Mathematical Modeling of Physical Systems: Applications of Fields, Circuits and Signal Processing 9811975574, 9789811975578

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Advances in Intelligent Systems and Computing 1436

Adhir Baran Chattopadhyay Shazia Hasan Snehaunshu Chowdhury

Mathematical Modeling of Physical Systems Applications of Fields, Circuits and Signal Processing

Advances in Intelligent Systems and Computing Volume 1436

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland Advisory Editors Nikhil R. Pal, Indian Statistical Institute, Kolkata, India Rafael Bello Perez, Faculty of Mathematics, Physics and Computing, Universidad Central de Las Villas, Santa Clara, Cuba Emilio S. Corchado, University of Salamanca, Salamanca, Spain Hani Hagras, School of Computer Science and Electronic Engineering, University of Essex, Colchester, UK László T. Kóczy, Department of Automation, Széchenyi István University, Gyor, Hungary Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX, USA Chin-Teng Lin, Department of Electrical Engineering, National Chiao Tung University, Hsinchu, Taiwan Jie Lu, Faculty of Engineering and Information Technology, University of Technology Sydney, Sydney, NSW, Australia Patricia Melin, Graduate Program of Computer Science, Tijuana Institute of Technology, Tijuana, Mexico Nadia Nedjah, Department of Electronics Engineering, University of Rio de Janeiro, Rio de Janeiro, Brazil Ngoc Thanh Nguyen , Faculty of Computer Science and Management, Wrocław University of Technology, Wrocław, Poland Jun Wang, Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong

The series “Advances in Intelligent Systems and Computing” contains publications on theory, applications, and design methods of Intelligent Systems and Intelligent Computing. Virtually all disciplines such as engineering, natural sciences, computer and information science, ICT, economics, business, e-commerce, environment, healthcare, life science are covered. The list of topics spans all the areas of modern intelligent systems and computing such as: computational intelligence, soft computing including neural networks, fuzzy systems, evolutionary computing and the fusion of these paradigms, social intelligence, ambient intelligence, computational neuroscience, artificial life, virtual worlds and society, cognitive science and systems, Perception and Vision, DNA and immune based systems, self-organizing and adaptive systems, e-Learning and teaching, human-centered and human-centric computing, recommender systems, intelligent control, robotics and mechatronics including human-machine teaming, knowledge-based paradigms, learning paradigms, machine ethics, intelligent data analysis, knowledge management, intelligent agents, intelligent decision making and support, intelligent network security, trust management, interactive entertainment, Web intelligence and multimedia. The publications within “Advances in Intelligent Systems and Computing” are primarily proceedings of important conferences, symposia and congresses. They cover significant recent developments in the field, both of a foundational and applicable character. An important characteristic feature of the series is the short publication time and world-wide distribution. This permits a rapid and broad dissemination of research results. Indexed by DBLP, INSPEC, WTI Frankfurt eG, zbMATH, Japanese Science and Technology Agency (JST). All books published in the series are submitted for consideration in Web of Science. For proposals from Asia please contact Aninda Bose ([email protected]).

Adhir Baran Chattopadhyay · Shazia Hasan · Snehaunshu Chowdhury

Mathematical Modeling of Physical Systems Applications of Fields, Circuits and Signal Processing

Adhir Baran Chattopadhyay Department of Electrical Engineering Budge Budge Institute of Technology (B.B.I.T) Kolkata, West Bengal, India BITS, Pilani-Dubai Dubai, United Arab Emirates

Shazia Hasan Department of Electrical and Electronics Engineering Birla Institute of Technology and Science, Pilani, Dubai Campus, Dubai International Academic City Dubai, United Arab Emirates

Snehaunshu Chowdhury Department of Mechanical Engineering Birla Institute of Technology and Science, Pilani, Dubai Campus, Dubai International Academic City Dubai, United Arab Emirates

ISSN 2194-5357 ISSN 2194-5365 (electronic) Advances in Intelligent Systems and Computing ISBN 978-981-19-7557-8 ISBN 978-981-19-7558-5 (eBook) https://doi.org/10.1007/978-981-19-7558-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Dr. Adhir Baran Chattopadhyay Children (Rimpu, Riki), Wife (Pratima) and Sons-in-law (Anand and Aritra) Dr. Shazia Hasan Syed Aquil Manzar Hasan (Father) You have been a constant source of my inspiration & Sarfraj, Iqra, Sidra and Inaya for your unconditional love and support Dr. Snehaunshu Chowdhury Late Smt. Rina Chowdhury (mother) You had always been a source of constant inspiration for me and Dolon and Samadarshi for your unconditional love and support

Preface

Generally, it is felt that a new research book on classical foundation concepts should focus on a crystal-clear and reasonable point of view not highlighted by researchbased books already published and used by the readers. Based on this philosophy, this book has been written with an anticipation that the experience and vision of the authors are reflected. The authors of this book also expect that the readers who are not intellectually convinced by the above-said view point of thoughts can search elsewhere the suitable academic or research materials. The mode of presentations and contents of the book express our personal opinions in different ways, as follows.

The Place of Magnetic Levitation and Propulsion as a Major Application Area of Electromagnetic Field Theory Electromagnetic Field Theory is basically based on the concept of the interaction of a source quantity with the concerned field quantity. Again, based on the concepts of electromagnetic field theory, analytical discussions of some major applications may be presented in any research book. However, seeing the limitations of inclusion of the application-based topics, this book partially presents the mathematical modelling of a linear induction motor (with long stator and short rotor), which is the most important component of a commonly used magnetic levitation and propulsion system.

Applications in Fluid Mechanics The theory of irrotational flows (potential flows) plays a vital role in the study of fluid mechanics. Not only are such flows of historical importance but they are still used to understand a lot of complex flows by superposition of simple elementary flows. An effort is made to introduce the students to fluid flow and include the governing

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equations describing their flow. Care and emphasis have been made to limit the discussion to irrotational flows as the governing equations for inviscid, steady, irrotational flows are the Laplace equations. But due to the extremely limited nature of complex geometry in real systems, the Laplace equations are not always solved analytically. Therefore, the readers are introduced to various numerical procedures such as finite difference, finite volume and finite element techniques. The mathematical nature of the governing equations and their physical significance are discussed. Finally, several cases of the Laplace equations are solved with various boundary conditions using the finite element method demonstrating the different fields in which such solutions could play an important role.

Mathematical Modelling Using the Generalized Theory of Electrical Machines The authors believe that till date, the mathematical modelling of any composite engineering system (or, problem) involves the mathematical modelling of the electrical drives part, or electrical power generation part (in general), as a major share. Such necessity basically leads to a unified type of mathematical modelling of all types of electrical machines. This particular terminology, “unified type of mathematical modelling” is basically known as “Generalized Theory of Electrical Machines”. Chapter 7 is dedicated to the sharing of the rigorous analytical aspects of electrical machines using the generalized theory of electrical machines.

Mathematical Modelling of a Non-linear System Using Circuit Theory Approach and Linearization Technique In contrast to the “Field Theory Approach”, the “Circuit Theory Approach” bears some advantages, and these advantages are encashed for modelling a power electronic D.C drive system involving some non-linearities. Moreover, a non-linear state variable modelling approach has been implemented based on linearization techniques. Chapter 8 reflects these aspects.

Applications of Signal Processing to Certain Classes of Electrical Power System Problems It is well known that a major part of modelling a power system involves mathematical handling of different types of data. Such work, in the long run, merges to be a major responsibility in the area of signal processing. The power system data generally are

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analysed using Kalman Filter (KF), Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) which are being considered as the very effective methods in discrete domains. Chapters 9 and 10 of this book basically reflects the above-said modelling philosophy.

Organization This book may be conceived as divided into four conceptual (also chronologically placed) area as follows: • Mathematical modelling of Linear Induction Motors having different constructional features, based on “Electromagnetic Field Theory” (Chaps. 1–3). • Mathematical modelling of the selected systems using the concept of “Fluid Fields” (Chaps. 4–6). • Mathematical modelling of the selected systems using “Circuit Theory Approach” (Chaps. 7 and 8). • Mathematical modelling of the selected power system phenomena using “Signal Processing Approach” (Chaps. 9 and 10).

Suggestions for Using This Book (i)

This book can be partially tailored as a course work for the research programmes (leading to the Ph.D. degree) in the universities. (ii) This book also may be used as a helping tool for the researchers working in the field of mathematical modelling of physical systems. (iii) A book chapter may be separately written based on the concept of comparing the “Electromagnetic Field Theory” with the “Fluid Field Theory”.

Notable Features (i)

(ii)

Chapter 1: Sect. 1.2: This section actually gives a feeling to the researchers about formulating and solving a partial differential equation (PDE) problem in two dimensions. Chapter 1: Sects. 1.2 and 1.5: The difference in the philosophical aspects of these two sections works as an eye opener before any researcher from the view point of giving extension to one existing class of PDE formulations. In other words, such sections show how the number of PDEs can be increased while pursuing the mathematical modelling of a practical problem.

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Chapter 2: Sect. 2.2.1.1 and Appendix B: This particular direction of thinking clearly shows how the deep knowledge of “Theory of Complex Variable” or “Contour Integration” helps a researcher to solve the so-called difficult type of improper integrals involving an integrand whose convergence generally cannot be assured. (iv) Chapter 4: The continuity in this chapter is based on elementary fluid mechanics taught in undergraduate courses. After a brief introduction on how flow fields are described, the governing equations of fluid flow are presented. Subsequently, the potential flow equation, both in terms of velocity potential and stream function, is developed. The concept of elementary flows is introduced along with the idea of linear superposition of elementary flows to reproduce certain physical flow cases. Methods for solving the Laplace equation are also provided in this chapter. (v) Chapter 5: Introduction to the various numerical techniques to solve the Laplace equation is presented. Finite difference schemes, finite volume as well as finite element techniques are introduced. It should be noted that where analytical solutions are not possible either due to complex boundary conditions or geometry, numerical solutions could be used. These techniques, therefore, refer to a generalized way of solving PDEs by eventually converting them into a system of algebraic equations. These techniques are extremely powerful and both academia and industry have been extremely diligent in pushing the frontiers of these techniques. (vi) Chapter 6: This chapter demonstrates the use of finite element techniques to solve the Laplace equation numerically. The Laplace equation forms the governing equation in multiple disciplines spanning fluid mechanics, groundwater seepage, electrostatics, steady-state heat transfer, etc. The readers should grasp the beauty of the interdisciplinary nature of this technique and its application. (vii) Chapter 7: Sect. 7.2.3 and Reference literature (5): The authors feel that these directions will clear the ideas of a researcher about modelling any transient or sub-transient equivalent circuits. (viii) Chapter 9: Sects. 9.2.4 and 9.2.5: The in-depth studies of these two sections will throw light on the concept of the basic difference between “Extended Kalman Filter” and “Unscented Kalman Filter”. This study also will help in identifying the drawbacks of the “Linearization Technique” being used during the problem formulation in the area of signal processing. (ix) Chapter 10: Sect. 10.2.2: In power systems, due to sudden load change or short circuit, there may happen a drastic reduction in system frequency, and in turn, such reduction may affect the overall performance of the particular power system in a most negative manner. That is why “Frequency Estimation” becomes very much necessary as a part of the mathematical modelling work. Moreover, this particular section has another feature that a detailed application of “Z-Transform” is involved and such exercise may accelerate the mathematical modelling skill of a researcher.

Preface

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Appendix A: Appendix A puts light on the actual use of Fourier Series for dealing with a practical problem. Such analysis dictates that the finiteness of rotor width leads to the deviation in the conductivity or resistivity from the original value. Appendix C: The content of this chapter is not actually needed from the view point of mathematical modelling but it is very much needed for prototype development. The authors believe that a perfect combination of knowledge and skill of prototype development and the knowledge of mathematical modelling helps any researcher to create a very high level of research environment.

Kolkata, India Dubai, United Arab Emirates Dubai, United Arab Emirates

Adhir Baran Chattopadhyay Shazia Hasan Snehaunshu Chowdhury

Acknowledgements

(i)

By Dr. Adhir Baran Chattopadhyay: Several individuals have helped us in the preparation of this book. I am grateful to Dr. Shazia Hasan and Dr. Snehaunshu Chowdhury of BITS, Pilani-Dubai, Dubai, U.A.E, for helping me in different ways for the completion of this book. My thanks are also to Mr. Aninda Bose, Executive Editor, Springer Nature Group, New Delhi, India, for constant mental support in completing this book. It is understood that writing a book is an obsessively time-consuming activity, which causes much hardship for family members, where the spouse suffers the most. So, what can I say except “thank you” to my spouse, Pratima, for enormous but invisible sacrifices. (ii) By Dr. Shazia Hasan: It is my pleasure to acknowledge the assistance I received from my co-authors Prof. A.B Chattopadhyay and Dr. Snehashu. I also sincerely thank Mr. Aninda Bose, Executive Editor, Springer Nature for all his support in this rigorous process of publishing a Book. With the immense trust of my academician father Syed Aquil Manzar Hasan and father-in-law Dr. Hanif Mohammed today this book become reality. None of this would have become possible without the unconditional love and constant support of my family members including my better half Sarfraj and my Little ones Iqra, Sidra and Inaya who stood constantly by my side. I also would like to acknowledge the EEE department at BITS Pilani, Dubai, for all the support. (iii) By Dr. Snehaunshu Chowdhury: I would like to acknowledge the help and support I have received from my co-authors Prof. ABC and Dr. Shazia. I also sincerely acknowledge the moral support I have received from my departmental colleagues at BITS Pilani, Dubai. Credit is also due to several others who played a part in bringing out the chapters in its current form. Please accept my heartfelt and sincere appreciation to everybody who has contributed in any form. Without the continuous support of Dr. Aninda Bose, this manuscript would not have seen the light of the day. I would like to convey my deepest regards for your understanding and humane response in the entire process. xiii

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Acknowledgements

Book writing is a rigorous time-consuming exercise. I must thank two men, my father and father-in-law who helped me keep up my spirits during this ordeal. Needless to say, there were numerous occasions when my family must have felt that I was neglecting them. I am extremely thankful to my wife, Dolon, for her understanding and cooperation during this ordeal. Lastly, as a father, I must acknowledge the contribution of my son, Samadarshi, who has often unknowingly been wronged by me so that I can finish the manuscript on time. Sorry dear. You are a bundle of joy and you are the reason I get up every day with a smile on my face.

Contents

1

2

Applications of Field Theory: Analysis of a Single Sided Linear Induction Motor with Stator and Rotor of Infinite Length and Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Formulation for Fields and Currents . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Boundary Conditions and Determination of the Coefficients, C 1 , C 2 , C 3 and C 4 . . . . . . . . . . . . . . . 1.2.2 Calculation of Flux Density Components in the Rotor Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Calculation of Current Density in the Rotor Sheet . . . . . 1.3 Calculation of Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Calculation of Levitation Force . . . . . . . . . . . . . . . . . . . . . 1.3.2 Calculation of Propulsion Force . . . . . . . . . . . . . . . . . . . . . 1.4 Normalized Propulsion and Levitation Forces and Their Maximum Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Three-Region Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Determination of Coefficients ' ' ' ' ' C1 , C2 , C3 , C4 and C6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Calculation of Flux Densities and Current Density in the Rotor Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.4 Calculation of Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical Modelling of Electromagnetic Forces Due to Finite Width Effects of a Single-Sided Linear Induction Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 4 7 7 8 8 8 9 11 13 15 16 16 18 20

23 23 24

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Contents

2.2.1 2.3 2.4 2.5 2.6 2.7 3

4

Formulation for the Field Due to Stator Current Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution for Stream Function uy . . . . . . . . . . . . . . . . . . . . . . . . . . . . Steps for Algorithm for Numerical Solution for uy , By2 . . . . . . . . Calculation of Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Mathematical Modeling of Electromagnetic Forces Due to Finite Length and Finite Width Effects of a Single-Sided Linear Induction Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Formulation for the Field Due to Stator Current . . . . . . . 3.2.2 Formulation for the Current in the Rotor Sheet . . . . . . . . 3.3 Steps of Algorithm for Numerical Solution for u y , B y2 , Bx2 and Bz2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Calculation of Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fluid Flow Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Fluid Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Kinematic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Thermodynamics Properties . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Transport Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Miscellaneous Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Description of Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Lagrangian Description—Control Mass (CM) . . . . . . . . . 4.2.2 Eulerian Description—Control Volume (CV) . . . . . . . . . 4.2.3 Field View to Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Material Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Governing Equations of Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Conservation of Mass—Continuity Equation . . . . . . . . . . 4.4.2 Conservation of Linear Momentum . . . . . . . . . . . . . . . . . . 4.4.3 Conservation of Angular Momentum . . . . . . . . . . . . . . . . 4.4.4 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Irrotational Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Potential Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Potential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Stream Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.4 Laplace Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.5 Properties of Laplace Equation . . . . . . . . . . . . . . . . . . . . . 4.5.6 Uniqueness of the Solutions of Laplace Equation . . . . . . 4.5.7 Uniqueness for Infinite Domain . . . . . . . . . . . . . . . . . . . . . 4.5.8 Kelvin’s Minimum Energy Theorem . . . . . . . . . . . . . . . . .

24 34 35 42 42 46

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Contents

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4.6

Elementary Potential Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Uniform, Free Stream Flow . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Point Source or Sink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Line Source or Sink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.4 Line Irrotational Vortex (Free Vortex) . . . . . . . . . . . . . . . . 4.7 Linear Superposition of Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Dipole (Doublet Flow) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Planar Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Flow Past an Obstacle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1 Flow Past a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.2 Rankine Half Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.3 Flow Around a Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Force on a 2-D Object of Arbitrary Shape . . . . . . . . . . . . . . . . . . . . 4.10 General 3-D Potential Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Solution to Laplace Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11.1 Method of Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11.2 Method of Separation of Variables . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 79 80 81 83 84 85 86 86 87 87 88 89 91 92 92 94 96

5

Computational Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Need for CFD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Types of Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Region of Disturbance and Influence . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Discretization of the Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Discretization Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Finite Difference (FD) Method . . . . . . . . . . . . . . . . . . . . . 5.6.2 Finite Volume (FV) Method . . . . . . . . . . . . . . . . . . . . . . . . 5.6.3 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 CFD Solutions to Simple Potential Flows . . . . . . . . . . . . . . . . . . . . 5.7.1 Flow Through a Duct with Changing Area . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97 97 97 98 99 100 100 100 101 101 101 103 104 107 107 111

6

Finite Element Formulation of Field Problems . . . . . . . . . . . . . . . . . . . 6.1 Two-Dimensional Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Governing Differential Equations . . . . . . . . . . . . . . . . . . . 6.1.2 Integral Equations for Element Matrices . . . . . . . . . . . . . 6.1.3 Element Matrices: Triangular Elements . . . . . . . . . . . . . . 6.1.4 Element Matrices: Rectangular Elements . . . . . . . . . . . . . 6.2 Point Sources and Sinks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Derivative Boundary Conditions . . . . . . . . . . . . . . . . . . . . 6.2.2 Evaluation of Element Integrals . . . . . . . . . . . . . . . . . . . . . 6.2.3 Assembly of Element Matrices into Global Matrix . . . . .

113 113 113 114 116 117 118 119 121 124

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 7

8

Modelling Approach Using Generalized Theory of Electrical Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 The Foundation of the Generalized Theory of Electrical Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 The Idealized Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 The Circuit View of a Two-Winding Transformer-Explanation of Sign Convention and the Per-Unit System for Electrical Quantities . . . . . . 7.1.3 Magneto-Motive Force and Flux in the Rotating Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.4 Voltage-Balance and Torque-Balance Equations of the Machine-The Per Unit System for Mechanical Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Development of the Sub-Transient, Transient and Steady State Equivalent Circuits along Direct and Quadrature Axes, Separately, of a Three-Phase Salient Pole Synchronous Machine Using, “Constant Flux-Linkage Theorem” and “Theory of Small Perturbation” . . . . . . . . . . . . . . . 7.2.1 Concept and Mathematical Model of “Constant Flux-Linkage Theorem” . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Theory of Small Perturbation in Terms of Taylor’s Series Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Combined Effect of the Above-Said Two Ideas to Develop the Resultant Equivalent Circuit . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Digital Modeling Approach for Stability Analysis of Synchronous Motor Drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Development of Transfer Function in Continuous Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Development of Transfer Function in Discrete Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Stability Analysis of Discrete Time Systems . . . . . . . . . . . . . . . . . 8.3.1 Stability Analysis Using Pole-Zero Mapping . . . . . . . . . . 8.3.2 Stability Analysis Using Jury’s Test . . . . . . . . . . . . . . . . . 8.4 Impulse Response Analysis in Continuous and Discrete Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Impulse Response Analysis in S-domain . . . . . . . . . . . . . 8.4.2 Impulse Response Analysis in Z-domain . . . . . . . . . . . . . 8.5 Analysis of Parameter Perturbation on Stability . . . . . . . . . . . . . . . 8.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127 127 128

135 140

144

161 161 162 169 177 179 179 180 180 182 183 183 183 186 187 188 190 194 194

Contents

9

Unscented and Complex Unscented Kalman Filtering for Parameter Estimation of a Single and Multiple Sinusoids in the Area of Power and Communication Signals . . . . . . . . . . . . . . . . 9.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Outline of Basic Estimation Methods . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Cramer–Rao Lower Bound (CRLB) . . . . . . . . . . . . . . . . . 9.2.2 Maximum Likelihood Estimation (MLE) . . . . . . . . . . . . . 9.2.3 Linear Predictor (LP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.4 Extended Kalman Filter (EKF) . . . . . . . . . . . . . . . . . . . . . 9.2.5 Unscented Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Motivation Behind the Development of a New Model Taking Real Sinusoid into Account . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Algorithm Development for a Real Sinusoid . . . . . . . . . . 9.3.2 Harmonic Estimation of Real Sinusoid . . . . . . . . . . . . . . . 9.3.3 Rigorous Mathematical Modelling Multiple Sinusoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.4 Algorithm Development for Harmonic Estimation . . . . . 9.3.5 Outline of Experimental Setup . . . . . . . . . . . . . . . . . . . . . . 9.3.6 A Comparative Discussion on Simulation and Experimental Result . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Complex Unscented Kalman Filter for Signal Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Motivation Behind the Development of a New Model Taking the Complex Nature of the Signal into Account . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Algorithm Development . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.4 A Comparative Discussion on Simulation and Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Mathematical Modeling for Parameter Estimation of a Non Stationary Sinusoids in the Area of Power and Communication Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Signal Model in Complex Domain . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Multi Objective Gauss–Newton Algorithm . . . . . . . . . . . 10.2.2 Frequency Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 Amplitude and Phase Estimation . . . . . . . . . . . . . . . . . . . . 10.2.4 Performance Analysis of Single Sinusoid . . . . . . . . . . . . . 10.3 Algorithm Development for Multiple Sinusoids . . . . . . . . . . . . . . . 10.3.1 Frequency Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Amplitude and Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 Performance Analysis of Multiple Sinusoids . . . . . . . . . .

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197 197 199 201 202 203 204 207 209 210 213 214 215 216 216 220

221 225 229 229 232 235

239 239 240 240 241 243 244 245 246 248 248

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Contents

10.4 A Comparative Discussion on Simulation and Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Outline of Experimental Setup . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Single Sinusoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.3 Multiple Sinusoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

249 249 250 251 258 262

Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Appendix C: Brief Description of the Fabrication of a Single-Sided Linear Induction Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

About the Editors

Dr. Adhir Baran Chattopadhyay completed his Ph.D. (Electrical Engineering) from Indian Institute of Technology (IIT), Kharagpur, India, in 1999. He has worked as an Assistant Engineer (Quality Control) in M/S Crompton Greaves Limited, Ahmednagar, Maharashtra, India, from 1980 to 1985; as a faculty-member (Associate Professor) in National Institute of Technology, Jamshedpur, India (from 1985 to 2006), and as a Professor (EEE Department), BITS Pilani, Dubai, UAE (from 2006 to 2020). He is presently working as a Professor (Electrical Engineering Department), Budge Budge Institute of Technology (B.B.I.T), Kolkata-700137, since March, 2021. His research interests include electrical machines and drives, control systems, power electronics, power systems, electromagnetic fields and waves and applied mathematics. He has published more than 45 research papers in reputed international journals. He serves as a reviewer in many popular journals. He has been awarded (jointly) “Cogent Engineering Best Paper 2020” award, by Cogent Engineering Journal, Publisher—Taylor and Francis, U.K—Award declared on 7th May, 2021. He has published, as a first author, the book, “Advanced Electrical and Power System Design” in July, 2021 (Publisher: New Age International Publishers, New Delhi, London). Dr. Shazia Hasan has more than 15 years of teaching and research experience in reputed institutes in India and currently working as an Assistant Professor at BITS Pilani Dubai Campus. She received her Ph.D. in Engineering from Biju Patnaik University of Technology, India, in the year 2012. She won the ‘Young Scientist Award’ from VIFRA in 2015. She has published more than 35 research papers in peer-reviewed international journals/conferences. She has served as a reviewer for various reputed Journals like as IEEE, Elsevier, Springer and IET. She is a Senior Member of IEEE. She has served as convenor/co-convenor for several international conferences. She is the recipient of “Academic achievement for university professor award” Leadership Excellence for Women Awards (LEWAS) 2020. Her research interest includes Digital Signal Processing, signal processing application to Power systems, renewable energy integration, and statistical filter design.

xxi

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About the Editors

Dr. Snehaunshu Chowdhury received his B.E. degree in 2001 from Regional Engineering College (currently National Institute of Technology), Trichy, India; M.S.M.E. degree from Indiana University Purdue University Indianapolis (IUPUI), Indiana, USA, in 2004; M.E. degree from Colorado State University, Fort Collins, USA, in 2005; and a Ph.D. degree from the University of Maryland College Park, USA, in 2012, all in Mechanical Engineering. He then worked as a postdoctoral fellow in the “Clean Combustion Research Center” at King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia, from 2013 to 2016. Currently, Dr. Chowdhury is working as an assistant professor in the department of BITS Pilani, Dubai Campus, Dubai, UAE, since 2017 after a short stint at B.C. Roy Engineering College, Durgapur, India. Dr. Chowdhury’s research interests include nanoparticle characterization and combustion, soot particle and flame speed measurements and computational fluid dynamics. He has authored many papers and serves as a reviewer of internationally reputed journals.

Chapter 1

Applications of Field Theory: Analysis of a Single Sided Linear Induction Motor with Stator and Rotor of Infinite Length and Width

1.1 Introduction Linear Induction motors (LIMs) have several practical applications [1–13] and they have been analysed by various researchers [14–23]. However, the bulk of literature on LIMs deals with the double-sided, short primary and long secondary LIM which is considered as one of the most suitable means of propulsion for high-speed ground vehicles [15, 24–26]. There is an interesting work on “High-Temperature Superconducting(HTS) LIM” [27] which gives a good view on the coordination between HTS and LIM, leading to a realistic practical application of Single-sided Linear Induction Motor (supported with back iron) to the rail system. However, in extra high-speed transportation, it is needed that the vehicle is to be free from wheel-rail friction, and therefore, it is necessary to provide both lift and propulsion to the vehicle, and in such context, preferably an optimum design may be needed [28]. Also, it is desirable to have a completely contactless and lightweight vehicle. Such requirements can be met by using a single-sided linear induction motor (SLIM) with long primary and short secondary (without back iron support for the secondary) and its analysis and performance (in different stages) form the subject matter of the first three chapters of the Part-A of this book. A single-sided LIM has an open-sided, infinitely long stator in which the flux is forced through long air paths and the rotor consists of simply a thin sheet of aluminium of finite length and width placed over the stator. When the stator winding is excited by three-phase currents, the induced currents in the rotor produce propulsion and levitation forces. However, it may be noted that the magnetic circuit of such a machine is not efficient and the construction is highly expensive compared to double-sided LIM. The design of a single-sided linear induction motor calls for an analysis of the flux distribution on the open-side of a long primary supported by back iron. Such an analysis has been first carried out by West and Hesmondhalgh [29]. Representing the stator current distribution by a sinusoidal current sheet and assuming the stator to be © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. B. Chattopadhyay et al., Mathematical Modeling of Physical Systems, Advances in Intelligent Systems and Computing 1436, https://doi.org/10.1007/978-981-19-7558-5_1

1

2

1 Applications of Field Theory: Analysis of a Single Sided Linear …

infinitely long and wide, Laplace’s equation in two dimensions was solved by them to find tangential and normal components of flux density. They also analysed the current distribution in a slab of conductor placed over the stator winding assuming that the slab was infinitely thick and extending to infinity both along length and width. These assumptions are too idealistic and unrealistic for practical applications. Therefore, an analysis based on more realistic assumptions is needed. The analysis in this particular direction is taken up in two stages and those stages constitute the next two chapters of Part-A of this book. The analysis of a single-sided linear induction motor(SLIM) is based on the analysis of electromagnetic fields in the open air gap of the machine subject to the condition that a thin rotor slab of finite length and finite width is placed in the air-gap zone, in the proximity of the stator iron block. The finiteness in the rotor dimensions gives rise to so-called “Longitudinal End Effects” and “Transverse Edge Effects”, respectively. In order to simplify our analysis, a SLIM without any end effect, either in a longitudinal or in a transverse direction is being considered. In other words, a SLIM with an infinitely long and wide stator and rotor is considered for preliminary analysis [30]. Figure 1.1 shows such an idealized model with a coordinate system fixed relative to the rotor and the origin lying just above the stator surface. The rotor sheet is of infinitely small thickness and is made of non-magnetic material. The sheet is placed over the stator surface with its longitudinal edges running parallel to the stator. The length of the gap between the surface of the stator and the rotor sheet along the y-axis direction is “h”. As per the construction of the SLIM in Fig. 1.1, conceptually there exist three regions in the y-direction where the field quantities are to be mathematically derived. Those regions include one particular region which is the rotor sheet itself. But practically, as the thickness of the rotor sheet is very small and as also the rotor is magnetically equivalent to air, only two regions (region I and region II) are considered for field calculations. Region I consists of the air gap between the stator surface and the rotor sheet and region II consists of all the space from the rotor surface to infinity in the y-direction. Thus, the problem becomes a two-region problem. With the configuration of the SLIM shown in Fig. 1.1, calculations of the following are hereby proposed: (i) Fields in the two regions and the current in the rotor sheet (ii) Electromagnetic forces exerted on the rotor sheet.

1.2 Formulation for Fields and Currents Three-phase distributed windings are laid on laminated stator iron. With reference to Fig. 1.1, the stator windings are approximated by a surface current sheet having sinusoidal current distribution along the length and time. This is similar to a travelling wave or travelling field. It is well known that for the practical purpose of developing a distributed winding, there should exist slots and teeth on the stator iron surface. It is also very interesting to notice that if a plane sheet (made of aluminium or

1.2 Formulation for Fields and Currents

3

Fig. 1.1 A SLIM with thin rotor sheet. Note x, y, z-coordinate system is fixed relative to the rotor for purpose of analysis and the origin is just above the stator surface

iron) is placed above such a slotted structure of the iron sheet, the air-gap length between the stator and rotor will have a non-uniform variation along the length of the stator structure (or, along the distribution of the winding). However, in practice, such variation may not affect the performance of the SLIM, much. Ignoring such variation of the air gap and also the effect of unbalances in three-phase windings (if any), with the coordinate system fixed to the rotor, the equivalent stator current sheet can be represented by [ ] jz = j0 e j(sωs t−kx) ,

(1.1)

where j0 s k ωs t j

peak value of the linear current density of the stator winding slip of the rotor wave number = πτ , where τ = pole pitch supply frequency in electrical radians per second time instant in seconds√ imaginary number = −1.

As the stator winding system is considered to be infinitely wide, only one component of the stator linear current density ( jz ) exists. Similarly, because of the assumption of an infinitely wide rotor, only one component of the induced current (which flows in the z-direction) exists. Since the primary and the secondary currents have only z-components, the magnetic vector potential at any field point will also become an − → − → one-dimensional vector, A z 1z , where 1z stands for unit vector in the z-axis direction. The field is uniform in the z-direction so that all the field quantities are independent of “z”. As regions I and II are current-free zones, Laplace’s equation is satisfied by A z in both the zones. The relevant equation is

4

1 Applications of Field Theory: Analysis of a Single Sided Linear …

∂2 ∂2 + (A ) (A z ) = 0 z ∂x2 ∂ y2

(1.2)

Within the gap between the stator surface and rotor sheet (region I) and in the region beyond the rotor sheet (region II), the solution of Eq. (1.2) can be expressed as )[ ] ( A z1 = C1 eky + C2 e−ky e j(sωs t−kx)

(1.3)

)[ ] ( A z2 = C3 eky + C4 e−ky e j (sωs t−kx)

(1.4)

and

respectively, where C1 , C2 , C3 and C4 are unknown coefficients. The usual method of finding the unknown coefficients using the associated boundary conditions of the differential equation follows in the next section.

1.2.1 Boundary Conditions and Determination of the Coefficients, C1 , C2 , C3 and C4 Region II extends up to infinity in the positive y-direction, A z2 must vanish at y = ∞. Therefore, from Eq. (1.4), we obtain C3 = 0

(1.5)

)[ ] ( A z2 = C4 e−ky e j(sωs t−kx)

(1.6)

Hence,

In connection with Eq. (1.2), the boundary conditions to be satisfied are as follows: (i) On the surface of the stator current sheet (at y = 0), ∂ A z1 = −μ0 jz . ∂y

(1.7)

In Eq. (1.7), μ0 (= 4π × 10−7 ) is the permeability of air. The stator surface is taken as a y = 0 plane. (ii) At the interface of regions I and II: (a) Normal component of flux density is continuous, i.e. at y = h,

1.2 Formulation for Fields and Currents

( −

5

∂ Az1 ∂x

) y=h

) ( ∂ A z2 = − ∂ x y=h

(1.8)

(b) Considering a rectangular contour (in the x–y plane), enclosing a section of the rotor sheet as shown in Fig. 1.1 and taking the line integral of H, we get, for a thin conducting rotor sheet, (Hx )bottom − (Hx )top = jzr .

(1.9)

In Eq. (1.9), (Hx )bottom and (Hx )top are the tangential components of magnetic field intensity at the bottom and top surfaces of the rotor sheet (respectively) and jzr is the linear rotor current density (in the z-direction). From Eqs. (1.3), (1.6) and (1.8), it yields C1 ekh + C2 e−kh = C4 e−kh

(1.10)

The rotor linear current density, jzr can be expressed as jzr = Jzr (d) = σ (E z ).

(1.11)

In Eq. (1.11), ‘ jzr ’ is the rotor current density in A/m2 and ‘d’ is the thickness of the rotor sheet and ‘σ ’ and ‘E z ’ are the conductivity and electric field intensity in → =∇ → × E→ = − d B→ , we can write, the rotor, respectively. From the relation, Curl( E) dt for an infinitely wide rotor, Ez = −

] d[ (A z2 ) y=h dt

(1.12)

In Eq. (1.12), (A z2 ) y=h represents the z-axis component of the magnetic vector potential in region II, at a height (y-value) of “h”. Equations (1.11), (1.12) and (1.6) give the final expression for jzr as [ ] jzr = − j σ s(ωs )(d)C4 e−ky e j (sωs t−kx)

(1.13)

From Eqs. (1.3) and (1.6), we obtain (Hx )bottom and (Hx )top as ( (Hx )bottom = ( = and

1 μ0

)[(

∂ A z1 ∂y

]

) y=h

) )[ ] k ( C1 ekh − C2 e−kh e j(sωs t−kx) μ0

(1.14)

6

1 Applications of Field Theory: Analysis of a Single Sided Linear …

)[( ) ] ∂ A z2 1 = μ0 ∂ y y=h ( ) )[ ] k ( =− C4 e−kh e j(sωs t−kx) μ0 (

(Hx )top

(1.15)

Substituting Eqs. (1.14), (1.15) and (1.13) in Eq. (1.9), it yields (

] ) [ ( ) ) k ( k C1 ekh − C2 e−kh = − + j σ s(ωs )(d) C4 e−kh μ0 μ0

(1.16)

From Eqs. (1.10) and (1.16), we obtain (after C4 being eliminated) [( )] )] [ ( C1 ekh − C2 e−kh σ s(ωs )(d)μ0 ( ) =− 1+ j k C1 ekh + C2 e−kh

(1.17)

The quantity “ σ s(ωsk)(d)μ0 ” is non-dimensional and may be designated as Magnetic Reynold’s number(R). Thus, we can write σ s(ωs )(d)μ0 k = μ0 σ d(svs ) = μ0 σ dvrel

R=

(1.18)

In Eq. (1.18), ‘vrel ’ is the relative velocity of the travelling field with respect to the rotor. Equations (1.17) and (1.18) give ( ) C1 ekh − C2 e−kh ( ) = −[1 + j R] C1 ekh + C2 e−kh

(1.19)

The boundary condition in Eq. (1.7), with the help of Eqs. (1.1) and (1.3), gives −(μ0 )( j0 ) = k(C1 − C2 )

(1.20)

From Eqs. (1.19), (1.20) and (1.10), it yields ( C1 =

C2 = and

(μ0 )( j0 ) k

)( ) − j Re−2kh

(2 + j R) + j Re−2kh ) ( (μ0 )( j0 ) (2 + j R) k (2 + j R) + j Re−2kh

(1.21)

(1.22)

1.2 Formulation for Fields and Currents

C4 =

2 (μ0 k)( j0 ) (2 + j R) + j Re−2kh

7

(1.23)

Based on the values of C1 , C2 and C4 , the magnetic vector potential ( A z ) in regions I and II can be computed.

