Fractional-Order Electrical Circuit Theory (CPSS Power Electronics Series)

✍ Scribed by Bo Zhang, Xujian Shu

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 310
• Category: Library

✦ Synopsis

This book presents a concise and insightful view of the knowledge on fractional-order electrical circuits, which belongs to the subject of Electric Engineering and involves mathematics of fractional calculus. It offers an overview of fractional calculus and then describes and analyzes the basic theories and properties of fractional-order elements and fractional-order electrical circuit composed of fractional-order elements. Therein, the fundamental theorems, time-domain analysis, steady-state analysis, complex frequency domain analysis and state variable analysis of fractional-order electrical circuit are included. The fractional-order two-port networks and generalized fractional-order linear electrical circuits are also mentioned. Therefore, this book provides readers with enough background and understanding to go deeper into the topic of fractional-order electrical circuit, so that it is useful as a textbook for courses related to fractional-order elements, fractional-order electrical circuits, etc. This book is intended for students without an extensive mathematical background and is suitable for advanced undergraduate and graduate students, engineers and researchers who focus on the fractional-order elements, electrical circuits and systems.

✦ Table of Contents

Preface
Acknowledgements
Contents
About the Authors
1 Introduction to Fractional Calculus
1.1 Special Functions of Fractional Calculus
1.1.1 Gamma Function
1.1.2 Beta Function
1.1.3 Mittag-Leffler Function
1.2 Definitions and Basic Properties of Fractional Calculus
1.2.1 Grünwald-Letnikov Fractional Calculus
1.2.2 Riemann-Liouville Fractional Calculus
1.2.3 Caputo Fractional Calculus
1.2.4 Comparison of Three Definitions of Fractional Calculus
1.2.5 Geometric and Physical Interpretation of Fractional Calculus
1.3 Methods of Solving Fractional Differential Equations
1.3.1 Laplace Transform Method
1.3.2 Integer-Order Approximation of Fractional-Order Transfer Function
1.3.3 Numerical Solutions
1.4 Matlab-Based Numerical Tools for Fractional Calculus
1.5 Summary
References
2 Fractional-Order Components and Their Basic Circuits
2.1 Fractional-Order Components
2.1.1 Fractional-Order Capacitor
2.1.2 Fractional-Order Inductor
2.1.3 Fractional-Order Mutual Inductor
2.2 Fractional-Order Circuits
2.2.1 Circuits of Fractional-Order Capacitors
2.2.2 Circuits of Fractional-Order Inductors
2.2.3 Circuits of Fractional-Order Inductors and Capacitors
2.3 Summary
References
3 Theorems of Fractional-Order Circuits
3.1 Superposition Theorem
3.2 Substitution Theorem
3.3 Thevenin’s and Norton’s Theorems
3.4 Tellegen’s Theorem
3.5 Reciprocity Theorem
3.6 Duality Theorem
3.7 Compensation Theorem
3.8 Bisection Theorem
3.9 Summary
References
4 Time-Domain Analysis of Fractional-Order Circuits
4.1 Characteristics of Differential Equations for Fractional-Order Circuits
4.2 Zero-Input Response of Fractional-Order RCα and RLβ Circuits
4.2.1 Fractional-Order RCα Circuit
4.2.2 Fractional-Order RLβ Circuit
4.3 Zero-State Response of Fractional-Order RCα and RLβ Circuits
4.3.1 Fractional-Order RCα Circuit
4.3.2 Fractional-Order RLβ Circuit
4.4 Complete Response of Fractional-Order RCα and RLβ Circuits
4.4.1 Fractional-Order RCα Circuit
4.4.2 Fractional-Order RLβ Circuit
4.5 Step and Pulse Response of Fractional-Order RCα and RLβ Circuits
4.5.1 Step Response
4.5.2 Pulse Response
