Abstract-A multi-scale (MS) approach combined to the generalized equivalent circuit (GEC) modeling is applied to compute the input impedance of pre-fractal structures with incorporated PIN diodes. Instead of treating the whole complex problem at once, the MS method splits the complex structure into a set of scale levels to be studied separately. The computation is done gradually from the lowest level. Each scale level is artificially excited by N modal sources to compute its input impedance matrix. The MS method is based on converting this input impedance matrix into an impedance operator to achieve the transition toward the subsequent level. The PIN diodes were easily integrated in the MS approach thanks to their surface impedance model. The main advantage of the MS-GEC method is the significant reduction of the problem's high aspect ratio since fine details are studied separately of the larger structure. Consequently, the manipulated matrices are well conditioned. Moreover, the reduced size of matrices manipulated at each level leads to less memory requirement and faster processing than the MoM. Values obtained with the MS-GEC approach converge to those given by the MoM method when a sufficient number of modal sources are used at each scale level. For frequencies between 1 GHz and 6.8 GHz, the agreement between the two methods is conspicuous.

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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.

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