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Progress In Electromagnetics Research B
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THE RENORMALIZATION GROUP THEORY COMBINED TO THE MS-GEC METHOD TO STUDY ACTIVE FRACTAL STRUCTURES WITH INCORPORATED PIN DIODES

By S. Mili, C. Larbi Aguili, and T. Aguili

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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.

Citation:
S. Mili, C. Larbi Aguili, and T. Aguili, "The Renormalization Group Theory Combined to the Ms-GEC Method to Study Active Fractal Structures with Incorporated PIN Diodes," Progress In Electromagnetics Research B, Vol. 29, 43-62, 2011.
doi:10.2528/PIERB11013105

References:
1. Mandelbrot, B., Les Objets Fractals, 4th Ed., Flammarion, 1995.

2. Werner, D. H. and R. Mittra, Frontiers in Electromagnetics, IEEE Press, NJ, 2000.

3. Skrivervik, A. K., J. F. Zurcher, O. Staub, and J. R. Mosig, "CS antenna design: The challenge of miniaturization," IEEE Transactions on Antennas and Propagation, Vol. 43, No. 4, 12-27, August 2001.

4. Gianvittono, J. and Y. Rahmat Samii, "Fractal antennas: A novel antenna miniaturization technique and applications," IEEE Antennas Propag. Mag., Vol. 44, No. 1, 20-36, 2002.
doi:10.1109/74.997888

5. Tennant, A. and B. Chambers, "A single-layer tuneable microwave absorber using an active FSS," IEEE Microwave and Wireless Components Letters, Vol. 14, No. 1, 46-47, January 2004.
doi:10.1109/LMWC.2003.820639

6. Chang, K., S. I. Kwak, Y, and J. Yoon, "Equivalent circuit modeling of active frequency selective surfaces," IEEE Radio and Wireless Symposium, 663-666, 2008.
doi:10.1109/RWS.2008.4463579

7. Bossard, J. A., D. H. Werner, T. S. Mayer, and R. P. Drupp, "A novel design methodology for reconfigurable frequency selective surfaces using Genetic Algorithms," IEEE Transactions on Antennas and Propagation, Vol. 53, No. 4, 1390-1400, April 2005.
doi:10.1109/TAP.2005.844439

8. Mili, S., C. Larbi Aguili, and T. Aguili, "Study of fractal-shaped structures with pin diodes using the multi-scale method combined to the generalized equivalent circuit modeling," Progress In Electromagnetics Research B, Vol. 27, 213-233, 2011.

9. Wilson, K. G., "The renormalization group and critical phenomena," Nobel Lecture, December 8, 1982.

10. Fisher, M. E., "Renormalization group theory: Its basis and formulation in statistical physics," Reviews of Modern Physics, Vol. 70, No. 2, April 1998.

11. Wilson, K. G. and J. Kogut, "The renormalization group and the ε expansion," Physics Reports (Section C of Physics Letters), Vol. 12, No. 2, 75-200, 1974.
doi:10.1016/0370-1573(74)90023-4

12. Wilson, K. G., "Renormalisation group and Kadanoff Scaling picture," Physics Rev. B, Vol. 4, 3174, 1971.
doi:10.1103/PhysRevB.4.3174

13. Fisher, M. E., "The renormalization group and the theory of critical behaviour," Reviews of Modern Physics, Vol. 46, 597, April 1998.

14. Pelissetto, A. and E. Vicari, "Critical phenomena and renormalization group theory," Physics Reports, April 2002.

15. Larbi Aguili, C., A. Bouallegue, and H. Baudrand, Utilisation d'un processus de renormalisation pour l'étude électromagnétique des structures fractales bidimensionnelles, Annales des Télecommunications, Vol. 60, No. 7--8, 1023-1050, Juillet-Août, 2005.

16. Aguili, T., Modélisation des composants S.H.F planaires par la méthode des circuits équivalents généralisés, Thesis, National Engineering school of Tunis ENIT, May 2000.

17. Baudrand, H., Representation by equivalent circuit of the integral methods in microwave passive elements, European Microwave Conference, Vol. 2, 1359-1364, Budapest, Hungary, September 10--13, 1990.

18., L'Electromagnétisme par les Schémas Equivalents, Cepaduès Éditions, 2003.

19. Kiani, G. I., K. P. Esselle, A. R. Weily, and K. L. Ford, Active frequency selective surface using PIN diodes, Antennas and Propagation Society International Symposium, 4525-4528, Honolulu, HI, June 9--17, 2007.

20. Wilton, D. R. and C. M. Butler, "Efficient numerical techniques for solving pocklington's equation and their relationship to other methods," IEEE Transactions on Antennas and Propagation, 83-86, January 976.


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