1.2.2 Calculation of Flux Density Components in the Rotor Sheet The longitudinal component of the flux density, Bxr , in the rotor sheet is not uniform along its thickness (in the y-direction) because of the rotor current. As seen from Eq. (1.9), the values of H x at the bottom and top surfaces of the rotor sheet differ by a finite quantity equal to the rotor linear current density as decided by the current enclosed in the same sheet. Therefore, Bxr can be considered as the average of its values at the bottom and top surfaces of the sheet. Based on this consideration, Bxr can be expressed as Bxr =

] [ 1 (μ0 ) (Hx )bottom + (Hx )top 2

The above-said expression can be simplified to ( ) )[ ] k ( −2C2 e−kh e j (sωs t−kx) Bxr = 2

(1.24)

The normal component of flux density in the rotor, Byr , being continuous at the interface of regions I and II, can be expressed as ) ( ∂ A z1 Byr = − ∂ x y=h )[ ] ( = jk C1 ekh + C2 e−kh e j(sωs t−kx)

(1.25)

In the equation the unknowns C1 and C2 are known from Eqs. (1.21) and (1.22).

1.2.3 Calculation of Current Density in the Rotor Sheet From Eqs. (1.11) and (1.13), we get [ ] Jzr = − j σ s(ωs )C4 e−kh e j (sωs t−kx)

(1.26)

8

1 Applications of Field Theory: Analysis of a Single Sided Linear …

In the equation the unknown C4 is known from Eq. (1.23). From J zr , the propulsion and levitation forces on the rotor can be calculated using the relation, F→ = J→ × B→ as explained below.

1.3 Calculation of Forces In any system, mechanical forces can be produced by so many ways. Such forces also can be produced by using the ever-known principle of magnetic flux–current interaction and they are known as electromagnetic forces. In the present problem, two-dimensional electromagnetic forces, namely “levitation force” and “propulsion force” are produced. The computation methods for the calculation of these forces are explained in the subsequent sections.

1.3.1 Calculation of Levitation Force The time average of the levitation force on the rotor sheet, per unit width and unit length, Fy is expressed as Fy =

( )} 1{ Real Jzr B∗xr {(1)(1)(d)} 2

(1.27)

∗ In Eq. (1.27), Bxr indicates the complex conjugate of Bxr . The expression for Fy finally simplifies to

Fy =

{ }] 1[ Real (σ sdωs k)jC1 C∗2 2

(1.28)

Substituting for C1 and C2∗ , it finally yields [ 1 2 Fy = μ0 ( j0 ) 2 1+

R 2 −2kh e 2 R2 4

( )2 1 + e−2kh

] (1.29)

1.3.2 Calculation of Propulsion Force The time average of the propulsion force on the rotor per unit width and length, Fx is given by

1.4 Normalized Propulsion and Levitation Forces and Their Maximum Values

Fx =

( )} 1{ Real −Jzr B∗yr {(1)(1)(d)} 2

9

(1.30)

∗ In Eq. (1.30), Byr stands for the complex conjugate of Byr . The above expression finally simplifies to

[ 1 2 Fx = μ0 ( j0 ) 2 1+

Re−2kh ( )2 R2 1 + e−2kh 4

] (1.31)

1.4 Normalized Propulsion and Levitation Forces and Their Maximum Values From the expression for the propulsion force, Fx in Eq. (1.31), the normalized propulsion force, FxN can be expressed as Fx N = =

Fx μ0 ( j0 )2 ⎡ 1 ⎢ R⎣ 2

⎤ −2π ( τh )

4e ⎥ ( )2 ⎦ −2π ( τh ) 2 4+ R 1+e

(1.32)

Similarly based on Eq. (1.29), the normalized levitation force, Fy N can be expressed as Fy μ0 ( j0 )2 ⎡ ⎤ ( 2) R −2π ( τh ) e 2 1⎢ ⎥ = ⎣ ( )2 ⎦ 2 R2 −2π ( τh ) 1+ 4 1+e

Fy N =

(1.33)

With reference to the above-said expressions for the normalized forces, one point is noticeable that the parameter “ τh ” has a dominant role. Based on this observation, two cases are discussed as follows: Case (i):

h τ

=0

The expressions for propulsion and levitation forces reduce to Fx N 0

) ( R 1 = 2 1 + R2

(1.34)

10

1 Applications of Field Theory: Analysis of a Single Sided Linear …

and

Fy N 0

⎧ ( 2) ⎫ R 1⎨ 2 ⎬ = 2 ⎩ 1 + R2 ⎭

(1.35)

It can be shown that Fx N 0 reaches its maximum value of 41 at R = 1 and Fy N 0 reaches its maximum value of 41 at R = ∞. The maximum values of normalized propulsion and normalized levitation forces for τh = 0 are equal though they occur at different Reynold’s numbers. Case (ii):

h τ

/= 0

It can be shown that Fx N reaches its maximum value of Fx N (max) =

[ −2kh ] e 1 2 1 + e−2kh

(1.36)

Equation (1.36) corresponds to a value of R as given by R=

2 1 + e−2kh

(1.37)

Similarly, Fy N reaches its maximum value of [ Fy N (max) =

e−2kh

]

( )2 1 + e−2kh

(1.38)

Equation (1.38) corresponds to a value of R as given by R = ∞. The normalized values of the propulsion and levitation forces are calculated and plotted against Reynold’s number in Figs. 1.2 and 1.3, respectively, for different values of “ τh ”. So far we have considered a thin rotor sheet in which the attenuation of the rotor fields and currents with respect to the sheet thickness has not been taken into account. Therefore, for better assessment of the electromagnetic forces, we consider a rotor of finite thickness as shown in Fig. 1.4 and analyse the field quantities including their attenuation within the rotor. Thus, the problem becomes a three-region problem.

1.5 Three-Region Problem

11

Fig. 1.2 Variation of propulsion force with speed without end effects at different heights of the rotor

1.5 Three-Region Problem With reference to Fig. 1.4, regions I and III are current-free zones where Laplace’s equation is satisfied by A z . Therefore, the solutions for A z in regions I and III are given by )[ ] ( A z1 = C1' eky + C2' e−ky e j(sωs t−kx)

(1.39)

[ ] A z3 = C6' e−ky e j (sωs t−kx)

(1.40)

and

In Eqs. (1.39) and (1.40), C1' , C2' and C6' are the coefficients to be determined.

12

1 Applications of Field Theory: Analysis of a Single Sided Linear …

Fig. 1.3 Variation of levitation force with speed without end effects at different heights of the rotor

Fig. 1.4 A SLIM with thick rotor

1.5 Three-Region Problem

13

The region II consists of the rotor sheet having thickness “d” with conductivity “σ ”. The magnetic vector potential A z within the rotor sheet satisfies the equation given by ∂2 ∂2 + (A ) ( A z ) = −μ0 Jzr z ∂x2 ∂ y2

(1.41)

In Eq. (1.41), Jzr is the rotor current density and it can be expressed as Jzr = − j σ sωs A z

(1.42)

From Eqs. (1.41) and (1.42), we have ∂2 ∂2 + (A ) (A z ) = j μ0 σ sωs A z z ∂x2 ∂ y2

(1.43)

The solution for A z in region II satisfying Eq. (1.43) is given by )[ ] ( A z2 = C3' eδy + C4' e−δy e j (sωs t−kx)

(1.44)

In Eq. (1.44), δ=

/( ) k 2 + jμσ sωs '

(1.45)

'

Also in Eq. (1.44), the coefficients C3 and C4 are the coefficients to be determined for boundary conditions, and in Eq. (1.45), “μ” stands for the magnetic permeability of the rotor material. As the rotor material of a SLIM generally is considered to be non-magnetic (aluminium), μ equals to “μ0 ”.

1.5.1 Boundary Conditions In connection with Laplace’s equation in regions I and III and the Diffusion equation in region II, the boundary conditions to be satisfied are as follows: (1) On the surface of the stator current sheet: (

∂ A z1 ∂y

)

[ ] = −μ0 jz = −μ0 j0 e j(sωs t−kx)

(1.46)

y=0

Equation (1.46) and the expression for A z1 in Eq. (1.39) yield C1' − C2' = −

(μ0 )( j0 ) k

(1.47)

14

1 Applications of Field Theory: Analysis of a Single Sided Linear …

(2) At the interface of regions I and II: (a) Continuity of normal components of flux density gives ) ) ( ( ∂ A z1 ∂ A z2 − =− ∂ x y=h ∂ x y=h

(1.48)

The expressions for A z1 and A z2 in Eqs. (1.39) and (1.44), respectively, and Eq. (1.48) yield C1' ekh + C2' e−kh = C3' eδh + C4' e−δh

(1.49)

(b) Continuity of tangential components of magnetic field intensity gives ( ) ( ) 1 ∂ A z1 1 ∂ A z2 = μ0 ∂ y y=h μ ∂ y y=h ( ) 1 ∂ A z2 = μ0 ∂ y y=h

(1.50)

Equations (1.39), (1.44) and (1.50) give ) ) k ( ' kh δ ( ' δh C1 e − C2' e−kh = C3 e − C4' e−δh μ0 μ0

(1.51)

(3) At the interface of regions II and III:

( ) (a) Continuity of normal components of flux density B→ mathematically takes the shape as given by ) ) ( ( ∂ A z2 ∂ A z3 − =− ∂ x y=h+d ∂ x y=h+d

(1.52)

The above condition leads to C3' eδ(h+d) + C4' e−δ(h+d) = C6' e−k(h+d)

(1.53)

( ) (b) Continuity of tangential components of magnetic field H→ requires ( ( ) ) 1 ∂ A z2 1 ∂ A z3 = μ0 ∂ y y=h+d μ0 ∂ y y=h+d The above condition finally leads to

(1.54)

1.5 Three-Region Problem

15

δ ' δ(h+d) δ −k ' −k(h+d) − C4' e−δ(h+d) = C e C3 e μ μ μ0 6

'

'

'

'

(1.55)

'

1.5.2 Determination of Coefficients C1 , C2 , C3 , C4 and C6 Equations (1.47), (1.49), (1.51), (1.53) and (1.55) can be arranged in the form of a matrix equation given by [ ] [ ] [M] C1' = F f

(1.56)

In Eq. (1.56), the matrix variables are given as follows: [M] = 1 −1 0 0 0 ekh e−kh −eδh −e−δh 0 k kh δ −δh e − μk0 e−kh − μδ0 eδh e 0 μ0 μ0 δ(h+d) −δ(h+d) −k(h+d) 0 0 e e −e δ δ(h+d) −δ −δ(h+d) k −k(h+d) 0 0 e e e μ μ μ0

(1.57)

and '

C1 ' C2 [ '] ' C1 = C3 ' C4 ' C6

(1.58)

and (−μ0 )( j0 ) k

[

] Ff =

0 0 0 0

(1.59)

Inverting the matrix [M] of Eq. (1.57), the unknown coefficients C1' , C2' , C3' , C4' and C6' can be computed numerically.

16

1 Applications of Field Theory: Analysis of a Single Sided Linear …

1.5.3 Calculation of Flux Densities and Current Density in the Rotor Sheet The normal and tangential components of flux density, Byr and Bxr , respectively, in the rotor sheet, can be expressed as Byr = −

∂ A z2 ∂x

(1.60)

and Bxr =

∂ A z2 ∂y

(1.61)

Based on Eqs. (1.60) and (1.61) and the expression for A z2 in Eq. (1.44), we get )[ ] ( Byr = jk C3' eδy + C4' e−δy e j(sωs t−kx)

(1.62)

)[ ( ' ] ' Bxr = δ C3 eδy − C4 e−δy e j (sωs t−kx)

(1.63)

and

Using the expression for Jzr in Eq. (1.42) and for A z2 in Eq. (1.44), it can be shown that )[ ] ( Jzr = − j(sσ ωs ) C3' eδy + C4' e−δy e j(sωs t−kx)

(1.64)

1.5.4 Calculation of Forces 1.5.4.1

Calculation of Levitation Force

The time average of levitation force on the rotor sheet per unit width and unit length, Fy is given by (h+d Fy =

( )}] 1 [{ Real Jzr B∗xr dy. 2

h ∗ In Eq. (1.65), Bxr stands for the complex conjugate of Bxr .

(1.65)

1.5 Three-Region Problem

17

[( )] ∗ ∗ (I I) (δ+δ )(h+d) − e(δ+δ )(h) sσ ωs ∗ I 'I 2e Real − j δ C3 Fy = 2 (δ + δ ∗ ) )] [( ∗ ∗ −(δ+δ )(h+d) (I I) − e−(δ+δ )(h) sσ ωs ∗ I 'I 2e − Real − j δ C4 2 −(δ + δ ∗ ) [( )] ∗ ∗ −δ)(h+d) (δ ( )∗ e − e(δ −δ)(h) sσ ωs Real − j δ ∗ C3' C4' + 2 (δ ∗ − δ) [( )] ∗ ∗ ( )∗ e(−δ +δ)(h+d) − e(−δ +δ)(h) sσ ωs Real j δ ∗ C4' C3' + . 2 (−δ ∗ + δ)

1.5.4.2

(1.66)

Calculation of Propulsion Force

The time average of the propulsion force on the rotor sheet per unit width and unit length, Fx is given by (h+d [{ ( )}] 1 Real −Jzr B∗yr dy. Fx = 2

(1.67)

h ∗ In Eq. (1.67), Byr stands for the complex conjugate of Byr . Using the expressions for Jzr and Byr in Eqs. (1.64) and (1.62), respectively, after simplification, Eq. (1.67) leads to [( )] ∗ ∗ (I ' I)2 e(δ+δ )(h+d) − e(δ+δ )(h) sσ ωs k I I C3 Real Fx = 2 (δ + δ ∗ ) [( )] ∗ ∗ −(δ+δ )(h+d) (I ' I)2 e − e−(δ+δ )(h) sσ ωs k I I C4 Real + 2 −(δ + δ ∗ ) )] [( ∗ ∗ ( ' )∗ ' e(δ −δ)(h+d) − e(δ −δ)(h) sσ ωs k Real C3 C4 + 2 (δ ∗ + δ) [( )] ∗ ∗ ( ' )∗ ' e(−δ +δ)(h+d) − e(−δ +δ)(h) sσ ωs k Real C4 C3 . (1.68) + 2 (−δ ∗ + δ)

Just as in the case of the two-region problem, the expressions for Fx and Fy can be divided by the term “μ0 ( j0 )2 ” to obtain the normalized propulsion and levitation forces, Fxn andFyn .The values of Fxn andFyn , thus obtained, may be plotted against Reynold’s number (which is a measure of the slip speed) for different values of “ τh ”. These plots are based on a rotor of thickness, d = 3.08 mm (which corresponds to the rotor of the experimental model). The results obtained for the assumed rotor thickness are almost the same as those obtained using a two-region model. This is because the rotor thickness is quite small compared to the classical depth of penetration even at s = 1. For example, the classical

18

1 Applications of Field Theory: Analysis of a Single Sided Linear …

depth of penetration at 50 Hz for aluminium rotor with σ = 2.42 × 107 Ω−1 m−1 works out to be 14.4 mm. Therefore, the actual rotor thickness, 3.08 mm, is only about 20 percent of the “depth of penetration”, even at unity slip. That is why these plots are not shown here. The application of the results obtained in this chapter to evaluate the performance of the experimental model is not straightforward because the actual stator and rotor have a finite width and further the rotor has a finite length. The finite width and finite length effects are treated in the next chapters. However, at this stage, some suitable correction factor to the rotor resistivity can be formulated( in the line of the analysis carried out by Russel and Northsworthy [31]) to take finite width effects into account, and this correction factor can be applied to approximately predict the force–speed characteristics of the experimental model using the results obtained in this chapter. The derivation for the rotor resistivity correction factor, K r has been presented in Appendix A and K r is given by [

{∞ 1 = Kr

n=1,3,5...

{∞

(an )2 ) ( 2 2 1+ n τ 2 (Wr )

2 n=1,3,5... (an )

] .

(1.69)

In Eq. (1.69), an represents the Fourier coefficient of the nth space-harmonic component obtained after carrying out the Fourier series analysis of the distribution of the normal component of the flux density with respect to the width and “Wr ” represents the rotor width. Using the data of the experimental model, K1r works ( ) out to be 0.5 and this value has been used to obtain σeff σeff = K1r (σ ) . Using the normalized force versus speed characteristics shown in Figs. 1.2 and 1.3, and using σeff for calculating Reynold’s number, the actual force–speed characteristics (for both propulsion and levitation forces), for τh = 0.75 (which corresponds to the data of the experimental model) are obtained and shown in Fig. 1.5 Even after applying the correction factor, as seen from Fig. 1.5, the predicted propulsion and levitation forces are still much higher than the corresponding experimental values (at unity slip). Here, it will be relevant to mention that the experimental values of levitation and propulsion forces for a stator current of 12 amps (r.m.s) and for a height of 2.1 cm are found to be 26.7 N and 31.1 N, respectively. Therefore, a method of analysis taking directly the finite width and finite length effects into account is needed and the subsequent chapters deal with these aspects.

1.6 Conclusions In this chapter, the analysis of a SLIM having no end effects either in the longitudinal or transverse direction has been presented. As the stator and rotor sheets are assumed to have infinite length and infinite width, only A z -component of magnetic vector potential exists. This vector potential

1.6 Conclusions

19

(a) Propulsion Force; (b) Levitation Force Fig. 1.5 Force–slip characteristics (for correction factor

h τ

= 0.075) after incorporating the rotor resistivity

has been computed by solving Laplace’s Equation in the air-gap region and the Diffusion Equation in the rotor region. As the first stage of analysis, considering the rotor to be very thin, the solution for A z has been obtained by solving Laplace’s Equation as a two-region problem. The important contribution of this chapter is to evaluate lift and propulsion forces as a function of speed in normalized parameters. It is also shown that the maximum possible lift force is the same as the maximum propulsion force. However, the maximum propulsion force occurs for R = 1 whereas the maximum levitation force occurs for R = ∞, asymptotically. These results help us in deciding the practical limitation of a linear propulsion and levitation system using the induction principle. For taking the effects of attenuation of fields and currents within the rotor sheet into account, a three-region model of the machine has also been considered and subsequently analysed. It is observed that for rotor thickness small compared to the classical depth of penetration, the results obtained using the two-region model are almost as accurate as those given by the three-region model. Therefore, in the subsequent chapters where finite width and finite length effects are considered, the rotor is considered to be thin enough so that it need not be considered as a separate region for the purpose of analysis.

20

1 Applications of Field Theory: Analysis of a Single Sided Linear …

References 1. Mc Lean GW (1988) Review of recent progress in linear motors. IEE Proc Part-B 135(6):380– 416 2. Laithwaite ER, Tipping D, Hesmondhalgh DE (1960) The application of induction motors to conveyors. IEE Proc Part-A 107:284–294 3. Shiabo S, Niioka M (1983) The application of linear induction motor to steal making process. In: Proceedings of the international power electronics conference, Tokyo 4. Eastham JF (1985) Linear machines: present status and future potential. In: Proceedings of the conference on “DRIVES/MOTORS/CONTROLS 85”, London 5. Laithwaite ER (1995) Adapting a linear induction motor for the acceleration of large mass to high velocities. IEE Proc Electr Power Appl 142(4):262–268 6. Poporich N, Pyzhov V, Ostroverhov N (1994) Linear induction motors drive for coil telescoping control. In: Proceedings, 20th international conference on industrial electronics, control and instrumentation, vol 1, pp 617–622, Italy 7. Fujisaki K, Nakagawa J, Misumi H (1994) Fundamental characteristics of molten metal flow control by linear induction motor. IEEE Trans. Magn 30(6):4764–4766 8. Watson DB, Watson DM, Bate GJ (1994) Surface winding for moving small conducting particles. IEE Proc-Sci Meas Technol 141(3):229–232 9. Morizane T, Masada E (1993) Study on the feasibility of application of linear induction motor for vertical movement. IEEE Trans Magn 29(6):2938–2940 10. Sato Y, Iga M, Yamada H (1992) Light-weight type linear induction motor and its characteristics. IEEE Trans Magn 28(5):3003–3005 11. Saifkhani F, Waltace AK (1993) A linear brushless doubly-fed machine drive for traction applications. In: Fifth European conference on power electronics and applications, conference publication No. 377, vol 5, pp 344–48 12. Tipping D (1964) The analysis of some special purpose electrical machines. Ph. D thesis. Manchester University, U.K 13. Laithwaite ER (1966) Induction machines for special purposes. Newness, London 14. Dukowicz, JK (1977) Analysis of linear induction machines with discrete windings and finite iron length. IEEE Trans Power Appar Syst PAS-96(1):66–72 15. Jansen PL, Li LJ, Lorenz RD (1995) Analysis of competing technologies of linear induction machines for high speed material transport systems. IEEE Trans Ind Appl 31(4):925–932 16. Cardoso JR, Benites PA (1995) Finite element method with BiCG solver applied to moving linear induction motors. IEEE Trans Magn 31(3):1888–1891 17. Im DH, Kim CE (1994) Finite element force calculation of a linear induction motor taking account of the movement. IEEE Trans Magn 30(5):3495–3498 18. Lu C, Eastham TR, Dawson GE (1993) Transient and dynamic performance of a linear induction motor. In: Conference record of the 1993 IEEE industry applications conference, vol. I, pp 266–273 19. Idir K, Dawson GE, Eastham AR (1993) Modelling and performance of linear induction motor with saturable primary. IEEE Trans Ind Appl 29(6):1123–1128 20. Iwamoto M, Ohno E, Itoh T, Shinryo Y (1973) End-effect of high-speed linear induction motor. IEEE Trans Ind Appl IA-9(6):632–638 21. West JC, Hesmondhalgh DE (1962) The analysis of thick cylinder induction machine. Proc IEE Monograp No. 477U 109(C):172 22. Chattopadhyay AB (1997) Some experimental and theoretical investigations on the propulsion, levitation and lateral guidance forces in a long stator, short rotor, single sided linear induction motor. Ph.D thesis, Department of Electrical Engineering, Indian Institute of Technology, Kharagpur 721302, India, pp 42–66 23. Russel RL, Northsworthy KH (1958) Eddy currents and wall losses in screened-rotor induction motors. Proc IEE105(A):163–175

References

21

24. Smolyanov I, Shmakov E, Gasheva D (2019) Research of linear induction motor as part of driver by detailed equivalent circuit. In: 2019 international Russian automation conference (RusAutoCon), IEEE, Russia. https://doi.org/10.1109/RUS1UTOCON.2019.8867757 25. Seo K-Y, Park C-B, Jeong G, Lee J-B, Kim T, Lee H-W (2020) A study on the design of propulsion/levitation/guidance integrated DSLIM with non–symmetric structure. AIP Adv 10(2):1–7 26. Caprio MT, Pratap SB, Wall WA, Zowarka RC (2001) Linear electric motors for aerospace launch assist. In: Sixth international symposium on magnetic suspension technology,Torino, Italy 27. Liu X, Gao L, Liu X, Peng X, Yang G, Mou S (2019) Research and analysis of electromagnetic thrust of variable pole distance linear induction motor. In: IOP conference series: materials science and engineering, vol 677, pp 1–8. IOP Publishing. https://doi.org/10.1088/1757-899X/ 677/5/052006 28. Liu B, Fang J, Cao J, Chen J, Shu H, Sheng L (2017) Electromagnetic performance calculation of HTS linear induction motor for rail systems. In: Proceeding, 29th international symposium on superconductivity; Journal of Physics: Conference Series, vol 871:012097 doi :https://doi. org/10.1088/1742-6596/871/1/012097 . 29. Bazghaleh AZ, Naghashan MR, Meshkatoddini MR (2010) Optimum design of single-sided linear induction motors for improved motor performance. IEEE Trans Magn 46(11):3939–3947 30. Park CB, Lee HW, Lee J (2012) Performance analysis of the linear induction motor for the deep-underground high-speed GTX. J Electr Eng Technol 7(2):200–206. https://doi.org/10. 5370/JEET.2012.7.2.200 31. Lv G, Li Q, Liu Z, Fan Y, Li GG (2008) The analytical calculation of the thrust and normal force and force analyses for linear induction motor. In: 9th international conference on signal processing (EI), pp 2795–2799

Chapter 2

Mathematical Modelling of Electromagnetic Forces Due to Finite Width Effects of a Single-Sided Linear Induction Motor

2.1 Introduction In the preceding chapter, the analysis of a SLIM with a stator and a non-magnetic rotor, both of infinite length and infinite width, has been presented. But in a practical machine, the width of the stator winding as well as the width of the supporting back iron must be finite. The width of the rotor sheet placed over the stator winding of finite width can be made very large but it has no practical advantage beyond a certain extent. From the experimental results, it is observed that the rotor width should be equal to the stator width, including the winding overhang for good performance. Therefore, the rotor width also should be finite. As far as the length of the rotor sheet is concerned, it must also be finite because to think of a rotor of infinite length is absurd. So, for all practical purposes, a rotor sheet of finite length and finite width should be considered. But to analyse the problem in stages, for the present, a rotor of infinite length and finite width is considered in this chapter (see Fig. 2.1a), while a rotor of finite length and finite width will be considered in the next chapter. With reference to Fig. 2.1a, we propose to calculate the following: (a) The field at any point in free space due to the stator winding having a sinusoidal distribution of linear current density along the length. (b) The induced currents in the rotor sheet considering the field due to the stator as well as the rotor currents. (c) The propulsion, levitation and lateral forces on the rotor. It may be noted that the coordinate system used is fixed relative to the rotor, the origin being placed just above the stator surface. The rotor is assumed to be thin and lying parallel to the x–z-plane at a height of “h” from the stator surface. It is also assumed that the stator back iron of width “Ws ” is infinitely permeable and perfectly laminated.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. B. Chattopadhyay et al., Mathematical Modeling of Physical Systems, Advances in Intelligent Systems and Computing 1436, https://doi.org/10.1007/978-981-19-7558-5_2

23

24

2 Mathematical Modelling of Electromagnetic Forces Due to Finite Width …

(a)

(b) Fig. 2.1 a Model of SLIM with coordinate system used for the analysis of finite width effects (Coordinate axes are fixed relative to the rotor). b Variation of “f (z)” with “z”

2.2 Problem Formulation 2.2.1 Formulation for the Field Due to Stator Current Sheet As the stator winding has a finite width, the stator current will have not only the axial component of linear current density, jz (which would have the only component for a stator winding of infinite width) but also the peripheral component of linear current

2.2 Problem Formulation

25

density, jx . The latter component may be viewed as one due to the end current of a winding having an overhang. From physical considerations, the axial component of linear current density can be assumed to be constant over the stator iron width. For diamond-shaped coils, the variation of jz with the overhang width can be considered as linear. Therefore, in a coordinate system fixed to the rotor, jz can be expressed as ] [ jz = Iz f (z) e j(sωs t−kx)

(2.1)

In Eq. (2.1), the symbols are as follows: s k ωs t j f (z)

slip of the rotor. wave number = πτ , where τ pole pitch. supply frequency in electrical radians per second. time instant in seconds. √ imaginary number = −1. Mathematical function of the dimension of the stator along the z-axis.

The function f (z) in Eq. (2.1) can be expressed as (refer to Fig. 2.1b) f (z) = 1 f or W1 < z < W2 ( =1− ( =1+

z − W2 C z − W1 C

(2.2)

) f or W2 < z < (W2 + C)

(2.3)

f or (W1 − C) < z < W1

(2.4)

)

where C = Stator overhang length (in the z-direction) W1 = −

Ws − W0 2

(2.5)

where Ws = width of the stator iron block and W0 = offset of the rotor sheet (width-wise) with respect to the stator. and W2 =

Ws − W0 2

(2.6)

It is necessary to assume a certain amount of offset for the rotor (W0 ), with respect to the stator to enable calculation of the lateral force. The continuity equation of current in the stator winding can be expressed as d jx d jz + =0 dz dx

(2.7)

26

2 Mathematical Modelling of Electromagnetic Forces Due to Finite Width …

As an infinitely long three-phase balanced distributed stator winding energized by three-phase balanced currents can be represented by a current sheet having a sinusoidal current distribution along the length and with respect to time, like jz expressed in Eq. (2.1), jx also can be expressed as [ ] jx = jx0 e j(sωs t−kx)

(2.8)

where jx0 is a function of “z” only. From Eqs. (2.1) and (2.8), we have ] [ d jz = Iz f ' (z) e j (sωs t−kx) dz

(2.9)

[ ] d jx = (− jk) jx0 e j(sωs t−kx) dx

(2.10)

and

respectively, where f ' (z) is the derivative of f (z) with respect to “z”. Equations (2.7), (2.9) and (2.10) yield ( jx0 = − j

Iz k

)

f ' (z)

(2.11)

Based on Eqs. (2.8) and (2.11), we can finally express jx as (

Iz jx = − j k

)

] [ f ' (z) e j(sωs t−kx)

(2.12)

Vector potential due to stator current The vector potential due to stator current at any field point (x ' , y ' , z ' ) can be expressed as μ0 −→ AI 1 = 4π

z=W ( 2 +C (+∞

z=W1 −C x=−∞

[(

) ( )] jx →i jz k→ + d x dz r r

(2.13)

where, →i, →j, k→ are the unit vectors along x, y and z-directions, respectively r=

/{

(x − x ' )2 + (0 − y ' )2 + (z − z ' )2

}

(2.14)

In the right-hand side of the Eq. (2.13), the component associated with the unit vector, →j, is absent. The parameter “r” in Eq. (2.14) indicates the distance of the field point (x ' , y ' , z ' ) from the source point (x, 0, z). The effect of the stator iron is taken

2.2 Problem Formulation

27

into account by considering the “image” of “ jz ” over the width of the iron block, Ws , which will contribute an additional component of vector potential given by μ0 −→ AI 2 = 4π

z=W ( 2 (+∞

(

) jz k→ d x dz r

(2.15)

z=W1 x=−∞

The “image” of jx is not taken into account as it follows in the overhang beyond the stator iron width. It may be noted that the method of images to take the stator iron boundary is not rigorously correct but reasonably good approximation. Thus, − → the vector potential at any field point A I is expressed as − → −→ −→ AI = AI 1 + AI 2

2.2.1.1

(2.16)

− → Derivation for Expression for A I

−→ −→ Based on Eqs. (2.13) and (2.15), A I 1 and A I 2 , respectively, are expressed as ⎧ z=W +C ⎞ ⎫⎤ ⎡ ⎛ +∞ ( ) (2 ( ⎬ − jkx e −→ ⎣ μ0 − j Iz jsωs t ⎨ e d x ⎠dz ⎦→i AI 1 = f ' (z)⎝ ⎭ ⎩ 4π k r ⎡

z=W1 −C

⎧ z=W +C ⎨ (2

μ0 + ⎣ (Iz )e jsωs t ⎩ 4π

⎛

f (z)⎝

z=W1 −C

x=−∞

(+∞

x=−∞

⎞ ⎫⎤ e− jkx ⎠ ⎬⎦→ d x dz k ⎭ r

(2.17)

and ⎡

⎧ z=W ⎨( 2

−→ ⎣ μ0 AI 2 = (Iz )e jsωs t ⎩ 4π

⎛ f (z)⎝

z=W1

(+∞

x=−∞

e

− jkx

r

⎫⎤ ⎬ d x ⎠dz ⎦→ k ⎭ ⎞

(2.18)

( +∞ ( − jkx ) The preceding Eqs. (2.17) and (2.18) involve the integral −∞ e r d x which can be rewritten, using the expression for “r” in Eq. (2.14), as (+∞( −∞

) (+∞ e− jkx e− jkx dx = { }1 dx r ' )2 + (0 − y ' )2 + (z − z ' )2 2 − x (x x=−∞

Equation (2.19) can be simplified as

(2.19)

28

2 Mathematical Modelling of Electromagnetic Forces Due to Finite Width …

(+∞( −∞

) (+∞ e− jkx e− jkp − jkx ' dx = e ( ) 1 dp r 2 + b2 2 p −∞

(2.20)

where p = x − x'

(2.21)

and b= The integral

+∞ (

e− jkp 1

−∞ ( p2 +b2 ) 2

[(

z − z'

)2

( )2 ] 21 + y'

(2.22)

dp, appearing in the right-hand side of Eq. (2.20) can be

simplified and converted to a real integral by the method of contour integration as, (+∞ ( −∞

e− jkp p 2 + b2

(∞ ) 1 dp = 2 2

1

e−kbt1 (2 ) 1 dt1 t1 − 1 2

(2.23)

The detailed derivation for the above transformation is presented in Appendix B. ( ∞ −kbt1 The numerical evaluation of the integral 1 e 1 dt1 is much faster as compared t 2 −1 2 ( +∞ e−(jkx1 ) to the evaluation of the original integral, −∞ r d x, which converges rather slowly. From Eqs. (2.19) and (2.23), we have. (+∞ −∞

e− jkx ' d x = 2e− jkx r

(∞ 1

e−kbt1 ) 1 dt1 (2 t1 − 1 2

(2.24)

− → Based on Eqs. (2.16), (2.17), (2.18) and (2.24), the net vector potential, A I , at any field point due to the stator current can be expressed finally as ⎡ − → ⎣ μ0 AI = 4π

(

⎧ W +C ⎛ ∞ ⎞ ⎫⎤ ) (2 ( ⎬ ⎨ −kbt1 − j Iz jsωs t − jkx ' e ⎠dz ⎦→i e e f ' (z)⎝2 ( 1 dt1 ) ⎭ ⎩ k t2 − 1 2 W1 −C

1

1

⎛ ∞ ⎞ ⎫ ⎡⎧ W +C (2 ( ⎬ ⎨ −kbt1 μ0 e ' ⎠ + dt f (z)⎝2 ( dz (Iz )e jsωs t e− jkx ⎣ )1 1 ⎭ ⎩ 4π t12 − 1 2 1 W1 −c ⎧ ⎛ ∞ ⎞ ⎫ ( ⎨( W 2 ⎬ −kbt1 e ⎠dz k→ f (z)⎝2 ( + 1 dt1 ) ⎩ W1 ⎭ t2 − 1 2 1

1

(2.25)

2.2 Problem Formulation

29

The integration over “z” can be carried out numerically without any further transformation because of the finite limits of “z”.

2.2.1.2

Calculation of Flux Density Due to Stator Current

− → From the vector potential, A I expressed in Eq. (2.25), the flux density vector at any → and the field point can be computed numerically using the relation B→ = Curl A, numerical algorithms are presented in the later sections.

