Open Access Open Access  Restricted Access Subscription Access

Bipolar Elements with Fractal Impedance and Their Application in Radio Engineering and Communications

Ushakov P.A., Baboshkin G.D., Stoichev S.V., Gravshin V.G.


Modern radio engineering also includes the fractals theory, the theory of fractional integro-differentiation operators, and fractal interpretation of various problems arising in different fields of science and technology. New terms “fractal radiophysics”, “fractal radar”, and “fractal radioelectronics” reflect a brand new approach to representing the components of electrical signals and the electromagnetic field.

This review also includes the definition of fractional calculus main operators: the derivative and integral of fractional orders, which are necessary for description and study fractal systems and processes.

Analog circuit simulation is practiced on a wide scale to solve practical problem when it has difficult mathematical formulation or when its analytical solution is not required, but when you need to know only a dynamic system reaction to a certain effect. However, in the case of fractional-order systems, the implementation of such simulation requires the presence of specific bipolar passive elements which relationship between current and voltage is described by a fractional-order differential equation. The impedance of such elements is different from the impedance of conventional capacitive or inductive elements. The frequency on which it depends has a fractional power a (0 < a <1). Therefore, we called them elements with fractal impedance (FOE).

The objective of this work is to bridge the gaps in informing about FOE, because information about FOE, their characteristics, design options and principles for implementing fractal impedance, also about using FOE to improve the functional devices of radio engineering and communications is practically unknown to Russian researchers, engineers, young scientists, etc.

This paper also includes the classification of FOE, some existing design and technological options for FOE and their characteristics. According to a comparative analysis of the characteristics of different FOEs, most of them are currently not suitable for manufacture on an industrial scale to satisfy the needs of scientists and engineers. However, FOEs based on a multilayer resistive-capacitive medium with controlled geometric and electrophysical parameters of the medium are quite promising. These elements are constructed in the form of integrated structures manufactured using standard technologies as well as for creating film or semiconductor microcircuits. Mathematical models, algorithms, and programs have been developed for its analysis and synthesis. This allows to design the FOE constructions with a given a in a given limited frequency band. Samples of FOE that were industrially manufactured and a comparison of the characteristics realized in practice with those synthesized at the design stage are also demonstrated in this work.

Physical FOE samples allow for evaluating the potential advantages of their application, possible limitations and design methods, taking into account those features that distinguish FOE from traditional passive elements used in circuitry and analog modeling.

The work considers the principles of building integrators and differentiators of fractional order and the results of integration and differentiation of signals with devices using FOE based on a resistive-capacitive medium (FOE based on one-dimensional homogeneous resistive-capacitive elements with a layer structure of R-C-NR). There is an example that shows the possibility of creating an analog processor using analog integrators and fractional order differentiators to solve fractional differential equations.

There is also an example of the fractional-order PID control implementation for constructing an automatic control system for an antenna-rotary device. The results of the designed device were checked using circuit modeling, in which their mathematical models were used as FOEs. It was shown that a fractional order regulator allows to create a control system with better control characteristics than the classical control system.

It is described how using FOE in creating chaos generators (which are the basis of systems characterized by deterministic chaos) allows to change the nature of chaotic signals and the shape of attractors without changing the initial conditions or switching chaos formation systems.

The work also includes a description of constructing fractional-order frequency-selective filters principles and features of the frequency characteristics of various types of filters. Besides that, the possibilities are presented for controlling filter parameters and characteristics by using an additional degree of freedom in the form of the parameter α as an FOE parameter.

Since frequency-selective filters are an integral part of oscillators of electrical oscillations, using fractional filters also introduces serious differences in the parameters of fractional oscillators in comparison with the parameters of their classic prototypes. The main difference is that the generation frequencies are several orders of magnitude higher than the generation frequencies of prototypes with the same time constants of phasing circuits. Beyond that an additional degree of freedom allows you to independently control the frequency and phase of the signal at the output of the generator and build multiphase generators.


fractional calculus, elements with fractal impedance, RC-elements with distributed parameters, devices for fractional differentiation and integration, fractional order PID controllers, fractional order generators, fractional order filters

Full Text


PDF (Русский)
References References

Потапов А. А. Фракталы в радиофизике и радиолокации : Топология выборки. М. : Университетская книга, 2005. 848 с.

Потапов А. А., Гильмутдинов А. Х., Ушаков П. А. Фрактальные радиоэлементы и радиосистемы : Физический аспект : монография / под ред. А. А. Потапова. М. : Радиотехника, 2009. 200 с.

Учайкин В. В. Метод дробных производных. Ульяновск : Артишок, 2008. 512 с.

