350 rub
Journal Radioengineering №3 for 2009 г.
Article in number:
Analogue Realization of Semi-Integrator
Keywords: fractional order integrator frequency-band semi-integrator uniform RC network Cauer network thin-film deposition photolithography
Authors:
S.V. Krupenin, V.V. Kolesov
Abstract:
Analogue realization of the semi-integrator based on distributed RC components is presented. The semi-integrating circuit is represented by uniform Cauer network: series-R and shunt-C ladder. As shown by Oldham and Spanier, the bandwidth of this semi-integrating circuit depends on RC constant and number of network elements. Thin-film realization of the distributed RC components is represented by microstrip line with modulated width. Microstrip line is arranged in zig-zag style in order to obtain sample compactness. Samples of the semi-integrator are fabricated by means of high-vacuum thin-film deposition and photolithography. The circuit containing 400 RC elements is implemented on silicon substrate and located within 6x8 mm2. Materials used in thin-film deposition are titanium, silicon dioxide and chromium. The upper conducting layer of microstrip structure (line with modulated width) is formed by means of photolithography procedure. In-plane sizes of resistive and capacitive elements are 10x100 and 100x100 square microns, respectively. Frequency-band semi-integration is verified by experimental results and numerical modeling. Experimental bandwidth of semi-integrator is one octave: 800-1600 MHz. Possible applications of the semi-integrator are as follows: fractal system modeling (for example, fractal noise generation) and synthesis of fractional solving circuits
Pages: 114
References
  1. Мандельброт Б. Фрактальная геометрия природы: Пер. с англ. А.Р. Логунова. - М.: Институт компьютерных исследований, 2002.
  2. Bardou F., Bouchaud J.-P., Aspect A., Cohen-Tannoudji C. Lévy Statistics and Laser Couling. How Rare Events Bring Atoms to Rest. - New York: Cambridge University Press., 2003.
  3. Liouville J. Mémoire sur le changement de la variable dans le calcul des différentielles à indices quelconques. - J. Ecole Polytech., 1835, vol. 15, pp.17-54.
  4. Riemann B. Versuch einer Auffassung der Integration und Differentiation. - Leipzig: Teubner, 1847.
  5. Oldham K.B., Spanier J. The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. - San Diego: Academic Press, 1974.
  6. Самко С.Г., Килбас А.А., Маричев О.И. Интегралы и производные дробного порядка и некоторые их приложения. - Минск: Наука и техника, 1987.
  7. Miller K.S., Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations. - New York: John Wiley & Sons, 1993.
  8. Freund J.A., Pöschel Th. (eds.). Stochastic Processes in Physics, Chemistry and Biology. - Berlin, Heidelberg, New York: Springer-Verlag, 2000.
  9. Нахушев А.М. Дробное исчисление и его применение. - М.: Физматлит, 2003.
  10. West B.J., Bologna M., Grigolini P. Physics of Fractal Operators. - New York: Springer, 2003.
  11. Sabatier J., Agrawal O.P., Tenreiro J.A. Machado (eds.). Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. - Dordrecht: Springer, 2007.
  12. Потапов А.А. Фракталы в радиофизике и радиолокации: Топология выборки. - М.: Университетская книга, 2005.
  13. Cole K.S., Cole R.H. Dispersion and absorbtion in dielectrics, alternation current characterization. - J. Chem. Physics, 1941, vol. 9, pp.341-351.
  14. Davidson D.W., Cole R.H. Dielectric relaxation in glycerol, propylene glycol and n-propanol. - J. Chem. Physics, 1951, vol. 19, pp.1484-1490.
  15. Sun H., Charef A., Tsao Y.Y., Onaral B. Analysis of Polarization Dynamics by Singularity Decomposition Method. - Ann. Biomed. Eng., 1992, vol. 20, pp.321-335.
  16. Warburg. Ueber das Verhalten Sogenannter Unpolarisirbarer Electroden gegen Wechselstrom. Ann. Physik. - Chemie, 1899, vol. 67, pp.493-499.
  17. Liu S.H. Fractal model for the AC response of a rough interface. - Phys. Rev. Lett., 1985, vol. 55, pp.529-532.
  18. Коверда В.П., Скоков В.Н. Критическое поведение и 1/f-шум при пересечении двух фазовых переходов в сосредоточенных системах. - ЖТФ, 2000, т. 70, № 10, с.1-7.
  19. Скрипов В.П., Виноградов А.В., Скоков В.Н., Решетников А.В., Коверда В.П. Капля на горячей плите: появление 1/f-шума при переходе к сфероидальной форме. - ЖТФ, 2003, т. 73, №6, с.21-23.
