On the symmetry features of some electrical circuits

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 -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. 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 '-circuits' introduced in Reference (1) (and for a general circuit theory background see also References (2, 3) )a re shown here to be the most 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