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The driving point immittance (impedance or admittance) function is commonly used in electrical characterization of polarized materials and interfaces. The immittance function typically attenuates following a power function dependence on frequency. This fact has been recognized as a macroscopic dynamical property manifested by strongly interacting dielectric, viscoelastic and magnetic materials and interfaces between different conducting substances. Linear interfacial polarization processes which occur at metal electrode-electrolyte interfaces have been represented by the Fractional Power Pole [FPP] function in single or multiple stages. The FPP function is referred to as the Davidson-Cole function in the dielectrics literature. A related function widely used in mathematical modeling of dielectric and viscoelastic polarization dynamics is the Cole-Cole function. The fractional power factor which parametrizes the FPP or the Davidson-Cole function has been shown earlier to equal the logarithmic ratio of the locations of the pole-zero singularities. In this paper we first review a modified form of the singularity decomposition of the FPP function accomplished within a prescribed error range. The distribution spectrum and the corresponding simulation by a cascade R-C network, as opposed to the synthesis by a ladder R-C network, are readily obtained as the next step in the simulation. The method is then applied to decompose the Cole-Cole function; the pole-zero placement of the singularity function is determined and the equivalent cascade R-C network is synthesized.

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The magnitude spectral density of many physical phenomena such as electrical noise, the relaxation of polarized dielectrics, viscous and magnetic materials, and the interface between two dissimilar conducting materials attenuate following a fractional power function dependence on frequency. Such systems are recognized as fractal systems distinguished by the Iff-type attenuation in the magnitude spectrum and characterized by global parameters such as fractal dimension and global corners. Fractal systems which relax to steady state by a distribution of purely real exponentials are recognized as fractal relaxation systems. This is the first part of a series of planned articles, each focusing on a particular aspect of fractal relaxation systems. In this part, the singularity structure model is proposed to represent the steady state frequency response of fractal relaxation systems in the linear range. The singularity structure model of fractal relaxation systems can be mathematically represented by a rational system function with simple poles and zeros arranged in a self-similar pattern in the complex frequency plane. We show that the local singularity structure, i.e. the placement of poles and zeros, can be generated by a recursive procedure following a simple rule which relates the global parameters to local ones. To illustrate the approach and test the model, we first synthesize a fractal R-C circuit; then, we demonstrate the strength of the method by analyzing experimental impedance data collected from a polarized metal electrodeelectrolyte interface over six decades of frequency. The standard singularity structure is also introduced to generalize the concept of singularity structure.Forthcoming articles in the series will focus on the analysis of fractal relaxation systems using the discrete distribution of relaxation times and on the concepts of 'structure' and 'view' scales. We promote the distribution of relaxation times as a powerful method to analyze fractal systems and note that it carries information identical to the singularity structure. In fractal systems, the observer becomes part of the modelling process. This is due to the self-similar property of fractal systems: the structure scale of the singularity function is determined according to the view scale chosen by the observer. The implication of this property on the conventional model order concept is elaborated. INDEX TERMS:Fractal relaxation systems, fractional power attentuation, Iff-type attenuation, power-law behavior, self-similarity, singularity structure. location ratio, structure scale, view scale.