On the symmetry features of some electrical circuits

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1 INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS Int. J. Circ. Theor. Appl. 2006; 34: Published online 26 September 2006 in Wiley InterScience ( On the symmetry features of some electrical circuits Emanuel Gluskin 1,2,, 1 Holon Institute of Technology, Holon 58102, Israel 2 Electrical Engineering Department, Ben-Gurion University, Beer-Sheva 84105, Israel SUMMARY The use of symmetry of some nonlinear circuits, composed of similar resistive (more generally, algebraic) elements, is considered for the analysis of the input resistive function of such a circuit. The focus is on recursively obtained ( fractal -type) 1-ports, analysed using the concept of α-circuit introduced by Gluskin. The methods under study should be of interest for the analysis and calculation of complicated nonlinear resistive (algebraic) 1-port structures, e.g. grid cuts for different symmetry conditions. Copyright q 2006 John Wiley & Sons, Ltd. Revised 20 July 2006 KEY WORDS: recursion; power-law circuits; fractals 1. INTRODUCTION The symmetry argument is widely used in basic physics, and is cultivated in its education. This argument may be useful in circuit theory too, and we consider, in this spirit, the calculation of the input (conductive) characteristic of some electrical 1-port circuits. The simplification that may be provided by the use of symmetry is most helpful in those nonlinear cases in which calculation may be difficult, but even then choice of the nonlinear models is still important for making methodologically effective treatment. The α-circuits introduced in Reference [1] (and for a general circuit theory background see also References [2, 3]) areshownheretobethemost suitable analytical tool. The relevant symmetry may be of different kinds. Besides the classical symmetries of reflection and periodicity, modern science considers recursively realized fractal structures as also exhibiting both a visual-realization and some analytical symmetry. The point noted here is that the very concept of the 1-port is inherently associated with the creation of a fractal structure. The recursive solvability of some fractal 1-ports, at least in the finite-circuit versions, becomes thus immediately clear, and the input conductivity function of such 1-ports is the focus of the study here. More Correspondence to: Emanuel Gluskin, Electrical Engineering Department, Holon Institute of Technology, Holon 58102, Israel. gluskin@ee.bgu.ac.il Copyright q 2006 John Wiley & Sons, Ltd.

2 638 E. GLUSKIN a b + v in - i in =F(v in ) + - { f(.) } Figure 1. The 1-port (the f F circuit ) of a given topology, composed of similar conductors f (.). Below, the terminal b will always be the common ground. The nodal voltages (potentials) are all measured with respect to b. regular reflection symmetry of a 1-port is touched briefly in Section 4 where, in particular, a very general scaling law for calculation of the input conductivity in the case of this symmetry is suggested. Our 1-port is shown in Figure 1. For the illustrating analytical side, we shall use the power-law model for the algebraic elements: f (.) = D(.) α (1) that includes some positive finite constants D and α independent of v. Such one-ports were named in Reference [1] α-circuits, and the existence and uniqueness of the solution is proved in Reference [1] for any structure of the α-circuit. The case of α = 1 gives linear elements and a linear circuit. Here one can read Equation (1) in terms of electrical current (i) and voltage (v), i.e. i v α. 2. RECURSIVELY SOLVABLE FRACTAL 1-PORTS Let us note that the concept of a 1-port is inherently associated with the concept of a fractal, because each 1-port s branch is also a 1-port, and thus the recursive (sequential) creation of a 1-port by means of similar 1-ports taken as its branches, is immediate. Solvability (when it exists, as in the case of f (t) given by Equation (1)) of the given 1-port is automatically kept at each recursive step, and it provides solvability of the sequentially appearing more complicated (fractal) 1-ports. We thus deal, for the power-law f (.), with solvable fractal 1-ports, a subclass of solvable 1-ports. Using the basic map defined by the given topology of the 1-port f F, we can symbolically write the one-step-fractal circuit as f F F. This is illustrated in Figure 2. One can similarly present each of the branches of the 12 small 1-ports of the latter circuit, making the fractal-circuit more and more detailed, and thus sequentially obtain the map f F F F, for n recursive steps, with the same map in each step. This procedure and the very fractal-type circuit may be denoted as f F n F or as f (F ) n F.