2.2.1.3

Formulation for Currents and Fields in the Rotor Sheet

With reference to Fig. 2.1a, the rotor sheet is placed parallel to the surface of the stator with its longitudinal edges running parallel to the length of the stator. As the rotor sheet is of finite width like the stator, the induced currents in the sheet must have closed paths. Therefore, the axial component of the rotor current density (Jz ) must gradually turn into the peripheral component of the current density (Jx ). The rotor sheet has a small( thickness and hence it does not allow any normal component of the ) current density Jy to flow. In other words, the rotor current is planar in nature. At any instant, in the sheet, the divergence of current density must vanish to maintain the continuity of current flow. Therefore, we can write Div J→ = 0

(2.26)

As the divergence of the curl of any vector is zero, J→ can be expressed in terms of a stream function, u→ given by J→ = curl u→

(2.27)

As planar current flows in the sheet, the stream function possesses only one component, u y , normal to its surface. From Eq. (2.27), we can write → × J→ = ∇ → ×∇ → × u→ ∇ ( ) → ∇ → · u→ − ∇ 2 (→ =∇ u)

(2.28)

As u x = 0, u z = 0 and u y does not vary with “y” → · u→ = 0 ∇

(2.29)

From Eqs. (2.28) and (2.29), we obtain ( )] [ → × J→ = 0→i + −∇ 2 u y →j + 0k→ ∇

(2.30)

30

2 Mathematical Modelling of Electromagnetic Forces Due to Finite Width …

In Eq. (2.30), →i, →j , k→ are the unit vectors along x-, y- and z-directions, respectively. From the relations, J→ = σ E→

(2.31a)

and curl E→ = −

∂ B→ ∂t

(2.31b)

− → where “ E ” is the electric field vector and “σ ” is the conductivity of the rotor sheet material, we get → → × J→ = −σ d B ∇ dt ) ( d By → = 0→i + −σ j + 0k→ dt

(2.32)

In Eq. (2.32), “B y ” is the total flux density (normal to the sheet) at the field point (x ' , y ' , z ' ) on the sheet. As “B y ” has the sinusoidal variation with respect to time, we can write d By = jsωs B y dt Substituting for

d By dt

(2.33)

from Eq. (2.33) in Eq. (2.32), we obtain ) ( → × J→ = − j σ sωs B y →j ∇

(2.34)

Equating the right-hand side terms of Eqs. (2.30) and (2.34), we obtain ( ) ∇ 2 u y = j σ sωs B y

(2.35)

Expanding the left-hand side terms of Eq. (2.35), we obtain ∂ 2u y ∂ 2u y + = jσ sωs B y ∂(x ' )2 ∂(z ' )2

(2.36)

As the rotor sheet is of infinite length, like stator linear current density components, rotor linear current density components also will have sinusoidal variation in the longitudinal direction. Therefore, at any point (x ' , y ' , z ' ) in the sheet, “u y ” can be expressed as ( ( )) ' u y = u y z ' e− jkx

(2.37)

2.2 Problem Formulation

31

( ) where u y z ' is a function of “z ' ” only. From Eq. (2.37), we obtain ( ( )) ∂ 2u y ' = −k 2 u y z ' e− jkx 2 ' ∂(x )

(2.38)

( ( )) d 2 u y z ' − jkx ' ∂ 2u y = e ∂(z ' )2 d(z ' )2

(2.39)

and

Equations (2.36), (2.38) and ((2.39) yield [

( ( )) d 2 u y z' d(z ' )2

] ( ( ' )) − jkx ' e − k uy z = j σ sωs B y 2

(2.40)

along with the boundary conditions given as ( )I u y Iz ' =− Wr = 0

(2.41)

( )I u y Iz ' = Wr = 0

(2.42)

2

2

where “Wr ” is the width of the rotor sheet. The boundary conditions in Eqs. (2.41) and (2.42) indicate that current cannot escape from the rotor sheet along its width. “B y ” in Eq. (2.40) can be expressed as B y = B y1 + B y2

(2.43)

In Eq. (2.43), B y1 is the normal component of flux density due to the stator current sheet (obtained as explained in Sect. 2.2.1.2) and B y2 is the normal component of flux density due to the rotor current taking into account the proximity of stator iron. The formulation and calculation of the quantity B y2 , based on the vector potential due to the rotor current, are presented in the following section.

2.2.1.4

Vector Potential Due to Rotor Current

− → The vector potential, A2 at any field point (x ' , z ' ) in the rotor sheet due to currents in the same sheet, taking into account the proximity of the stator back iron, is given by − →( ' ' ) −→ −→ A2 x , z = A21 + A22 where,

(2.44)

32

2 Mathematical Modelling of Electromagnetic Forces Due to Finite Width … + W2r +∞[( ( (

μ0 −→ A21 = 4π

− W2r

−∞

) ( )] jxr →i jzr k→ + d x dz r1 r1

(2.45)

and μ0 −→ A22 = 4π

(W2 (+∞[( W1 −∞

) ( → )] jzr k j xr→ı + d x dz r2 r2

(2.46)

where r1 =

[(

)2 ] 21 ( + z − z'

(2.47)

)2 ( ( )2 ] 21 + z − z' + 4 y'

(2.48)

x − x'

)2

and r2 =

[(

x − x'

)2

where “y ' ” is the height of the rotor sheet above the stator iron surface. −→ −→ With reference to the Eqs. (2.45) and (2.46), A21 and A22 are the vector potentials due to the rotor current and its image over the stator iron width, respectively. “ jxr ” and “ jzr ” are the longitudinal and lateral components of rotor linear current density and “Wr ” is the width of the rotor sheet. Vector potential due to rotor currents has similar expressions as those due to stator current, except that the rotor current density components are unknown unlike the components in the stator winding. “ jxr ” and “ jzr ” can be expressed as jxr = d Jxr

(2.49)

jzr = d Jzr

(2.50)

and

With reference to Eqs. (2.49) and (2.50), Jxr and Jzr are the rotor current density components (in A/m2 ) and “d” is the thickness of the rotor sheet. From the relation J→ = Curl u→, Jxr and Jzr can be expressed as Jxr = −

∂u y ∂z

(2.51)

and Jzr =

∂u y ∂x

(2.52)

2.2 Problem Formulation

33

From Eqs. (2.49), (2.50), (2.51) and (2.52), we obtain jxr = −d

∂u y ∂z

(2.53)

and jzr = d

∂u y ∂x

(2.54)

From Eq. (2.37), we can write ∂u y du y − jkx = e ∂z dz

(2.55)

∂u y = (− jk)u y e− jkx ∂x

(2.56)

and

With reference to Eqs. (2.55) and (2.56), u y appearing on the right-hand side is a function of “z” only. Based on Eqs. (2.53)–(2.56), we can write jxr = −d

du y − jkx e dz

(2.57)

and jzr = d(− jk)u y e− jkx

(2.58)

) − →( From Eqs. (2.44), (2.45), (2.46), (2.57) and (2.58), A2 x ' , z ' can be expressed as − → − → − → A2 = A2x i + A2z k

(2.59)

where ⎡

A2x

⎫ ⎤ ⎧ )⎨ (+∞ ( ⎬ − jkx du y e (μ0 )d ⎢ ⎥ = d x − ⎣ { } 1 ⎭dz ⎦ 4π dz ⎩ 2 2 2 ' ' −∞ (x − x ) + (z − z ) − W2r ⎫ ⎤ ⎧ +∞ ⎡W )⎨ ( (2 ( ⎬ − jkx du y e (μ0 )d ⎣ dz ⎦ − + 1 dx { } 4π dz ⎩ (x − x ' )2 + (z − z ' )2 + 4(y ' )2 2 ⎭ + W2r

(

W1

−∞

(2.60) and

34

2 Mathematical Modelling of Electromagnetic Forces Due to Finite Width …

⎡

A2z

⎧ +∞ ⎫ ⎤ ( ⎨ ⎬ − jkx ( ) e (μ0 )d ⎢ ⎥ = (− jk) u y d x dz ⎦ ⎣ 1 { } ⎩ ⎭ 4π ' 2 ' 2 2 −∞ (x − x ) + (z − z ) − W2r ⎧ +∞ ⎫ ⎤ ⎡W (2 ( ⎨ ⎬ − jkx ) ( e (μ0 )d ⎣ (− jk) u y d x dz ⎦ + { }1 ⎩ 4π (x − x ' )2 + (z − z ' )2 + 4(y ' )2 2 ⎭ + W2r

(

−∞

W1

(2.61) The integrals,

( +∞

e− jkx

−∞

1

{(x−x ' )2 +(z−z ' )2 } 2

d x and

( +∞ −∞

e− jkx 1

{(x−x ' )2 +(z−z ' )2 +4(y ' )2 } 2

d x,

appearing in the expressions for A2x and A2z in Eqs. (2.60) and (2.61) respectively, can be reduced to rapidly converging form using the method of contour integration as was done in the case of fields due to stator currents in Sect. 2.2.1.1. Thus A2x and A2z can be simplified to the expressions given by ⎡ A2x =

2(μ0 )d − jkx ' ⎢ e ⎣ 4π

+ W2r

(

− W2r

⎫ ⎤ ⎧ )⎨(∞ −k |(z−z ' )|t1 ⎬ ( du y e ⎥ − (2 ) 21 dt1 ⎭dz ⎦ dz ⎩ t −1 1

1

⎫ ⎤ ⎧ }1 { )⎪ (W2 ( (∞ −k (z−z ' )2 +4( y ' )2 2 t1 ⎪ ⎬ ⎨ du y e 2(μ0 )d − jkx ' ⎢ ⎥ dz ⎦ (2.62) e dt + ⎣ − 1 (2 ) 21 ⎪ ⎪ 4π dz ⎩ ⎭ t1 − 1 1 W ⎡

1

and A2z

⎫ ⎤ ⎧ ⎡ ( Wr ⎨( ∞ e−k |(z−z ' )|t1 ⎬ 2(μ0 )d − jkx ' ⎣ + 2 ⎦ e = (− jk)u y (2 ) 21 dt1 ⎭dz ⎩ 1 4π − W2r t1 − 1 ⎫ ⎤ ⎧ ⎡ ( W2 ( ⎨ ∞ e−k{(z−z ' )2 +4( y ' )2 ) 21 t1 ⎬ 2(μ0 )d − jkx ' ⎣ e + dt1 dz ⎦ (− jk)(− jk)u y (2 ) 21 ⎭ ⎩ 1 4π W1 t −1 1

(2.63)

2.3 Solution for Stream Function uy The main equation to be satisfied by u y is Eq. (2.64) which is reproduced below for convenience as ] [ ( ( )) ( ( ' )) − jkx ' d 2 u y z' 2 e − k uy z = j σ sωs B y (2.64) d(z ' )2

2.4 Steps for Algorithm for Numerical Solution for uy , By2

35

With reference to Eq. (2.64), the total normal component for flux density, “B y ” has two components namely “B y1 ”, the component due to stator current and “B y2 ”, the component due to rotor current. At any point on the rotor sheet, “B y1 ” can be − → calculated from the vector potential, “ A I ” due to known stator current as given in Eq. (2.25). Similarly, “B y2 ” can be obtained from “A2x ” and “A2z ” (Eqs. (2.62) and (2.63)) which are integrals involving the unknown “u y ” and its partial derivatives with appropriate kernels. Equation (2.64) has on its left-hand side, the unknown function, “u y ” and its second derivative while the right-hand side involves a known function, “B y1 ” and the integral expression involving “u y ” and its partial derivatives with suitable kernels. Thus, Eq. (2.64) is in fact an integro-differential equation in “u y ”. It is perhaps possible to solve such an equation analytically using some sort of integral transforms. However, the kernel involved in this equation is singular and difficult to handle analytically. Hence, it has been decided to adopt the numerical solution to the problem. The next section, therefore, gives a step-by-step numerical procedure for obtaining the solution for “u y ”.

2.4 Steps for Algorithm for Numerical Solution for uy , By2 Step 1: The rotor sheet is discretized along its width (in the z-direction) into a large number of elements each of width “dz”. Discretizing the rotor sheet along the length (in the x-direction) is not necessary. The coordinates of the grid point along the width are noted. Step 2: Calculation of flux density due to stator current: (B y1 ) Rearranging the expression for the vector potential due to the stator current sheet, − → A I in Eq. (2.25), we obtain, for a fixed value of “y ' ”, ( ) ( ) ] − → [ ' A I = A1x z ' →i + A1z z ' k→ e− jkx e jsωs t

(2.65)

( ) ( ) With reference to Eq. (2.65), A1x z ' and A1z z ' can be expressed as follows: ( ) μ0 A1x z ' = 4π

(

⎧ Wr ⎛∞ ⎞ ⎫ ⎪ )⎪ (2 ( ⎬ ⎨ −kbt1 − j Iz e ' ⎝ ⎠ 2 f (z) dt dz 1 (2 )1 ⎪ ⎪ k ⎭ ⎩ t −1 2 − W2r

and

1

1

(2.66)

36

2 Mathematical Modelling of Electromagnetic Forces Due to Finite Width …

⎡⎧ Wr ⎛∞ ⎞ ⎫ ⎪ ⎪ (2 ( ⎬ ⎨ −kbt 1 ( ') e μ0 ⎢ ⎝ ⎠ A1z z = 2 f (z) dt dz (Iz )⎣ 1 1 (2 ) ⎪ ⎪ 4π ⎭ ⎩ Wr t1 − 1 2 +

⎧W ⎨( 2 ⎩ W1

−

1

2

⎛∞ ⎞ ⎫⎤ ( ⎬ −kbt1 e ⎠ ⎦ 2 f (z)⎝ ( dt dz ) 21 1 ⎭ 2 t − 1 1 1

(2.67)

The flux density (By1 ) at any field point due to stator current can be expressed as [ ( )] ' B y1 = B y1 z ' e− jkx

(2.68)

where ( )) ( ( ) ( ' ) d A1x z ' + jk A1z z ' B y1 z = ' dz

(2.69)

− → The grid points ( ' ) obtained after discretizing the rotor sheet are used to calculate A I and then B y1 z using Eqs. (2.65) and (2.69), respectively. The required numerical differentiation [ ] has been done by the method of forward difference. Thus, a column matrix B y1 is formed. Step 3 Calculation of normal component of flux density due to rotor current: (B y2 ) The expressions for the components of the vector potential, A2x and A2z , at any point in the sheet, due to rotor current, can be rewritten, based on Eqs. (2.62) and (2.63), respectively, as ( ) ' A2x |(x ' ,y ' ,z ' ) = A2x z ' e− jkx

(2.70)

( ) ' A2z |(x ' ,y ' ,z ' ) = A2z z ' e− jkx

(2.71)

and

( ) ( ) With reference to Eqs. (2.70) and (2.71), the variables A2x z ' and A2z z ' can be expressed as ⎡ A2x (z ' ) =

2(μ0 )d ⎢ ⎣ 4π

+ W2r

(

− W2r

⎡

+

2(μ0 )d ⎢ ⎣ 4π

⎫ ⎤ ⎧ )⎨(∞ −k |(z−z ' )|t1 ⎬ ( du y (z) e ⎥ − (2 ) 21 dt1 ⎭dz ⎦ ⎩ dz t −1

+ W2r

(

− W2r

1

1

⎫ ⎤ ⎧ )⎨(∞ −k{(z−z ' )2 +4( y ' )2 } 21 t1 ⎬ ( du y e ⎥ − dt1 dz ⎦ (2 ) 21 ⎭ dz ⎩ t −1 1

1

(2.72)

2.4 Steps for Algorithm for Numerical Solution for uy , By2

37

and ⎡

⎫ ⎤ ⎧∞ ( −k |(z−z ' )|t1 ⎬ ⎨ e 2(μ0 )d ⎢ ⎥ (− jk)u y A2z (z ' ) = ⎣ (2 ) 21 dt1 ⎭dz ⎦ ⎩ 4π t1 − 1 1 − W2r ⎧ ⎫ ⎤ ⎡ Wr { }1 ⎪ (2 (∞ −k (z−z ' )2 +Δ( y ' )2 2 t1 ⎪ ⎨ ⎬ e 2(μ0 )d ⎢ ⎥ dz ⎦ + (− jk)u y dt ⎣ 1 1 ( ) ⎪ ⎪ 4π ⎩ ⎭ t12 − 1 2 −Wr 1 + W2r

(

(2.73)

2

( ) ( ) The expressions for A2x z ' and A2z z ' in Eqs. (2.72) and (2.73), respectively, are once again discretized, and the following matrix equations are obtained: [

du y [A2x ] = [H1 ] dz

] (2.74)

and [

] [ ] A2z = [H2 ] u y

(2.75)

where the elements of matrices [H1 ], [H2 ] are known in terms[ of ]μ0 , d and grid [ ] [ ] du point coordinates. The column matrices [A2x ], A2z , u y and dzy have for their elements the corresponding values at the grid points. Step 4: Using the method ] central difference for simulating the differential [ of du y d operators “ dz ”, the matrix dz can be expressed as [

du y dz

]

[ ] = [S1 ] u y

(2.76)

With reference to Eq. (2.74), [S1 ] is the coefficient matrix in terms of the grid coordinates along the z-direction. While forming the matrices, the boundary lines of the rotor sheet along its width are considered as u y = 0 lines for satisfying the connected with the differential Eq. (2.64). conditions in Eqs. (2.41) and ] [ (2.42), du y Step 5: Substituting for dz from Eq. (2.77) in Eq. (2.75), [A2x ] can be expressed as [ ] [A2x ] = [H1 ][S1 ] u y [ ] = [M1 ] u y

(2.77)

In Eq. (2.77), the matrix [M1 ] is expressed as [M1 ] = [H1 ][S1 ]

(2.78)

38

2 Mathematical Modelling of Electromagnetic Forces Due to Finite Width …

Step 6: B y2 can be expressed as I B y2 I

(

(x ' ,y ' ,z ' )

) ( ) I I ∂ A2x II ∂ A2z II = − ∂z ' I(x ' ,y ' ,z ' ) ∂ x ' I(x ' ,y ' ,z ' ) ) ( ( ( ' )) ( ') − jkx ' d A2x z + jk A2z z =e dz '

(2.79)

I ( ) ( ) In Eq. (2.79), A2x z ' and A2z z ' are functions of z ' only. Again, B y2 I(x ' ,y ' ,z ' ) can be expressed as I { ( )} ' B y2 I(x ' ,y ' ,z ' ) = B y2 z ' e− jkx

(2.80)

( ) where B y2 z ' is a function of “z” only. From Eqs. (2.79) and (2.80), we can write ( )) ( ( ' ) d A2x z ' ( ) B y2 z = + jk A2z z ' (2.81) ' dz ( ) After discretizing the expression for B y2 z ' in Eq. (2.81), we obtain a matrix equation as [

B y2

]

[

d A2x = dz '

] + jk[A2x ]

(2.82)

Step7: Using the method[ of forward difference for simulating the differential ] A2x d ”, the matrix ddz can be expressed as operator, “ dz ' [

d A2x dz '

] = [S2 ][A2x ]

(2.83)

where [S2 ] is a known coefficient matrix. Substituting for [A2x ] from Eq. (2.77) in Eq. (2.83), we obtain the following matrix equation as [

d A2x dz '

]

[ ] = [S2 ][M1 ] u y [ ] = [M2 ] u y

(2.84)

where [M2 ] = [S2 ][M1 ]

(2.85)

2.4 Steps for Algorithm for Numerical Solution for uy , By2

39

[ ] Step 8: Based on Eqs. (2.74), (2.81) and (2.83), we obtain the expression for B y2 as [

] [ ] [ ] B y2 = [M2 ] u y + jk[H2 ] u y [ ] = [H ] u y

(2.86)

With reference to the matrix Eq. (2.85), the coefficient matrix [H ] is expressed as [H ] = [M2 ] + jk[H2 ]

(2.87)

Step 9: From Eq. (2.64), we have ( ( )) d 2 u y z' d(z ' )2

( ( )) ( ) − k 2 u y z ' = j σ sωs B y z '

(2.88)

( ) ' B y = B y z ' e− jkx

(2.89)

where

( ) [ ] The discretized form of B y z ' can be expressed as a column vector B y where [

] [ ] [ ] B y = B y1 + B y2

(2.90)

2

The operator “ dzd ' 2 ” is simulated using the method of central differences. Thus, discretizing the left-hand side of the Eq. (2.65), the obtained matrix form is given by [

] [ ] d 2u y 2 − k u y = [LAPLACIAN] u y dz ' 2

(2.91)

where the coefficient matrix [LAPLACIAN] is expressed in terms of the discretized forms of the second-order differential operators. From Eqs. (2.81), (2.90) and (2.91), we get the matrix equation given by [[ ] [ ]] [ ] [LAPLACIAN] u y = j σ sωs B y1 + B y2

(2.92)

[ ] Step 10: Substituting for B y2 from Eq. (2.86) in Eq. (2.92), it yields [ [ ] ][ ] [LAPLACIAN] − jc' [H ] u y = jc' B y1 [ ] [ ] or [ p] u y = jc' B y1

(2.93)

With reference to Eq. (2.93), [ p] and c' can be expressed as [ ] [ p] = [LAPLACIAN] − jc' [H ]

(2.94)

40

2 Mathematical Modelling of Electromagnetic Forces Due to Finite Width …

and c' = σ sωs

(2.95)

[ ] [ ] u y = [ p]−1 jc' B y1

(2.96)

From Eq. (2.93), we get

[ ] [ ] From u y , B y2 can be obtained using Eq. (2.86). Step 11: Calculation of Bx and Bz components: The longitudinal and lateral components of flux density in the rotor sheet due to both stator and rotor currents are Bx and Bz , respectively, and they can be expressed as Bx = Bx1 + Bx2

(2.97)

Bz = Bz1 + Bz2

(2.98)

With reference to Eqs. (2.97) and (2.98), Bx1 , Bz1 are the components of flux density due to the stator current sheet, and Bx2 , Bz2 are the components of flux density due to rotor current. As the rotor sheet is of infinite length, Bx2 and Bz2 at ' any field point can be expressed (for a fixed value of “y ”) as { ( )} ' Bx2 |(x ' ,y ' ,z ' ) = Bx2 z ' e− jkx

(2.99)

{ ( )} ' Bz2 |(x ' ,y ' ,z ' ) = Bz2 z ' e− jkx

(2.100)

and

( ) ( ) With reference to Eqs. (2.99) and (2.100), Bx2 z ' and Bz2 z ' are functions of “z ' ” only. It is known that vector potential at any field point due to the stator current sheet has a sinusoidal distribution with respect to the length of the stator winding, as explained → the flux density components also will have sinusoidal in Sect. 2.2.1.1. As B→ = curl A, distribution with respect to length. Therefore, Bx1 and Bz1 can be expressed as { ( )} ' Bx1 |(x ' ,y ' ,z ' ) = Bx1 z ' e− jkx

(2.101)

{ ( )} ' Bz1 |(x ' ,y ' ,z ' ) = Bz1 z ' e− jkx

(2.102)

( ) ( ) With reference to Eqs. (2.101) and (2.102), Bx1 z ' and Bz1 z ' are functions of → we can write “z ' ” only. From the relation, B→ = curl A,

2.4 Steps for Algorithm for Numerical Solution for uy , By2

Bx1 =

∂ A1z ∂ y'

Bz1 = − Bx2 =

41

(2.103)

∂ A1x ∂ y'

(2.104)

∂ A2z ∂ y'

Bz2 = −

(2.105)

∂ A2x ∂ y'

(2.106)

Step 12: If the rotor sheet is placed parallel to the stator surface at a height of y ' , from the latter, by the method of central difference, Bx1 and Bz1 (at that height) can be calculated as { } { } A1z (i )| y ' =y ' + dely' − A1z (i )| y ' =y ' − dely' 2 2 Bx1 (i )| y ' =y ' = (2.107) dely ' { } { } A1x (i )| y ' =y ' + dely' − A1x (i )| y ' =y ' − dely' 2 2 Bz1 (i )| y ' =y ' = − (2.108) dely ' With reference to Eqs. (2.107) and (2.108), the symbol “(i)” indicates the “i-th” discretized point in the z-direction and “dely ' ” indicates the mesh (grid) size in the ydirection used for numerical computation of the derivative of a mathematical function at a given point. For obtaining the mean values of the flux density components, Bx and Bz , in a thin rotor sheet, “dely ' ” can be taken as “d” where “d” is the thickness of the rotor sheet. Step 13: For calculation of Bx2 (i ) and Bz2 (i ), we proceed in the following lines: { Bx2 (i )| y ' =y ' =

Bz2 (i )| y ' =y ' = −

} { } A2z (i )| y ' =y ' + dely' − A2z (i )| y ' =y ' − dely' 2

2

dely ' { } { } A2x (i )| y ' =y ' + dely' − A2x (i)| y ' =y ' − dely' 2

2

dely '

(2.109)

(2.110)

Again for a thin rotor sheet, “dely ' ” can be taken as the thickness of the rotor sheet, “d”. It may be noted that Bx2 and Bz2 have a discontinuity across the rotor sheet. The values calculated above give the average of the values at the top and bottom surfaces of the rotor.

42

2 Mathematical Modelling of Electromagnetic Forces Due to Finite Width …

2.5 Calculation of Forces Once the fields and currents in the rotor sheet are known, all the force components − → − → are calculated using the established relation, F = J→ × B .Total time average of propulsion, lateral and levitation forces, Fxt , Fzt and Fyt, respectively, for a unit length of the infinitely long rotor is calculated by using the algorithms given by ) ( { } d(delz) {n Fxt = − Real Jzr (i)B y∗ (i ) i=1 2 ) ( { } d(delz) {n Real Jxr (i )B y∗ (i ) Fzt = i=1 2 ) n ( { } d(delz) { Fyt = Real Jzr (i )Bx∗ (i) − Jxr (i)Bz∗ (i) 2 i=1

(2.111) (2.112)

(2.113)

With reference to the Eqs. (2.111)–(2.113), the superscript “∗,” associated with a particular variable represents its complex conjugate.

2.6 Results and Discussion The above numerical method of solution is applied to predict the various components of flux density and forces for the experimental model, the details of which are given in Appendix 3. However, for ready reference, the main details of the experimental model are reproduced as follows: Ws = 10.6 cm; Wr = 27 cm; ωs = 314 rad/s; L = 76 cm; σ (at 75 ºC) = 2.42 ×× 107 Ω−1 m−1 ; τ = 28 cm; d = 3.08 mm; length of stator winding overhang (C) = 8.2 cm; Iz = 360 A/cm corresponding to a stator current of 12 A/phase (r.m.s). Flux density distribution For the stator current of 12 amp/phase, the total flux density distribution, B y on the rotor surface (y ' = 2.1 cm) for unity slip is shown in Fig. 2.2. From this figure, it can be seen that B y -distribution is similar to B y1 -distribution but has undergone some attenuation due to the presence of rotor currents. Distribution of Bx1 with “z” for a stator phase current of 12 amps (r.m.s) is shown in Fig. 2.3. In the same figure, the variation of flux density, Bx (due to both stator and rotor currents) at unity slip is also shown. Once again Bx -distribution is similar to Bx1 -distribution but it has undergone some attenuation. The distribution of Bz1 at a height of 0.725 cm, against “z” is shown in Fig. 2.4. The corresponding measured values are also plotted in the same graph. It can be seen that there is a good correlation between the two values. The total flux density, Bz at a height of 2.1 cm at s = 1 is also shown in the same figure.Bz -distribution is

2.6 Results and Discussion

43

Fig. 2.2 Distribution of By against “z” (with rotor)

Fig. 2.3 Distribution of Bx1 and Bx against z. Notes (i) Bx1 at y = 2.1 cms (ii) Bx at y = 2.1 cms (iii) slip = 1.0

once again similar to Bz1 -distribution but very much attenuated due to (i) increased distance from stator surface and (ii) rotor currents. Rotor current distribution u y -values are calculated at different grid points on the rotor sheet at any chosen slip (say, s = 1) for a rotor height of 2.1 cm above the stator surface. The values of u y (non-vector quantity expressed as a complex number) are multiplied by the term, “e j(sωs t−kx) ” and the instantaneous value, u c of u y is obtained as ] [ u c = Real u y (z)e j(sωs t−kx)

(2.114)

Choosing the time instant as zero, u c -values are calculated at different grid points and constant u c -contours are plotted and shown in Fig. 2.5 over a distance of two pole pitches along the rotor length.

44

2 Mathematical Modelling of Electromagnetic Forces Due to Finite Width …

Fig. 2.4 Distribution of Bz1 and Bz against “z”

= W= Rotor width

(ii)

=

=

L= Rotor length

Fig. 2.5 Typical current distributions in the rotor sheet of infinite length and finite width at a certain instant

2.6 Results and Discussion

45

Forces The propulsion and levitation forces per unit length are calculated using Eqs. (2.110) and (2.112), respectively, for the experimental model. The above values are multiplied by the actual rotor length (0.76 cm) to get the total force on the rotor of the experimental model. It should be remembered that by doing so, actually we are ignoring the discontinuity of the rotor in the longitudinal direction. The forces are calculated for “slip values” varying from 0 to 1 and assuming no rotor offset with respect to the stator width. The results are plotted in Fig. 2.6. The experimental values which could be measured only at s = 1 are also shown in the same figure. The experimental value of the levitation force is in good agreement with the calculated value while the experimental value of the propulsion force is less than its theoretical value by about 17%. The lateral force per unit length is calculated from Eq. (2.111) for different rotor offsets in the range of 0–7 cm at any chosen value of slip (s = 1). The above values are multiplied by the actual rotor length (0.76 cm) to get the total lateral force for the rotor of the experimental model and are plotted in Fig. 2.7.

Fig. 2.6 Variation of propulsion and levitation force with slip based on a rotor model of infinite ––Experimental value of levitation force at unity slip length and finite width. ––Experimental value of propulsion force at unity slip

46

2 Mathematical Modelling of Electromagnetic Forces Due to Finite Width …

Fig. 2.7 Variation of lateral force against rotor offset (at unity slip)

2.7 Conclusions In this chapter, the analysis of a SLIM with a stator and rotor of finite width has been presented. The finite width of the rotor leads to the development of peripheral currents in addition to the axial currents which are simultaneously reduced in length. The mathematical formulation and calculation of the fields due to infinitely long but finitely wide stators have been done using a special function (Hankel function) for faster numerical convergence. An integro-differential equation involving the stream function (u y ) has been formulated due to the induced currents in the rotor. As the rotor, like the stator, is assumed to be infinitely long, the same Hankel function approach has been used for quick solution of the variable, u y . Based on u y -values, flux density components due to rotor current have been calculated. Finally, the analysis leads to the evaluation of flux density distributions, current contours on the rotor sheet and the propulsion, levitation and lateral forces on the rotor. The observed flux density distributions and the measured electromagnetic forces (at unity slip) in the experimental model have shown reasonably good agreement with the corresponding calculated values. The lateral force, which is zero for zero offsets, increases with rotor offset indicating that the rotor is inherently unstable in the lateral direction.

Chapter 3

Mathematical Modeling of Electromagnetic Forces Due to Finite Length and Finite Width Effects of a Single-Sided Linear Induction Motor

3.1 Introduction The analysis of a single-sided linear induction motor having a stator winding of infinite length and finite width and a non-magnetic rotor sheet of infinite length and finite width has been presented in the preceding chapter. But in practice, a rotor of infinite length is not realistic. Therefore, in the present chapter, the analysis of a SLIM with a rotor sheet of finite length and finite width is going to be proposed. As before, it is assumed that the rotor is placed over the stator with its longitudinal edges running parallel to those of the stator. This arrangement is shown in Fig. 3.1. The discontinuity in the rotor in the longitudinal direction further constricts the rotor current flow affecting propulsion, levitation and lateral forces to some extent. With reference to the model in Fig. 3.1, the following propositions regarding the calculations are made: (i)

The field at any point in free space due to the stator winding having a sinusoidal current distribution along the length. (ii) The induced currents in the rotor sheet considering the field due to the stator as well as the rotor currents. (iii) The propulsion, levitation and lateral forces exerted on the rotor sheet. As at the end of this chapter, the analytical results have been compared with the experimental results, and a brief description of the fabrication of single-sided linear induction motor has been presented in Appendix C.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. B. Chattopadhyay et al., Mathematical Modeling of Physical Systems, Advances in Intelligent Systems and Computing 1436, https://doi.org/10.1007/978-981-19-7558-5_3

47

48

3 Mathematical Modeling of Electromagnetic Forces Due to Finite Length …

Fig. 3.1 Coordinate system used for the analysis of the SLIM with the rotor of finite length and finite width

3.2 Problem Formulation 3.2.1 Formulation for the Field Due to Stator Current Since there is no change in the stator configuration of the model presented in Chap. 2 (Part-A) and the model in the present chapter, the analysis already presented earlier for calculating the fields due to stator currents remains valid here also.

3.2.2 Formulation for the Current in the Rotor Sheet The SLIM under consideration has a rotor sheet whose placement over the stator surface and geometrical shape are the same as described in the preceding chapter except that its length is finite. Nevertheless, the concept of stream function, u y , in the rotor sheet discussed in Chap. 2, still holds good since the rotor currents are planar i.e. only jxr and jzr components exist. Therefore, the partial differential equation (PDE) satisfied by ‘u y ’, as derived in Eq. (2.36) of Chap. 2, is still valid and is reproduced as follows: ∂ 2u y ∂ 2u y + = jσ sωs B y ∂(x ' )2 ∂(z ' )2

(3.1)

With reference to Eq. (3.1), the symbols σ, s, ωs and B y have their usual meanings as stated in the previous chapter. Since current cannot escape from the rotor sheet, the boundary conditions for the PDE in (3.1) are given by

3.2 Problem Formulation

49

( )I u y Iz ' =− Wr = 0

(3.2)

( )I u y Iz ' = Wr = 0

(3.3)

( )I u y Ix ' =0 = 0

(3.4)

( )I u y Ix ' =L = 0

(3.5)

2

2

where Wr and L are the width and length of the rotor sheet, respectively. Similar to the treatment, dealt with in Chap. 2, the total normal component of the flux density,B y , at any field point in the rotor sheet, can be split into two components, one being the normal component of flux density due to the stator current only, designated as ‘B y1 ’, and the other being the normal component of the flux density due to the current induced in the rotor, designated as ‘B y2 ’. Therefore, we can express B y as, B y = B y1 + B y2

(3.6)

From Eqs. (3.1) and (3.6), we can write ( ) ∂ 2u y ∂ 2u y + = jσ sωs B y1 + B y2 2 2 ' ' ∂(x ) ∂(z )

(3.7)

− → With reference to Eq. (3.7), B y1 can be calculated from the vector potential, A I , due to stator current, as described in the preceding chapter. Although ‘B y1 ’, which can ' be considered as the main forcing function, varies as ‘e− jkx ’, u y cannot be expressed − jkx ' ’ because of the discontinuity of the rotor in the x-direction. varying simply as ‘e Thus u y and B y2 have unknown distributions, both along the x- and z-directions.

3.2.2.1

Formulation for ‘B y2 ’ in Terms of ‘u y ’

The variable, ‘B y2 ’, can be calculated from rotor current (which is yet to be determined) either by applying Biot-Savart’s law or from the vector potential approach. The disadvantage of the vector potential approach is that numerical differentiations of the vector potential components are needed to calculate the flux densities. On the other hand, it is also possible to apply Biot-Savart’s law to calculate directly the flux density components in the rotor sheet due to rotor currents. We will use this approach here. Applying Biot-Savart’s law and taking the presence of stator back iron, B y2 can be expressed as B y2 = B y21 + B y22

(3.8)

50

3 Mathematical Modeling of Electromagnetic Forces Due to Finite Length …

With reference to Eq. (3.8), ‘B y21 ’ and ‘B y22 ’ are the normal components of the flux density at any point (x ' , y ' , z ' ) on the rotor sheet due to the rotor current and its image over the stator iron boundary. ‘B y21 ’and ‘B y22 ’ can be expressed as

B y21 =

(μ0 )d 4π

Wr 2

(

− W2r

(L 0

⎡(

)⎤ ) ∂u y ( ' ) ∂u y ( ' ⎢ ∂ x x − x + ∂z z − z ⎥ ⎢ ⎥ ⎣ {( }− 23 ⎦d x dz ) ( ) 2 2 × x ' − x + z' − z

(3.9)

and

B y22 =

(μ0 )d 4π

(W2 (L W1 0

⎡(

) ⎤ ) ∂u y ( ' ) ∂u y ( ' x z − x + − z ⎢ ∂x ⎥ ∂z ⎢ ⎥d x dz ⎣ {( 3 )2 ( ' )2 ( ' )2 }− 2 ⎦ ' × x − x + z − z + 2y

(3.10)

In Eq. (3.10), the parameter, “d”, indicates the thickness of the rotor sheet. The partial differential Eq. (3.7) along with Eqs. (3.8), (3.9) and (3.10) forms an integrodifferential equation to be solved for ‘u y ’ in terms of ‘B y1 ’, the normal component of the flux density due to stator current.