Tenreiro Machado J., Kiryakova V., Mainardi F. Recent history of fractional calculus. Communications in Nonlinear Science and Numerical Simulations, 2011, no. 3, pp. 1140-1153. doi: 10.1016/j.cnsns.2010.05.027

Hilfer R. Applications of Fractional Calculus in Physics. Singapore, World Scientific Publ. Company, 2000. ISBN: 978-981-02-3457-7.

Magin R. Fractional Calculus in Bioengineering. Redding, Begell House Publ., 2006. ISBN: 978-1567002157.

Mainardi F. Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. London, Imperial College Press, 2010. ISBN: 978-1-84816-329-4.

Monje C., Chen Y., Vinagre B., Xue D., Feliu V. Fractional Order Systems and Controls: Fundamentals and Applications. London, Springer, 2010. ISBN: 978-1849963343.

Zaslavsky G. Hamiltonian Chaos and Fractional Dynamics. New York, Oxford University Press, 2005. ISBN: 978-0198526049.

Podlubny I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, vol. 198, New York, London, Sydney, Tokyo and Toronto, Academic Press, 1999.

Caponetto R., Dongola G., Fortuna L., Petráš I. Fractional Order Systems: Modeling and Control Applications, World Scientific Publ. Co. Pt. Ltd., 2010, 195 p.

Podlubny I. Fractional Differential Equations. San Diego, Academic Press, 1999.

Тетельбаум И. М., Шнейдер Ю. Р. Практика аналогового моделирования динамических систем : справочное пособие. М. : Энергоатомиздат, 1987. 383 с.

Кампе-Немм А. А. Решение инженерных задач на электронных моделирующих машинах. М. : Энергия, 1970. 96 с.

Dingyu Xue, Yang Quan Chen. Dingyu Solving applied mathematical problems with MATLAB, Chapman & Hall/CRC, Taylor & Francis Group, 2009. 433 p.

Нигматуллин Р. Ш. Общее уравнение и электрический аналог электролитической ячейки со сферическим стационарным микроэлектродом // Докл. АН СССР. 1963. N. 151, № 6. С. 1383–1388.

Карамов Ф. А. Суперионные проводники. Гетероструктуры и элементы функциональной электроники на их основе. М. : Наука, 2002. 303 с.

Biswas K., Sen S., Dutta P. K. Realization of a constant phase element and its performance study in a differentiator circuit. IEEE Trans. Circuits Syst. II, 2006, vol. 53, pp. 802-806.

Riccardo Caponetto, Salvatore Graziani, Fulvio L. Pappalardo, Francesca Sapuppo. Experimental Characterization of Ionic Polymer Metal Composite as a Novel Fractional Order Element, Corporation Advances in Mathematical Physics, Hindawi Publ., 2013, Article ID 953695, 10 p.

Agambayev A., Patole S.P., Farhat M., Elwakil A., Bagci H., Salama K.N. Ferroelectric Fractional-Order Capacitors. Chem. Electro Chem, 2017, vol. 4, pp. 2807-2813.

Haba T., Loum G., Zoueu J., Ablart G. Use of a component with fractional impedance in the realization of an analogical regulator of order ½. J. Appl. Sciences, 2008, vol. 8, no. 1, pp. 59-67.

Biswas K. Fractional-Order Devices. Springer, 2017, 111 p. doi: 10.1007/978-3-319-54460-1

Gil’mutdinov A.Kh., Ushakov P.A., El-Khazali R. Fractal Elements and their Application. Springer, 2017, 252 p. doi: 10.1007/978-3-319-45249-4

Кайзер Г., Кастро Р., Николс А. Схемы с распределенными параметрами на тонких пленках // Зарубежная радиоэлектроника. 1967. Т. 9, № 4. С. 112–123.

Колесов Л. Н. Введение в инженерную микроэлектронику. М. : Советское радио, 1974. 280 с.

Гильмутдинов А. Х., Ушаков П. А. Неоднородные резистивно-емкостные элементы с распределенными параметрами. Классы и анализ // Методы моделирования : тр. Казанского науч. семинара / под ред. В. А. Райхлина. Вып. 3. С. 233–252. Казань : Изд-во КГТУ, 2007.

Гильмутдинов А. Х. Резистивно-емкостные элементы с распределенными параметрами : Анализ, синтез и применение. Казань : Изд-во КГТУ, 2005. 350 с.

Гильмутдинов А. Х., Потапов А. А., Ушаков П. А. Математические и алгоритмические основы синтеза фрактальных элементов на основе ОСН RC-ЭРП // Фракталы и дробные операторы / предисл. акад. Ю. В. Гуляева, чл.-кор. РАН С. А. Никитова ; под общ. ред. А. Х. Гильмутдинова. Казань : Изд-во Академии наук РТ, 2010. 488 с. ISBN 978-59690-0123-4.