3 ON THE SYMMETRY FEATURES OF SOME ELECTRICAL CIRCUITS a - b Figure 2. An example of f F F 1-port. Fractal 1-port structures may be created using (starting from) any 1-port, not necessarily such a (geo)metrically symmetric (i.e. having some lines of reflection symmetry). If the basic structure includes m elements, then the recursively obtained fractals include mm = m 2, m 2 m 2 = m 4,..., i.e. m 2n 1 elements. Here, m = 12 and n = 2 (the second circuit in the sequence) If the basic f F circuit is solvable, then the f (F ) n F circuit is also solvable, obviously. Remark 1 It may be assumed that if, for a non-monotonic f (.), the f F circuit is unsolvable, then the f (F ) n F circuit is also such, n. Remark 2 The recursive procedure may be even made more sophisticated by changing the number of series elements (1-ports) in the branches at each successive step, etc., i.e. may be somewhat changed in the recursive steps, but the solvability is kept. Equation (2) suggests that a rescaling of D with the steps should be also considered. For the α-circuits, the constructive (formulae) solvability is very simple. It is proved in Reference [1] that for f (.) given by Equation (1), F(.) is obtained as F(.) = Dφ(α)(.) α where φ(α) is defined by the circuit topology, and can always be found taking D = 1 in (1). With the first step of the creation of the recursive (fractal) structure (this relates to n = 2below, i.e. to the second circuit in the sequence of the circuits which we consider), the above F(.) becomes, in terms of the same basic structure, the new f (.), and for the new circuit composed of such elements, we obtain (note that φ(α) is unchanged) the next F(.) as F(.) = D[φ(α)] 2 (.) α

4 640 E. GLUSKIN + a - b Figure 3. The basic structure of the circuit of Figure 2. In order to determine φ(α), this basic circuit can be split along the line a b, with reduction to parallel connection of two circuits, including elements f (.) and f (.)/2,whichleadsustoFigure4. For the next step, D[φ(α)] 2 is the new D, and F(.) = D[φ(α)] 2 [φ(α)] 2 (.) α,andaftern 1such recursive steps, we obtain (consider induction on n) F(.) = D[φ(α)] 2n 1 (.) α (2) i.e. i in = F(v in ) = D[φ(α)] n (v in ) α.thatis,φ(α) of the realization obtained after n 1 recursive steps, is 2 n 1 th degree of the φ(α) of the basic structure. Thus, F(.) of the circuit in Figure 2 is obtained using φ(α) of the circuit in Figure 3 (see Equation (7)), by one step (n = 2). It thus appears that for the α-circuit, we need only to calculate φ(α) for the basic structure, in order to find the input current F(v in ) of the fractal circuit. Remark 3 If, starting from the second recursive step, one uses (substitutes) the already obtained structure not in its branches, but in the branches of the basic structure (e.g. the circuit of Figure 2 is thus substituted instead of the branches of Figure 3, etc.), then for the circuit number n, F(.) = Dφ n (α)(.) α,and for φ(α) = 1, 1nF(v in ) is linear by n. Such a specific type of fractals (in which the number of the elements is increased exponentially, i.e. relatively slowly) may be more physical for 2D structures. The case of n (unlimited increase in structure) is of methodological interest. If φ(α)>1, then from Equation (2) lim F lim φ2n 1 (α) = (3) n n which in terms of the conductivity characteristic means that the input resistance is zero. If, on the contrary, φ(α)<1, then lim F lim φ2n 1 (α) = 0 (4) n n which means that the input resistance is infinite for any v in.