3.2.2.2

Formulation for Total Longitudinal and Lateral Components of Flux Density in the Rotor Sheet

While the knowledge of B y is sufficient to calculate the propulsion force, it is necessary to know Bx and Bz components to calculate the levitation and lateral forces. The total longitudinal and lateral components of flux-densities, Bx and Bz , respectively, can be expressed as Bx = Bx1 + Bx2

(3.11)

Bz = Bz1 + Bz2

(3.12)

With reference to Eqs. (3.11) and (3.12), Bx1 ,Bz1 are the longitudinal and lateral components of the flux density, at any field point, due to stator current while Bx2 , Bz2 are similar components of flux density, due to rotor current. Bx1 and Bz1 can be computed numerically using the relation, B→ = Curl A→ as given by Bx1 = and

∂ AI z ∂ y'

(3.13)

3.3 Steps of Algorithm for Numerical Solution for u y , B y2 , Bx2 and Bz2

Bz1 = −

51

∂ AI x ∂ y'

(3.14)

With reference to Eqs. (3.13) and (3.14), A I x and A I z are the components of vector potential due to stator current only. These vector potentials have already been expressed in convenient forms in Eq. (2.25) in Chap. 2. The flux-density components, Bx2 , Bz2 at any field point (x ' , y ' , z ' ), on the rotor sheet, can be obtained from the rotor current using Biot-Savart’s law as ⎡ Bx2

(μ0 )d = 4π

(L

⎤

) ( ( ) ∂u y −2y ' ∂x

⎥ (W2 ⎢ ⎢ ⎥ ⎢0 ⎥dz ⎢ ⎥ ⎣ {( ⎦ 3 } − ) ) ( ) ( W1 2 2 2 2 × x ' − x + z ' − z + 2y ' dx

(3.15)

and ⎡ (μ0 )d Bz2 = 4π

(L

⎤

) ( ( ') ∂u y 2y − ∂z

⎥ (W2 ⎢ ⎢ ⎥ ⎢0 ⎥dz ⎢ ⎥ ⎣ {( ⎦ 3 } − ) ) ( ) ( W1 2 2 2 2 × x ' − x + z ' − z + 2y ' dx

(3.16)

Equations (3.15) and (3.16) indicate that the flux-density components, Bx2 and Bz2 are produced due to the image of rotor current only. As long as the source point and the field point are on the same rotor sheet (i.e. they lie on the [ same ]plane), the J→×→ r) ( μ 0 ∫ dV, retains term, “ →j × r→”, in the expression for Biot-Savart’s law, B→ = 3 4π

r

only the normal component and no other planar component. It may be noted that the integral in the above-said Biot-Savart’s law indicates a volume integral only. Thus, the rotor current itself does not contribute to any longitudinal and lateral component of flux densities in the rotor sheet. It produces only the normal component of the flux density. It should be noted that the flux-density components, Bx andBz , have discontinuity across the thin rotor sheet. The values given by Eqs. (3.15) and (3.16) correspond to the average of the top and bottom values of the rotor sheet. These are the values to be used for force calculation.

3.3 Steps of Algorithm for Numerical Solution for u y , B y2 , Bx2 and Bz2 Step 1: The rotor sheet is discretized into a large number of meshes and the coordinates of the grid points are stored. Discretization is done both along the length(x-direction) and width (z-direction) of the sheet.

52

3 Mathematical Modeling of Electromagnetic Forces Due to Finite Length …

− → Step 2: The grid points obtained are used to calculate A I at these points using Eq. (2.25) of Chap. 2. B y1 -values at different field points are calculated from the equation, B y1 =

∂ AI x ∂ AI z − ∂ z' ∂x'

(3.17)

In Eq. (3.17), the method of forward difference was used for simulating the differential operators, ‘ ∂∂x ' ’ and ‘ ∂∂z ' ’. Step 3: Again the expression for B y2 in Eqs. (3.8), (3.9) and (3.10) is discretized. ∂u ∂u Arranging ‘ ∂ xy ’ and ‘ ∂ zy ’ at different grid points as column matrices, a matrix equation is obtained in the form, [

[ [ ] ] ] ∂u y ∂u y + [q2 ] B y2 = [q1 ] ∂x ∂z

(3.18)

In Eq. (3.18), the matrices, [q1 ], [q2 ] are the coefficient matrices in terms of μ0 ,d and the grid point coordinates obtained from Eqs. (3.9) and (3.10). Step 4: Using the method of central for simulating the differential [ ] difference [ ] ∂u y ∂u y ∂ ∂ operators, ∂ x and ∂ z , the matrices ∂ x , ∂ z can be expressed as [ [

∂u y ∂x ∂u y ∂z

] ]

[ ] = [ p1 ] u y

(3.19)

[ ] = [ p2 ] u y

(3.20)

where [ p1 ] and [ p2 ] are the coefficient matrices expressed in terms of the grid coordinates. While forming these coefficient matrices, the boundary lines of the rotor sheet along the z-direction and x-direction, have been considered as u y = 0 lines for satisfying the boundary conditions given in Eqs. (3.2)–(3.5). Step 5: The term involving the second-order partial derivatives appearing in the left-hand side of Eq. (3.1) is also discretized following similar lines in Chap. 2. However, the discretization is done here both in z- and x-directions. Thus, a matrix equation is formed as [[ ] [ ]] [ ] [LAPLACIAN] u y = jc' B y1 + B y2

(3.21)

With reference to Eq. (3.21), the matrix, [LAPLACIAN] is expressed in terms of the discretized forms of the second-order differential operators and “c' ” is a known constant. [ ] Step 6: From Eqs. (3.18) to (3.20), we obtain B y2 as, [

] [ ] B y2 = [[q1 ][ p1 ] + [q2 ][ p2 ]] u y

(3.22)

3.4 Calculation of Forces

53

[ ] where, the coefficient matrix pre-multiplying u y is a known matrix in terms of μ0 , d and the grid point coordinates and grid dimensions. Step 7: For ease of computation, we substitute [M] = [[q1 ][ p1 ] + [q2 ][ p2 ]]

(3.23)

Then, the Eq. (3.22) takes the shape [

] [ ] B y2 = [M] u y

(3.24)

Step8: From Eqs. (3.21) and (3.24), it yields [ ] [ ] [ p] u y = jc' B y1

(3.25)

where, the coefficient matrix, [p] can be expressed as, [ ] [ p] = [LAPLACIAN] − jc' [M]

(3.26)

[ ] this matrix [ p], u y can be calculated based on the Eq. (3.25). From [ By ] [inverting ] u y , B y2 can be obtained. [ ] Step 9: The expressions for [Bx2 ] and Bz2 in Eqs. (3.15) and (3.16) respectively are discretized. Following a similar procedure as given in step 3, the following equations are obtained: [ ] [Bx2 ] = [M1 ] u y [

] [ ] Bz2 = [M2 ] u y

(3.27) (3.28)

[ ] where, [ ] [M1 ] and [M2 ] are known matrices. As u y is known from step 8, [Bx2 ] and Bz2 can be computed. Step 10: The quantities Bx1 ,Bz1 at the grid points are computed using Eqs. (3.13) and (3.14) respectively and hence the total longitudinal and lateral components of flux densities, Bx and Bz are calculated.

3.4 Calculation of Forces Once the fields and currents in the sheet are known, all the force components are calculated using the relation, F→ = J→ × B→ as explained in Chap. 2.

54

3 Mathematical Modeling of Electromagnetic Forces Due to Finite Length …

3.5 Results and Discussion The above numerical method of calculation for computing the fields, currents and forces has been applied to the experimental model of the SLIM, the details of which are reproduced below for convenience. Ws = 10.6 cm; Wr = 27 cm; ωs = 314 rad/s; L = 76 cm; σ (at 75 °C) = 2.42 × 107 Ω−1 m−1 ; τ = 28 cm; d = 3.08 mm; length of stator winding overhang (C) = 8.2 cm; I z = 360 A/cm corresponding to a stator current of 12 A/phase (r.m.s). Flux density distributions The flux density distributions, “Bx ”, “B y ” and “Bz against “z” are calculated using the algorithm given in Sect. 3.3. ( The)distribution of B y on the rotor surface at s = 1 at the middle of the length x = L2 is plotted in Fig. 3.2. In the same graph, B y distribution without rotor is also shown. It can be seen that B y -distribution undergoes considerable attenuation due to rotor current. Variation of “B y ” against “z” (at unity slip) at the entry and exit ends of the rotor (x = 3.6 cm and x = 72.4 cm) is plotted in Figs. 3.3a, b, respectively. Also B y -distribution on the rotor surface along the rotor length at s = 1 at the middle of the rotor width (z = 0) is plotted in Fig. 3.4. The rise of B y at the rotor ends (x ∼ = 0 and x ∼ = L) is due to a decrease in rotor currents at the ends. The “Bx ”—and “Bz ”-distributions at x = L2 and at s = 1 are shown in Fig. 3.5. Rotor current distribution u y -values are calculated at different grid points on the rotor sheet at any chosen slip (say, s = 1), when the rotor is at a height of 2.1 cm above the stator surface. The values (non-vector quantity expressed as a complex number) of u y are multiplied by the term, “e j(sωs t) ” and the instantaneous value, u c of u y is obtained as, ] [ u c = Real u y (x, z)e j (sωs t−kx) .

( ) Fig. 3.2 Flux density B y distribution along the rotor width at x = L2 =38.0 cm and slip = 1.0 [I-without rotor, II-with rotor]

3.5 Results and Discussion

55

Fig. 3.3 Flux density (B y ) distributions (with rotor) along the rotor width at x = 3.62 cm (near the entry end) and x = 72.38 cm (near the exit end)

Fig. 3.4 Variation of flux density (B y ) distributions (with rotor) with rotor length at z = 0

Choosing the time instant as zero, u c -values are calculated at different grid points and constant u c -contours are plotted in Fig. 3.6. Comparing with u c -contours in Fig. 2.6 of Chap. 2, it may be noted that due to the finite length of the rotor, the contours are confined to the rotor sheet along the length (in addition to the width). Electromagnetic Forces The propulsion and levitation forces, with a symmetrically placed rotor (zero offset), are computed for different values of slip in the range of 0–1 and are plotted in Fig. 3.7. Comparing these plots with those of Fig. 2.7 of Chap. 2, it is observed that these electromagnetic forces, in general, are reduced due to finite-length effects. The reduction in levitation force is very small (1.4% at s = 1). However, there is a considerable reduction in propulsion force (40% at s = 1). Similarly, the lateral forces are computed with different rotor offsets in the range of 0–5 cm at any chosen value of slip (s = 1) and are plotted in Fig. 3.8. Once again comparing this result

56

3 Mathematical Modeling of Electromagnetic Forces Due to Finite Length …

Fig. 3.5 Flux density (“Bx ” and “Bz ”) distributions (with rotor)

Fig. 3.6 Typical current distributions in the rotor sheet of finite length and finite width at a certain instant

3.6 Conclusions

57

Fig. 3.7 Variation of propulsion and levitation forces with slip for a rotor of finite length and finite width

(i)

Propulsion Force

(ii)

Levitation Force

Fig. 3.8 Variation of lateral force against rotor offset

shown in Fig. 3.8 with that of Chap. 2 (Fig. 2.8), we find that lateral force is reduced (by 53% for rotor offset = 1 cm) due to finite length effect.

3.6 Conclusions In this chapter, a realistic model of a SLIM, with a rotor of finite width and finite length, has been analysed. The analysis leads to the evaluation of the flux density distribution, current contours in the rotor and the electromagnetic forces on the rotor, namely propulsion, levitation and lateral forces.

58

3 Mathematical Modeling of Electromagnetic Forces Due to Finite Length …

Table 3.1 Table showing experimental and theoretical values of levitation and propulsion forces Experimental (N)

Theoretical Using model in Chap. 1 (N)

Using model in Chap. 2 (N)

Using model in this chapter (N)

Levitation force

26.7

40

27.1

25.7

Propulsion force

31.1

65

37.4

22.9

The levitation and propulsion forces calculated using models of Chaps. 1 and 2 and in this chapter, along with the experimental results are shown in the following table (Table 3.1), at unity slip, for a stator current of 12 A (r.m.s) per phase and with the rotor at a height of 2.1 cm from the stator surface (with σ (at 75 °C) = 2.42 × 107 Ω−1 m−1 ). From the results in Table 3.1, it appears that the experimental values lie in between those predicted by models in Chaps. 2 and in this chapter while the model in Chap. 1 gives unduly optimistic values. There is a much better agreement between the experimental value and those predicted by models in Chaps. 2 and in this chapter, regarding lift force as compared to propulsion force. The following may be the possible reasons for the discrepancy between the experimental and predicted values: (i)

Coarse mesh sizes are used particularly in this chapter due to (computer) memory space and computational time limitations. Since discretization is to be done both along length and width directions (along x- and z-axis, respectively), the total number of grid points is to be restricted. In connection with the discretization, the number of points used in the x- and z-directions are 20 and 18, respectively. Therefore, the total number of grid points used is 360. Also, since the numerical differentiations are involved, they too can give rise to some errors. (ii) Imperfect modelling for the stator iron: The image method is used to take care of stator iron. The stator iron is considered to have infinite permeability (a reasonable assumption) but the images of the stator and rotor currents are assumed to be confined strictly to the stator width. Beyond this stator width, the effect of the iron body is abruptly terminated. In other words, the effect of the iron body for overhang currents of the stator and for rotor current beyond the stator width is completely ignored. Some kind of conformal mapping technique may be applied to model the iron boundary more accurately but it appears to be rather too complicated for easy implementation. (iii) The slotting effect of the stator has been completely ignored, i.e. no correction has been made to the air gap length between the rotor and stator surface.

Chapter 4

Fluid Flow Representation

Fluid flow has been a matter of human curiosity since human civilizations existed. Ancient civilizations started beside water bodies as they were readily dependent on the water bodies to meet their daily needs. For several civilizations, they were also a source of food. Water, one of the essential ingredients of life, normally exists as a liquid. Similarly, the oxygen we breathe from the atmosphere exists in gas phase. Like water and oxygen, there are several other liquids and gases which are of everyday use in our lives and, therefore, are matters of human curiosity and interest. Collectively, liquids and gases are called fluids. However, from a scientific perspective, a more formal definition of a fluid is preferred. All materials known could be classified into three phases: solids, liquids and gases. When a shear force stress is applied on a solid, the solid deforms. This deformation is in the form of shear strain. The action (shear force) causes readjustments internal to the solid body and the shear strain (deformation) is the response of the solid to the applied shear force (cause). Typically for solids, this deformation is fixed for a given shear force and the solid will not deform further. At this point in time, the molecules of the solid have re-structured to counter the shear force. For fluids and gases, howsoever small this shear force may be, they continue to strain under the shear force. Thus, any substance which cannot resist a shear force (stress), howsoever small it may be, is known as a fluid. Needless to say, that both liquids and gases fall in this category and are commonly called fluids. For further clarification on the definition of a fluid, the reader is encouraged to any standard text book on fluid mechanics [1–4].

4.1 Fluid Properties Although we commonly call different fluids by different names, they could also be uniquely identified by a set of values which we call properties. No two fluids will have identical set of properties. Therefore, it is imperative to know what properties are © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. B. Chattopadhyay et al., Mathematical Modeling of Physical Systems, Advances in Intelligent Systems and Computing 1436, https://doi.org/10.1007/978-981-19-7558-5_4

59

60

4 Fluid Flow Representation

used to characterize a fluid. These properties are grouped together as they belong to a common type. There are broadly four different types of fluid properties as outlined below.

4.1.1 Kinematic Properties Kinematic properties are those properties which are due to the flow of the fluid itself. Truly speaking, these properties are not intrinsic properties of the fluid. As such, the same fluid may have different kinematic properties when subjected to different flow conditions. Examples of kinematic properties of a fluid include linear and angular velocity, acceleration, vorticity of the fluid and strain rate. The definition of each of these properties is provided in Sect. 4.2 after we introduce the formal way of describing fluid flow.

4.1.2 Thermodynamics Properties The origin of these properties could be traced back to or derived from classical/statistical thermodynamics. Such properties would include pressure, temperature, entropy, density/specific gravity, internal energy (e), enthalpy (h), Gibbs free energy, specific heats and bulk’s modulus. It may be noted that several of these fundamental properties are fundamental variables in classical thermodynamics (such as pressure, temperature, entropy) while others act as potentials (for e.g., internal energy, enthalpy, Gibbs free energy). Properties such as specific heats are described in terms of the rate of change of the thermodynamic potentials with temperature.

4.1.3 Transport Properties These properties include various properties like mass diffusivity, viscosity (μ) and thermal conductivity. Such properties may be derived using methods of statistical mechanics. These properties will not be discussed here as they would merely act as coefficients in the conservation equations that we later develop in Sect. 4.4.

4.1.4 Miscellaneous Properties These properties include surface tension, diffusivities due to turbulence in the flow, vapor pressure etc. Interested readers can refer to the [5] for further understanding.

4.2 Description of Fluid Flow

61

4.2 Description of Fluid Flow Before we proceed further, let’s introduce the concepts of control mass (CM) and control volume (CV). A control mass (CM) consists of a chosen set of particles of interest to us. If we can put a tag on each of the particles, say, 1, 2, . . . ., n, on each one of them, then the CM will always consist of these n particles. A flow description of these particles will entail where each of these n particles are, in a given coordinate system, at any given instant of time, t. On the other hand, a control volume (CV) consists of a given region in space where we are interested. Since fluids can flow, there may be fluid particles entering the CV and leaving the CV. Therefore, particles constituting the CV can change with time. For a rigid body, which could be described as a CM without any deformation under the action of the acting forces, describing/predicting the trajectory is easy as long as all the forces acting on the body are known. Newton’s second law can be used to describe such a trajectory using the following equation: ∂ 2 r→ { − → Fi = ∂t 2 i

(4.1)

Here r→ refers to the radius vector denoting the location of the object, t refers to the − → time and {i Fi refers to the overall summation of each of the individual forces acting on the object. This differential equation can then be solved subject to the initial condition → r→ = − r0

(4.2)

→ r0 denoting the location of the rigid body at time t = 0 in a given reference with − frame. Note that for a Cartesian coordinate system, r→ = r→(x, y, z, t), where x, y and z refer to mutually orthogonal coordinate axes. Describing how a fluid flows (movement) is difficult by using the traditional concept of describing the trajectory of a fluid particle using Newton’s second law of motion. The differences between the description of motion of a rigid body with that of a fluid are outlined in the subsequent sections.

4.2.1 Lagrangian Description—Control Mass (CM) The Lagrangian description is the description of motion of control mass. In this formulation, the motion of the CM is described in terms of the Eqs. 4.1 and 4.2 as mentioned before. The simplicity of this equation is that it is easier to formulate but the solution of this equation is difficult as this entails knowing each of the individual forces acting on the object. The reader should, however, be aware that Lagrangian description is used in some critical multiphase applications, such as, tracking of

62

4 Fluid Flow Representation

droplets of diesel fuel (liquid) through compressed air as is the case in diesel engines. The necessity in such cases arises as each individual fuel droplet needs to be tracked to see how the droplet breaks down, vaporizes, mixes and eventually reacts with air in the internal combustion engines.

4.2.2 Eulerian Description—Control Volume (CV) Unlike a rigid body, fluid flow involves a large number of particles and therefore, in order to describe the flow, we need to describe the trajectory of each of those fluid particles. That means, we would end up with a very high number of equations of the form shown in Eq. 4.1, with each equation describing the trajectory of an individual fluid parcel. Solving such an exorbitantly high number of equations is absurd, and therefore, we need to forego such an attempt. Therefore, an alternate approach is taken where we choose a certain volume or region in space, and only focus on the fluid particles flowing through this region. This space is called control volume and the surface enclosing this region is known as the control surface. In this form of fluid flow description, the individual fluid parcels are not of interest as they come in and flow out of the control volume. Rather the parcels that exist inside the control volume at any instant of time, is of interest. This means that the fluid parcels of interest inside the control volume, if tagged as 1, 2, 3 . . . etc., then the tagging number denoting the identity of the parcels would, in general, change with time. This is because, some of the fluid parcels may have gone out of the CV (say, 3, 7, 10, etc.) while some other new fluid parcels may have come in (say number 1000, 1004, 1010..) etc. And we need to describe the motion of the fluid parcels that are inside the control volume at any instant of time. This is therefore referred to as the field view or field or Eulerian representation of fluid flow.

4.2.3 Field View to Fluid Flow The CV representation or formulation is also known as the field view or field representation of fluid flow. This is so as in this formulation/description, we are interested in finding out the behaviour (motion, velocity and other derived quantities such as force or stress) of parcels inside the control volume. We are not worried about the identity (tag number of the particles) but we are analysing the behaviour of any parcel that is inside the control volume at any instant of time. This means we are only monitoring the parcel inside the region of interest (control volume) as it suits us and hence the Eulerian representation is also known as the field view. Needless to say, that in this formulation, in general, quantities of interest will vary from point to point inside the control volume as well as with time. Hence, quantities such as velocity, V→ , are represented analytically as

4.3 Material Derivative

63

d r→ = V (→ r , t) V→ = dt ) ( V→ = f i u, j v, k z /\

/\

(4.3)

/\

(4.4)

where r→ = (x, y, z) refers to a specific point in space in the Cartesian coordinate and t denotes the time. i , j , k refer to the unit vectors along the (x, y, z) axes which are orthogonal in space. Note that this location (x, y, z) is chosen to be a point inside the control volume. If the velocity is known analytically, then quantities such as acceleration, a→ could easily be found out from /\

/\

/\

a→ =

D V→ Dt

(4.5)

The above equations are a clear testimony that the quantities of interest are functions of space and time which is why it is known as the field view or field formulation. The operator D/Dt is known as the material derivative and is described below.

4.3 Material Derivative A material derivative is a formal term used to mean the rate of change of a property (variable) as it would be observed if the fluid flow is being represented in a Lagrangian manner. In other words, material derivative refers to the rate of change of a property if we are moving along with the fluid parcel or looking at the same set of fluid parcels, viz. a control mass description. Let us assume that there exists a property A. We ought to find out how would A = A(x, y, z, t) change if we are following the same particle. The value of the independent variables, x, y, z and t of the dependent variable A, for this particle of interest would change as it flows and so would time t. Thus, we can use multivariate calculus to determine what would be the rate of change in A due to each of these mutually independent variables. The total change in A, ΔA, due to changes in the independent variables simultaneously, is given as the sum of the changes of A with each individual variable while the other variables are held constant ) ) ) ) ( ( ( ( ∂A ∂A ∂A ∂A ΔA = Δt + Δx + Δy + Δz (4.6) ∂t x,y,z ∂ x y,z,t ∂ y x,z,t ∂ z x,y,t In the above expression, Δx, Δy and Δz represent finite changes in the respective orthogonal projections as the fluid parcel moved in a span of time, Δt. To calculate the time rate of change of A, we divide both sides of Eq. 4.6 by the time interval Δt.

64

ΔA = Δt

4 Fluid Flow Representation

(

∂A ∂t

)

( + x,y,z

∂A ∂x

(

) y,z,t

Δx Δt

)

( +

∂A ∂y

(

) x,z,t

Δy Δt

)

( +

∂A ∂z

(

) x,y,t

) Δz Δt (4.7)

Applying the limit Δt → 0 to both sides, and acknowledging the fact that the time derivative of any variable is defined as ddtf = lim f leads to Δt→0

( ) ( ) Δx ∂A ∂A dA ∂A = + + lim dt ∂t x,y,z ∂ x y,z,t Δt→0 Δt ∂ y x,z,t ( ( ) ) ( ) Δy Δz ∂A + lim lim Δt→0 Δt ∂z x,y,t Δt→0 Δt (

)

)

(

(4.8)

Each of the limit terms on the right-hand side represents instantaneous velocities (u, v, w) along the respective axes (x, y, z). ( u = lim

Δt→0

) ) ( ( ) Δx Δy Δz , v = lim , w = lim Δt→0 Δt Δt→0 Δt Δt

(4.9)

Using Eq. 4.9, and dropping the subscripts on the partial derivatives, Eq. 4.8 could finally be written as dA = dt

(

∂A ∂t

)

( +u

∂A ∂x

)

( +v

∂A ∂y

)

( +w

∂A ∂z

) (4.10)

In fluid mechanics, the term on the left-hand side is known as the total derivative or material derivative or substantial derivative as it has been computed for a fluid parcel. This is commonly written as is written as DDtA . The right-hand side is(usually ) split into two terms, known as the local and convective part. The first term, ∂∂tA denotes the change in the property A, at a fixed point (x, y, z) in space with time t. Therefore, this is the local rate of change of A. The other three terms are collectively known as convective term as they denote the rate of change of A along the respective coordinate axes with the instantaneous velocities as a multiplier indicating that the flow velocities can affect the change. In the above derivation, A has been chosen as an arbitrary property of the control volume. In fluid mechanics, this could be any property mentioned above in Sect. 4.1. A case of particular interest is when the property, velocity (V→ ) is chosen. The material → derivative in this case, DDtV , would then yield the acceleration of the parcel which is either due to the variation of velocity locally or due to convective flow. The material derivative could be written using vector notation using the gradient → It is an elementary exercise to figure out that operator ∇. → =u V→ · ∇

(

∂ ∂x

)

( +v

∂ ∂y

)

( +w

∂ ∂z

) (4.11)

4.4 Governing Equations of Fluid Flow

65

and therefore, DA = Dt

(

∂A ∂t

)

→ + (V→ · ∇)A

(4.12)

( ) → .∇ → A represents the rate V represents local rate of change of A while ∂t of change of A due to convective flow. The beauty of the vector notation is that it is coordinate system independent, i.e., the expression on the right-hand side of → in Eq. 4.12 is true in any coordinate system as long as we use the expression for ∇ the coordinate system we choose to use. For steady flows, the local term is zero, (∂A)

(

∂A ∂t

) =0

(4.13)

4.4 Governing Equations of Fluid Flow Motion of fluids obeys certain fundamental laws based on the conservation of certain physical quantities such as mass, linear momentum, angular momentum and energy. All the physical laws stating the conservation of the physical quantities were classically derived for control masses, i.e., the same set of material packets/particles were under observation and the physical quantity of interest is monitored. However, since fluid flow is described in the Eulerian form where a predetermined region of interest in space is kept under observation (where parcels/packets of fluids can flow through), the conservation equations are written in a different form. It is worthwhile to introduce, at this point, the concept of an infinitesimally small control volume as well as a finite control volume. An infinitesimally small CV is a differential volume such that d V → 0, where V stands for the volume. For a finitesized control volume, the volume V is finite. Based on whether an infinitesimal or a finite CV is considered, the resulting governing equations will be differential or integral in nature. From a physical standpoint, a differential form of the governing equation is to be solved (with known boundary conditions) to get the fluid properties at a region in space for the infinitesimally small CV. The same equations need to be solved at all locations to solve for the entire flow-field. On the contrary, the integral form of the governing equations considers an infinitesimal CV and then integrates over the entire finite region in space taking into consideration the possibility of the variations in its properties. These differences will become clearer in the next section where the forms of all the governing equations are described. Since the common formulation for fluid flow is the Eulerian description (field view), we need a way to relate between the properties as measured in the Eulerian and the Lagrangian formulation. The relation between a quantity measured in the

66

4 Fluid Flow Representation

Eulerian description is converted to the Lagrangian description using the famous Reynolds transport theorem (RTT). For any property A, (

DA Dt

) CM

⎞ ⎛ ( ( ∂⎝ = α ρ d V ⎠ + α ρ V→ · d A→ ∂t CV

(4.14)

CS

) ( In the above equation, α = mA is the corresponding mass specific property of the extensive property A, ρ denotes the density at a location inside the CV and ( V = C V d V is the total volume of the CV obtained by integrating the differential volume element over the entire CV. V→ represents the velocity of the fluid as it flows out of (or into) the boundary of the CV defined by the control surface (CS) and d A→ → represents the differential I I area vector at the boundary (CS). The area vector, d A, is I I defined as d A→ = n Id A→I with nˆ being the outward directed unit normal vector locally. The physical interpretation of the above statement is that the rate of the change of the extensive property A for a CM could be expressed as the summation of the time rate of change of A happening inside the CV (1st term on the right-hand side) and the net exchange of A between the surroundings and the CV across the control surface (2nd term on the right-hand side). A proof of the RTT is not given here. However, the readers are encouraged to refer to [1, 2] for a complete description of RTT. The reader should note that we have used the material derivative D/Dt as it pertains to describing CM. The next four sections introduce the various conservation laws both in the differential as well as the integral form. For the sake of brevity, these equations are mentioned here without derivations. Interested readers can look in [1–3] for the same. /\

4.4.1 Conservation of Mass—Continuity Equation The equation for the conservation of mass is popularly known as the continuity equation when it is applied to fluid flow. Differential form: ∂ρ → · (ρ V→ ) = 0 +∇ ∂t

(4.15)

For incompressible flow, the condition is → · V→ = 0 ∇

(4.16)

4.4 Governing Equations of Fluid Flow

67

Integral form: Note here, that the conserved extensive property of interest is mass and the corresponding intensive property, α, is 1. Therefore using Eq. 4.14, the continuity equation could be written as ⎞ ⎛ ( ( ∂⎝ ρd V ⎠ + ρ V→ · d A→ = 0 (4.17) ∂t CV

CS

The zero on the RHS is essentially a statement that mass cannot be created or destroyed.

4.4.2 Conservation of Linear Momentum The principle of the conservation of linear momentum as stated in Newton’s second law of motion is valid for a control mass. In the absence of any forces, the linear momentum of a system remains unchanged and the body moves in a straight line (Newton’s first law of motion). Newton’s second law of motion relates the change in linear momentum of the body as a consequence to the resultant forces acting on the body to the acceleration of the body. By analogy, the momentum equation for fluid flow must relate the net force acting on the fluid to the acceleration experienced by it. It is important to identify the various forces that act on fluids in motion. These are volumetric and surface forces. Volumetric forces act throughout the control volume while the surface forces act on specific surfaces in certain directions. The weight of the fluid itself is a volumetric force. The surface forces are also conveniently classified into normal and tangential or shear forces. Pressure is an example of normal force as it always acts normal to any surface. On the other hand, shear forces act on particular surfaces and act in certain directions. Avoiding unnecessary details, we would mention the momentum equation here only for incompressible flows assuming that the viscosity of the fluid, μ, is constant. Differential form: ρ

D V→ → P + μ∇ 2 V→ = ρ g→ − ∇ Dt

(4.18)

→ ·∇ → is called the Laplacian operator. Depending on whether the flow where ∇ 2 = ∇ is 2-D or 3-D, and the coordinate system used for describing the flow, appropriate expression should be used for the Laplacian operator. The left hand side (LHS) in the above equation represents the net force per unit volume acting on a fluid. The first term on the right hand side (RHS) denotes the body force, the second term denotes the net pressure force and the third term denotes

68

4 Fluid Flow Representation

the net viscous forces acting on the fluid, all on a volumetric basis. Again, the readers should note that this equation is only valid for incompressible fluid flow, given by the condition mentioned in Eq. 4.16 and is a subset of the generalized Navier-Stokes equations. Integral form: For the momentum equation, A→ = m V→ and therefore, α = Eq. 4.14, this could be written as

A→ m

= V→ . Again, utilizing

⎞ ⎛ ( ( ∂⎝ → ⎠ V ρd V + V→ ρ V→ · d A→ = F→C V ∂t CV

(4.19)

CS

Essentially, this equation says that the rate of change of linear momentum inside the CV and the net rate of momentum exchange through the boundary is equal to the net force acting on the control volume. Please note that each of the terms is vectorial in nature and has both magnitude and direction.

4.4.3 Conservation of Angular Momentum The principle of the conservation of angular momentum is trivial for a differential CV. It merely states that under equilibrium, the shear forces in the adjacent areas are equal, i.e. τxy = τyx . τxy , τyx are the shear stresses (forces). The first subscript on τxy is x and refers to the surface on which this stress acts. The surfaces given by x is normal to the x axes. The second subscript,y, refers to the direction in which this s.ear stress acts which is in the y direction. Differential form: τxy = τyx

(4.20)

Integral form: For the integral form stating the principle of the conservation of angular momentum, → A = r→ × m V→ and consequently, α = mA = r→ × V→ . Note that in the integral form, the rate of change of angular momentum is equated to the net torque acting on the control volume due to the various forces mentioned in the linear momentum equation. ⎞ ⎛ ( ( ( ∂⎝ r→ × V→ ρd V ⎠ + r→ × V→ ρ V→ · d A→ = r→ × F→s + r→ × g→ρd V + T→shaft ∂t CV

CS

CV

(4.21)

4.4 Governing Equations of Fluid Flow

69

In this equation, r→ is the radius vector joining the origin of the coordinate system to the point(s) in space where various forces are acting and the torque due to those forces need to be calculated. The LHS denotes the net rate of change of angular momentum inside the CV and due to fluid flow across the boundary. The RHS is the net torque acting on the CV which has been subdivided into torque due to the net − → surface force F s (first term on the RHS), the distributed body force (second term on the RHS) and a shaft term (last term on the RHS). All of these three terms may or may not be present depending on the choice of the CV.

4.4.4 Conservation of Energy This essentially is the statement of the first law of thermodynamics as it applies to fluid flow. Differential form: For an incompressible flow with constant properties, the energy equation is ρ

Dh = k∇ 2 T + Φ Dt

(4.22)

) ( Here, h = e + ρP represents the enthalpy of the fluid, k is the thermal conductivity of the fluid while Φ is the irreversible degradation of energy due to viscous forces, often called as the viscous dissipation function [6]. In a nutshell, the LHS is the net time rate of change of enthalpy and this is due to thermal conduction and viscous dissipation represented by the first and second term, respectively, on the RHS. All these changes are on a per unit volume basis. Note that in the above equation, no shaft work has been considered. Integral form: For the integral form, A = U , the internal energy, and therefore, α = e = ⎛ ∂⎝ ∂t

(

CV

⎞ eρd V ⎠ +

( (

U . m

) V→ · V→ + gz ρ V→ · d A→ = Q˙ − W˙ shaft − W˙ S − W˙ other h+ 2

CS

(4.23) In this case, the first term on the LHS is net rate of change of specific internal energy inside the control volume and the second term denotes the net energy exchange rate due to net(fluid) exchange across the control surface, considering it has both → → kinetic energy V2·V and potential energy (gz) with respect to the chosen reference frame. This net exchange rate of energy of the control volume is a result of several interactions shown on the right-hand side. The first one is due to net rate of heat

70

4 Fluid Flow Representation

exchange, and the rest of the terms are the rate of work done by the various forces ( ) that act on the control volume. Often, for many cases, the shaft power W˙ shaft and ( ) the other power terms W˙ other are non-existent or are not considered for the sake of a simplistic analytical solution.