4.6 Zero-Input Response of Fractional-Order RLβCα Circuits
4.6.1 Series Fractional-Order RLβCα Circuit
4.6.2 Parallel Fractional-Order RLβCα Circuit
4.7 Zero-State Response of Fractional-Order RLβCα Circuits
4.7.1 Series Fractional-Order RLβCα Circuits
4.7.2 Parallel Fractional-Order RLβCα Circuits
4.8 Complete Response of Fractional-Order RLβCα Circuits
4.8.1 Series Fractional-Order RLβCα Circuits
4.8.2 Parallel Fractional-Order RLβCα Circuits
4.9 Step and Pulse Response of Fractional-Order RLβCα Circuits
4.9.1 Step Response
4.9.2 Pulse Response
4.10 Response to Arbitrary Excitations
4.11 Summary
References
5 Sinusoidal Steady-State Analysis of Fractional-Order Circuits
5.1 Preliminaries
5.1.1 Phase Difference
5.1.2 Effective Values
5.1.3 Phasor Representation
5.2 Phasor Representation of Fractional-Order Components and Circuits
5.2.1 Fractional-Order Capacitor
5.2.2 Fractional-Order Inductor
5.2.3 Fractional-Order Mutual Inductor
5.2.4 Fractional-Order Circuit
5.3 Fractional-Order Resonant Circuits
5.3.1 Resonance in Series Fractional-Order RLβCα Circuit
5.3.2 Resonance in Parallel Fractional-Order RLβCα Circuit
5.4 Fractional-Order Mutual Inductance Coupling Circuit
5.5 Fractional-Order Filter Circuits
5.6 Summary
References
6 Sinusoidal Steady-State Analysis of Three-Phase Fractional-Order Circuits
6.1 Preliminaries
6.1.1 Connection Types
6.1.2 Relationship Between Phase and Line Voltages
6.1.3 Relationship Between Phase and Line Currents
6.2 Three-Phase Fractional-Order Circuits
6.2.1 Symmetrical Three-Phase Fractional-Order Circuits
6.2.2 Complex Symmetrical Three-Phase Fractional-Order Circuits
6.2.3 Asymmetric Three-Phase Fractional-Order Circuits
6.3 Power Measurement of Three-Phase Fractional-Order Circuits
6.4 Summary
References
7 Non-sinusoidal Periodic Steady-State Analysis of Fractional-Order Circuits
7.1 Preliminaries
7.1.1 Harmonic Impedance
7.1.2 Effective Values
7.1.3 Power
7.2 Fractional-Order RLβCα Circuits
7.3 Symmetrical Three-Phase Fractional-Order Circuits
7.4 Summary
References
8 Two-Port Fractional-Order Networks
8.1 Y-parameter
8.1.1 Y-parameter Equation
8.1.2 Physical Meaning of Y-parameter
8.1.3 Equivalent Circuit of Y-parameter
8.2 Z-parameter
8.2.1 Z-parameter Equation
8.2.2 Physical Meaning of Z-parameter
8.2.3 Equivalent Circuit of Z-parameter
8.3 H-parameter and G-parameter
8.3.1 H-parameter
8.3.2 G-parameter
8.3.3 Reciprocity and Symmetry of H-parameter and G-parameter
8.4 Transmission Parameters and Propagation Characteristics
8.4.1 Forward Transmission Parameters
8.4.2 Reverse Transmission Parameters
8.4.3 Propagation Characteristics of Two-Port Symmetric Fractional-Order Networks
8.5 Fractional-Order Mutators
8.5.1 Basic Concepts
8.5.2 Fractional-Order Cα-R Mutator
8.5.3 Fractional-Order Lβ-R Mutator
8.5.4 Fractional-Order Mα,β-R Mutator
8.6 Connection Types of Multiple Two-Port Fractional-Order Networks
8.6.1 Cascade
8.6.2 Series Connection
8.6.3 Parallel Connection
8.6.4 Series-Parallel Connection
8.6.5 Parallel-Series Connection
8.7 Summary
References
9 State-Variable Analysis of Fractional-Order Circuits
9.1 Basic Concepts
9.1.1 State Variables
9.1.2 State Space and State Vector
9.1.3 General Form of State Equation
9.2 State Equation of Fractional-Order Circuits
9.2.1 Normal Fractional-Order Circuit
9.2.2 Ill-Conditioned Fractional-Order Circuit
9.2.3 Normal Circuit with Fractional-Order Coupled Inductor
9.3 Solutions to State Equation of Fractional-Order Circuits
9.3.1 State Equation with the Same Order
9.3.2 State Equation with Different Orders
9.4 Summary
References