An estimation of the input conductivity characteristic of some resistive (percolation) structures composed of elements having a two-term polynomial characteristic

A specific circuit connection is suggested for the calculation of the input conductivity (current caused by voltage) characteristic iin=F(vin) of a 1-port named “f-circuit”, composed of similar conductors described by conductive characteristic of the type f(v)=Dmvm+Dnvn. It is assumed that such circuit models can be relevant to the theory of percolation structures [S.W. Kenkel, J.P. Straley, Percolation theory of nonlinear circuit elements, Phys. Rev. Lett. 49(11)(1982)767–770; J.P. Straley, S.W. Kenkel, Percolation theory for nonlinear conductors, Phys. Rev. B 29(11)(1984)6299–6305; R. Rammal, A.-M.S. Trembley, Resistance noise in nonlinear resistor networks, Phys. Rev. Lett. 58(4)(1987)415–418; Ch.P. Hatsell, A quasi-power theorem for bulk conductors: comments on Rheoencephalography, IEEE Trans. Biomed. Eng. 38(7)(1991)665–669; L. Venkataraman, Y.S. Hong, P. Kim, Electron transport in a multichannel one-dimensional conductor: molybdenum selenide nanowires, Phys. Rev. Lett 96(076601)(2006)076601-1–076601-4; A.A. Snarskii, K.V. Slipchenko, A.A. Sevryukov, Critical behavior in two-phase, highly inhomogeneous composites, J. Exp. Theor. Phy. 89(4)(1999)788–799; A.A. Snarskii, M. Zhenirovskiy, Effective conductivity of non-linear composites, Physica B 322(2002)84–91], in a state not far from the percolation threshold, even though the main result seems to be sufficiently interesting by itself and one can expect that other interesting applications for such conductive structures will be found. The f-circuit is constructed here from two power-law “α-circuits”, f(v)∼vα, of the same topology [E. Gluskin, One-ports composed of power-law resistors, IEEE Trans. CAS-II 51(9)(2004)464–467], but having different α. In this construction, named “f-connection”, the respective nodes of the α-circuits are short-circuited, and the respective branches become connected in parallel, which causes f(.) to be an additive function; thus f(v)=Dmvm+Dnvn is obtained for each element of the connection. It appears that iin of the connection is close to the sum of the input currents of the independent α-circuits, connected to the same voltage source. Since precise calculation of the input current of the connection is extremely difficult, this result seems to be valuable for calculation of F(.). The reason for the unexpectedly high precision of the approximation is explained in basic circuit terms, and will be given in more detail in a separate circuit-theory publication.