4.5 Irrotational Flow Before we proceed with what are irrotational flows, let us first define the angular →) of a fluid element that is related to the rotation of a fluid. If velocity ( vector, ω, → as found V→ = i u, j v, k w represents the linear velocity in the flow field, then ω, in standard text books on fluid mechanics, is defined as I I I i j k I I I 1( ( ) ) 1I ∂ I → × V→ = 1 curl V → ω → = I ∂∂x ∂∂y ∂z (4.24) ∇ I= I 2 2I 2 I u v wI /\

/\

/\

/\

/\

/\

Another important terminology often associated with rotational flows is vorticity, − → ζ , which is defined as twice the angular velocity. → × V→ ζ→ = 2ω → =∇

(4.25)

If ω → /= 0 over the entire flow domain, then the flow is rotational and the vorticity could be found out from Eq. 4.30. Therefore, a flow is considered to be irrotational → = 0, i.e., the cross product of the del operator, ∇ and the velocity if and only if ω vector, V→ = (u, v, w) is zero. Consequently, ζ→ = 0 for an irrotational flow field.

4.5.1 Potential Flow A flow is known as potential flow when it is devoid of any rotationality, i.e., the flow is completely irrotational (ω → = 0) in any part of the flow field. Additionally, we will → · V→ = 0 limit ourselves to only incompressible flows, the condition for which is ∇ as mentioned earlier in Eq. 4.16. Before we take any more strides, let us introduce an important terminology used in this area of fluid mechanics. We will be dealing only with simply connected regions or simply connected domains. The definition of a simply connected region is rather mathematical in nature. A simply connected region in space is that “wherein every closed path forms the edge of a family of hypothetical surfaces called capping surfaces which do not cut through the physical boundaries of the flow” [3]. Any flow domain that does not satisfy this definition is known as multiply connected regions. The figure below

4.5 Irrotational Flow

71

Fig. 4.1 a Simply and b Multiply connected domain

illustrates this concept. It is seen that in the case of Fig. 4.1a, the path is simply connected as any closed path inside the domain leads to a capping surface which does not cut across the domain boundaries. On the other hand, for Fig. 4.1b, a closed path shown has to cut through the boundary of the donut and hence is a multiply connected region.

4.5.2 Potential Function → × V→ = 0, The condition of irrotationality for an incompressible flow is given by ∇ as stated before. Note that this means the cross product of the velocity vector V→ with → is zero. Also, it is useful to remember that the cross product the del operator is ∇ is normal to the plane of V→ . Therefore, from vector calculus, there always exists a scalar function φ, such that, → V→ = ∇φ

(4.26)

where φ is known as the velocity potential. If (u, v, w) are the scalar components − → of V in the three orthogonal Cartesian directions respectively, then it could easily be noted that for a potential flow field, the velocity components are related to the potential function through the following relations. u=

∂φ ∂φ ∂φ ,v = ,w = ∂x ∂y ∂z

(4.27)

For 2-D flows, only u, v are relevant as no flow exists in the z direction. If the flow is incompressible and steady, then this velocity potential function will always satisfy the incompressibility equation given by

72

4 Fluid Flow Representation

→ · V→ = ∇ → · ∇φ → = ∇2φ = 0 ∇

(4.28)

Hence for inviscid, incompressible, irrotational flows, the velocity potential function satisfies the Laplace equation.

4.5.3 Stream Function → − → − Let us closely look at the condition for incompressible flow which is ∇ · V = 0. Let us limit ourselves to two-dimensional (2D) steady flows. In Cartesian coordinates, − → where the velocity could be represented as V = (u, v), this means ∂v ∂u + =0 ∂x ∂y

(4.29)

If we define the magnitude of both u and v in a manner such as u=

∂ψ ∂ψ ,v = − ∂y ∂x

(4.30)

then the Eq. 4.29 is automatically satisfied as ∂ ∂x

(

∂ψ ∂y

)

( ) ∂ ∂ψ + − =0 ∂y ∂x

(4.31)

ψ is known as the stream function. It is an alternate description of the flow field where a single variable ψ is used to describe the incompressible flow field instead of a two-component description (u, v) for the velocity field. One can easily find out the velocity components from the stream function by using the relations mentioned in Eq. 4.30. But this single variable description comes with a penalty. Note that the condition for incompressibility is a second order differential equation in ψ, whereas in terms of (u, v) the condition for incompressibility is a first order differential equation. Therefore, additional boundary conditions are required to solve the flow field when using the stream function formulation. Lines of constant ψ are called streamlines. Streamlines on a flow field are such that the tangent to the streamline at any point gives the direction of the velocity vector at that point. The physical interpretation of the stream function ψ is that the difference (ψ1 − ψ2 ) between two streamlines ψ1 &ψ2 gives the volumetric flow rate of the fluid in the 2-D flow field that it describes. Interested readers can look at references [1–3] for a proof. Further, in the stream function formulation, if the flow is irrotational then,

4.5 Irrotational Flow

73

( 2 ) ∂ ψ ∂ 2ψ → → = −k ∇ 2 ψ = 0 + ∇ × V = −k ∂x2 ∂ y2 /\

/\

(4.32)

which means that for steady, 2-D inviscid, incompressible, irrotational flows ∇2ψ = 0

(4.33)

Thus, the Laplacian of the stream function ψ is zero. Or, in other words, the stream function ψ has to satisfy the Laplace equation which is the topic of the next section.

4.5.4 Laplace Equation → · ∇, → acting on a variable is zero, When the value of the Laplacian operator, ∇ 2 = ∇ then such an equation is known as the Laplace equation. Thus, ∇2 A = 0

(4.34)

is the Laplace equation for the variable A. This notation of ∇ 2 , although is the same for all coordinate systems, the expression for ∇ 2 would depend on the choice of the coordinate system. From the preceding discussion in Sect. 4.5, we have seen that it is possible to represent steady, inviscid, incompressible fluid flows in terms of a velocity potential function, φ, or, stream functions ψ such that the Laplacian of these functions is zero. In other words, both φ and ψ satisfy the Laplace equation.

4.5.4.1

Cartesian Coordinate System /\

/\

/\

→ = i ∂ + j ∂ + k ∂ and therefore, for three-dimensional In the Cartesian system, ∇ ∂x ∂y ∂z (3D) flows, the Laplacian operator is → ·∇ → = ∇2 = ∇

∂2 ∂2 ∂2 + + ∂x2 ∂ y2 ∂z 2

(4.35)

For 2D flows, the Laplacian operator is 2 2 → ·∇ → = ∂ + ∂ ∇2 = ∇ ∂x2 ∂ y2

(4.36)

It is worthwhile to note that for inviscid, incompressible, irrotational flows, the Laplacian operator acting on φ or ψ yields zero. This means that for irrotational flows, the flow field could entirely be described in terms of the single variable φ or ψ. The reader should be aware at this point of time, that the potential function can be

74

4 Fluid Flow Representation

used for 3-D flows, whereas the applicability of the stream function is functionally restricted to 2-D domains. Such descriptions are only for inviscid, incompressible flow-fields only.

4.5.4.2

Cylindrical Coordinate System

In cylindrical systems, a point in space is defined with (r, θ, z) where r is the distance of the point from the origin, θ is the polar angle measured from the x-axis in a counterclockwise fashion and z refers to the distance of the point from the x − y → In this coordinate system ∇ → defined as plane. The del operator, ∇, → = δ→r ∂ + δ→θ 1 ∂ + δ→z ∂ ∇ ∂r r ∂θ ∂z

(4.37)

( ) where δ→r , δ→θ , δ→z refers to the unit vectors in the (r, θ, z) direction. Please note that depending on the location of the point in description, the direction of δ→r , δ→θ changes. The Laplacian operator ∇ 2 in the cylindrical coordinate system is defined as [6] ∇2 =

4.5.4.3

( ) 1 ∂ ∂2 ∂ 1 ∂2 r + 2 2+ 2 r ∂r ∂r r ∂θ ∂z

(4.38)

Spherical Coordinate System

In cylindrical systems, a point in space is defined with (r, θ, φ) where r is the distance of the point from the origin, θ is the polar angle and φ is the angle between the z axis and the radius vector containing both the z-axis and the radius vector. The del operator is defined as ( ) 1 ∂ ∂ 1 ∂ → → → → + δθ + δφ ∇ = δr ∂r r ∂θ sin φ ∂φ

(4.39)

) ( where δ→r , δ→θ , δ→φ refers to the unit vectors in the (r, θ, φ) direction. Please note that depending on the location of the point in description, the direction of δ→r , δ→θ &δ→φ can change. Also, the value of φ ranges between [0, π ]. The square brackets imply that both values are inclusive. The Laplacian operator ∇ 2 is defined as [6] ( ) ( ) 1 ∂ ∂ 1 ∂2 1 ∂ 2 ∂ r + 2 sin θ + 2 2 ∇ = 2 r ∂r ∂r r sin θ ∂θ ∂θ r sin θ ∂φ 2 2

(4.40)

4.5 Irrotational Flow

75

4.5.5 Properties of Laplace Equation The Laplace equation possesses some properties of its own. Before we mention those properties in the next two sections, we need to look at some vector calculus identities as these will be necessary to establish the properties of uniqueness of the solutions of the Laplace equation. If g(x) is a function defined in a simply connected domain, V with boundary S, then from vector calculus, we know that I I2 ( ) → II → · ∇g → = g∇ 2 g + II∇g → · g ∇g → ∇ = g∇ 2 g + ∇g

(4.41)

Therefore, (

( ( ( ) 2 → → ∇ · g ∇g d V = g∇ gd V +

V

V

I I2 I→ I I∇g I d V

(4.42)

V

The LHS of Eq. 4.42 contains the volume integral of the divergence of a function. A volume integral could be converted into a surface integral using the divergence theorem. This gives ( ( S

( ( ) 2 → g ∇g · n dS = g∇ gd V + /\

V

I I2 I→ I I∇g I d V

(4.43)

V

a result that will be of significant consequence in the next section.

4.5.6 Uniqueness of the Solutions of Laplace Equation The section is relevant as we are going to prove that the solution of the flow field, in terms of φ or ψ. is unique for an inviscid, steady, incompressible and irrotational flow field. This means, that there exists one and only one solution to φ or ψ as the case may be. This could be obtained by solving the Laplace equation ∇ 2 φ = 0 or ∇ 2 ψ = 0. These methods are described later in Sect. 4.11. Given the value of the normal component of fluid velocity on the bounding surface, S, shown in Fig. 4.2, there exists a unique flowfield in terms of φ or ψ satisfying the → · V→ = 0 and ∇ → × V→ = 0 incompressibility and the irrotationality condition given by ∇ respectively. To prove our conjecture, let’s take the case of the Laplace equation in terms of the velocity potential function φ. Going by the method of contradiction, let us assume that → 1 it is possible to have another solution, i.e., the solution is not unique. Let V→1 = ∇φ → 2 be the two solutions to the Laplace equation ∇ 2 φ = 0. We can and V→2 = ∇φ then easily construct another function, g, which is the difference of the two potential

76

4 Fluid Flow Representation

Fig. 4.2 A simply connected region V bounded by the surface S

functions, g = φ2 − φ1 . Then, since both φ1 and φ2 represent potential functions, ∇ 2 g = ∇ 2 φ2 − ∇ 2 φ1 = 0

(4.44)

→ − → → · n = ∇φ → 2 · n − ∇φ → 1·n =− and ∇g V2 · n − V1 · n = 0 /\

/\

/\

/\

/\

(4.45)

Both the above expressions have a value of zero over the entire domain V. Now → · n=0 let us return back to Eq. 4.43. The LHS of that equation is zero because ∇g and the first term on the RHS is zero as ∇ 2 g = 0 over the entire domain V. This leads us to ( I ( I ( I2 I2 I→ I→ I 2 → → I |∇g| d V = I∇(φ2 − φ1 )I d V = I∇φ (4.46) 2 − ∇φ1 I d V = 0 /\

V

V

V

Since the domain V has been completely arbitrary in definition, and the integrand is a square term, the only possibility to satisfy the equality condition in Eq. (4.47) is → 1 = V→2 − V→1 = 0 → 2 − ∇φ ∇φ

(4.47)

which proves the uniqueness as V→2 = V→1 . A similar proof could be devised for the steady, incompressible, irrotational flow field defined in terms of ψ, bearing in mind that this is generally restricted to 2-D flows.

4.5.7 Uniqueness for Infinite Domain The previous section has laid out the proof of the uniqueness of the solution of the flow field based on a single variable that could be used to represent the incompressible flow field. This section deals with the domain of the flow field which is infinite. This is really the case with many flow situations, such as external flow around an obstacle. Consider the flow of air around an object kept stationary by some means. How large

4.5 Irrotational Flow

77

Fig. 4.3 An infinite domain as an extension of a simply connected domain

is the domain? Theoretically, infinite; even though the object may not have any effect on the incident free stream flow. This issue could be addressed similar to the previous section, except that, in this case, we would treat the boundary S to be at infinity, i.e., S → ∞ as shown in the Fig. 4.3. The result is not derived here and is left as an exercise to the readers.

4.5.8 Kelvin’s Minimum Energy Theorem Kelvin’s minimum energy theorem identifies the type of flow necessary for the kinetic energy of the fluid to be minimum as the flow happens inside the control volume. This theorem assumes importance as in many cases, it is not possible to have any flow visualization inside the control volume while the flow behavior at the boundary is known and observable. Kelvin’s minimum energy theorem states that for the kinetic energy of the fluid during steady incompressible flow through a simply connected region happens only when the flow is irrotational in the region. It is assumed that the flow behaviour at the control surface encompassing the control volume is known, i.e., the normal component of velocity at the boundary is known. Let u 1 be the velocity of an irrotational incompressible fluid and u be the velocity of any other incompressible fluid motion (rotational or irrotational), such that the normal components of both u 1 and u are the same. Mathematically, this implies that u→1 · n→ = u→ · n→

(4.48)

→ n represents the unit normal at the boundary encompassing the domain. The where − difference between the kinetic energy of the fluid under those two different motions is given by

78

4 Fluid Flow Representation

ΔK .E. = K .E.2 − K .E.1 =

1 ρ 2

(

( 2 ) u − u 21 d V

(4.49)

V

where ρ and V represent the density and the associated volume of the incompressible fluid for which we compute the change in kinetic energy. The above equation could easily be written as ΔK .E. =

1 ρ 2

( [ ( )2 ) →] ( → → u→ − − u 1 + 2 u→ − − u1 d V u1 · − V

1 = ρ 2

(

( )2 → u→ − − u1 d V + ρ

V

(

( ) → → u→ − − u1 · − u1 d V

(4.50)

V

As we mentioned in Sect. 4.5.3, that for irrotational flow fields, we can always define a scalar velocity potential function φ, such that − → → u1 = ∇φ

(4.51)

→ u 1 in the second integral, it becomes, Using this for − ( ρ

( ) → → u→ − − u1 · − u1 d V = ρ

V

(

( ) → → V u→ − − u 1 · ∇φd

(4.52)

V

( ) → → are vectors. Therefore, using the vector identity Note that both →−− u 1 and ∇φ ( u) → − B∇ → where A→ is a vector and B is a scalar, the right-hand →B =∇ → · AB → · A., A→ · ∇ term of (Eq. 4.52) becomes ( ρ

⎡ ⎤ ( ( ( ( ) ) ) ( → → → → → · u→ − − → · u→ − − u1 d V ⎦ u→ − − u1 · − u1 d V = ρ⎣ ∇ u 1 φd V + φ ∇

V

V

V

(4.53) − → → → → − Since both ) u 1 and u→ refer to incompressible flow fields, ∇ · u 1 = ∇ · u→ = 0 = ( − → → ∇ · u→ − u 1 , meaning that ( ρ V

⎤ ⎡ ( ( ) ) ( → → → → · u→ − − u→ − − u1 · − u1 d V = ρ⎣ ∇ u 1 φd V ⎦

(4.54)

V

Finally, the volume integral could be written in terms of area integral by applying divergence theorem,

4.6 Elementary Potential Flows

( ρ

79

( ) → → u→ − − u1 · − u1 d V = ρ

V

(

( ) → u→ − − u1 · nd A /\

(4.55)

A

The right-hand side of Eq. 4.55 is identically equal to zero as both fields had the same normal component at the boundary given by Eq. 4.48. Then it necessarily means that ( ( )2 1 → u→ − − u1 d V ≥ 0 ΔK .E. = ρ (4.56) 2 V

with the equality sign holding true if and only if u→ = u→1 , i.e., the flow field is steady, incompressible and irrotational in nature. This proves the theorem. Reference [7] provides further information for flows through multiply connected regions in space. An extension of the Kelvin’s minimum energy theorem to incompressible flow with open domains could be found in [8].

4.6 Elementary Potential Flows The concept of a potential flow stems from that of the definition of the potential function φ. Potential flows are necessarily irrotational. There are plenty of applications where the flow is irrotational, i.e. potential in nature. However, here, we will consider certain very basic form of potential flows only for inviscid, incompressible cases. These are discussed in the subsequent sections.

4.6.1 Uniform, Free Stream Flow For a uniform free stream flow, say in the x direction, in 2-D flow, both the velocity potential φ and the stream function ψ could be found. In this case the velocity could be represented as V→ = i u and this scalar component u (a known value) is related to φ&ψ through the following equation of irrotationality and incompressibility. /\

→ = ı ∂φ → u = ∂φ , v = 0 V→ = ı u = ∇φ ∂x ∂x /\

/\

(4.57)

Similarly, −u =

∂ψ ∂ψ ,v = − =0 ∂y ∂x

(4.58)

80

4 Fluid Flow Representation

Fig. 4.4 Streamlines (solid) and potential lines (dotted) for free uniform flow

We can integrate the u component equations, and would find φ&ψ respectively as, φ = ux, ψ = −uy

(4.59)

During the integration, we neglected the constants of integration as they do not affect the velocities in any way. The results indicate that the potential lines have a fixed potential along vertical lines (x = constant) and that the streamlines are lines of constant ψ along horizontal lines y = constant. Note that the streamlines and the potential lines are orthogonal. This is shown in Fig. 4.4 where the flow is towards the positive x-direction. The equipotential lines, lines of constant φ, are, therefore, straight lines in the vertical direction (x = constant) but having values of φ1 = ux1 , φ2 = ux2 etc. The streamlines, lines of constant ψ, are in the horizontal direction (y = constant), but different values of ψ1 = uy1 , ψ2 = uy2 etc. as is found out in Eq. 4.59.

4.6.2 Point Source or Sink In this case, we assume that the flow is outwards or inwards from/to a single point. This point is called a source if the flow originates and is outwards from this point and is called a sink, if the flow is towards this point. Naturally, this is a case of spherical symmetry and hence in this case we have to use the potential equation in the spherical coordinate system. Also, because of symmetry, there are no variations along the θ or φ direction and accordingly, any derivative with respect to θ or φ is zero. Without any loss of generality, we can assume that, the point source or sink is at the origin of the coordinate system. Thus, the potential function in this case is given by φ = φ(r ). Accordingly, the Laplace equation becomes, ∇2φ =

( ) 1 d 2 dφ r =0 r 2 dr dr

(4.60)

4.6 Elementary Potential Flows

81

Fig. 4.5 Streamlines (solid) and potential lines (dashed) for a point, a source and b sink. The flow is spherically outwards/inwards from/to the origin as indicated by the arrowheads for a point source/sink

Readers should note that the Laplace equation is now an ordinary differential equation (ODE) rather than a partial differential equation as φ varies only in the radial direction. The solution to the ODE in Eq. 4.60 is given by φ(r ) = −

m +C r

(4.61)

where m and C are constants of integration. m is known as the strength of the source/sink and r is the radial distance of any point in the field from the same. The radial velocity is then given by → = m δr V→ = ∇φ r2 /\

(4.62) /\

Note that here we have used the unit vector δr as the velocity is radially inward/outward in nature. For a source, the velocity is directed radially outwards Q˙ and hence, m > 0. For a sink, m < 0. As a convention, the constant m = 4π . Q˙ ˙ represents the net volumetric flux through the sphere of radius r. Q is positive for outward flow and negative for inward flow. For a point source, the potential lines are spherical surfaces with the value of φ increasing (less negative as r ↑) while for a point sink, the potential lines are spherical surfaces with decreasing values with increase in r as illustrated through Fig. 4.5.

4.6.3 Line Source or Sink These are also referred to as 2-D sources and sinks. Their geometry is axisymmetric and hence we would look at the velocity potential in an axisymmetric manner, i.e., in a cylindrical coordinate system where φ = φ(r ) and the symmetry around the axis leads to the vanishing of partial derivatives in the θ and z direction, i.e., ∂θ∂ = ∂∂z = 0. The Laplace equation, therefore, becomes

82

4 Fluid Flow Representation

∇2φ =

( ) dφ 1 d r =0 r dr dr

(4.63)

dφ m = dr r

(4.64)

leading to

where m is an integration constant and r is the modulus of the radius vector. Solving the ODE, φ(r ) = m ln r + C

(4.65)

− → where C is another constant of integration. The velocity field V is then given by → = d (φ)δr = m δr V→ = ∇φ r dr /\

/\

(4.66)

For a source, m > 0 while for a sink, m < 0. m is known as the strength of the vortex Q˙ and is by convention defined as 2π , Q˙ being the volumetric flow rate. Figure 4.6 shows equipotential potential lines for such types of flow. Flow through radial perforations on a hose can be approximated using such potential flows. The stream functions are given by ψ = mθ

(4.67)

Fig. 4.6 Streamlines (solid) and potential lines (dashed) for a line source and b line sink. For a point source, the flow is radially outward from the origin as indicated by the arrowhead and vice versa for a sink

4.6 Elementary Potential Flows

83

4.6.4 Line Irrotational Vortex (Free Vortex) A free vortex or a line irrotational vortex is the one which has only velocity component in the θ direction, Vθ = f (r ). The radial component of velocity in this case is Vr = 0. In a free vortex, the velocity of the fluid increases continuously towards the centre of the vortex. This increase is velocity in terms of the principle of ( can be explained ) → conservation of angular momentum r→ × massV . As r→ decreases, V→ increases as the angular momentum of the fluid remains constant in the absence of any external torque. Consider a potential function, φ = φ(θ ). Then, as in the previous section, the Laplacian of the potential function when equated to zero leads to 1 dφ m = r dθ r

(4.68)

φ(θ ) = mθ + C1

(4.69)

solving which,

and the stream function is, → = 1 d (φ)δ θ = m δ θ V→ = ∇φ r dθ r /\

/\

(4.70)

where m is the strength of the vortex. Note that this predicts that the velocity decreases as the radial distance decreases. It is interesting to note that, in Eq. 4.69, ln r is not defined at the point represented by r = 0, and hence, this point serves as a singular point. This is also evident in Eq. 4.70 as the relationship is not defined for r = 0. Figure 4.7 illustrates a free vortex. The equipotential lines look like the spokes of the wheel while the streamlines are the circles. The streamlines always perpendicular to the equipotential lines. The streamlines are a family of circles given by the equations ψ = −m ln r + C2

4.6.4.1

(4.71)

Circulation

The free vortex brings us to another interesting concept which is called circulation. Circulation is defined as the line integral of the tangential component of the velocity along a chosen closed path at any given time instant t. Thus if C denotes circulation about any closed path c as shown in Fig. 4.8, then ( C= C

V→ · d→s

84

4 Fluid Flow Representation

Fig. 4.7 Streamlines (solid) and potential lines (dashed) for an irrotational (or line) vortex. The direction of rotation is shown with the arrowhead

Fig. 4.8 Illustration of the concept of circulation about a closed path c

For a free vortex, the circulation about any path not including the origin could be shown to be zero (Fig. 4.8).

4.7 Linear Superposition of Flows The linear superposition of two flows is a powerful technique to approximate/reproduce several real flows in terms of two or more potential flows. The basis of the principle of superposition is the mathematical fact that, if φ1 and φ2 are two potential functions obeying the Laplace equation, i.e., ∇ 2 φ1 = ∇ 2 φ2 = 0, then any linear combination of φ1 and φ2 is also necessarily a solution to the Laplace equation. ∇ 2 (A1 φ1 + A2 φ2 ) = ∇ 2 φ1 + ∇ 2 φ2 = 0 The application of this method is shown in the remainder of the Sect. 4.7.

(4.72)

4.7 Linear Superposition of Flows

85

4.7.1 Dipole (Doublet Flow) A dipole consists of a line source and a sink of equal strength,m, placed at (−L , 0) and (L , 0). The separation distance between ( / them is,)therefore, 2L. If we pick any arbitrary point P(x, y), at a distance r = x 2 + y 2 from the origin, then we can calculate the potential at this point which will be the sum of the potential at P due to the individual point source and sink. Let r1 &r2 be the distance between P and the source and sink, respectively. Then we can write that, r12 = r 2 + L 2 + 2r L cos θ r22 = r 2 + L 2 − 2r L cos θ

) (4.73)

The combined potential at the location is then the sum of the potential due to the sink and the source, given using Eq. 4.65, and could then be written as φ=

m |(ln r1 − ln r2 ) 2

(4.74)

Substituting the expressions from Eq. 4.73, and with some mathematical manipulation, this expression could be written as φ=

) ( )] [ ( 2r L cos θ 2r L cos θ m − ln 1 − ln 1 + 2 4 r + L2 r2 + L2

(4.75)

( cos θ ) < 1 for the point P everywhere in It is to be noted that the term 2rr 2L+L 2 ( cos θ ) < 1, the two the domain except when P is at the source or sink. Since 2rr 2L+L 2 logarithmic terms in the square brackets could be expanded as a pow series, which would result in ] [ ) ( m 4r L cos θ 2 2r L cos θ 3 φ= + + ... (4.76) 4 r2 + L2 3 r2 + L2 If we now bring the source and the sink together, i.e., L → 0 but increase the strength of their strength m to infinity, such that m L is finite, then the potential could be found by ] [ )3 ( m L 4r cos θ 2 2r L 2/3 cos θ 2m L cos θ φ = lim + + ... = 2 2 2 2 L→0 4π r +L 3 r +L r

(4.77)

The product, 2m L, is termed as the dipole moment, and μ, called dipole strength, is defined as μ = m(2L)

(4.78)

86

4 Fluid Flow Representation

Fig. 4.9 Streamlines (solid) and potential lines (dashed) are shown for a doublet when a line source and sink is superimposed, each of infinite strength, are kept extremely close to each other

Therefore, φ=

μ cos θ r

(4.79)

It could easily be shown that the stream functions are given by ψ =−

μ sin θ r

(4.80)

The lines of constant φ and ψ are shown in Fig. 4.9.

4.7.2 Planar Flow When we combine a line source with another line sink, both of equal strength, then what we get is a planar flow.

4.8 Flow Past an Obstacle In this section, we are going to evaluate the case of potential flow around certain solid obstacles. We would see that such flows could be assumed as the superposition of two or more elementary potential flows described in an earlier section.

4.8 Flow Past an Obstacle

87

Fig. 4.10 Streamlines showing flow around a sphere

4.8.1 Flow Past a Sphere The irrotational, incompressible flow around a sphere of radius a could be explained by means of the superposition of a 3D doublet and a uniform flow, u, expressed in the axisymmetric terms. It could be easily found from [3], that at a point located at a distance R from the origin, ψ =−

) ( ua 3 sin2 β u sin2 β a 3 u R 2 sin2 β + = − R2 2 2R 2 R

(4.81)

3

with μ defined as μ = ua2 . It could easily be noticed, that for ψ = 0, there exists two angles β = 0 and β = π . These two points are, therefore, stagnation points. This condition of ψ = 0 is also satisfied for R = a. Therefore, for ψ = 0, β = 0orπ (flow direction) and R = a. Thus, there is no flow along the surface of the sphere. The other streamlines around the sphere would like what is shown in Fig. 4.10

4.8.2 Rankine Half Body A half-body oval shape appears when a uniform stream is superimposed with a line source kept at the origin. The superposed stream function is then expressed as ψ = ur sinθ + mθ r=

ψ − mθ u sin θ

(4.82) (4.83)

For any chosen value of ψ at any given angle θ , the value of r could be found from the above equation as u and m are known quantities. Changing the value of θ for the same ψ changes r . Thus, by keeping a constant value of ψ and varying θ , we can

88

4 Fluid Flow Representation

Fig. 4.11 Streamlines showing the flow superimposed due to a uniform flow and a line source kept at the origin. The Rankine-half body is shown in pink. The maximum velocity in the domain is 1.26u, where u is the free stream velocity

generate a curve as shown in Fig. 4.11. The shape of a half-body appears (pink line) and this separates the two elementary flows. This is named after William Rankine. The maximum velocity is found to be 1.26u at an angle of 63◦ [1]. The ha width of the body far downstream of the flow is π m/u. Please note that this body is given by the streamlines ψ = ±π m. The stagnation points on these streamlines are at (−a, 0) as could be verified from the velocity components.

4.8.3 Flow Around a Cylinder In this section, we will show that the superposition of a doublet and a uniform flow represents the flow over a cylinder. The combined stream function for this combination is going to be ψ = ψdoublet + ψuniform flow = −

μ sin θ + ur sin θ r

(4.84)

Note that here we have written Ur sin θ , which the reader will recognize immediately by comparing with Eq. (4.59), that y = r sin θ . The velocity potential is φ = φdoublet + φuniform flow =

μ cos θ + ur cos θ r

(4.85)

These two equations can then be integrated and the expressions for ψ and φ could be found out.

4.9 Force on a 2-D Object of Arbitrary Shape

89

Fig. 4.12 Flow around a cylinder without circulation. The lines shown are the streamlines, having constant ψ. A. and B are stagnation points in the flow and along the line AB, ψ = 0. Potential lines at any point othe domain would be normal to the streamlines

/ ) ( μ a2 ψ = ur 1 − 2 sin θ, a = = const r u

(4.86)

Similarly, / ) ( μ a2 = const φ = ur 1 + 2 cos θ, a = r u

(4.87)

The profiles of the floware shown in Fig. 4.12 below which is the flow around a cylinder. Note that at any point P, the velocity is tangential to the streamline shown. A and B are stagnation points in the flow and along the line AB. The potential lines at P are normal to the streamlines at P.

4.9 Force on a 2-D Object of Arbitrary Shape In this section, we are going to look at the force exerted on a 2D object which is placed in free stream flow. In particular, this is of significant interest to the aerodynamic community as this forms the basis of the concept of airfoils generating lift. Of particular interest, in airfoils there are two forces known as lift (FL ) and drag (FD ) which are forces acting in the vertical direction and against the motion of the fluid. Note that, in this case too, we are considering only incompressible, irrotational flows and hence there is no flow separation from the body. Consider an airfoil of arbitrary shape as shown in Fig. 4.13. The pressure force acts normal to the surface element d→s . The pressure force can be resolved into the lift and drag components along the vertical and the horizontal axes and they are given by d FL = Pd x d FD = −Pdy

) (4.88)

90

4 Fluid Flow Representation

Fig. 4.13 A 2D airfoil of arbitrary shape being placed in uniform flow. Taken from ref [10]

We can now form a complex conjugate vector for the differential force on this airfoil surface over d→s . Thus, if z = x + i y, z = x − i y & dz = d x − idy, we get, d(Fd − i FL ) = −Pdy − i Pd x = −i (Pdz)

(4.89)

We can find the lift and the drag on the airfoil if we carry out a contour integral over the entire surface in a counterclockwise fashion. The pressure may be calculated from Bernoulli’s equation as the flow is incompressible in nature. 1 dF dF 1 1 P = P0 − ρW 2 = P0 − ρW W = P0 − ρ 2 2 2 dz dz

(4.90)

Here, F = φ +i ψ denotes the complex potential and W = ddzF = ∂φ −i ∂ψ = u −i v, ∂x ∂y is the conjugate of the complex velocity. Using this expression for pressure in Eq. 4.89 leads to 1 dF dF d(FD − i FL ) = −i P0 dz + iρ 2 dz

(4.91)

Note that the surface contours in this case would denote various streamlines, ψ = constant. Since = φ + i ψ, (see [1–3]), equating the real parts we get, d F = dφ = d F. Hence, ⎡ FD − i FL = i ⎣−

(

ρ P0 dz + 2

C

(

⎤ W 2 dz ⎦

(4.92)

C

It is a known fact that the integral of a constant around a closed contour is zero. Consequently, FD − i FL = i

ρ 2

( W 2 dz C

(4.93)

4.10 General 3-D Potential Flows

91

The above formula is known as Blausius theorem. If W does not contain any singularities, which it does not, then we can extend this contour to z → ∞. Contour integrals in the complex space is evaluated using the residue theorem, i

ρ 2

( W 2 dz = 2πi

{

Rk

(4.94)

k

C

where Rk are the residues of the function W 2 , found out using Laurent series expansion of the function W 2 . Since we want to approximate the flow using a fixed free stream velocity, a source of strength m with circulation C and a doublet in arbitrary direction, the complex potential is defined as F = Uz +

1 1 (m + iC) ln z + (a + ib) + · · · 2π z

(4.95)

Taking derivative, W =

iC 1 dF =U+ + other terms dz 2π z

(4.96)

with W2 = U2 + i

CU + other terms πZ

(4.97)

and, therefore, we get, Corresponding residue of W 2 = i CU π FD = 0 FL = ρU C

) (4.98)

Thus, the drag force on any airfoil for potential flow is zero, while the lift is given by ρU C. For fixed ρ&U , the lift could be increased if the circulation is higher, i.e. if the motion is such that it generates more vortices. This is known as the KuttaJoukowski law. Readers are encouraged to read reference [9] for further discussion on the topic.

4.10 General 3-D Potential Flows The generalized theory of 3D potential flows is well developed dealing with flow around bodies generated by revolution around a specific axis. It is possible to come up with approximate flow solutions for flows around bodies of revolution using elementary flows of some form of arrangement of line sources and sinks. The idea here is that when we combine the effects of uniform flow and the line sources and

92

4 Fluid Flow Representation

sinks, we should be able to form a stream surface that represents the shape of the body. The number of sources and sinks used in the arrangement depends on the accuracy of analysis desired. The entire body is divided into smaller lengths and then the field at any location P(x, y) is approximated. Interested readers can refer to [3] for further discussion on this topic.

4.11 Solution to Laplace Equation There are several methods of solving the Laplace equation. They are (a) Analytical means (i) Method of images (ii) Method of separation of variables These two methods will be the subject of the subsequent sections of this chapter. (b) Numerical techniques Several numerical techniques have been devised to solve the PDEs by discretizing the domain (dividing the domain into smaller cells/zones) and then solving the governing equations for each cell as it has to be valid over the entire domain. A description of how this discretization is done is provided in the next chapter. (c) Conformal mapping This method deals with a complex analysis and mapping the domain. The name originates from the fact that at any local point (x, y, z) inside the domain, the angle (form) is preserved but not necessarily the area of the domain. We will not discuss this method in this book.

4.11.1 Method of Images Method of images is a way to solve the Laplace equation with the assumption of a similar flow field along a line of symmetry. In this case, the boundary conditions remain the same on the extended domain (image domain) too. This technique is generally utilized to calculate the flow field produced due to singularities near the boundary. An example of how this technique works is shown below. Let us consider a single source of strength m(> 0) placed at a location (l, y, z) near a plane wall which is taken to be at x = 0. In the absence of the wall, the potential due to the source at any other location (x, y, z) is given by (Fig. 4.14) (

m

φ=− / (x − l)2 + y 2 + z 2

) (4.99)

4.11 Solution to Laplace Equation

93

Fig. 4.14 Problem statement for method of images. A single source kept on one side of the wall would not produce the x-component of velocity u to zero. The image introduced is shown by the dotted lines

Here, we have used Eq. 4.61 assuming C = 0 and used the fact/that the distance ( ) to a point with coordinates (x, y, z) is given by the relation r = x 2 + y2 + z2 , / and therefore, this specific point of interest is at a distance r = (x − l)2 + y 2 + z 2 from the point source. The x-component of velocity at this point is given by u=

m(x − l) ∂φ =[ ]3 ∂x (x − l)2 + y 2 + z 2 2

(4.100)

If we put x = 0, the location of the solid wall, then the x-component of velocity at the wall, as calculated from Eq. 4.100 will be non-zero. This is not feasible as u = 0 due to no-slip boundary condition. Note thathe other velocity components v&w need not be zero at the wall. To overcome this, a source is of equal strength m is added outside of the given domain, diametrically opposite to the actual point source at (−l, −y, −z). The combined potential at any specific location due to the point sources is then given by ( φ=− /

m (x − l)2 + y 2 + z 2

)

( −

m

)

/ (x + l)2 + y 2 + z 2

(4.101)

and correspondingly, u=

m(x − l) ∂φ m(x + l) =[ ] 23 + [ ]3 ∂x 2 (x − l) + y 2 + z 2 (x + l)2 + y 2 + z 2 2

(4.102)

94

4 Fluid Flow Representation

At the wall (x = 0), u = 0, which is the correct boundary condition. Because of symmetry, the hypothetical point source added generates a field where the xcomponent of velocity is equal in magnitude but opposite in direction to the real point source and hence the combined effect is u = 0 at the wall. The y&z components of the velocity is obtained through v=

2my ∂φ =[ ]3 ∂y 2 (x − l) + y 2 + z 2 2

(4.103)

w=

2mz ∂φ =[ ]3 ∂z 2 (x − l) + y 2 + z 2 2

(4.104)

It should be noted that even though u = 0, the fluid may slip along the y&z directions along the wall as both v&w are non-zero at the wall where x = 0.