  20. Коверда В.П., Скоков В.Н. Масштабные преобразования 1/f флуктуаций при неравновесных фазовых переходах. - ЖТФ, 2004, т. 74, № 9, с.4-8.
  21. Klafter J., Blumen A., Shlesinger M.F. Stochastic pathways to anomalous diffusion. - Phys. Rev. A, 1987, vol. 35, no. 7, pp.3081-3085.
  22. Fogedby H.C. Lévy Flights in Random Environments. - Phys. Rev. Lett., 1994, vol. 73, no. 19, pp.2517-2520.
  23. Hilfer R., Anton L. Fractional master equations and fractal time random walks. - Phys. Rev. E, 1995, vol. 51, no. 2, pp.R848-R851.
  24. Allegrini P., Grigolini P., West B.J. Dynamical approach to Lévy processes. - Phys. Rev. E, 1996, vol. 54. no. 5, pp. 4760-4767.
  25. Compte A. Stochastical foundations of fractional dynamics. - Phys. Rev. E, 1996, vol. 53, no. 4, pp.4191-4193.
  26. West B.J., Grigolini P., Metzler R., Nonnenmacher Th.F. Fractional diffusion and Lévy stable processes. -Phys. Rev. E, 1997, vol. 55, no. 1, pp.99-106.
  27. Shlesinger M.F., West B.J., Klafter J. Lévy Dynamics for Enhanced Diffusion: Application to Turbulence. - Phys. Rev. Lett., 1987, vol. 58, no. 11, pp.1100-1103.
  28. Glöckle W.G., Nonnenmacher Th.F. A Fractional Calculus Approach to Self-Similar Protein Dynamics. - Biophys. J., 1995, vol. 68, pp.46-53.
  29. Grenness M., Oldham K.B. Semiintegral Electroanalysis: Theory and Verification. - Anal. Chem., 1972, vol. 44, no. 7, pp.1121-1129.
  30. Heaviside O. Electromagnetic Theory. - London: The Electrician, 1899.
  31. Рехвиашвили С.Ш. Моделирование фликкер-шума с помощью дробного интегро-дифференцирования. - ЖТФ, 2006, т. 76, № 6, с.123-126.
  32. Montroll E.W., Shlesinger M.F. On 1/f noise and other distributions with long tails. - Proc. Natl. Acad. Sci. USA, 1982, vol. 79, pp.3380-3383.
  33. Mathieu B., Melchior P., Oustaloup A., Ceyral Ch. Fractional differentiation for edge detection. - Signal Processing, 2003, vol. 83, no. 11, pp.2421-2432.
  34. Oustaloup A., Levron F., Mathieu B., Nanot F.M. Frequency-Band Complex Noninteger Differentiator: Characterization and Synthesis. - IEEE Trans. Circuits Syst.-I, 2000, vol. 47, no. 1, pp.25-39.
  35. Morrison R. RC Constant-Argument Driving-Point Admittances. - IRE Trans. Circuit Theory, 1959, vol. CT-6, pp.310-317.
  36. Carlson G.E., Halijak C.A. Approximation of Fixed Impedances. - IRE Trans. Circuit Theory, 1962, vol. CT-9, pp.302-303.
  37. Lerner R.M. The Design of a Contant-Angle or Power-Law Magnitude Impedance. - IEEE Trans. Circuit Theory, 1963, vol. CT-10, pp.98-107.
  38. Steiglitz K. An RC impedance approximant to s-1/2. - IEEE Trans. Circuit Theory, 1964, vol. CT-11, pp.160-161.
  39. Carlson G.E., Halijak C.A. Approximation of Fractional Capacitors 1/s1/n by a Regular Newton Process. - IEEE Trans. Circuit Theory, 1964, vol. CT-11, pp.210-213.
  40. Halijak C.A. An RC impedance approximant to 1/s-1/2. - IEEE Trans. Circuit Theory, 1964, vol. CT-11, pp.494-495.
  41. Dutta Roy S.C. On the Realization of a Constant-Argument Immitance or Fractional Operator. - IEEE Trans. Circuit Theory, 1967, vol. CT-14, no. 3, pp.264-274.
  42. Jiang C.X., Carletta J.E., Hartley T.T. Implementation of Fractional-order Operators on Field Programmable Gate Arrays. In: J. Sabatier et al. (eds.). Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, 2007, §5, pp.333-346.
  43. Крупенин В.А., Паволоцкий А.Б., Прохорова И.Г., Снегирев О.В. Технология изготовления и характеристики диэлектрических слоев тонкопленочных RC фильтров для джозефсоновских и одноэлектронных уcтройств. - Письма в ЖТФ, 1996, т. 22, № 2, с.19-27.