The Renormalization Group Theory Combined to the MS-Gec Method to Study Active Fractal Structures With Incorporated Pin Diodes

Abstract

Abstract-The renormalization group theory (RGT) is used in this paper to develop an extension of the multi-scale approach (MS-GEC), previously developed by the authors, in order to enable the study of fractal structures at infinite iterations. In this work, we focused on active fractal structures with incorporated PIN diodes but the developed concept can be applied to a wide variety of fractals. The MS-GEC method deals with fractal-shaped objects as a set of scale levels. The processing is done gradually, one scale at each step, from the lowest scale till the highest one. To compute the input impedance of fractal-shaped structures using the MS-GEC method, we demonstrated that the input impedance of any scale level is generated from the input impedance of the previous scale level. When the iteration of fractal tends toward infinity, the structure contains an unknown number of levels. Since the atomic level cannot be defined, a critical point is reached limiting then the scope of the MS-GEC and of the existing classical methods. Based on RGT concepts, if the relation between the input impedances of two consecutive levels can be rewritten independently of the critical parameter (which is in our case the scale level), a transformation called "renormalization group" is generated. Consequently, the input impedance of the infinite active fractal structure approaches the fixed point of the defined transformation independently of the system details at the atomic level. The MS-GEC method combined to the RGT is a very powerful technique since it profits from the advantages (rapidity and reduced memory requirements) of the MS-GEC method and from the ability of the RGT to solve problems at their critical point.

Multiscale Auxiliary Sources for Modeling Microwave Components

“…This scheme integrates the environment of the dipole in the measurement process. In particular, the capacitance between the terminals is taken into account and should be removed to reach the intrinsic (or nude) impedance (Figure 18a): If this precaution is not taken, the capacitance C will be integrated into the localized impedance or in the equivalent medium [19] but, in the calculation of the global device, there will be the presence of external field lines (Figure 18a) in the electromagnetic resolution. These field lines contribute to capacity C, and they have already been taken into account to establish the equivalent scheme.…”