5 ON THE SYMMETRY FEATURES OF SOME ELECTRICAL CIRCUITS 641 Since, when having the basic structure composed only of one conductor we would have F(.) = f (.) (F(v in ) f (v in )), i.e. φ(α) = 1, it is obvious that if the basic structure includes a conductor directly connecting the input nodes a and b, and some other conductor(s), then φ(α)>1. Thus the simplest illustration for Equation (3) is the case of the basic structure given as two parallel conductors, when F(v in ) 2 f (v in ), φ(α) = 2>1. With n iterations of the structure, we obtain in this case 2 2n 1 parallel conductors, which obviously means an unlimited increase in the input conductivity. Respectively, the simplest illustration to Equation (4) is the case when the basic structure includes two series conductors. Then F(v in ) f (v in /2), which means φ(α) = 1/2 α <1, and it is obvious that with the iterations we obtain in this case 2 2n 1 series conductors which unlimitedly increases the input resistance, asn. If, as in the circuit shown in Figure 3, there are two series conductors directly connecting a and b, and some other conductors, then whether φ(α)<1, or φ(α)>1, depends on α. Observe from Equation (2) that for α : φ(α) = 1, F(.) = f (.), and is independent of n. Thisis true also for the fractals mentioned in Remark CALCULATION OF φ(α) FOR THE CIRCUIT IN FIGURE 2 The above treatment related to any basic structure/topology, without requiring any geometric (reflection) symmetry. However, considering that the circuit in Figure 2 is also symmetric in the simple geometric sense, we shall calculate φ(α) of the basic structure given in Figure 3, using the circuit of Figure 4. The use of Figure 4 is also a preparation for Section 4 where the simple reflection symmetry is the focus. We thus present the structure shown in Figure 3 as two structures of Figure 4, connected in parallel, changing some f (.) to f (.)/2. In Figure 4, the conductors a o and o b have the characteristics f (.)/2, and the rest of the conductors remained f (.). It is obvious that two such circuits in parallel give the needed basic structure, and that F(.) of the reduced circuit is F(.)/2 of the basic structure of Figure 3. Since v b = 0 and v a = v in, and since, by the symmetry, v o = v in /2 and v d = v in v c, i.e. v c v d = v c (v in v c ) = 2v c v in, the nodal equation at c, (v a v c ) α = (v c v d ) α + (v c v o ) α becomes (v in v c ) α = (2v c v in ) α + (v c v in /2) α = (2 α + 1)(v c v in /2) α Taking the power 1/α of both sides, we easily obtain v c = (2α + 1) 1/α 1 + (2 α + 1) 1/α v in (5) and then (see Figure 4 again) the KCL equation at node a, F(v in ) = (1/2)(v in /2) α + (v in v c ) α, quickly leads to the final (already for Figure 3, i.e. after adding the factor 2) F = 2(1 + 2α ) +[1 + (2 α + 1) 1/α ] α 2 α [1 + (2 α + 1) 1/α ] α vin α (6)

6 642 E. GLUSKIN a + c v in o d b - Figure 4. The auxiliary circuit for calculation of φ(α) for the basic structure of the previous circuit. The conductors a o and o b have the characteristics f (.)/2, and the rest of the conductors remained f (.). It is obvious that two such circuits in parallel give the basic structure of Figure 3, i.e. F(.) of the present circuit is F(.)/2 of the basic structure. In the calculation of φ(α), weset f (.)=(.) α, i.e. take D=1. That is, φ(α) = 2(1 + 2α ) +[1 + (2 α + 1) 1/α ] α 2 α [1 + (2 α + 1) 1/α ] α (7) Using Equation (7) and n = 2 in Equation (2), one obtains F(v in ) of the circuit in Figure 2. From Equation (7), φ(1) = 10/8>1, and φ(2) 0.489<1, with the respective conclusions (Equations (3) and (4)) regarding taking n in F of the n-order fractal circuit, which we already know. Since φ(α) is a continuous function, such an 1<α<2 can be found that φ(α) = 1, and, regarding F, the whole structure is equivalent for this α to a single conductor, i.e. F(v in ) = f (v in ). We shall not calculate this α ad hoc here, but note that, in general, the specific value of α, given by the equation φ(α) = 1, is a characteristic of the basic structure in the context of creation from it of the large fractal circuits. For such α, the whole fractal conducts, for any n, as one of its single power-law conductors. We turn now to a brief consideration of some more usual symmetry aspects, sufficiently interesting by themselves and helpful for the analysis of some fractal circuits. 4. A GENERAL SCALING RELATION FOR THE CASE OF THE REGULAR REFLECTION SYMMETRY Considering Figure 3 again, we see two features that, however, are fully relevant for any circuit having the reflection symmetry; thus we could, e.g. also directly refer to the circuit in Figure 2, or a higher-order fractal realization. That the symmetry remains with the recursive steps, is obvious. The first feature is given by the already used relation between Figures 3 and 4. The second feature is to use the equality of the potentials of the symmetric nodes (of the symmetric wings of the circuit) in Figure 3 and introduce the respective short-circuiting connections. Figurally speaking, this is equivalent to representing the given circuit as a book, or a butterfly, that is being closed. We thus obtain from Figure 3 a circuit similar to that in Figure 4, but now the conductors on the line a b remain f (.), while the rest become 2 f (.). The function F(.) is not changed with the reduction of the structure.