4.11.2 Method of Separation of Variables In this method, we assume that the dependent variable, φ, could be expressed as a product of the functions of individual independent variables x, y or z. For the sake of simplicity, we will show how this method works with two variables only. Assume that φ = φ(x, y) in Cartesian coordinate system. To move ahead with the method of separation of variables, we write φ(x, y) = X (x)Y (y)

(4.105)

where X (x) and Y (y) are functions only of the independent variables x&y, respectively. Differentiating Eq. 4.105 partially with respect to x&y yields ∂ 2φ d2 X =Y 2 ∂x dx2

(4.106)

∂ 2φ d 2Y = X ∂ y2 dy 2

(4.107)

Note that in the above two equations, we have dropped indicating the independent variables of functions X, Y . Using these results in the original potential equation, ∇ 2 φ = 0, we get, Y

d 2Y d2 X + X =0 dx2 dy 2

which when divided by X (x)Y (y), yields

(4.108)

4.11 Solution to Laplace Equation

95

1 d2 X 1 d 2Y + =0 X dx2 Y dy 2

(4.109)

This could be rewritten as −

1 d2 X 1 d 2Y = = λ2 (say) X dx2 Y dy 2

(4.110)

with λ2 , being known as the separation constant. This leads to two independent ODE’s d2 X + λ2 X = 0 dx2

(4.111)

d 2Y − λ2 Y = 0 dy 2

(4.112)

Note that the sign on the second terms are opposite in the above equations. Let us summarize the general solutions of the two above ODE’s in terms of the value of λ2 . Table 4.1 shows that the solution φ(x, y) could be, with some algebraic manipulations, ⎧ ⎪ ⎨ Ax y + Bx + C + D φ(x, y) = Aeλy sin(λx) + Be−λy sin(λx) + Ceλy cos(λx) + De−λy cos(λx) ⎪ ⎩ Aeλx sin(λy) + Be−λx sin(λy) + Ceλx cos(λy) + De−λx cos(λy)

λ2 = 0 λ2 > 0 λ2 < 0

(4.113)

Thus, there are 4 constants that need to be found out from the boundary conditions in order to solve the potential equation. Once the potential, φ(x, y), at any location is found, the velocity components at that location could easily be found out. The above derivation has been done for a rectangular or Cartesian coordinate system. The same exercise could be carried out in a cylindrical coordinate system. We assume that the velocity potential, φ = R(r )Θ(θ ). In this case, Table 4.1 General solutions to the ODEs based on the value of λ2 . A, B, C&D are constant that could be found from the given boundary conditions O D Es

Solutions

λ2

X

Y

0

d2 X dx2

=0

d2Y dy 2

>0

d2 X dx2

+ λ2 X = 0

d2Y dy 2

4 AC. In this case, there are no real characteristic curves as ddyx is imaginary. Steady, incompressible, irrotational flows are represented by the following equation, with F = φ or ψ. ∂ 2F ∂ 2F + =0 ∂x2 ∂ y2

(5.9)

( ) Clearly, for such equations, A = 1, C = 1, B = 0 and hence B 2 − 4 AC < 0. These potential flow equations are thus elliptic equations.

100

5 Computational Fluid Dynamics

5.3.2 Parabolic Equations The is zero, i.e. ( 2 PDE is) classified as a parabolic equation if the discriminant B − 4 AC = 0. In this case, there exists only one solution to ddyx given by Eq. (5.8). Consider the PDE in F (x, t) given by ∂ 2F ∂F − α 2 = 0, α > 0 (5.10) ∂t ∂x ( ) Here, A = −α, B = C = 0. Therefore, B 2 − 4 AC = 0. Therefore, this is a parabolic equation.

5.3.3 Hyperbolic Equations ( ) The PDE is classified as a hyperbolic equation if the discriminant B 2 − 4 AC > 0. In such cases, there are two separate real roots of ddyx given by Eq. (5.8) along which the first-order derivatives vanish. These are called the characteristic curves. Consider the second-order wave equation, 2 ∂ 2F 2∂ F − a =0 ∂t 2 ∂x2

(5.11)

) ( Like before, A = −a 2 , B = 0, C = 1, and therefore, B 2 − 4 AC = 4a 2 > 0. Equatio (5.11) is thus a hyperbolic equation.

5.4 Region of Disturbance and Influence Based on the real or imaginary nature of the discriminant, we can identify the entire domain in certain regions. If a disturbance originates from a fixed location, it could only reach certain regions downstream. The set of all possible points in space (region) whose signal reaches downstream is known as the region of disturbance/dependence. Similarly, the region downstream of the disturbance source where such a signal can reach is known as the region of influence. Since the characteristics of elliptic equations are imaginary in nature, the whole field can feel the disturbance. For a parabolic equation, only one characteristic line exists. Therefore, the disturbance cannot cross this specific characteristic line. For hyperbolic equations, the zone of influence and zone of disturbance must be contained in the region contained between the two real characteristic lines/surfaces Fig. 5.1.

5.6 Discretization Methods

101

Fig. 5.1 Domain of dependence shown for a hyperbolic, b parabolic and c elliptic equations. From Veerstag and Malalasekera [4]

5.5 Discretization of the Domain Discretization refers to the process of dividing the domain into smaller elements/zones such that we can apply different numerical methods to convert the governing partial differential equation into a series of algebraic equations. These methods are discussed in the next section.

5.6 Discretization Methods There are primarily three main methods of discretizing the domain and thereafter proceeding to seek a solution to the governing equations over the domain. These methods differ in their approach although almost all of them, for real applications, seek solutions to algebraic equations in an iterative manner. These methods are discussed below.

5.6.1 Finite Difference (FD) Method The finite difference technique is a method of discretizing the governing equations based on the concept of Taylor’s series expansion [1]. We will demonstrate this with respect to the Laplace equation given in (Eq. 5.9) but in terms of φ. ∂ 2φ ∂ 2φ + =0 ∂x2 ∂ y2

(5.12)

102

5 Computational Fluid Dynamics

Fig. 5.2 A schematic of a 2D domain discretized into several nodes separated from each other by Δx and Δy in the x- and y-directions. If P is named as the (i, j) node, then the nomenclature of the neighbouring nodes is also shown

The domain is divided into several small elements subdividing it into smaller domains by establishing what are called nodes. The nodes along x-axis and y-axis are separated from each other by Δx and Δy, respectively. This is shown in Fig. 5.2 To replace the second-order partial derivatives with algebraic expressions in terms of nodes, we resort to the Taylor series expansion of φ in terms of x and y. Assume the φ is a function of (x, y). A Taylor’s series expansion of φi+1, j about the node (i, j) gives ( φi+1, j = φi, j +

∂φ ∂x

)

Δx + i, j 1!

(

)

∂ 2φ ∂x2

Δx 2 + i, j 2!

(

)

∂ 3φ ∂x3

Δx 3 + ... i, j 3!

(5.13)

from which we can find the expression of the first derivative of φ with respect to x (

∂φ ∂x

) i, j

φi+1, j − φi, j − = Δx

(

∂ 2φ ∂x2

)

Δx 2 − i, j 2!

(

∂ 3φ ∂x3

)

Δx 3 + ... i, j 3!

(5.14)

If we neglect the second and higher order terms, then (

∂φ ∂x

) ≈ i, j

φi+1, j − φi, j Δx

(5.15)

This is known as the finite difference quotient and should be used to get algebraic expressions. The terms ignored represent what are called truncation errors. When the domain is sufficiently discretized, truncation errors are below a threshold value and do not influence the final result. In a similar manner, we can write ( φi−1, j = φi, j −

∂φ ∂x

)

Δx + i, j 1!

(

∂ 2φ ∂x2

)

Δx 2 − i, j 2!

(

∂ 3φ ∂x3

)

Δx 3 + ... i, j 3!

(5.16)

Adding (Eqs. 5.13 and 5.16) together, and some after mathematical manipulation,

5.6 Discretization Methods

(

103

∂ 2φ ∂x2

) ≈ i, j

φi+1, j − 2φi, j + φi−1, j Δx 2

(5.17)

Proceeding exactly the same way, we can also show that ( (

∂ 2φ ∂ y2

∂φ ∂y )

)

(5.18)

φi, j+1 − 2φi, j + φi, j−1 Δy 2

(5.19)

i, j

≈ i, j

φi, j+1 − φi, j Δy

≈

We can now use the expressions in Eqs. 5.12 and 5.19 in Eq. 5.12 yielding what is known as the nodal equation for interior nodes. φi+1, j − 2φi, j + φi−1, j φi, j+1 − 2φi, j + φi, j−1 + =0 2 Δx Δy 2

(5.20)

which, for Δx = Δy, could be written as φi, j =

) 1( φi+1, j + φi−1, j + φi, j+1 + φi, j−1 4

(5.21)

Thus, we can sweep over all the interior nodes and would have one such equation. If there are n nodes, there would be n equation. Therefore, we end up with a system of n equations which are usually solved iteratively as such solution schemes are computationally less expensive. Interested readers can look into references [1–3] for further details. A problem showing the discretization method and the process of obtaining these equations is given in Sect. 5.7.

5.6.2 Finite Volume (FV) Method A primary difference between the FD and the FV method is that the latter uses an integral form of the governing equation to discretize the domain, meaning that the control volume is of finite size. The readers should be aware at this point that the differential equations are a result of applying the conservation laws on a differential control volume [1]. As mentioned in reference [2], this method of formulation can either be used in the FD way or the finite element (FE) way. A lot of details on this method are given in reference [4]. We will not elaborate any further on this technique here in this book.

104

5 Computational Fluid Dynamics

5.6.3 Finite Element Method This is another method of discretizing the domain and finding solutions to the governing PDEs. In this approach, an integral approach is adopted and a weak form of the governing equation is used to arrive at the nodal equations. Let us now discuss about how to discretize and arrive at a set of equations for the irrotational, incompressible flow system given by Eq. (5.12) Since this is a 2D domain, hence we need to use two-dimensional elements. We could use the following two types of elements. (a) Linear triangular elements When a linear triangular element is used for discretizing the domain as shown in Fig. 5.3, we assume that φ, the variable we ought to solve for, on the element would depend on a linear polynomial given by φ(x, y) = α1 + α2 x + α3 y

(5.22)

This is known as the interpolation function and the point (x, y) here lies inside the triangle. To evaluate the values of α1 , α2 and α3 , we have to use the nodal (vertices) values of the triangle, given by φi = Φi , i ∈ [i, j, k]

(5.23)

where Φi ’s are nodal values. This information yields Φi = α1 + α2 X i + α3 Yi , i ∈ [i, j, k]

(5.24)

From these three equations, we can express α1 , α2 and α3 in terms of Φi , Φ j and Φk as follows. α1 =

( ) ) ]⎫ X k Y j Φ)i + (X k Yi − X i Yk )Φ j (+ X i Y j)− X] j Yi Φk ⎬ X j Yk − [( Y j − Yk) Φi + (Yk − Yi )Φ j + (Yi − Y j Φ α2 = 21[( A )k ] ⎭ α3 = 21A X k − X j Φi + (X i − X k )Φ j + X j − X i Φk (5.25)

1 2A

[(

with I I I 1 X i Yi I I I 2 A = II 1 X j Y j II = 2(area of the triangle) I1 X Y I k k

(5.26)

Substituting these values in (5.6.3.3) gives φ(x, y) = Ni Φi + N j Φ j + Nk Φk

(5.27)

5.6 Discretization Methods

105

where the shape functions, Ni' s at any location (x, y) are given by ⎫ Ni = 21A[[ai + bi x + ci y]] ⎬ N j = 21A a j + b j x + c j y ⎭ Nk = 21A [ak + bk x + ck y]

(5.28)

The values of these ai' s, bi' s and ci' s appearing in (Eq. 5.28) are given in Table 5.1 (b) Bilinear rectangular elements. A typical bilinear element chosen in 2D domains is shown in Fig. 5.4. This is of length 2b and width 2a. Note that there are several coordinate axes shown in the same figure. The location of the vertices of the element would be different in different systems (Fig. 5.4). The interpolation function φ(x, y), in terms of local coordinates (s, t), is given as φ = C1 + C2 s + C3 t + C4 st

(5.29)

Again, going through a similar exercise of figuring out Ci ’s and then finding out the shape functions corresponding to the four nodes of the element, we get Table 5.1 Coefficients of the linear polynomial for the shape functions

i j k

Fig. 5.3 Geometry and φ(x, y) over a linear triangular element. From [5]

a ( ) X j Yk − X k Y j

b ( ) Y j − Yk

c (

(X k Yi − X i Yk ) ( ) X i Y j − X j Yi

(Yk − Yi ) ( ) Yi − Y j

(X i − X k ) ( ) X j − Xi

Xk − X j

)

106

5 Computational Fluid Dynamics

Fig. 5.4 Al and φ(x, y) over a bilinear triangular element. Taken from [5]

)( )⎫ ( s 1 − 2at ⎪ Ni = 1 − 2b ⎪ ( ) ⎬ s 1 − 2at N j = 2b st ⎪ Nk = 4ab ( ) ⎪ ⎭ s t Nm = 2a 1 − 2b

(5.30)

The relation between the (s, t) and the (q, r ) coordinates is s = (b + q) & t = (a + r )

(5.31)

which could be used to get the shape functions in terms of the centroidal axes (q, r ) [5, 6] ( )( )⎫ Ni = 41 1 − qb 1 − ar ⎪ ( )( )⎪ ⎬ N j = 14 1 + qb 1 − ar ( )( ) Nk = 14 1 + qb 1 + ar ⎪ ( )( )⎪ ⎭ Nm = 41 1 − qb 1 + ar

(5.32)

Now that we have computed the various shape functions associated with discretizing the domain, we would postpone our discussion of using the integral approach to arrive at individual nodal equations until the next chapter. What follows next is an example showing the use of the FD methods before moving on to the next chapter dedicated for FE methods.

5.7 CFD Solutions to Simple Potential Flows

107

5.7 CFD Solutions to Simple Potential Flows As we are mainly concerned about potential flows, let us look at a simple example of how CFD can be used to solve the flow field in such cases. In particular, we will use the FD technique as this is fairly simple and still can yield meaningful results. As mentioned earlier, this technique is based on replacing the governing equation with finite difference equations (algebraic equations) as outlined in Sect. 5.6.1. The governing equation for such steady, irrotational, inviscid flow, in terms of stream function, is the Laplace equation ∂ 2ψ ∂ 2ψ + =0 ∂x2 ∂ y2

(5.33)

In addition to the governing equation, we would need a knowledge of the value of ψ on the boundary, either directly or as a derivative to arrive at the exact flow solution. A simple case demonstrating the solution of a simple potential flow, in terms of ψ, through a duct with a finite area change over a certain length is presented in the next section.

5.7.1 Flow Through a Duct with Changing Area The geometry of the duct along with the relevant information is shown in Fig. 5.5. There are three sections of the duct, each with a length of 1 m. The width of the duct is 1 m at the inlet, and 2 m at the outlet. The expansion in the width of 1 m occurs over the central 1 m length of the duct. Inlet velocity is specified as 5 m/s. Let us now discretize this domain. Let us introduce nodes in the x- and y-directions all along the flow domain. Furthermore, let us choose the increments in a manner such

Fig. 5.5 Geometry of the duct along with boundary conditions at the inlet and outlet

108

5 Computational Fluid Dynamics

Fig. 5.6 Finite difference discretization of the flow domain by placing interior and boundary nodes. Note that Δx = Δy = 0.2m for the model developed here. The stream function values along the boundary could be calculated from the stream function definition

that Δx = Δy = 0.2m. This would place 6 nodes in y-direction at the inlet, 16 nodes in the flow direction (x-direction) and 11 nodes in the y-direction at the outlet. All such nodes are shown in Fig. 5.6. Altogether, we would find that there are 91 internal nodes (inside the flow passage and not on the boundary). Additionally, we have placed 45 boundary nodes placed along the inlet, outlet and the lower and upper walls of the duct. Let us choose to identify any node by (i, j ) where i is the number in the x-direction counted from the inlet towards the right and j is the number in the y-direction counted from bottom to the top. Hence, the variable i varies from 1 to 16 and j varies from 1 to 11. From the definition of the stream function, we can figure out what the value of the stream function would be on the boundary nodes. Stream function values represent the volume flow rate per unit depth. Assuming the duct to be of unit depth and arbitrarily assigning a value of ψ = 0 to the lower wall of the duct, we can compute the values( of stream function for all the boundary nodes. Since the flow rate per unit ) depth is 10 ms (1 m) = 10 m2 /s, the boundary nodes on the top wall should have 2 a value of ψ = 10 ms . Left are the nodes on the inlet. Arguing that the increase in width leads to the increased volume flow rate in a linear fashion; hence, the ψ values on the inlet nodes will undergo a linear increase, i.e. ψ will increase from ψ = 0 at node (1, 6) to ψ = 10 at the node (1, 11) at the top wall. This means an increase of 2 2 ms for each node in the y-direction. A similar argument could be put forth for the 2 boundary nodes at the outlet and the change in the value of ψ would be 1 ms at the outlet nodes. This completes the definition of the boundary in terms of the stream function. Our problem definition is now complete and we can move to find the solution. This means that we need to find the value of ψ for all the interior nodes subject

5.7 CFD Solutions to Simple Potential Flows

109

to satisfying the Laplace equation for ψ along with the boundary conditions just described in the previous paragraph. As mentioned in Sect. 5.6.1, the partial derivatives are approximated based on the numerical evaluation of derivatives depending on the ψ values. At any interior node (i, j), (

∂ψ ∂x

)

ψi+1, j − ψi, j Δx

(5.34)

≈

ψi+1, j − 2ψi, j + ψi−1, j Δx 2

(5.35)

≈

ψi, j+1 − 2ψi, j + ψi, j−1 Δy 2

(5.36)

≈ i, j

and (

∂ 2ψ ∂x2

) i, j

Similarly, (

∂ 2ψ ∂ y2

) i, j

Using the above two equations in the governing equation (Eq. 5.33), we get ψi+1, j − 2ψi, j + ψi−1, j ψi, j+1 − 2ψi, j + ψi, j−1 + =0 Δx 2 Δy 2

(5.37)

which simplifies to ψi, j =

) 1( ψi+1, j + ψi−1, j + ψi, j+1 + ψi, j−1 4

(5.38)

This expresses the value of ψi, j at any of the interior node (i, j). The solution is usually sought through an iterative process as a direct solution proves to be extremely costly. Direct methods of solving is cost and memory effective only for low number of nodes. As a general method, the iterative solution is used. This calculation starts with initial guesses of ψi, j at all interior nodes. Note that the initial guesses do not matter as we use a convergence criteria for all nodes ensuring that the results will converge. However, the number of iterations necessary to converge may change depending on the initial guess. By sweeping over all nodes in the x- and then in the y-direction, ψi, j could be found out. Calculations are stopped when the difference in ψ value for the (k +1)th and the kth iteration for any node in the domain falls below a pre-defined criterion, say 10−2 . The iterative process is not shown here. We will produce the final results in a tabular form in Table 5.2. The row numbers and column numbers shown in the table correspond to the location of the nodes from the bottom left and top left respectively. Once we know the value of ψi, j , the velocity components u and j could easily be found out. It must be noted that the error in such computations is proportional to the

2

0

7

6

1

2

3

4

5

6

4

9

8

10

8

1

10

Row (i )

11

0

2.02

4.03

6.03

8.02

10

2

Column ( j )

0

2.05

4.07

6.06

8.04

10

3

0

2.09

4.13

6.12

8.07

10

4

0

2.2

4.26

6.22

8.12

10

5

0

2.44

4.48

6.37

8.20

10

6

0

1.33

3.08

4.84

6.58

8.3

10

7

0

1.0

2.22

3.69

5.24

6.82

8.41

10

8

10

0

1.42 0.63

0.80

2.37

3.45

4.65

5.93

7.26

8.62

10

0

1.77

2.92

4.22

5.61

7.05

8.52

10

9

0

0.44

1.09

1.9

2.83

3.87

5.0

6.19

7.44

8.71

10

11

0

0.66

1.4

2.24

3.18

4.19

5.28

6.41

7.59

8.79

10

12

Table 5.2 Values of the stream function ψ found out using FD technique at all nodes. Adapted from reference [7] 13

0

0.79

1.61

2.5

3.45

4.45

5.5

6.59

7.71

8.85

10

14

0

0.87

1.77

2.7

3.66

4.46

5.69

6.74

7.82

8.91

10

15

0

0.94

1.89

2.86

3.84

4.84

5.85

6.88

7.91

8.95

10

16

0

1

2

3

4

5

6

7

8

9

10

110 5 Computational Fluid Dynamics

References

111

square of the distance between the elements, i.e. Δx 2 . If we choose a lower value of Δx, we would get a much more accurate solution. The downside is that there will be an increase in the number of nodes leading to increased computational time and resources. In general, CFD analysts try to strike a balance between the accuracy and the cost involved. This is a matter of experience and is non-trivial in nature [7].

References 1. Anderson JD (1995) Computational fluid dynamics: the basics with applications, Intl. edn., McGraw-Hill International Edition, Singapore 2. Chung TJ (2002) Computational fluid dynamics. Cambridge University Press, Cambridge, U.K. 3. Pletcher RH, Tanenhill JC, Anderson DA (2012) Computational fluid mechanics and heat transfer, 3rd edn. CRC Press, Boca Raton, U.S.A 4. Veerstag HK, Malalasekera W (2007) An introduction to computational fluid dynamics: the finite volume method, 2nd edn. Prentice Hall, Harlow, U.K 5. Segerlind L (1984) Applied finite element analysis. Wiley 6. Reddy JN (1993) An introduction to the finite element method, 2nd edn. McGraw-Hill, Singapore 7. White FM (2011) Fluid mechanics, 7th edn. McGraw-Hill, India

Chapter 6

Finite Element Formulation of Field Problems

In this chapter, we will look into the numerical solution of the Laplace equation mainly by using the finite element discretization scheme which was introduced in the last chapter. This represent a powerful tool for solving the Laplace equation when analytical solutions do not exist or the geometry and physics of the problem preclude analytical solutions.

6.1 Two-Dimensional Field Equations In general, the 2-D field equations refer to the field definition where the entire field could be described in terms of one single value, like the velocity potential φ or the stream function ψ. This formulation allows us to calculate the velocity in terms of the stream function or potential function as mentioned in Chap. 4. In this chapter, we will look at the finite element formulation and solution of several governing equations. It will be noticed later that the Laplace equation serves as the governing equation for many physical phenomena like irrotational flow of fluids, electrostatic field due to charges, groundwater flow, steady state heat transfer and so on. The significance of the terms in the Laplace equation will be introduced and explained at appropriate times.

6.1.1 Governing Differential Equations As mentioned in the earlier chapters, the governing equation for irrotational flow in terms of the velocity potential is given by

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. B. Chattopadhyay et al., Mathematical Modeling of Physical Systems, Advances in Intelligent Systems and Computing 1436, https://doi.org/10.1007/978-981-19-7558-5_6

113

114

6 Finite Element Formulation of Field Problems

∂ 2φ ∂ 2φ + =0 ∂x2 ∂ y2

(6.1)

We would employ the finite element technique to find a numerical solution to this partial differential equation in 2-D.

6.1.2 Integral Equations for Element Matrices The main aspect of the FE method is to discretize the domain (divide the domain into small elements), convert the given PDE into the weak form and apply the weak form over all the smaller individual elements. The element contribution is given by {

} R (e) = −

(

{ [N ]T A

) ∂ 2φ ∂ 2φ + dA = 0 ∂x2 ∂ y2

(6.2)

Here [N ] denotes a row vector containing the shape functions of the elements and φ = φ(x, y) is known as the interpolation function which need not be continuous between the elements due to the weak form. We can remove the second-order derivatives in (Eq. 6.2) using the product rule of differentiation. ( ) ∂ ∂ 2φ ∂[N ]T ∂φ T ∂φ = [N ]T 2 + [N ] ∂x ∂x ∂x ∂x ∂x

(6.3)

Using this result in (Eq. 6.3) to evaluate the first term on the right-hand side of (Eq. 6.2)gives { −

∂ 2φ dA = − [N ] ∂x2

{

T

A

A

( ) { ∂ ∂[N ]T ∂φ T ∂φ dA + dA [N ] ∂x ∂x ∂x ∂x

(6.4)

A

The first area integral on the right in (Eq. 6.4) could be expressed in terms of line integral along the boundary by applying Green’s theorem, and is expressed as { − A

( ) { ∂φ ∂ T ∂φ d A = − [N ]T cos θ dΓ [N ] ∂x ∂x ∂x

(6.5)

Γ

Therefore { −

[N ]T A

Similarly

∂ 2φ dA = − ∂x2

{ [N ]T Γ

∂φ cos θ dΓ + ∂x

{ A

∂[N ]T ∂φ dA ∂x ∂x

(6.6)

6.1 Two-Dimensional Field Equations

{ −

∂ 2φ dA = − [N ] ∂ y2

115

{

A

∂φ sin θ dΓ + [N ] ∂y

{

T

T

Γ

A

∂[N ]T ∂φ dA ∂y ∂y

(6.7)

Putting back (Eq. 6.6) and (Eq. 6.7) in (Eq. 6.2), we get {

} R (e) = −

(

{ [N ]T Γ

) ) { ( ∂φ ∂[N ]T ∂φ ∂φ ∂[N ]T ∂φ cos θ + sin θ dΓ + + dA ∂x ∂y ∂x ∂x ∂y ∂y A

(6.8) If we now introduce the following function, } { φ (e) = [N ] φ (e)

(6.9)

then (Eq. 6.8) could be rewritten as {

R

(e)

}

(

{ =−

[N ] Γ

{ ( + A

T

∂φ ∂φ cos θ + sin θ ∂x ∂y

) dΓ

) { } ∂[N ]T ∂[N ] ∂[N ]T ∂[N ] + d A φ (e) ∂x ∂x ∂y ∂y

(6.10)

This is usually written as {

} { } [ ]{ } R (e) = I (e) + k (e) φ (e) = 0

(6.11)

where {

} I (e) = −

) ∂φ ∂φ cos θ + sin θ dΓ ∂x ∂y

(6.12)

) ∂[N ]T ∂[N ] ∂[N ]T ∂[N ] + dA ∂x ∂x ∂y ∂y

(6.13)

[N ]T Γ

[ (e) ] k =

(

{

{ ( A

{ (e) } Here, is the contribution due to the line integral over the entire boundary while [ (e) ] I k denotes contributions internal to the domain. The gradient vector of φ could be written as [

such that

∂φ ∂x ∂φ ∂y

]

[ =

∂[N ] ∂x ∂[N ] ∂y

]

{

} { } φ (e) = [B] φ (e)

(6.14)

116

6 Finite Element Formulation of Field Problems

[ [B] =

∂[N ] ∂x ∂[N ] ∂y

] (6.15)

Then the second integrand in equation (Eq. 6.10) and mentioned in (Eq. 6.13) could easily be expressed as [ (e) ] k =

{ ( A

) { ∂[N ]T ∂[N ] ∂[N ]T ∂[N ] + d A = [B]T [B]d A ∂x ∂x ∂y ∂y

(6.16)

A

6.1.3 Element Matrices: Triangular Elements If we choose to use triangular elements over the domain, then over the triangular region represented by each element, φ is defined as ]{ } [ φ e = Ni N j Nk φ (e)

(6.17)

These shape functions, for a triangular element, are given by [1–3] ⎫ Ni = 21A((ai + bi x + ci y)) ⎬ N j = 21A a j + b j x + c j y ⎭ Nk = 21A (ak + bk x + ck y)

(6.18)

where the coefficients a, b, c are given in Chap. 5. The gradient vector, denoted by [B], is then [ [B] =

∂[N ] ∂x ∂[N ] ∂y

]

[ =

∂ Ni ∂ N j ∂ N k ∂x ∂x ∂x ∂ Ni ∂ N j ∂ N k ∂y ∂y ∂y

]

] [ 1 bi b j bk = 2 A ci c j ck

This matrix is thus a constant, and therefore { { [ (e) ] T T k = [B] [B]d A = [B] [B] d A = [B]T [B]A A

(6.19)

(6.20)

A

The product of quantities on the right-hand side is mentioned below [

k

] (e)

⎤ ⎡ 2 bi + ci2 bi b j + ci c j bi bk + ci ck 1 ⎣ bi b j + ci c j b2j + c2j b j bk + c j ck ⎦ = [B]T [B]A = 4A bi bk + ci ck b j bk + c j ck bk2 + ck2

(6.21)

6.1 Two-Dimensional Field Equations

117

6.1.4 Element Matrices: Rectangular Elements Instead of using triangular elements, if we use rectilinear elements, then we ought to go through the exercise mentioned in the previous section for rectilinear elements. Note that, in the last chapter, care was taken to derive the rectilinear form of the equation in an orthogonal st coordinate system rather than the x y coordinate system, which presents a problem as all the integrals here are with respect to the x y system with d A = d xd y. However, the readers should be aware that the elements could be mapped back and forth between the two coordinate systems as presented below. As the st system is parallel to the x y system, unit lengths in s & x direction are the same. The same would be true in the t & y direction. This would lead to {

{ f (x, y)d xd y = A

f (s, t)dsdt

(6.22)

A

Since x&s are mapped, therefore,

∂s ∂x

=

∂t ∂y

= 1. This leads to the fact that

∂ Nβ ∂s ∂ Nβ ∂ Nβ = = ∂x ∂s ∂ x ∂s

(6.23)

∂ Nβ ∂t ∂ Nβ ∂ Nβ = = ∂y ∂t ∂ x ∂t

(6.24)

and

The shape functions for a rectilinear element are (s) (t ) − + Ni = 1 − 2b ( s ) 2ast − 4ab N j = 2b st Nk = 4ab (t ) st Nm = 2a − 4ab

st 4ab

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

(6.25)

Once we have the shape functions, the gradient matrix [B] is [ [B] =

∂ Ni ∂ N j ∂ N k ∂ N m ∂x ∂x ∂x ∂x ∂ Ni ∂ N j ∂ N k ∂ N m ∂y ∂y ∂y ∂y

]

[ ] 1 −(2a − t) (2a − t) t −t = 4ab −(2b − s) −s s (2b − s)

(6.26)

Here we have made use of the fact that the derivatives are the same as mentioned in (6.23) and (6.24). Noting that in this case, the area of the element is A = 4ab, we can then compute the product of [B]T [B]A and the final result is shown below.

118

6 Finite Element Formulation of Field Problems

⎡

2 ⎢ [ (e) ] a −2 ⎢ k = [B]T [B]A = 6b ⎣ −1 1

−2 2 1 −1

−1 1 2 −2

⎡ ⎤ 1 2 1 ⎢ 1 2 b −1 ⎥ ⎢ ⎥+ 2 ⎦ 6a ⎣ −1 −2 2 −2 −1

−1 −2 2 1

⎤ 2 −1 ⎥ ⎥ 1 ⎦ 2

(6.27)

6.2 Point Sources and Sinks Point sources and heat sinks are often encountered in fluid flows and heat transfer problems. These are called point sources/sinks as they act over an extremely small area as compared to the overall domain of interest. The governing equation, under such circumstances, is altered to ∂ 2φ ∂ 2φ + +Q=0 ∂x2 ∂ y2

(6.28)

Going by the method developed in earlier sections, the element contribution, in this case, would be given by {

} R (e) = −

(

{ [N ]T A

) ∂ 2φ ∂ 2φ + + Q dA ∂x2 ∂ y2

(6.29)

which yields {

R

(e)

}

{

= I

(e)

}

[

]{ (e) } + k φ − (e)

{ Q [N ]T d A

(6.30)

A

{ } As}mentioned earlier, I (e) is due to the specified boundary conditions while the { (e) f is an integral defined as {

} f (e) =

{ Q[N ]T d A

(6.31)

A

{ } Clearly, f (e) denotes the response of the elements to the point source or since. Note that the way (Eq. 6.28) is written, Q > 0, represents a sink, while Q < 0, represents{ a source. } {In the } next two sections, we will look at the evaluation of the matrices I (e) and f (e) . But before we deal specifically with point sources or sinks, we need to look into the way boundary conditions are specified for various problems.

6.2 Point Sources and Sinks

119

6.2.1 Derivative Boundary Conditions There are usually three types of boundary conditions occurring in 2-D field problems. The first kind, known as Dirichlet boundary conditions, specify the dependent variable, φ, directly on the boundary. The second type, known as the von Neumann or ∂φ on the boundary. boundary condition, specify the derivative of the variable, ∂φ ∂x ∂y A third kind, known as the mixed kind, is where the derivative of the dependent variable is proportional to the difference between the value of the unknown dependent variable at the boundary and a known value of the same variable far from the boundary. For the first kind, φb = φ is known on the boundary. It is to be noted that the second and the third kind of boundary conditions could be formalized as ∂φ ∂φ cos θ + sin θ = −Mφb + S ∂x ∂y

(6.32)

∂φ = −Mφb + S ∂n

(6.33)

or,

Here, φb is unknown at the boundary. This refers to the mixed boundary or the derivative boundary condition. When M = 0, this refers to the von Neumann boundary condition. From (Eq. 6.34) {

I

(e)

}

(

{ =−

[N ]

T

Γ

) ∂φ ∂φ cos θ + sin θ dΓ ∂x ∂y

(6.34)

with the line integral being carried out in an anticlockwise manner. If rectangular elements are used, then the integral in (Eq. 6.34) is the sum of four integrals along the four sides of the rectangle. Each of those four integrals will further consist of the sum of the integral of two terms, one for each term shown (Eq. 6.34). We can write them as different components, {

} { (e) } { (e) } + Ii I (e) = Ibc

(6.35)

where {

(e) Ibc

}

(

{ =−

[N ] Γbc

T

) ∂φ ∂φ cos θ + sin θ dΓ ∂x ∂y

(6.36)

refers to the integral {evaluated } over the element sides bc where the boundary condition (e) then refers to the integral with no boundary condition is specified (Γbc ). Ii specified on them. Therefore, there must be interelement continuity that has to be

120

6 Finite Element Formulation of Field Problems

maintained at the junction where there is boundary condition specified on one side and no boundary condition specified on the other side. For the element sides with boundary condition specified on them as given in (Eq. 6.32), the integral in (Eq. 6.37) becomes {

} { (e) Ibc =

Γbc

[N ]T (Mφb − S)dΓ

(6.37)

When φb = φ is specified directly on the boundary, then, { { {

(e) Ibc

}

} { (e) Ibc = } { (e) Ibc = {

{ =

Γbc

Γbc

Γbc

[N ]T (Mφ − S)dΓ

(6.38)

{

Mφ[N ]T dΓ − { Mφ[N ]T dΓ −

S[N ]T dΓ

(6.39)

S[N ]T dΓ

(6.40)

Γbc

Γbc

} [ ] { } (e) (e) Ibc = Mφ k (e) M − fS

(6.41)

with [ ] { k (e) = M

[N ]T dΓ

(6.42)

} { f S(e) =

S[N ]T dΓ

(6.43)

Γbc

and, {

Γbc

} { When φb is unspecified on the boundary, then, φb = [N ] φ (e) , and (Eq. 6.44) becomes { } { ( { } ) (e) Ibc = (6.44) [N ]T M[N ] φ (e) − S dΓ {

(e) Ibc

}

Γbc

({ =

T

M[N ] [N ]dΓ Γbc

) {

} φ (e) −

{ S[N ]T dΓ

(6.45)

Γbc

which is usually written as {

Therefore,

} [ ]{ } { (e) } (e) (e) Ibc = k (e) φ − fS M

(6.46)

6.2 Point Sources and Sinks

121

Fig. 6.1 A rectangular element of length 2b and width 2a, showing the various coordinate systems of measuring the length and width. In this case, the boundary consists of each of the four sides i j, jk, km & mi. The outward unit normals’ and the angles they make with the x-axis is also shown for the sides jk and km

[ ] { k (e) = M

M[N ]T [N ]dΓ

(6.47)

S[N ]T dΓ

(6.48)

Γbc

and, {

} { f S(e) =

Γbc

It is assumed that M and S are known for a problem. For rectangular elements, the boundary condition represented through (Eq. 6.32) takes various forms along each of the four sides. This is due to the fact that the value of θ is the angle between the x−axis and outward normal to the side of the rectangles measured in an anticlockwise manner as shown in Fig. 6.1. Along the side jk of the rectangular element, θ = 0◦ , cos θ = = 1, sin θ = 0. Therefore, the boundary condition specified on the side jk is ∂φ ∂x ∂φ −Mφb + S. Along the side km, the boundary condition will be ∂ y = −Mφb + S, as θ = 90◦ for this side.