Ушаков П. А., Максимов К. О. Разработка генетического алгоритма для синтеза конструкций фрактальных элементов на основе резистивно-емкостной среды со структурой слоев вида R-C-NR // Вестник ИжГТУ. 2012. № 3 (55). С. 104–108.

Happ W.W., Castro P.S., Fuller W.D. Synthesis of Solid-state distributed parameters functions. IRE Int. Conv. Rec., 1962, vol. 10, pt. 6, pp. 262-278.

Максимов К. О. Решение задачи обеспечения заданных параметров фрактальных радиоэлементов на основе резистивно-емкостной среды : автореф. дис. канд. техн. наук, Ижевск, 2013. 28 с.

Adhikary A., Sen S., Biswas K. Practical Realization of Tunable Fractional Order Parallel Resonator and Fractional Order Filters. IEEE Trans. on Circuit and Systems, August 2016, vol. 63, no. 8, pp. 1142-1151.

Гильмутдинов А. Х., Ушаков П. А. Физическая реализация элементов с фрактальным импедансом: Состояние и перспективы // Радиотехника и электроника. 2017. Т. 52, № 5. С. 1–14.

Krishna B.T. Studies on fractional order differentiators and integrators: A survey. Signal Processing, 2011, vol. 91, pp. 386-426. doi: 10.1016/j.sigpro.2010.06.022

Ушаков П. А., Шадрин А. В. Схемотехническое моделирование аналогового процессора для решения дифференциальных уравнений дробного порядка // Интеллектуальные системы в производстве. 2013. № 1. С. 56–58.

Podlubny I., Petras B., Vinagre P., Leary O., Dorcak L. Analogue realizations of fractional order controllers. Nonlinear Dynam., 2002, vol. 29, pp. 281-296.

Monje A. Fractional-order Systems and Controls: Fundamentals and Applications. Springer, 2010, 414 р. doi: 10.1007/978-1-84996-335-0

Tepljakov A. ractional-order modeling and con-trol of dynamic systems. Springer, 2017, 184 p. doi: 10.1007/978-3-319-52950-9

Ушаков П. А., Бабошкин Г. Д. Моделирование системы управления дробного порядка с высокоинерционным объектом управления на примере системы стабилизации антенно-поворотного устройства // Вестник концерна ВКО «Алмаз-Антей». 2019. № 3. С. 41–51.

Tarasov V.E. Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Beijing, Higher Education Press, 2010, 522 p.

Petráš I. Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Beijing, Higher Education Press, 2010, 235 р.

Mekkaoui T., Hammouch Z., B.M. Belgacem F.B.M., Abbassi A.E. Fractional-order Nonlinear Systems: Chaotic Dynamics, Numerical Simulation and Circuits Design. Degruyter Publ., 2016, pp. 343-356. doi: 10.13140/RG.2.1.3974.4082

Агуреев К. И. Применение детерминированного хаоса для передачи информации // Изв. Тульского гос. ун-та. 2014. Вып. 4. С. 1281–1310.

Леонов К. Н., Потапов А. А., Ушаков П. А. Использование инвариантных свойств хаотических сигналов в синтезе систем передачи информации // Радиотехника и электроника. 2014. Т. 59, № 12. С. 1209–1229.

Леонов К. Н., Потапов А. А., Ушаков П. А. Математическое моделирование системы передачи данных на основе хаотических сигналов с фрактальной размерностью // Физика волновых процессов и радиотехнические системы. 2010. Т. 13, № 3. С. 47–53.

Ушаков П. А., Леонов К. Н. Инвариантный способ передачи информации в системах с хаотическими сигналами // Вестник Ижевского государственного технического университета. 2010. № 4. С. 92–96.

Спротт Д. К. Элегантный хаос: алгебраические простые хаотические потоки. М. ; Ижевск : Ижевский ин-т компьютерных исследований, 2012. 328 с.

Белослудцев В. Н., Ушаков П. А. Исследование генератора хаоса дробного порядка, построенного на системе Нозе – Гувера // Приборостроение в XXI веке – 2017 : Интеграция науки, образования и производства : материалы XIII Междунар. науч.-техн. конф. (Ижевск, 22–24 ноября 2017 г.). Ижевск : Изд-во ИжГТУ имени М. Т. Калашникова, 2018. С. 17–23.

Elwakil A.S. Fractional-order circuits and systems: an emerging interdisciplinary research area. IEEE Circuits and Systems Magazine, 2010, vol. 10, no. 4, pp. 40-50.

Tsirimokou G., Psychalinos C., Elwakil A. Design of CMOS Analog Integrated Fractional-Order Circuits, Applications in Medicine and Biology. Springer, 2017, 114 р.

