Singularity structure analysis of fractal relaxation systems

Abstract

The fractional slope frequency spectrum has been observed in many physical phenomena such as electrical noise, the relaxation of polarized dielectrics, viscous and magnetic materials as well as the interface between two dissimilar conducting materials. Such systems are recognized as fractal systems and several empirical mathematical models have been suggested to model their dynamical behaviour. The singularity structure model to characterize the steady-state dynamics of fractal systems in the linear range is proposed. The approach incorporates fractal concepts applied to classical system theory. The singularity structure model of fractal systems can be mathematically represented as a rational system function. Differing from the standard system function, the poles and zeros of fractal systems function are located very close to each other in the complex plane and are distributed in a special manner. The local singularity structure of fractal systems, i.e. the placement of poles and zeros, can be constructed by a recursive procedure following a simple rule which relates the global parameters to local ones, using the self-similarity property. The standard singularity structure is also introduced to generalize the singularity structure of fractal systems. A fractal circuit is constructed to test the model and the synthesized singularity structure for the impedance of the electrode-electrolyte interface is compared with experimental data.