6.2.2 Evaluation of Element Integrals The integrals represented using (Eqs. 6.42 and 6.43), needs to be evaluated for the elements. Let us compute them for rectangular elements as they are easy. When S is specified over any side i j of a rectangular element (see Fig. 6.1) of length 2b and width 2a, with unit thickness, then the integral could be computed as {

} f S(e) =

⎤ Ni b ⎢ N ⎥ j ⎥ S[N ]T dΓ = ∫ S ⎢ ⎦dq ⎣ N −b k Nm ⎡

{ Γbc

(6.49)

122

6 Finite Element Formulation of Field Problems

The shape functions are described using the orthogonal qr coordinate system placed at the centroid of the element (see Fig. 6.1) with s = b + q & t = a + r . Since Nk = Nm = 0 along the side i j, the above integral is ⎧ ⎤ 1 b−q ⎪ ⎪ ⎨ { } b S ⎢ b+q ⎥ S L 1 ij ⎢ ⎥ f S(e) = ∫ ⎣ 0 ⎦dq = 2 ⎪ 0 −b 2b ⎪ ⎩ 0 0 ⎡

⎧ ⎫ 1⎪ ⎪ ⎪ ⎨ ⎪ ⎬ 1 = Sb ⎪0⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎭ 0 ⎫ ⎪ ⎪ ⎬

(6.50)

Along the other three sides jk, km & mi, these integrals are ⎧ ⎫ ⎧ 0⎪ ⎪ ⎪ ⎪ ⎪0 ⎪ { } S L jk ⎨ 1 ⎬ { (e) } S L km ⎨ 0 (e) fS = = , fS 1⎪ 1 2 ⎪ 2 ⎪ ⎪ ⎪ ⎩ ⎪ ⎩ ⎭ 0 1

⎧ ⎫ ⎪ ⎪1⎪ ⎪ } { S L mi ⎨ 0 ⎬ (e) = & fS (6.51) ⎪ 0⎪ 2 ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ ⎭ 1 ⎫ ⎪ ⎪ ⎬

[ ] In a similar manner, we can also compute the k (e) M matrix by integrating along the four sides of the rectangular element. A 4 × 4 matrix is formed for the product ⎡ ⎢ ⎢ [N ]T [N ] = ⎢ ⎣

Ni2 Ni N j Ni N k Ni N m

Ni N j N 2j N j Nk N j Nm

Ni N k N j Nk Nk2 Nk Nm

Ni N m N j Nm Nk Nm Nm2

⎤ ⎥ ⎥ ⎥ ⎦

(6.52)

and therefore, ⎡ [ ] { = k (e) M

Γbc

⎢ ⎢ M⎢ ⎣

Ni2 Ni N j Ni N k Ni N m Ni N j N 2j N j Nk N j Nm Nk Nm Ni Nk N j Nk Nk2 Ni Nm N j Nm Nk Nm Nm2

⎤ ⎥ ⎥ ⎥dΓ ⎦

(6.53)

Along the side i j of the element, Nk , = Nm = 0. Therefore, for a fixed value of M, ⎡ ⎤ Ni2 Ni N j 0 0 ⎢NN [ ] { N 2j 0 0 ⎥ ⎢ i j ⎥ k (e) = M (6.54) ⎢ ⎥dq M ⎣ 0 0 0 0⎦ Γbc 0 0 0 0 Since r = −a along the side i j,

6.2 Point Sources and Sinks

123

b

b

−b

−b

∫ Ni2 dq = ∫

b

b

−b

−b

∫ Ni N j dq = ∫

Li j 2b (b − q)2 dq = = 4b2 3 3

Li j 2b (b − q)(b + q) dq = = 2 4b 6 6

(6.55) (6.56)

and b

b

−b

−b

∫ N 2j dq = ∫

Li j 2b (b + q)2 = dq = 4b2 3 3

(6.57)

Hence for the side i j, ⎡

2 [ ] ⎢ M L i j ⎢1 k (e) Mi j = 6 ⎣0 0

1 2 0 0

0 0 0 0

⎤ 0 0⎥ ⎥ 0⎦ 0

(6.58)

[ ] For the other sides jk, km & mi, the k (e) M matrices are given respectively by ⎡

0 [ ] M L jk ⎢ (e) ⎢0 k M jk = 6 ⎣0 0 ⎡ 0 [ ] M L km ⎢ (e) ⎢0 k Mkm = 6 ⎣0 0 ⎡ 2 [ ] M L mi ⎢ (e) ⎢0 k Mmi = 6 ⎣0 1

⎤ 0 0⎥ ⎥ 0⎦ 0 ⎤ 0 0 0 0 0 0⎥ ⎥ 0 2 1⎦ 0 1 2 ⎤ 0 0 1 0 0 0⎥ ⎥ 0 0 0⎦

0 2 1 0

0 1 2 0

0 0

(6.59)

(6.60)

(6.61)

2

[ ] If the value of M is specified, then we can find the k (e) matrix for the entire M [ ] (e) ' element by summing up the k Mi j s over all the sides (i.e., four sides for a rectangular element). The effect of the point source or sink is taken care through the integral represented { } by f Q(e) as mentioned in (Eq. 6.31). Since the applied source or sink is concentrated to a point (x0 , y0 ), therefore it is expressed in terms of a Dirac delta function

124

6 Finite Element Formulation of Field Problems

Q = Q ∗ δ(x − x0 )δ(y − y0 )

(6.62)

Therefore, the integral becomes {

}

f Q(e) =

{

⎤ Ni ⎢ Nj ⎥ ⎥ Q[N ]T d A = Q ⎢ ⎣ Nk ⎦ ∗ δ(x − x0 )δ(y − y0 ) d x d y A Nm {

A

⎡

where the Dirac delta function is defined as { 0 x /= x0 δ(x0 ) = 1 x = x0

(6.63)

(6.64)

The finite element equation at the element level is then given by (from (Eq. 6.30)) {

} } { } [ ]{ } { R (e) = I (e) + k (e) φ (e) − f Q(e) = 0

(6.65)

or, {

} [ ]{ } } { (e) Ibc + k (e) φ (e) − f Q(e) = 0

or, [ (e) ]{ (e) } [ (e) ]{ (e) } { (e) } { (e) } k φ + kM φ − fS − fQ = 0

(6.66)

[ ] [ ] In this equation, k (e) and k (e) M could be computed from (Eq. 6.27) and summing { { } } up (Eqs. 6.58–6.61) respectively, f S(e) from (Eq. 6.51) and f Q(e) from (Eq. 6.62) for a rectangular element with known sides 2a & 2b.

6.2.3 Assembly of Element Matrices into Global Matrix } { [ ] [ ] { (e) } Once the individual element matrices, k (e) , k (e) and f Q(e) are found, M , fS then the global matrix is formed by adding appropriate terms. The original governing equation mentioned in (Eq. 6.1) or (Eq. 6.28) is true for the entire domain, i.e. an assembly of all the elements. Therefore, we need to combine/assemble all the element matrices and come up with a global matrix [RG ] and then solve for [RG ] = 0. This assembly is not a simple addition of the element matrices as the nodes of one element are not always shared with the other adjacent elements. Thus, the assembly of the individual element matrices have to be done into a global form, keeping in mind the

6.2 Point Sources and Sinks

125

Fig. 6.2 a Global element assembly using rectangular elements and global node numbers. b Numbering of individual rectangular elements along with the node numbers (element wise) in the global assembly. Element numbers are mentioned at the center of the individual elements. while the global and local node numbers, named anticlockwise, are shown at the nodes in (a) and (b) respectively

nodes that are shared by adjacent elements. This concept of the assembly of element matrices into a global form is explained below with the help of Fig. 6.2a, b using rectangular elements. Consider a domain as shown in Fig. 6.2, made up of only 4 rectangular elements. The global or the assembled domain consists of the four rectangular elements marked with numbers 1, 2, 3 & 4, inside the elements in Fig. 6.2a. The qr axes for each of the elements is also shown for clarity of understanding. The nodes of each of these four elements are numbered in a counterclockwise manner as shown in Fig. 6.2b. From Sect. 2.2.2, for element 1, side 1, in Fig. 6.2b, ⎡

2 [ ] ⎢ M L (1) (1) 12 ⎢ 1 kM = 6 ⎣0 0

1 2 0 0

0 0 0 0

⎤ 0 0⎥ ⎥ 0⎦ 0

(6.67)

0 2 1 0

0 1 2 0

⎤ 0 0⎥ ⎥ 0⎦ 0

(6.68)

Similarly, for element 3, side 2, in Fig. 6.2b, ⎡ 0 (3) ⎢ [ ] M L 23 ⎢ 0 = k (3) M 6 ⎣0 0

The superscript on k M and L i j refers to the element number in the global element assembly and the value of L i(e) j should be taken appropriately to represent the length of the side for the element number (e) at the node in an anticlockwise manner.

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6 Finite Element Formulation of Field Problems

It is easily seen in Fig. 6.2a, that the global element assembly has only 6 nodes and hence the global k M matrix will be a 6 × 6 matrix. It is also noticed that the side joining nodes 2 & 3 of element 1 is coincident with the side formed by joining node numbers 4 & 1 of element 4. Thus, when we assemble the global matrix, node 2 from element 1 will have its contribution added to node 1 of element 4.

References 1. Segerlind L (1984) Applied finite element analysis. Wiley 2. Reddy JN (1993) An introduction to the finite element method, 2nd edn. McGraw-Hill, Singapore 3. Thompson E (2005) Introduction to the finite element method theory, programming and applications. Wiley

Chapter 7

Modelling Approach Using Generalized Theory of Electrical Machines

7.1 The Foundation of the Generalized Theory of Electrical Machines It is well known that the magnetic flux cutting between the stator winding and the rotor winding (or, conductors) of a rotating electrical machine depends on the relative positions of those windings (either on the stator or rotor) and the flux-density established in the air-gap of any electrical machine basically acts as one of the variables to govern the “shaft torque” in “motor mode” or the “terminal voltage” in “generator mode”. Such a concept is visualized in all types of the electrical machine but the analytical methods take different shapes in different types of machines due to the following major factors: (i) Air gap saliency between the stator and rotor (ii) Nature of electrical supply (or, generated electrical power) (d.c or a singlephase a.c or three-phase a.c or two-phase a.c) to the (or from the) stator or rotor windings. The first factor can be looked upon as follows: Air gap length (radial), if varies between some maximum and minimum values, then the air-gap permeance can no longer be assumed as a constant. As a consequence, due to a given magneto motive force, the magnetic flux does not become a singlevalued function. Such findings make the analysis of the machine complicated because the so-called Cylindrical Rotor Theory does not become amenable to such analysis. The second factor also can be highlighted in the following manner: In the case of d.c machines, the magnetic field becomes uniform and steady and such nature do not lead to the development of any method to analyze the machine, based on the concept of equivalent circuits. In the case of a.c machines, depending on single phase or two phase or three phase natures of supply (or, generation) the nature of mmf (or, flux) fields become pulsating magnetic field or rotating magnetic field. As a consequence, the nature of the equivalent circuits become different for the machines © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. B. Chattopadhyay et al., Mathematical Modeling of Physical Systems, Advances in Intelligent Systems and Computing 1436, https://doi.org/10.1007/978-981-19-7558-5_7

127

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7 Modelling Approach Using Generalized Theory of Electrical Machines

based on a different type of a.c supply. For example, the nature of an equivalent circuit of a single-phase induction motor is not so simple. Before developing the equivalent circuit of a single-phase induction motor, it was necessary to decompose a pulsating magnetic field in terms of two rotating magnetic field components, one forward and another, backward component. Such work was not simple and it demanded a good extent of mathematical exercises. As a whole, the overall process of analyzing any electrical machine becomes cumbersome due to the varieties involved in the nature of the electrical power supply (or, generated) fed to (or, from) the stator and/or rotor windings (or, conductors). Based on the above facts and figures, the authors hereby propose to initiate the mathematical modeling of any electrical machine, based on the generalized theory of electrical machines [1, 2], on the following lines:

7.1.1 The Idealized Machine Figure 7.1 shows the common features of all types of rotating electrical machines. With reference to this figure, the outer member is stationary and the inner member gets rotation through the bearings fixed to the stator frame. Basically, this diagram in Fig. 7.1 leads to the conceptual realization that most of the rotating electrical machines can be thought of as an iron cylinder placed within another iron cylinder. The rotor is equivalent to a continuous magnet, and the stator is equivalent to a discontinuous (or, broken) magnet. This discontinuity exists because there exists an air gap between those two members. Magnetic flux (ϕ) lines pass through the air gap. So, the flux path may be described as: lower stator half-air gap-lower rotor halfvia the shaft, upper rotor half-air gap-upper stator half-flux (ϕ) being divided into two parts as ϕ2 and ϕ2 , passing though the left half of the stator outer frame and the right half of the stator outer frame, respectively. This is how the continuity of flux is maintained in a closed magnetic circuit. When the main flux splits into two halves, it indicates a parallel magnetic circuit and when the two halves unite to produce the same flux (ϕ), it again indicates a parallel magnetic circuit. The above-said figure does not give any particular information about the nature of the windings used in a particular type of electrical machine. Keeping this view point in mind, basic knowledge of the general types of windings used in electrical machines is imparted hereby as explained in the next subsection.

7.1.1.1

Classification of Machine Windings

The windings used in electrical machines are classified into three main types: (i)

Field pole winding: Generally, salient poles of d.c machines and salient poles of synchronous machines (salient pole type) function as field poles and windings used in such

7.1 The Foundation of the Generalized Theory of Electrical Machines

129

Captions: (1) Stator Core (2) Main Magnetic Flux ( ) (3)Stator Winding (4) Rotor Core (5) Rotor Winding (6) Bearing (7) Shaft Fig. 7.1 Diagrammatic representation of the common features of all types of rotating electrical machine

cases are concentric windings. These windings are also known as coil windings. However, it does not necessarily mean that field winding must be always of concentric type. For example, in a cylindrical rotor synchronous machine, the field winding is on the rotor and it is a distributed winding (distributed in slots). (ii) Poly-phase winding: Any coil is generally of hexagon type (approximately) and each coil-side of the coil, consisting of a number of conductors, is placed in the slots. When one coil side is placed, the other coil side runs through a particular slot depending on the designed value of the slot pitch. Suppose, phase-R of the winding starts from slot no.1. Then phase-Y (of a three-phase winding system following a phase sequence of R-Y-B (counter-clockwise)) must start from slot no. 25 (120° electrical) if the total no. of slots is considered as 72. Several circuits may be used for each phase. A group of conductors constitute a phase band in a poly-phase winding system. The very important parameter in such winding connections is “No. of slots/pole/phase (SPP)”. (iii) Commutator Winding: The part of a d.c machine, commutator winding, plays the role of a rectifier or inverter, as used in power electronic applications. In other words, in a d.c

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7 Modelling Approach Using Generalized Theory of Electrical Machines

generator, it works as a rectifier (converting a.c to d.c) and in a d.c motor, it works as an inverter (converting d.c to a.c). However, such type of winding is also used in a three-phase A.C Commutator Motor (Schrage Motor). The conductors are placed in slots and are connected to commutator segments in a continuous sequence. The current flows from an external source or a circuit into and out of the winding. This flow of current coming in or going out happens through the brushes placed on the commutator surface. In the case of a d.c machine, the commutator is equivalent to another cylindrical rotor of a smaller diameter (smaller as compared to that of the main rotor of the d.c machine). Figure 7.1 basically indicates an idealized electrical machine. But the complexity lies in the said figure with respect to the actual understanding when the number of magnetic poles of the machine is greater than two. Therefore, to have a better understanding of the functional aspects of any electrical machine in an easy way, it is very important to visualize any machine in the form of an “Idealized two-pole machine” which is explained in the next sub-section.

7.1.1.2

The Idealized Two-Pole Machine

Any electrical machine, in practice, is not straightforward, in construction and that is why its exact representation is also not straightforward. Hence to make the foundation of the generalized theory of electrical machines it was necessitated to develop the framework of an “idealized machine”, which is a two-pole machine. That is why the combination of the two terms, “idealized machine” and “two poles” constitute the nomenclature of the title of the present topic. Most of the windings of any practical machine are distributed windings and these windings are distributed in slots. But in the idealized two-pole machine, such windings and also concentric windings are represented by a single coil. To represent such a concept of a two-pole idealized machine in a diagrammatic form, some basic question hereby is automatically raised and that is a question in context to the exact location of the two elements of any electrical machine on the rotor or on the stator. The answer to such a query lies in the fact that for the purpose of analysis it is immaterial which of the two elements of the machine rotates and which is stationary since its operation depends only on the relative motion between them. As an example, an idealized three-phase salient-pole synchronous machine is shown in Fig. 7.2. In this figure, only one of the two salient poles is indicated. The outer member in Fig. 7.2 carries a field coil F and damper coils KD and KQ. The axis of the pole around which the field coil F is wound is called the ‘Direct Axis’ of the machine. Simultaneously the axis at the right angle (in the counterclockwise direction) with respect to the ‘direct axis’, is called the ‘Quadrature Axis’. At the instant considered, the axis of the coil representing the armature phase A winding makes an angle ‘θ ’ in a counter-clockwise direction with the direct axis. The instantaneous angular speed is expressed by the well-established equation as given by:

7.1 The Foundation of the Generalized Theory of Electrical Machines

131

Fig. 7.2 Diagram of an idealized synchronous machine with field and damper windings on the stator and three-phase armature windings on the rotor

ω=

dθ dt

(7.1)

The positive direction of the current in any coil is into the coil in the conductor nearer to the center of the diagram, as indicated by an arrow in Fig. 7.2. The positive direction of the magnetic flux linking a coil is radially outwards, as indicated by an arrow-head along the axis in the middle of the loop representing the coil (see phase C coil in the figure, shown as a sample example). The basic convention adopted is that the current in a positive direction in a coil sets up a magnetic flux in the positive direction of that axis. The cage damper winding of a practical synchronous machine consists of many circuits carrying different currents and would require a large number of coils for its exact representation. For many purposes, it is sufficiently accurate to use two lumped coils placed on two axes, in place of distributed coils and as an implementation of this concept, the damper windings placed on the direct axis and quadrature axis of the machine, KD- and KQ-coils are shown in Fig. 7.2. Strictly

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7 Modelling Approach Using Generalized Theory of Electrical Machines

speaking, based on the well-known armature m.m.f diagram of a d.c machine, we have the idea that any winding distributed in slots along the periphery of cylindrical stator or rotor, will have a stepped waveform of the mmf distribution in space. This waveform is basically a staircase type waveform and after applying Fourier series analysis, generally such waveform is converted to a sinusoidal distribution if the space harmonics are neglected. The positive peak of the spatial distribution of armature m.m.f in the case of a d.c machine merges with the quadrature(q-)axis. Therefore, in this case, we can assume as if some lumped coil is placed on the q-axis. Similarly, we can also assume as if some coil is placed on the direct (d-) axis. This is the explanation behind the existence of KD and KQ coils. As a second example, Fig. 7.3 is the diagram of an idealized cross-field direct current machine having salient poles on the stator and a commutator winding with main brushes on the direct axis and cross brushes on the quadrature axis. In the diagrams of commutator machines, the convention is adopted that a brush is shown at the position occupied by the armature conductor to which the segment under the brush is connected. The circuits through the brushes are named as D and Q, respectively. In the same diagram, F and G indicate main field winding and quadrature field winding, acting along direct and quadrature axes, respectively. That is why the electrical machine as shown in Fig. 7.3 can be named also as a “Cross Field D.C Machine with Dual Excitation”. The machine represented in Fig. 7.3 may be visualized alternatively by the four coils of Fig. 7.4 corresponding to the four separate circuits of Fig. 7.3. The theory given hereafter is more in the form of having a generalized approach and it leads Fig. 7.3 Diagram of an idealized cross-field d.c machine

7.1 The Foundation of the Generalized Theory of Electrical Machines

133

Fig. 7.4 Diagram of a primitive machine with four coils

to the development of a new machine analysis foundation, named as “Generalized Theory of Electrical Machines”. Hence such an approach is supposed to cover a wide range of electrical machines in a unified manner. A very important portion of this generalization can be applied to the two-axis theory. As per this theory, by means of an appropriate transformation, any machine can be represented by coils on the axes. Such transformations are often called as “Transformation from phase model to axis model”. As per Kron, the two-axis idealized machine, from which many others can be derived, can be called as a “Primitive Machine”. Figure 7.4 shows the diagram of such a primitive machine with one coil on each axis on each element (i.e., rotating and stationary members). F and G are on the stationary member and D and Q are on the rotating member. If the coils of any practical machine are permanently located on the axes, they correspond exactly to those of the primitive machine. On the contrary, if they are not permanently located on direct and quadrature axes, then it becomes necessary to convert the variables of the practical machine to the equivalent axis variables of the corresponding primitive machine, or vice versa. In the language of mathematical modelling, this conversion is termed as ‘the theory of transformation’ where as the “vice versa” is termed as ‘the theory of inverse transformation’. Any particular system of description of a machine is known as a “Reference Frame” and a conversion from one reference frame to another is known as “Transformation”. In order that the effects of rotation in a practical machine may be presentable in an equivalent form of the primitive machine, it is necessary that the windings placed on the moving element of the machine (i.e., rotor) should possess some typical properties. These properties are: (i) A current in the coil produces a m.m.f stationary in space (ii) Nevertheless, a voltage can be induced in the coil by rotation of the moving element.

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7 Modelling Approach Using Generalized Theory of Electrical Machines

In Fig. 7.4, D and Q are the coils which also should possess the said properties. Such a coil, located on a moving element but with its axis stationary, may be termed as a “Pseudo-stationary coil”. Any pseudo-stationary coil must possess the above-said two properties. In applying the two-axis theory to any type of rotating electrical machine the process may be summarized in steps as given below: Step 1: The diagram of the idealized two-pole machine is first set out using the smallest number of coils required to obtain a model of sufficient accuracy. Step 2: A primitive machine model is developed seeing the idealized two-pole model. Step 3: The development as stated in step 2 is possible either directly or using suitable transformation. Step 4: The theory is developed by deriving the voltage balance equations relating the voltages and currents of the coils placed on the primitive machine model and in addition, a torque balance equation relating the torque to the currents. Step 5: The speed appears as a variable in the torque balance equation as shown below (example of the machine in motor mode): Te − TL = J

dω dt

(7.2)

Where Te TL J ω dω dt

Electromagnetic torque offered by the motor. Load torque. Combined moment of inertia of the motor and load Motor speed in radians per second. Time derivative of angular velocity (or, motor speed in radians/second).

At this stage, it is needed to develop the theory involving the activities of the coils of the primitive machine model. For fulfilling this need, the details of magnetic flux in terms of useful flux and leakage flux and the sign conventions of voltages and currents in any coil of the primitive machine are very much needed to know. Now the fact is that even though a rotating machine differs from a static transformer because of its rotation, the study of a transformer from the perspective of the above said view-points (as specified in the preceding sentence) may also be quite helpful. This study is presented in the next section.

7.1 The Foundation of the Generalized Theory of Electrical Machines

135

7.1.2 The Circuit View of a Two-Winding Transformer-Explanation of Sign Convention and the Per-Unit System for Electrical Quantities In this section, a two-winding transformer is considered to explain certain facts and figures in relation to the analysis of rotating electrical machines. Basically, a two-winding transformer is constructed based on the principle of two magnetically coupled concentrated coils. In practice, this magnetic coupling is made strong using an iron core. However, for the sake of understanding the principle of mutual flux, the system containing two coils without an iron core is shown in Fig. 7.5. Based on Fig. 7.5, the concepts of main flux and leakage flux, and sign conventions of voltages and currents are explained in the subsequent sections.

7.1.2.1

Main Flux and Leakage Flux

With reference to Fig. 7.5, the magnetic flux in a transformer can be split into three parts: (a) Main magnetic flux ϕ linking both coils (b) Primary leakage flux ϕl1 , due to current i 1 , and linking coil-1 but not coil-2 (c) Secondary leakage flux ϕl2 , due to current i 2 , and linking coil-2 but not coil-1. This statement indicates the fundamental definition of the leakage flux. It is the flux due to the current in one coil, which links that coil but not the other. Actually, from the view point of electrical machine design, leakage flux path length mostly should

Fig. 7.5 Diagram of a transformer (without iron core)

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7 Modelling Approach Using Generalized Theory of Electrical Machines

comprise of air path length whereas the portion of the iron path is very minimum. This statement is obviously valid when the iron-air combination is considered for the calculation of the permeance of flux paths. Another critical concept in a practical transformer having an iron core is involved that it becomes very difficult to calculate the permeance of a leakage flux path (either in the primary side or in the secondary side). Such difficulty arises due to the fact that the traditional circuit theory does not give an accurate mathematical formulation to find the length of the path of leakage flux. However, such a problem is not generally encountered if electromagnetic field theory is applied to such design work.

7.1.2.2

Sign Conventions

The sign convention adopted for voltages and currents is as follows: (i) ‘u’ represents the voltage impressed on the coil from an external source. (ii) ‘i’ represents the current measured in the same direction as ‘u’. With the said convention, the instantaneous electrical power ‘ui’ flows into any electrical circuit from any external source, if both ‘u’ and ‘i’ are positive in sign. To take a simple example, the equation for a circuit with resistance ‘R’ and a capacitance ‘C’ (parallel circuit) is: i=

( ) du 1 u+C R dt

(7.3)

where (i) ‘i’ is the impressed current ’ is the induced current (current induced due to electrostatic field) (i) ‘−C du dt (ii) ‘−C du ’ is the internal current, and ( 1 ) dt (iii) ‘ R u’ is the ohmic current. 7.1.2.3

The Per-Unit System

In connection with the development of the theories of electrical machines and power systems, ‘per-unit system’ becomes more advantageous in many ways as compared to the ‘actual units’, to express the associated quantities. With reference to Fig. 7.5, the following discussions are put in steps centering the concept and implementation of per unit (p.u) quantities: Value Per-unit = Actual Base Value Number of turns in secondary winding We assume that Number of turns in primary winding = N The base value of secondary voltage = (N)× ) base value of primary voltage ( the The base value of the secondary current = N1 × the base value of the primary current (e) The base value of time is one second.

(a) (b) (c) (d)

7.1 The Foundation of the Generalized Theory of Electrical Machines

137

(f) In per-unit system, ϕ = L 12 (i 1 + i 2 )

(7.4a)

ϕl1 = L l1 i 1

(7.4b)

ϕl2 = L l2 i 2 ,

(7.4c)

where L 12 is the ‘mutual inductance’ and each of L l1 and L l2 is the ‘leakage inductance’. (g) The induced voltage in coil-1 is −

d (ϕ + ϕl1 ) dt

(h) The impressed voltage (on any coil) opposes fully to the induced voltage (in that coil). (i) The value of the impressed voltage on coil-1 and on coil-2 can be expressed, respectively as (in the time domain): u 1 = R1 i 1 + (L 12 + L l1 )

di 1 di 2 + (L 12 ) dt dt

(7.5)

u 2 = R2 i 2 + (L 12 + L l2 )

aι2 aι1 + (L 12 ) dt dt

(7.6)

and

(j) Applying Laplace Transform to the Eqs. (7.5) and (7.6), with initial conditions being relaxed, it yields: U1 (s) = R1 I1 (s) + (L 12 + L l1 )s I1 (s) + (L 12 )s I2 (s)

(7.7)

U2 (s) = R2 I2 (s) + (L 12 + L l2 )s I2 (s) + (L 12 )s I1 (s)

(7.8)

and

With reference to the above-said equations, U1 (s) = L[u 1 (t)]

(7.9a)

U2 (s) = L[u 2 (t)]

(7.9b)

I1 (s) = L[i 1 (t)]

(7.9c)

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7 Modelling Approach Using Generalized Theory of Electrical Machines

I2 (s) = L[i 2 (t)].

(7.9d)

where ‘L’ indicates the Laplace Transform operator. (k) From Eqs. (7.7) and (7.8), we have, in matrix form: [

] [ ] ][ U1 (s) R1 + (L 12 + L l1 )s I1 (s) (L 12 )s = R2 + (L 12 + L l2 )s U2 (s) I2 (s) (L 12 )s

(7.10)

The matrix representation in (7.10) may be expressed, in general form (involving the transformed transformer impedance matrix, [Z T (s)]) as given by: [U (s)] = [Z T (s)][I (s)].

(7.11)

where, ] U1 (s) U2 (s) ] [ I1 (s) [I (s)] = I2 (s) [

[U (s)] =

(7.12a)

(7.12b)

and [ [Z T (s)] =

R1 + (L 12 + L l1 )s (L 12 )s R2 + (L 12 + L l2 )s (L 12 )s

] (7.12c)

On the basis of this matrix Eq. (7.10), it is possible to view the self- inductance (or, complete inductance) and the leakage inductance parameters in some other interesting form (only when per-unit values are considered), and this discussion along with the necessary mathematical derivations are presented in the next section.

7.1.2.4

Complete Inductance and Leakage Inductance

If the induced voltage magnitude is ‘e’ and total flux linkage is ‘Ψ’, then as per the law of electromagnetic induction, it can be written that e=

dΨ = dt

(

dΨ di

)(

di dt

) (7.13)

where ‘i’ and ‘t’ represent the current and time instant, respectively. On the other hand, the inductance (L) of any coil is defined as the rate of change of magnetic flux linkage per unit current. Hence, we have

7.1 The Foundation of the Generalized Theory of Electrical Machines

L=

dΨ . di

139

(7.14)

Substituting Eq. (7.14) in Eq. (7.13), it yields e=L

di dt

(7.15)

Based on Eq. (7.15), the term ‘inductance’ can be defined as the voltage induced by the total flux set up when the time rate of change of current is unity. With reference to Fig. 7.5, applied to the primary coil (coil no.-2), this inductance would correspond to the flux (ϕ + ϕl1 ) when i 2 is zero. It is known as Complete Self-inductance and is represented by the symbol ‘L 11 ’. Now, in context to the same diagram (Fig. 7.5), it reveals that mutual flux (ϕ) must correspond to the mutual inductance, L 12 and the leakage flux associated with the primary coil (ϕl1 ) must correspond to its leakage inductance, L l1 . Therefore, we can write qualitatively (in per-unit), L 11 = L 12 + L l1

(7.16)

On the basis of same explanation, similarly for the coil no.-2, the complete selfinductance, L 22 can be expressed as, L 22 = L 12 + L l2

(7.17)

It may be noted that Eqs. (7.16) and (7.17) are only valid in per-unit, not in terms of the actual quantity. The logic behind this statement is quite straight forward because the magnetic circuit permeances are different for the mutual flux path and leakage flux path. Furthermore, the number of turns of two coils (in Fig. 7.5) may be different. That is why in terms of actual quantities, the mutual inductance term and the leakage inductance term should not be added up, conceptually. However, as the value of ‘per-unit quantity’ does not depend on the term, ‘number of turns’ and as the base value of permeance of the path of the “combination of mutual flux and leakage flux” may be approximated as equal to that of the mutual flux path as well as the leakage flux path (without loss of much accuracy), the ‘complete self-inductance’ may be decomposed into the terms, ‘mutual inductance’ and ‘leakage inductance’. In this context, it also may be noted that for a practical transformer, leakage inductance is much smaller than the self-inductance. Based on Eqs. (7.16) and (7.17), the transformed voltage balance equation of the transformer in Fig. 7.5, i.e., Eq. (7.10) may be modified as, [

] [ ] ][ U1 (s) R1 + (L 11 )s I1 (s) (L 12 )s = R2 + (L 22 )s U2 (s) I2 (s) (L 12 )s

(7.18)

At this stage, the question may arise what is the actual benefit of using per-unit values in calculation work because considerable amount of efforts have been put into

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7 Modelling Approach Using Generalized Theory of Electrical Machines

the treatment dealing with the per-unit quantities in this section and in the preceding section. The answer to this query is discussed in the next section.

7.1.2.5

Advantages of the Per-Unit System

(i)

The per-unit system is of great benefit in making design calculations for machines, because it makes the comparison between different machines much easier. Corresponding quantities are of the same order of magnitude even for widely different designs. (ii) In the formulation of the theory, the per-unit system has the great merit that the per-unit values are independent of the number of turns. (iii) Addition of mutual inductance and leakage inductance directly gives the selfinductance.

7.1.3 Magneto-Motive Force and Flux in the Rotating Machine The m.m.f. (magneto-motive force) in any winding or coil, basically is the product of current and number of turns. At a fixed time, this product fundamentally indicates a space-dependent function in a distributed winding as a part of a rotating electrical machine. The terminology, “space” in a rotating electrical machine basically refers to the position of a slot in the cylindrical coordinate system. In any machine, all the concerned windings carry currents (all of them may be basic currents or some of them may be induced currents) and the individual m.m.f.-s are developed. The sum of all these magneto-motive forces contributes to the establishment of the resultant flux. The responsibility of the magnetic flux in a rotating electrical machine (induction type) is a four-fold action: (i)

Firstly, due to a flux in one member, the voltage is induced in the other member and in turn, current flows in the second member as an induced current. (ii) This induced current produces its own flux in the second member. (iii) Interaction of the individual flux components sets up the resultant air gap flux (iv) Interaction of the resultant air-gap flux with the induced current leads to the development of the electromagnetic torque, exerted on the second member if it is mechanically free to move. The discussion in the preceding paragraph was just an introduction in nutshell, to the process of the resultant magneto-motive force and flux development, in a rotating electrical machine. But for having a clearer picture of the said process, there is a necessity of presenting the in-depth discussions on ‘Main air-gap flux’ and ‘Magneto-motive force’, separately and these are presented in the subsequent sections.