Radwan A.G., Elwakil A.S., Soliman A.M. On the generalization of second-order filters to the fractional-order domain. J. Circuits Syst. Comput, 2009, vol. 18, pp. 361-386.

Lahiri A., Rawat T. Noise analysis of single stage fractional-order low-pass filter using stochastic and fractional calculus. ECTI Trans. Elect Eng., Electron. Com-mun., 2009, vol. 7, pp. 136-143.

Maundy B., Elwakil A.S., Freeborn T.J. On the practical realization of higher-order filters with fractional stepping. Signal Processing, 2011, vol. 91, pp. 484-491.

Tripathy M.C., Mondal D., Biswas K., Sen S. Design and performance study of phase-locked loop using fractional-order loop filter. International J. of Circuit Theory and Applications, 2015, vol. 43, is. 6, pp. 776-792.

Freeborn T., Maundy B., Elwakil A.S. Approximated Fractional Order Chebyshev Lowpass Filters. Mathematical Problems in Engineering, 2015, vol. 2015, pp. 1-7. doi: 10.1155/2015/S3246S

Tripathy M.C., Biswas K., Sen S. A design example of a fractional-order Kerwin-Huelsman-Newcomb biquad filter with two fractional capacitors of different order. Circuits, Systems, and Signal Proc., 2013, vol. 32, no. 4, pp. 1523-1536.

Soltan A., Radwan A.G., Soliman A.M. Measurement Fractional Order Sallen-Key Filters. Interna-tional J. of Electrical, Electronics, Communication, Energy Science and Engineering, 2013, vol. 7, no.12, pp. 1088-1092.

Adhikary A., Sen S., Biswas K. Practical Realization of Tunable Fractional Order Parallel Resonator and Fractional Order Filters. Regular Papers, August 2016, vol. 63, no 8, pp. 1142-1151.

Князев А. В., Ушаков П. А. Сравнительный анализ характеристик фрактального параллельного колебательного контура // Молодые ученые – ускорению научно-технического прогресса в XXI веке : сб. тр. II Всерос. науч.-техн. конф. аспирантов, магистрантов и молодых ученых с междунар. участием. Ижевск : Изд-во ИжГТУ имени М. Т. Калашникова, 2013. С. 337–341.

Radwan A.G., Elwakil A.S., Soliman A.M. Fractional-order sinusoidal oscillators: design procedure and practical examples. IEEE Trans, on Circuits and Systems I: Regular Papers, 2008, vol. 55, pp. 2051-2063.

Elwakil A.S., Agambayev A., Allagui A., Sala-ma K.N. Experimental demonstration of fractional-order oscillators of orders 2.6 and 2.7. Chaos, Solitons & Fractals, 2017, vol. 96, pp. 160-164.

Agambayev A. Fractional-Order Hartley Oscilla-tor. Proc. 14th Conference on Ph.D. Research in Microelectronics and Electronics (2-5 July 2018). doi: 10.1109/PRIME.2018.8430336

Fouda M.E., Soltan A., Radwan A.G., Soli-man A.M. Fractional-order multiphase oscillators design and analysis suitable for higher-order PSK applications. Analog Integrated Circuits and Signal Proc., 2016, vol. 87, pp. 301-312.

Maundy B., Elwakil A., Gift S. On the realization of multiphase oscillators using fractional-order allpass filters. Circuits, Systems and Signal Processing, 2012, vol. 31, pp. 3-17.

Lahiri A. Low-frequency quadrature sinusoidal oscillators using current differencing buffered amplifiers. Indian Journal of Pure & Applied Physics, 2011, vol. 49, no. 6, pp. 423-428.

Pittala C.S., Srinivasulu A. Quadrature oscillator using operational transresistance amplifier. Applied Electronics, 2014, pp. 117-120.

Lobna S.A., Madian A.H., Radwan A.G., Soliman A.M. Fractional order oscillator with independent control of phase and frequency. Proc. 2nd International Conf. on Electronic Design (ICED) 19-21 Aug. 2014. doi: 10.1109/ICED.2014.7015803

Maundy B., Elwakil A., Gift S. On a multivibrator that employs a fractional capacitor. Analog Integr Circ Sig Proc., 2009, 62. DOI: 10.1007/s 10470-009-9329-3.

Sacu I.E., Alcy M. Electronically Controllable Fractional Multivibrator. IETE Journal of Research, 06 Dec 2018. doi: 10.1080/03772063.2018.1548909


Article Metrics

Metrics Loading ...

Metrics powered by PLOS ALM

Copyright (c) 2020 Bulletin of Kalashnikov ISTU

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

ISSN 1813-7903 (Print)
ISSN 2413-1172 (Online)