Fractal elements and their applications

• Main Authors: Gil'mutdinov, Anis Kharisovich (Author, Editor), Ushakov, Pyotr Arkhipovich (Author), El-Khazali, Reyad (Author)
• Format: Electronic eBook
• Language: English
• Published: Cham, Switzerland : Springer, 2017.
• Series: Analog circuits and signal processing series.
• Subjects: Fractals.; Fractional calculus.; Signal processing.; elektronické knihy; electronic books
• ISBN: 9783319452494; 9783319452487
• Physical Description: 1 online resource (xviii, 252 pages)

This book describes a new type of passive electronic components, called fractal elements, from a theoretical and practical point of view. The authors discuss in detail the physical implementation and design of fractal devices for application in fractional-order signal processing and systems. The con...

Table of Contents:

• Foreword; Contents; Abbreviations; Chapter 1: Modeling of Fractal Elements and Processes; 1.1 Concept of Fractals, Self-Similarity, and Scaling; 1.2 Dimension Types; 1.3 Regular Fractals; 1.4 Irregular Random Fractals; 1.5 Multifractals; 1.6 Fractal Signals; 1.7 Physical Meaning of Hurst Parameter; 1.8 Relation Between Fractality and Spectrum Response; 1.9 Examples of Signal Analysis; Chapter 2: Fractal Calculus Fundamentals; 2.1 Preliminaries; 2.2 Properties of Fractional-Order Integrals and Derivatives; 2.2.1 Riemann-Liouville Fractional-Order Integral and Derivative.
• 2.2.2 Grunwald-Letnikov Fractional-Order Derivative and Integral2.2.3 Properties of Fractional-Order Derivatives; 2.3 Laplace Transform of Fractional-Order Operators; 2.3.1 Fundamentals of Laplace Transform; 2.3.2 Laplace Transform of Fractional-Order Integrals; 2.3.3 Laplace Transform of Fractional-Order Derivatives; 2.4 Fourier Transform of Fractional-Order Operators; 2.4.1 Fundamentals of Fourier Transform; 2.4.2 Fourier Transform of Fractional-Order Integrals; 2.4.3 Fourier Transform of Fractional-Order Derivatives; 2.5 Dynamics of Fractional-Order Transfer Functions.
• 2.5.1 Fractional-Order Transfer Functions2.5.2 Mittag-Leffler Function; 2.5.3 Solving Fractional-Order Differential Equation (FoDEQ) Using Laplace Transform; 2.6 Fractional-Order Electrical and Electronic Systems; 2.6.1 Semi-infinite Transmission Line; 2.6.2 Electrochemistry; 2.6.3 Rough Surface Impedance; Chapter 3: Fractal Elements; 3.1 Fractal Impedances and Fractal Element; 3.2 Implementation of Fractal Impedances Using Electrochemical Converters; 3.2.1 Liquid Electrolyte Electrochemical Signal Converters; 3.2.2 Solid Electrolyte Electrochemical Signal Converters.
• 3.2.3 Fractal Element Fractor3.3 Implementation of Fractal Impedances (Immittances) Using RC Circuits with Lumped Parameters; 3.3.1 Preliminaries; 3.3.2 Properties of Input RC-Transfer Functions; 3.3.3 Circuit Realization Using Foster Forms; 3.3.4 Circuit Realization Using Cauer Forms; 3.3.5 Rational Approximation of Fractal-Order Impedances; 3.3.5.1 The Oustaloup Method of Rational Approximation; 3.3.5.2 Shareff Method of Rational Approximation; 3.3.5.3 El-Khazali Approximation of Fractional-Order Integro-Differential Operators; 3.3.6 Realization of Fractional-Order Inductors (FoIs).
• 3.3.7 Realization of Fractional-Order Capacitors (FoC)3.4 Realization of Fractal Impedances Using RC Circuits with Distributed Parameters; 3.5 Fabrication of Fractal Impedances Using Nanostructured Materials; 3.6 Comparison of Fractal Element Characteristics Using Different Fabrication Technology; Chapter 4: Design and Implementation of Thin RC-EDP Films; 4.1 Classification of Static Heterogeneous Characteristics of RC-EDP Films; 4.2 RC-EDP Film Design of Multilayer Structure; 4.3 Design Development of RC-EDP Films by Changing Layers Geometry.