7 ON THE SYMMETRY FEATURES OF SOME ELECTRICAL CIRCUITS 643 It is essential that in both cases the reduced topology is the same, and we introduce for it the following symbolic notation F 1/2 (z 1, z 2 ) where F 1/2 is F of the reduced structure/topology. The argument z 1 represents the dependence of F 1/2 on those of the conductivities of the conductors, which are placed on the symmetry line, and z 2 the dependence of F 1/2 on the conductivities of all other conductors. Comparing the two mentioned ways of calculating the same F, we immediately obtain that for any such symmetric circuit the function F 1/2 (.,.) satisfies the following scaling relation: F 1/2 ( f, 2 f ) = 2F 1/2 ( f/2, f ) (8) in which each side is F of the given circuit. The very presence of the symmetry is reflected here in the use of the function of two variables. Equation (8) should be helpful in many cases for determination of F of a given circuit. In particular, the infinite 2D grids considered in References [4, 5] can be reduced, using symmetry, to quarter of grid, which may result, e.g. in the possibility to make a simulation using only a part of the large grid cut employed in Reference [5]. Finally, let us note that there are some cases of symmetry when a 1-port not given as a series-parallel may be reduced as regards F(.) to such a structure, which ensures, as Reference [1] explains, a straightforward formulaic determination of F(.). 5. CONCLUSIONS As the main point, it is noted that a 1-port s structure composed of similar elements naturally leads to the creation of fractal-type circuits, because each branch is also a 1-port. If f (.) is the characteristic of the elements, and the input characteristic is F(.), and if the basic 1-port, f F, is solvable, then the 1-port f F F F is also solvable in the same way, and complicated solvable nonlinear circuits can be thus easily created. For the α-circuits introduced in Reference [1], the formulaic determination of F(.) is relatively simple. For the basic (and simplest) definition of resistive fractal, the number of its elements and the sensitivity of the F(v in ) to n are increased with an increase in n in an explosion manner, and another type of fractals is also suggested (Remark 3) to moderate these parameters. The condition φ(α) = 1definesα for which F(.) f (.). The Letter thus demonstrates one of the possible applications of the α-circuits. ACKNOWLEDGEMENTS I am grateful to the unknown reviewer for his overall helpful comments on the initially submitted version. As all my other works published or written in 2006, this Letter is dedicated to the 70 birthday of Professor Ben-Zion Kaplan. REFERENCES 1. Gluskin E. One-ports composed of power-law resistors. IEEE Transactions on Circuits and Systems II: Express Brief 2004; 51(9): Zemanian AH. A classical puzzle: the driving point resistance of infinite grids. IEEE Circuits and Systems Magazine March 1984; 7 9.

8 644 E. GLUSKIN 3. Chua LO, Desoer ChA, Kuh E. Linear and Non-linear Circuits. McGraw-Hill: New York, Gluskin E. Calculations and hypotheses relating to the input conductivity function of an infinite grid of similar non-linear resistors. International Journal of Systems Science 2001; 32(6): Gluskin E. An approximation for the input conductivity function of the non-linear resistive grid. International Journal of Circuit Theory and Applications 2001; 29:

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

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*