That is, we consider a finite circuit which includes three parallel branches, connected to the nodes P and Q. Two of the branches include, each, 3 resistors, and the central branch includes one resistor. Applying the voltage v to the points P and Q, and noting that in the branches which include 3 resistors the voltage fall is equally distributed between the resistors, we obtain the currents in the branches, respectively, as f v 3, f (v), f v John Wiley & Sons, Ltd.

3 LETTER TO THE EDITOR 209 Figure 1. The cut grid for the 2D-case. We connect the source of voltage v to the points P and Q. The circuit is composed of non-linear similar resistors (conductors) which are described by the function i" f (v) and the total input current of the circuit is i"ψ(v)"f (v)#2f v 3 "gv#εv#2gv 3 #2ε v 3 " 5 3 gv#11 9 εv"5 3 gv#11 15 εv Thus in the 2D-case, for the cut grid, ψ(v)&(gv# εv) with B"11/15+0)733, i.e. with the relative role of the nonlinear term weaker than in the individual characteristic (1a). The precision of this ψ(v), associated with cutting the grid, (we consider first A in (2)) is seen if we transfer to the linear case, setting ε"0. Then we obtain i" gv+(1)67/r)v, instead of i"2gv"(2/r)v, asin References 1 5 for an infinite linear grid. Certainly, the cutting causes also some imprecision in B. This imprecision cannot be precisely estimated in the approach taken, but we can see that successive (iterating) cutting influences A more strongly than B, and we can find a half-empirical correction to the value of B which corresponds to a cut grid, which leads to an estimation of B for the infinite grid. We are essentially using that ψ(v) can include only the powers of v which appear in f (v). This follows from the fact that using Kirchhoff s equations, we perform only linear operations with the voltages and currents, and thus the transformation f (v)pψ(v) may be associated only with scaling changes, as we have seen for the 2D cut grid, and as we shall also see in Section 3 for a 3D cut grid John Wiley & Sons, Ltd.

4 210 LETTER TO THE EDITOR 2.2 Estimation of ψ(v) for infinite grid Taking larger pieces of the grid we always have, as is easy to understand, an increase in A (more paths for the current) and a decrease in B (less current in each resistor, and thus a more weakly revealed non-linearity). Passing from the most cut grid which includes only one resistor between P and Q to the cut grid considered above which includes 7 resistors, we transfer from A"1 and B"1 toa"and B". The value of A is changed more strongly than the value of B: compared to i.e. A A " 5 3!1 :1"2 3 B B " 11 15!1 :1"4 15 A A B B "2)5. Now using the fact that the transfer from the grid with 7 resistors to the infinite grid is associated with the change in A from to 2, and assuming that the ratio ( A/A)/ B/B is less changed, with the iterative approximations of the infinite lattice by increasing cut grids, than the relative changes in B or A by themselves, let us estimate the value of B for the infinite grid, B, using the value 2)5 for this ratio, and the value B" for 7-resistor grid for finding the addition ΔB (ΔB(0): B R "B# B+B! B"B1! B B +B 1! 1 A 2)5 A " !2 5 2!5 3 : 5 3 " "0)675 Thus, for the infinite 2D grid our estimation is or For ε small the inverse resistivity characteristic ψ (v)+2(gv#0)675εv) v"ψ (i)+r 2 i!0)675εr R 2 i + R i!0)169εri, 2 v"ψ (i)"r 2 i#0)169αi where we returned to the notations R"g and α"!εr, for which the inverse characteristic of the individual resistor is (for α small) v"ri#αi John Wiley & Sons, Ltd.

5 LETTER TO THE EDITOR THE ESTIMATION FOR A 3D-LATTICE Somewhat extending the traditional borders of the problem, let us also consider a 3D- grid, which is an orthogonal lattice with one non-linear resistor connected between each of the two nodes which are closest to each other. Six resistors are now connected to each node. The solution for the linear version of the infinite circuit is R "R/3, or g "3/R, which is easily obtained by the superposition procedure, quite similarly to that in References 1 5 for the 2D case Calculation of ψ(v) for the cut lattice and a conclusion about the role of the dimension Consider the cutting, which here also leaves only two different kinds of branches, we have for the 3D case (see Figure 2) four similar branches, each with 3 resistors, which are closest to the resistor connected in parallel with the source of the voltage v. For the cut lattice the input current is i"f (v)#4 f v 3 "gv#εv#4g v 3 #4ε v 3 " 7 3 gv#13 9 εv"7 3 gv#13 21 εv For the linear case we set ε"0, obtaining i" gv+(2)33/r)v for the cut lattice instead of 3gv"(3/R)v for the infinite linear lattice, which here too shows an indifferent precision in A, associated with the cutting of the lattice, but, nevertheless, since, now ψ(v)&(gv#εv), there is an important difference from the 2D case: in the 3D case the role of the non-linearity of a single resistor is more weakly expressed in ψ(v), B"13/21+0)619 instead of the value 0)733 in the 2D case. Noting, regarding this point, that for a 1D case, when the grid is precisely an infinite chain of the resistors, the input characteristic measured between two closest connections ( nodes ) of the chain is, clearly, the individual characteristic, ψ(v)"f (v) (i.e. B"1), and considering all of the values for B obtained for the cut grids of different dimensions: 1, 0)733, 0)619, we come to the conclusion that for the model chosen (and, actually, for any f (v) with a monotonic non-linearity), with an increase in the number of the dimensions of the grid, the role of the non-linearity of a single element from which the grid is composed is monotonically decreased as seen from the input. For the model (1a) this decrease is not as strong as Figure 2. The cut lattice for the 3-D case 1998 John Wiley & Sons, Ltd.

