Constant-phase-angle and power-law regimes in the frequency response of a general determinate fractal circuit

A deterministic-fractal-circuit model for the impedance of heterogeneous systems is investigated and its ac response determined. The model, originally applied by Clerc and his co-workers to percolative systems, is generalized to include arbitrary numbers of elements in the irreducible circuit with an arbitrary composition, and is also extended into the subpercolative regime of filling factors. It is shown that, as the fraction of embedded conducting elements in the irreducible circuit approaches the critical value for percolation from either side, there is a crossover from a constant-phase-angle response to a power-law behavior, whenever the number of elements in each embedding is greater than about four.