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

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