6 212 LETTER TO THE EDITOR the increase in the dimension. Similar conclusions relate to the comparison of the estimated input conductivity characteristics of infinite grids/lattices. 3.2 Estimation of ψ(v) for the infinite lattice Following the approach of Section 2.2, we estimate ψ(v) for the infinite 3D-grid. We consider that for the 3D-grid the transfer from the most cut lattice with one resistor to the cut 3D-lattice with 13 resistors, analysed in the above, is associated with changes in A from 1 to, and in B from 1to, i.e. the relative changes of the absolute values relate as A A B B "7/3!1 1 : 13/21!1 " "7 2 "3)5 This ratio is assumed to be changed weakly with transfer to larger pieces of lattice. Since the precise value A "3 yields for the transfer to the infinite lattice ΔA/A"(3!)/", we thus estimate B R " ! " ! 49 4 " )569 or Thus, for the 3D infinite grid our estimation for ψ(v) is ψ (v)+3(gv#0)569εv) For a small ε the inverse resistivity characteristic is written, using the notations R"g and α"!εr,as v"ψ "R 3 i!0)569 εr R 3 i v+ R 3 i#0)063αi 4. CONCLUSIONS There is an interesting analytical problem of calculating the grid/lattice of non-linear resistors, which is a generalization of the linear problem of References 1 6. This problem seems to be relevant to both topics of circuit theory and physical measurements, deserving attention of both physicists and circuit specialists. The analysis, devoted, first of all, to the mapping of the individual characteristic ( f ) of the resistor to the input characteristic (ψ) of the grid/lattice, fpψ was focused on the relative weights of the non-linear terms in f (v) and ψ(v) in models with cut grids, and on the possibility to simply estimate ψ(v) for the infinite grid/lattice. We summarized some of the results of the investigation in the Table I which includes the resistivity characteristics related to the case when the nonlinear term is much smaller than the linear one. We see from the table that the corresponding decreasing of the relative weight of the non-linear term with increases in the dimension of the grids are 1D: 1 : 1"1; 2D: 0)169 : 0)5"0 338; 3D: 0)063 : 0)3333" John Wiley & Sons, Ltd.

7 LETTER TO THE EDITOR 213 Table I The input resistivity characteristic of the infinite grid The individual resistivity 1D-case 2D-case 3D-case characteristic of resistors ν"ri#αi Ri#αi R 2 i#0)169αi R 3 i#0)063αi (precisely equals the individual (approximately) (approximately) characteristic) These weight values are analogies of the parameter B in equation (2), now applied to the resistive characteristic. Contrary to the case of conductivity characteristic, for the resistivity characteristic the decrease in this parameter is larger than the corresponding increase in the dimension of the grid. Though some of the derived conclusions clearly relate to any monotonic non-linear characteristic, the analysis was, in general, limited by (a) certain analytical model of f (v); (b) the use of a cut grid/lattice in precise calculations; (c) the assumption (which is especially important for the transfer to an infinite grid), based on (see Section 2.2) transfer from one particular cut grid to another, of a relatively weak change in the ratio of the relative changes in the parameters A and B with increase in the cut grid; and (d) consideration of only time-independent (algebraic) problem, ignoring inductive and capacitive features of realistic elements, which would be revealed for the input voltage dependent on time. Regarding the use of the finite polynomial conductivity characteristic f (v), it needs to be noted that the finite polynomial is turned, in principle, into an infinite series of powers for the inverse individual resistivity function. Corresponding to this, for v not small, the input resistivity function of the grid/lattice must also be given by an infinite series of powers. If we were, on the contrary, to postulate the individual resistivity characteristic as a finite polynomial, for v not small then the inverse individual conductivity characteristic would be given by an infinite series, and the input conductivity function of the grid/lattice also would be given by an infinite series. This injustice with respect to the analytical form of one of the characteristics, either conductive or resistive, gives some support to the usual use of analytical models in realistic physical problems only in the cases of weak non-linearity, when we can ignore the high powers of the argument in each characteristic. This is in contrast to the case of a singular non-linearity which may be taken as a strong one in a realistic problem, not causing significant analytical difficulties. (See e.g. the examples in References 7, 8 and 9). REFERENCES 1. E. M. Purcell, Electricity and Magnetism, McGraw-Hill, New York, 1965, p L. O. Chua, C. A. Desoer, and E. S. Kuh, inear and Nonlinear Circuits, McGraw-Hill, New York, 1987, pp A. H. Zemanian, A classical puzzle: the driving point resistances of infinite grids, IEEE Circuits Systems Mag., 7 9 (1984). 4. A. H. Zemanian, Infinite electrical networks, Proc. IEEE, 64, 6 17 (1976). 5. B. van der Pol and H. Bremmer, Operational Calculus Based on the ¹wo-Sided aplace ¹ransform, Cambridge Univ. Press, Cambridge, G. Venezian, On the resistance between two points on a grid, Am. J. Phys. 62, (1994). 7. L. O. Chua, Introduction to Nonlinear Network ¹heory, McGraw-Hill, New York, E. Gluskin, On the theory of an integral equation, Adv. in App. Math., 15, (1994). 9. E. Gluskin, Nonlinear systems: between a law and a definition, Reports on Progress in Physics, Vol. 60, no. 10 (Oct. 1997) John Wiley & Sons, Ltd.