7.1 The Foundation of the Generalized Theory of Electrical Machines

7.1.3.1

141

Main Air-Gap Flux

A rotating electrical machine is considered to be mainly, a copper-iron-air combination, if the windings are wound using copper wires (which is usually followed, mostly in all the machines). Now when a current is flowing in any winding, the magnetic flux spreads throughout the whole machine, leaving the copper material. Thereafter, the flux-spreading practically becomes only through a path made of the iron-air combination. As it is assumed generally that the magnetic permeability (μ) of iron is very high (≈ ∞), to set up a specified amount of magnetic flux in the iron part, the m.m.f. required will be very much negligible. Therefore, effectively all the m.m.f. is consumed by the airgap to set up a particular amount of flux and this flux is generally known as ‘airgap flux’. The next point of the whole thought process is whether we should concentrate on the spatial distribution curve of flux-density around the air-gap circumference or, the flux-distribution curve. Now, conceptually, flux is a total effect of the flux-lines which take the shape of a fountain, approximately. Hence to find mathematically, the air-gap flux as a function of space angle, or slot position (θ ) will not be very much result oriented and therefore this approach is hereby rejected. Next remains the flux-density distribution (B − θ curve). Analytically, such a distribution profile will be very much needed because geometrically, the annular airgap has a particular cross-sectional area. For a machine with a uniform air-gap (say, induction machines), at each point on the air-gap periphery, ‘B’ will be uniform but in a machine with a non-uniform airgap (say, three-phase salient pole synchronous machine) along the direct-axis and the quadrature axis, the ‘B’-values will be different, due to the different values of air-gap permeances offered along the two axes. At any instant, the B − θ curve around the air-gap circumference may be of any form and is not necessarily sinusoidal. Now the difficulty arises of how to proceed with mathematical derivation for the electromagnetic torque with such non-sinusoidal spatial flux-density distribution. The simplest way bearing slight inaccuracy is to consider the fundamental component of the said distribution, obtained mathematically after applying Fourier Series Analysis. The main flux of an alternating current machine is defined to be that governed by the fundamental component of the B − θ distribution and the radial line where the peak value is shown is called the axis of the flux. Even though the fundamental component is basically a sine wave (say), the peak value of the air-gap flux density, converted to the RMS (root mean square) value at θ = 90◦ (electrical), ultimately looks like a vector (or, phasor). − → Such phasor is generally designated as B and the corresponding main flux is then completely defined by a magnitude and a direction.

7.1.3.2

Magneto-Motive Force

In order to calculate the flux due to a given system of currents it is necessary to find out the magneto-motive force (m.m.f.) due to the currents. Figure 7.6a is a developed diagram for a two-pole electrical rotating machine extending between space positions

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7 Modelling Approach Using Generalized Theory of Electrical Machines

θ = 0 and θ = 2π. The currents in the conductors of a coil are distributed in slots as indicated and form two bands, symmetrically distributed about the points, ‘A’ and ‘B’. The currents in the two bands flow in the opposite directions, and they are indicated by ‘CROSS’ and ‘DOT’ symbols. In Fig. 7.6a one big CROSS conductor and one big DOT conductor have been shown within one box. But it is only for indication. Actually, there may remain so many conductors distributed in slots and in Fig. 7.6b this equivalence has been shown. Since the current distribution is known, the m.m.f. round a path crossing the air-gap, such as ABCDFGHA (in Fig. 7.6a). From Stoke’s theorem as discussed in ‘Electromagnetic Field Theory’, it is known that: ( S

→ → × H→ ) · − (∇ dS =

( S

− → ( J→) · d S =

(

− → ( H→ ) · dl

(7.19)

C

Where H→ J→ − → dS − → dl

is the magneto-motive force intensity vector (in Amp-turns /meter), is the current density vector (in Amp /meter-square), is the elemental surface area vector, is the elemental length vector, a

b

Fig. 7.6 a Developed diagram showing the distribution of current and mmf. b Representation of the actual conductors in slots (with CROSS currents-for example), pertaining to Fig. 7.6a

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143

→ × H→ is the Curl of H→ ∇ and, I is the current enclosed. The special integral notation in the right-most term of the Eq. (7.19) indicates a line integral along a closed contour “C”, whereas the other two integrals indicate surface integrals. From the Eq. (7.19), it is understood that the current (or, m.m.f.) enclosed is → − fundamentally a line integral of the magnetomotive force intensity vector ( H ) round a closed path. But even then, a value can be associated with each point along the airgap circumference, giving the space distributed m.m.f. curve of the machine. Strictly speaking, the m.m.f. curve of the conductors distributed in slots becomes a stepped one. If the number of slots is very large within a given diameter of the stator or rotor, the above-said stepped waveform can be approximated to a triangular distribution. Furthermore, after the application of the Fourier Series to the said stepped waveform or the triangular distribution, we can neglect the harmonic m.m.f. components and finally we get a fundamental sine or cosine component. The radial line at the point of maximum m.m.f. (XX in Fig. 7.6a) is called the axis of the m.m.f. and since such m.m.f. orientation depends only on the conductor distribution, it is also the axis of the coil. The m.m.f. distribution in space dictates the spatial flux-density distribution because fundamentally magnetic flux and magnet-motive force are related by the equation: ϕ = (Pe)F,

(7.20)

where ‘Pe’ is the magnetic circuit permeance of the particular flux path and, ‘F’ is the magneto-motive force enclosed. Furthermore, it is known that the magnetic circuit permeance is a geometrydependent function and is given by: Pe =

μA , l

(7.21)

where ‘μ’ is the magnetic permeability, ‘A’ is the area of the cross section of the flux path. and, ‘l’ is the length of the magnetic circuit or the flux path. Substituting the expression for ‘Pe’ from Eq. (7.21), in Eq. (7.20), it yields, F ϕ =μ , A l The preceding equation leads to,

(7.22)

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7 Modelling Approach Using Generalized Theory of Electrical Machines

B = μH,

(7.23)

where ‘B’ is the flux density and ‘H’ is the magneto-motive force intensity. It is known that the concept behind the nature of the ‘B-H’ curve is basically dictated by the magnetic properties of matter and hence this curve becomes non-linear (due to magnetic saturation) if the excitation is increased beyond some fixed value. That is why when ‘μ’ is not changing, ‘B’ will be proportional to ‘H’. This fact is conceptually valid for a static electromagnet as well as for the airgap of a rotating electrical machine. Therefore, if the machine has a uniform air-gap, the length of the magnetic circuit, ‘l’ (which is the same as the radial air-gap length between the stator and rotor of the electrical machine), will be constant. As a consequence, from Eq. (7.23), we can conclude that the air-gap flux-density of the machine will be proportional to the magneto-motive force when magnetic saturation is neglected. On the other hand, in a salient-pole machine the ratio, “ Fϕ ” throughout the airgap periphery is not constant. In other words, the theory of constant reluctance fails in this type of machine. In order to calculate the flux, it becomes necessary to resolve the m.m.f. wave into the component waves along the direct and the quadrature axes. This is the starting concept of the well-known “Two Reaction Theory”. In the present and the preceding sections, a good number of discussions along with necessary mathematical treatments have been presented to understand the critical concepts behind the spatial distributions of flux-density and magneto-motive force when practically the annular air-gap in a rotating electrical machine behaves like an equivalent static electromagnet. Apparently, such concepts look simple but a tremendous amount of technical insight is involved to feel the essence of m.m.f. and flux-density distributions and also the essence of ‘Electromagnetic Field Theory’. Once this responsibility is over, there remain two important functional activities of a rotating electrical machine, which are generally mathematically modelled under a common category as, ‘The general voltage and torque equations of the machine’. Obviously, it is not possible to explain this aspect with necessary mathematical derivations under a single section because there are involved a lot of sub-concepts, each of which needs to be explained conceptually or, mathematically. The next section takes care of all these explanations presented under different sub-sections.

7.1.4 Voltage-Balance and Torque-Balance Equations of the Machine-The Per Unit System for Mechanical Quantities It is a well-established concept that in any machine, voltage is to be supplied and torque is to be developed for the motor mode of operation and voltage is to be generated and torque is to be provided for the generator mode of operation. As this particular chapter is devoted to deal with the generalized theory of electrical

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machines, it is necessary at this juncture, to develop the voltage equations and torque equations of a primitive machine which is the starting concept of a unified machine. Furthermore, as any machine can work under a steady state as well as under a transient state, it will be also necessary to put some special efforts on the mathematical treatment of this particular aspect. The conventional theory of electrical machines, mostly deal with steady-state phenomena but the generalized theory of electrical machine is naturally supposed to put mathematical light on both the steady and transient states. All these technical details are explained in the following subsequent sections.

7.1.4.1

Steady State and Transient State

In general, the steady condition of a system is tested by investigating the status of the system as time (t) approaches to infinity as counted from switching of the particular system “ON”. Mathematically, if f (t) is assumed as a time-varying function and if the small perturbation on f (t), i.e., Δ f (t) follows a relation as given by Δ f (t) = 0,

(7.24)

then it is stated that the system has reached the steady condition or, steady state. But, in the case of transients, just at the moment of the sudden disturbance imposed on the system, the status of that particular system is checked and hence it is not justified to wait up to infinite time (or, t → ∞, as is generally done, in the case of steady-state or steady conditions). During a transient condition, the voltages and currents along with the torque and speed of an electrical machine are behaving as a time-varying function. Under transient conditions, the equations of motors and generators appear as differential equations. Hence to solve such equations, it is, therefore, needed to find out the initial conditions or boundary conditions. The voltage balance and torque balance equations of the primitive machine model can be used to find out the performance of any particular type of electrical machine. Generally, linear differential equations (D.E) in currents or other suitable variables of the electrical machine may be solved either by classical method (method of particular integral and complementary functions) or by applying Laplace Transform to that particular differential equation. The advantage of finding the solution by applying the method of Laplace Transform, to such applications is that even sitting on the complex frequency domain (s-domain), applying the Final Value Theorem, one can find out the steady-state value of the current or other variables. Similarly, applying the Initial Value Theorem, one can find out the value of the transient at t = 0 (instant of switching or instant of the occurrence of the disturbance), sitting in the s-domain itself, without jumping to the time domain. Steady conditions are of two types: (i) d.c conditions (ii) a.c conditions. In type (i), quantities are not time-dependent and in type (ii), quantities represent sustained oscillations (sinusoidally varying with respect to time).

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7 Modelling Approach Using Generalized Theory of Electrical Machines

The traditional theory of electrical machines deals with a major portion of analysis which is basically, a steady-state analysis. The steady-state theories are developed in different forms for particular machines. For example, if we write the voltage balance (Kirchoff’s Voltage Law) equation for the armature circuit of a d.c motor, then it is presented as: V = Ra i a (t) + L a

di a (t) + kb i f (t)n(t), dt

(7.25)

where V t i a (t) i f (t) n(t) Ra La kb

d.c voltage applied to armature winding. Time instant in seconds. Current (time varying) through armature winding. Current (time varying) through field winding. Speed of the motor (not fixed rather varying with respect to time). Resistance of the armature winding of the motor. Leakage inductance of the armature winding of the motor. Back emf constant of the motor expressed in terms of “volts/ampere-revolution per second (rps)”.

With reference to the above-said voltage balance Eq. (7.25), if the steady state is considered, then naturally armature current, field current and speed become constant, a (t) ” vanishes. expressed generally as Ia , I f , N , respectively and hence the term, “ didt Therefore, based on this statement, the steady state armature voltage balance equation takes the shape as given by: V = Ra Ia + k b I f N .

(7.26)

Such steady-state analysis falls into the first category as stated in the preceding paragraph. The second category, i.e., a.c conditions strictly speaking, represent time varying quantities. But, practically speaking, such variables subjected to sinusoidal time variation, is considered as a steady state because the peaks of a sine wave are time bounded. As a consequence, it becomes a necessity to apply a sinusoidal steady-state approach using phasor quantities to develop the steady-state theory of the concerned alternating current (a.c) machine. For example, the armature voltage balance equation of a three-phase non-salient pole synchronous generator under a lagging power factor case may be expressed (using the phasor approach) under steady state, as E f (cos δ + jsin δ) = (V + j0) + {Ia (cos ϕ − jsin ϕ)}(Ra + j X s ), where E f Excitation emf in volts per phase. δ Torque angle or power angle or load angle (in radian or degree).

(7.27)

7.1 The Foundation of the Generalized Theory of Electrical Machines

V Ia ϕ Ra Xs

147

Generated emf in volts per phase (taken as reference phasor). Armature current magnitude in amperes per phase. Power factor angle of armature current. Resistance of the armature winding of the synchronous generator in ohms per phase. Synchronous reactance of the armature winding of the synchronous generator in ohms per phase.

The above-said equation again represents an equivalent circuit and also a phasor diagram can be developed based on this equation. Therefore, the steady-state theory of a.c machines can be analyzed using the equivalent circuit approach or phasor diagram approach. It may also be noted that the steady-state performance of a three-phase induction motor (which is generally considered as a work horse in a.c motor applications) is, generally analyzed by using the equivalent circuit approach. Solved Problems: Solved Problem (1): For a d.c separately excited motor with constant field excitation, the following are the data: Armature supply voltage (V ) = 110 V. Armature winding resistance (Ra ) = 0.9 ohm. Moment of inertia = 1.0 kg − m2 . Armature leakage inductance may be neglected. It is observed in a separate test that under a steady state, the motor moves with 1800 rpm on no load, with an armature current of 7.0 amperes at a 110 vols d.c supplied to the motor armature winding. When the motor is switched on to a 110 V d.c supply, find the time needed from zero speed, for the motor on no load to reach a speed which is 98% of the steady no-load speed. Apply any method for solution. Solution of the Solved Problem (1): The steady-state armature circuit voltage balance equation (with constant field excitation) on no load, is written as: V = Ra Ia + kω, where ω Motor speed in radians/second. k Back emf constant in volts/(radians/seconds). Hence, )) ( ( 2 × π × 1800 110 = (7.0 × 0.9) + k × 60

(7.28)

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7 Modelling Approach Using Generalized Theory of Electrical Machines

∴k=

103.7 = 0.550 (60 × 3.14)

Torque dynamic equation of any motor (without any frictional or damping torque): Te − TL = J

dω , dt

where Te i a (t) TL J dω dt

Electromagnetic torque offered by the motor = ki a (t) Armature current under transient or dynamic condition. Load torque. Moment of inertia of the motor plus load. Time derivative of angular velocity (or, motor speed in radians/second).

In this particular problem, as the load is absent, TL = 0 and hence, also, J = 1.0 kg m2 = Moment of inertia of the motor itself. From the preceding equations and statements, we can write the following torque balance and voltage balance equations under transients/dynamics as given by: V = Ra i a (t) + kω(t) ki a (t) = J

dω(t) , dt

where i a (t) and ω(t) indicate the instantaneous values of the concerned variables. Also, it may be noted that the voltage balance equation within the preceding set of equations, does not involve the “ dtd ” term as the armature leakage inductance has been neglected. Applying Laplace Transform to the preceding voltage balance and torque balance equations, it yields, V = Ra Ia (s) + kΩ(s) s and, k Ia (s) = J sΩ(s) − ω(0). In the above-said equation, ω(0) indicates the initial value of ω or, ω(t). In the present problem, ω(0) = 0. From the above said transformed equations, we get (after substituting the numerical value of the variables/parameters), ] (110) × 0.550 [ = {(0.9 × 1) × s} + (0.55)2 Ω(s) s

7.1 The Foundation of the Generalized Theory of Electrical Machines

∴ Ω(s) =

149

67.2 s(s + 0.336)

Now, separately treating (using the Partial Fraction method), ( ) 1 1 1 1 = − s(s + a) a s (s + a) where a = any constant. Hence, using the concept of Inverse Laplace Transform, we may write, ( ) ) ( 1 1 67.2 L −1 − ω(t) = L −1 [Ω(s)] = 0.336 s (s + 0.336) ] [ −0.336t . = 200 1 − e ) ( ×1800 98 × 2×π60 , then t =? When (t) = 100 Hence to find out the answer to the above-said query, 98 × 100

(

2 × π × 1800 60

)

[ ] = 200 1 − e−0.336t

) ( = 0.0768 Or, e−0.336t = 1 − 184.63 200 Taking the natural logarithm of both side terms of the preceding equation, we get, −0.336 t = ln(0.0768) = −2.566 Or, t =

2.566 0.336

= 7.64s →→→ Answer

Solved Problem (2): In the preceding problem (Solved Problem (1)) find the steady state value of motor speed (in rpm) using the Final Value Theorem (a topic under the mathematics book chapter, “Laplace Transform”). Verify your answer by other methods. Solution of the Solved Problem (2): 1st Part The final Value Theorem says that if a time-varying function, f (t), has Laplace Transform, F(s), then, limt→∞ f (t) = lims→0 s F(s)

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7 Modelling Approach Using Generalized Theory of Electrical Machines

With reference to the Solved Problem (1), we know that the steady-state speed of the motor (ωss in radians per second unit) can be expressed as, ωss = limt→∞ ω(t) = lims→0 sΩ(s) s(67.2) = lims→0 s(s + 0.336) (67.2) = lims→0 (s + 0.336) 67.2 = 200 radians/s = 0.336 The steady-state motor speed in the rpm unit will be, n ss = N ωss × 60 = 2×π 200 × 60 = 6.28 = 1910.828 rpm →→→ Answer 2nd Part ] [ ωss = limt→∞ ω(t) = limt→∞ 200 1 − e−0.336t = 200 radians/s. Hence, the answer is hereby verified Solved Problem (3): In Solved Problem (1), it was assumed that the initial motor speed was zero. Verify this assumption using the Initial Value Theorem (a topic under the mathematics book chapter, “Laplace Transform”). Solution of the Solved Problem (3): The initial Value Theorem says that if a time-varying function, f (t) has Laplace Transform, F(s), then, limt→0 f (t) = lims→∞ s F(s) With reference to the Solved Problem (1), we know that the initial speed of the motor (ω(0) in radians per second unit) can be expressed as

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151

ω(0) = lims→∞ sΩ(s) s(67.2) = lims→∞ s(s + 0.336) (67.2) = lims→∞ (s + 0.336) 67.2 =0 = ∞ Therefore, the assumption is hereby verified

7.1.4.2

The General Voltage Equation (After Applying Laplace Transform) and the Concept of Transformed Impedance Matrix

Let us consider the 4-coil primitive machine model shown in Fig. 7.4 for developing the concerned voltage-balance equation in the time domain and then to convert these equations to Laplace-domain (s-domain). This is a very bright side of the generalized theory of electrical machines because it gives a direct circuit theory view of the devices dealing with the principles of electromechanical energy conversion. The voltage balance equations of the four coils such as, F, D, Q and G- coils are developed in the time domain following a so-called generalized rule, expressed as: Impressed voltage (u) = Ohmic drop (Ri) + Transformer e.m.f (ut ) + Rotational e.m.f (ur ), or, in the form of a mathematical equation, it takes the shape: u = Ri + u t + u r ,

(7.29)

where ‘R’ is the resistance of the coil and ‘i’ is the current though that coil. In general, the transformer emf is the product of the particular coil inductance and the time rate change of the concerned coil current. On the other hand, the rotational emf calls for the involvement of the speed of the machine so that a coil (or, winding) being under motion (somehow) in a magnetic field of constant strength, generates a voltage. That is why in the expression for rotational emf, the coefficient of the term, ‘speed (ω)’ (expressed in radians per second) is not the actual inductance rather it is the so-called fictitious inductance termed as any one out of the symbols L rd , L rf , L rq , L rg appearing in the voltage balance equations of each of D, F, Q and G-coils. These voltage balance equations directly may be written in the complex frequency (s) domain but it is always advantageous to write the equation in the time domain and then to apply Laplace Transform to enter the ‘s’ domain. These time domain equations for a 4-coil primitive machine model (see Fig. 7.4) are given as follows: di f di d + Ld f dt dt

(7.30)

di f di d + Ld + L rq ωi q + L rg ωi g dt dt

(7.31)

u f = R f i f + L ff u d = Rd i d + L d f

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7 Modelling Approach Using Generalized Theory of Electrical Machines

u q = Rq i q + L qg

di g di q + Lq + (−)L rf ωi f + (−)L rd ωi d dt dt

(7.32)

dig diq + L gq dt dt

(7.33)

u g = Rg i g + L gg

The negative signs (under bracket) are appearing in Eq. (7.32), because the rotational emf induced in the coil placed along the quadrature axis, due to a flux along the direct axis and due to the positive speed (ω, in the counter clock wise direction), has a negative sign as per the generalized sign convention. After applying Laplace Transform to each of the above-said voltage balance equations (from (7.30) to (7.33)) and neglecting the initial conditions, a matrix equation is obtained and it looks like as follows: [U (s)] = [Z P (s)][I (s)].

(7.34)

⎤ U f (s) ⎢ Ud (s) ⎥ ⎥ [U (s)] = ⎢ ⎣ Uq (s) ⎦, Ug (s)

(7.35)

where ⎡

R f + sL f f s Ld f 0 0 s Ld f Rd + s L d L rq ω L rg ω [Z p (s)] = −L r f ω −L r d ω Rq + s L q s Lqg 0 0 s Lqg R f + s L f f ⎤ ⎡ I f (s) ⎢ Id (s) ⎥ ⎥ [I (s)] = ⎢ ⎣ Iq (s) ⎦

(7.36a)

(7.36b)

Ig (s)

The matrix [Z P (s)] in Eq. (7.36a) is known as the Transformed Primitive Impedance Matrix. Once the generalized form of the voltage balance equation of a primitive machine model has been derived in the time domain and Laplace (s-)domain, it is now needed to derive the expression for the general torque equation in the time domain for a primitive machine model. This aspect is dealt with in the next section.

7.1.4.3

The General Torque Equation in the Time Domain

Let us talk about the development of electromagnetic torque in a machine working in the generator mode. This is only taken as an example. Let ‘M t ’ be the instantaneous

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153

applied mechanical torque while any friction or mechanical damping torque can be included with M t , as a reasonable assumption. Once this external torque drives the machine and simultaneously the magnetic excitation (flux or current) is supplied to any rotating machine, the voltage is generated by the machine. Somehow current starts flowing in this member when the machine is loaded or short-circuited in itself. This current produces its own flux and this flux interacts with the parent or original flux. As a result, the resultant air-gap flux is set-up. This resultant flux interacts with the load current to produce a torque and finally this torque is named as “Electromagnetic Torque (Me )” which opposes the applied mechanical torque. As per Newton’s second law applied to the rotating system, the difference between ‘M t ’ and ‘M e ’ must be Inertia torque, as expressed mathematically as: Mt = Me + J

dω , dt

(7.37)

where J and ω have their usual meanings already stated. Even though, Eq. (7.37) indicates a torque balance equation for all types of electrical machines, practically it is mostly used in the case of synchronous generators in connection with the design and running of the large power plant units. As the turbo-generators, practically are built of large ratings in the range of some hundred mega-watts, the rotor diameters of those machines are also having large dimensions. This statement is basically based on the standard theory of “Electrical Machine Design”, which is well-known in the name of “Output Equation”, expressed as, ( ) Q = C0 D 2 Ln,

(7.38)

where the symbols such as Q, C0, D, L and n stand for output KVA of the machine, output coefficient, rotor diameter, core length of the machine and rated speed (in rpm) of the machine, respectively. The tendency of rotor diameters to be in the higher range for turbo-generators makes the actual moment of inertia of the synchronous generators high in the actual units. To handle such large figured numerical data in the case of design calculations or in other works like installation, testing etc., per-unit conversions become mostly necessary. In practical work, the moment of inertia of the machine is expressed by the “Inertia Constant”, denoted by ‘H’, which is a normalized quantity but differs from the per-unit moment of inertia, ‘J’. Similarly, the per-unit values of power, speed, torque, and time are to be defined separately. As a whole, it is seriously felt that some efforts should be put separately on the presentation of per-unit values of the mechanical parameters involved in the generalized torque balance Eq. (7.37). This is presented in the next sub-section.

154

7.1.4.4

7 Modelling Approach Using Generalized Theory of Electrical Machines

Mechanical Parameters in Per-Unit System

As it is known that per-unit is the ratio of the actual value to the base value, it is very important to define the base value of the different mechanical quantities and associated electrical quantities. These detailed discussions are presented in the following paragraphs: (i)

Expression for power in per-unit: In the per-unit system, it is advantageous to have a value of input power equal or close to unity when the voltage and currents in the main circuit as 1.0 per-unit (each). But in the case of three-phase machines, there are three main circuits because phase-R, phase-Y and phase-B windings are involved to generate or consume the electrical power. So, as per original definition as given above, total input power should be sum of 1unit,1unit and 1unit. Hence it should be 3 units. But in per-unit, power input cannot be more than 1 unit. To avoid this fallacy, the total input power(P), in per-unit is expressed as below: P = kp

3 {

u k ik ,

(7.39)

k=1

(ii)

(iii)

(iv) (v)

where ‘k p ’, ‘u’ and ‘i’ stand for ‘factor’, ’voltage impressed’ and ‘current flown’, respectively. For a three-phase system, the value of k p is 13 and the upper limit of the summation symbol in Eq. (7.39) is ‘3’.In general, k p is the reciprocal of the number of main circuits used in any electrical machine. Base value of power: The base value of power is defined for a single main circuit as the power corresponding to the base voltage and base current. In a system with more than one main circuit, the value of k p (as specified in Eq. (7.39)) should be chosen accordingly. Base power in alternating current machines: Based on the content of discussions put in point (ii), it becomes very interesting to justify the base power in a.c. machines. It is the power corresponding to the rated KVA. Suppose, we consider a three-phase synchronous generator of 1000 KVA at 0.8 power factor. The base power will be 1000 KW. Hence 800 = 0.8 per unit. the rated power will be 1000 × 0.8 = 800 KW, or 1000 Let us go into the depth that why such things happen. The answer to the such query is very simple in the sense that practically in a generalized machine, nobody bothers the power factor when current is flowing in any winding due to some voltage applied. That is why base power is thought in terms of the multiplication of only currents and voltages, in a particular circuit. Base value of angle: Radian is the base value of angle. Base value of time: The ‘second’ is the base value of time.

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155

Base value of speed: The base value of speed adopted in this book is ‘radians per second’. It is based on the statements made in points(iv) and (v). (vii) Nominal speed (ω0 ): In a direct current machine, the nominal speed is the normal rated speed of the equivalent two-pole machine in ‘electrical radians per second’. In an alternating current machine, the nominal speed is the synchronous speed of the equivalent two-pole machine. (viii) Base value of torque: The ‘base value of torque’ is defined as the torque which produces the base power at the nominal speed (ω0 ). (ix) Per-unit value of torque: Based on the definition given in point(viii), the per-unit torque can be derived as given below: Per-unit torque is Tpu (say). (vi)

Tpu =

Actual value of power × ω Actual value of torue = Base value of torque Base value of power × ω0

(7.40)

When ω = ω0 , Tpu =

(x)

Actual value of power × ω0 = Per - unit power Base value of power × ω0

(7.41)

Hence, in the per-unit system, torque is numerically equal to power when speed ‘ω’ equals the nominal speed, ‘ω0' . ‘Moment of Inertia’ expressed in per-unit: In practical work, the moment of inertia of the synchronous machine (in most of the cases) is expressed by the parameter, “Inertia Constant”, denoted by “H”. ‘H’ is a normalized quantity but it differs from the per-unit value of ‘J’. ‘H’ is defined by the following formula: H=

Stored Energy at synchronous speed in KW − sec Rated KVA

(7.42)

The ‘Base value of energy’ is given by base power acting for one second, and therefore, as seen in Eq. (7.42), ‘H’ is numerically equal to the per-unit stored energy. Hence, it can be written that, Actual Stored energy at nominal speed (ω0 ) Base value of energy (1) J (ω0 )2 2 = (Base value of torque) × (Base value of angle)

H |per-unit =

(7.43)

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7 Modelling Approach Using Generalized Theory of Electrical Machines

Based on the definitions given in point numbers (ii), (iv), (v) and (viii), Eq. (7.43) takes the shape as, (1)

H |per-unit =

J (ω0 )2 1 = Jω0 . (Base value of power) × (ω0 ) × (1) 2 2

(7.44)

From Eq. (7.44), it yields, ) ( 2 H |per - unit J= . ω0

(7.45)

Substituting for the value of ‘J’ from Eq. (7.45) in Eq. (7.37), we have the torque dynamic equation in per-unit, expressed as, ( ( )) 2 H |per-unit dω . Mt = Me + ω0 dt

(7.46)

The inertia constant ‘H’ is used mainly for synchronous generators. One may notice that there is no clear consistency in the inertia constants for a specific type of technology or size. The amount of inertia of a power plant is thus very much case specific and strongly depends on the design of both the generator and turbine [3]. However, in [4] it is stated that with respect to P A G E | 12/72 thermal power plants, units equipped with a four-pole synchronous generator (1500 rpm in a 50 Hz power system) generally have larger inertia constants than units using a two-pole generator running at a higher speed (3000 rpm in a 50 Hz power system). For large thermal units using a four-pole generator, the inertia constant can even exceptionally reach values up to 10 s. The possible reason behind such higher value, from the view point of the electrical machine design, is that compared to two poles, four poles will demand for the larger rotor diameter. Hence, the moment of inertia will increase considerably, leading to the larger value of the inertia constant. With respect to gas turbines, there is often also a clear distinction in the inertia constants of aero derivative gas turbines and heavy-duty ones. Flexible aero-derivative turbines generally weigh considerably less than the heavy-duty ones which results in an inertia constant at the low end of the range of 2–9 s. The inertia constant of heavy-duty gas turbines is more in the order of 4–6 s. Finally, it should be noted that current-day turbines and generators are generally lighter than the one developed in the ‘70 s and ‘80 s, resulting in a lower H.

7.1.4.5

Derivation for Expression for Electrical Torque Based on the Expression for Transformed Impedance Matrix

If four numbers of windings or coils are considered, then the total instantaneous electrical power input (Pe ) can be expressed as:

7.1 The Foundation of the Generalized Theory of Electrical Machines

Pe = (u 1 i 1 ) + (u 2 i 2 ) + (u 3 i 3 ) + (u 4 i 4 )

157

(7.47)

where the symbols involving “u” and the symbols involving “i” indicate the voltages impressed across and the currents in the coils, respectively. Based on this expression, for the four-coil primitive machine model in Fig. 7.4, Pe can be written as [ ]T [ ] Pe = u dq i dq ,

(7.48)

[ ]T [ ] where u dq indicates the transpose of the column matrix u dq and [

] [ ] u dq = u f u d u q u g ,

(7.49)

[ ] [ ]T i dq = i f i d i q i g .

(7.50)

and

On the basis of the existence matrix ([Z ]) of a primitive machine ] the impedance [ ] [ of model, the column vectors u dq and i dq can be related as [

] [ ] u dq = [Z ] i dq .

(7.51)

Furthermore, [Z ] may be decomposed as, [Z ] = [R] + [L p] + [ωL r ],

(7.52)

[ ] ] u dq = [[R] + [L p] + [ωL r ]] i dq

(7.53)

leading to: [

Strictly speaking it will be more accurate to name this [Z] as Transformed Impedance Matrix in place of Impedance Matrix because the operator, “ dtd ” or “p” has been used. With reference to the Eq. (7.52), the right-hand side terms may be expressed as, Rf 0 [R] = 0 0

0 Rd 0 0

0 0 Rq 0

0 0 0 Rg

L f f p Ld f p 0 0 Ld f p Ld p 0 0 [L p] = 0 0 L q p L qg p 0 0 L qg p L gg p

(7.54a)

(7.54b)

158

7 Modelling Approach Using Generalized Theory of Electrical Machines

0 0 0 0 0 0 −ωL rq −ωL rg [ωL r ] = −ωL r f −ωL r d 0 0 0 0 0 0

(7.54c)

On the basis of Eqs. (7.48)–(7.54c), Pe may be expressed as, [ [ ]]T [ ] i dq Pe = [Z ] i dq [ ] [ ]T T = i dq [Z ] i dq ][ ] [ ]T [ = i dq [R]T + [L p]T + [ωL r ]T i dq

(7.55)

Now, Pe can be written in the form of, Pe = Pe1 + Pe2 + Pe3 .

(7.56)

With reference to Eq. (7.56), the power components, Pe1 , Pe2 and Pe3 are expressed as ][ ] [ ]T [ Pe1 = i dq [R]T i dq

(7.57a)

][ ] [ ]T [ Pe2 = i dq [L p]T i dq

(7.57b)

][ ] [ ]T [ Pe3 = i dq [ωL r ]T i dq

(7.57c)

As in the above-said expressions, the power component, Pe3 involves the speed term (ω), the term, “ Pωe3 ” must indicate the expression for electromagnetic torque. ][ ] [ ]T [ Pe3 = i dq [ωL r ]T i dq

(7.58)

From the Eqs. (7.54c) and (7.58), we have,

Pe3 = i f i d i q i g

= i f id iq i g

0 0 −ωL r f 0 0 −ωL r d 0 ωL rq 0 0 ωL rg 0 −ωL r f i q −ωL r d i q ωL rq i d ωL rg i d

0 0 0 0

if id iq ig

(7.59a)

7.1 The Foundation of the Generalized Theory of Electrical Machines

159

From the Eq. (7.59a), the component of the power, Pe3 can further be simplified as, } } { { Pe3 = −ω L rf i f + L rd i d i q + ω L rq i q + L rg i g i d

(7.59b)

The expression for Pe3 in the preceding equation, can be expressed in another form as given by: } { Pe3 = ω ψq i d − ψd i q ,

(7.60)

ψq = L rq i q + L rg i g

(7.61)

ψd = L rf i f + L rd i d .

(7.62)

where,

and

Hence, the generalized expression for the electromagnetic (electrical) torque (Te ), can be written as, Te =

Pe3 = ψq i d − ψd i q . ω

(7.63)

In this expression for torque, ψd and ψq indicate the terms similar to magnetic flux linkages along direct (d-) and quadrature (q-)axes, respectively. Another { important point is to note that ψd and ψq , basically indicate the term expressed as Li, where “L” should represent “inductance” and “i” should represent “coil current”. In this section, our main objective was to derive the generalized expression for the electromagnetic (electrical) torque in a primitive machine model. However, it is not also desirable to give less emphasis, at this stage, to the simplifications of the other two components (Pe1 and Pe2 ) of the total electrical input power (Pe ). Such mathematical simplifications are presented as follows: Mathematical expansion for the expression for Pe1 in Eq. (7.57a) and its physical interpretation: [ ] Based on the expressions for [R], i dq and Pe1 in Eqs. (7.54a), (7.50) and (7.57a), respectively, we get, ][ ] [ ]T [ Pe1 = i dq [R]T i dq = i f id iq i g

Rf 0 0 0

0 Rd 0 0

0 0 Rq 0

0 0 0 Rg

if id iq ig

(7.64)

160

7 Modelling Approach Using Generalized Theory of Electrical Machines

So, finally Pe1 comes out to be as given by, ( )2 ( )2 ( )2 Pe1 = i f R f + (i d )2 Rd + i q Rq + i g Rg .

(7.65)

From the above-said expression for Pe1 in Eq. (7.65), the physical interpretation can be stated [as follows: ]T [ T ] [ ] i dq represents the summation of the copper (ohmic) The term i dq [R] losses in all the windings (coils). Mathematical expansion for the expression for Pe2 in Eq. (7.57b) and its physical interpretation: [ ] Based on the expressions for [L p], i dq and Pe2 in Eqs. (7.54b), (7.50) and (7.57b), respectively, we get, ][ ] [ ]T [ Pe2 = i dq [L p]T i dq = i f id iq i g

L f f p Ld f p 0 0 Ld f p Ld p 0 0 0 0 L q p L qg p 0 0 L qg p L gg p

if id iq ig

(7.66a)

Equation (7.66a) takes the form as given by, Pe2 = L ff i f pi f + L d i d pi d + L q i q pi q + L gg i g pi g + L df i f pi d + L df i d pi f + L qg i g pi q + L qg i q pi g

(7.66b)

So, finally Pe2 comes out to be as ( ) ( ) ( ) (( ) ) ( )2 ( )2 ( )2 d 1 1 1 1 L ff i f + L d (i d )2 + L q iq + L gg i g dt 2 2 2 2 } d{ (7.67) L df i d i f + L qg i q i g + dt

Pe2 =

From the above-said expression for Pe2 in Eq. (7.67), the physical interpretation can be stated [as follows. ]T [ ][ ] The term i dq [L p]T i dq represents the summation of the time derivative of the stored energy in all the windings (coils).

7.2 Development of the Sub-Transient, Transient and Steady State …

161

7.2 Development of the Sub-Transient, Transient and Steady State Equivalent Circuits along Direct and Quadrature Axes, Separately, of a Three-Phase Salient Pole Synchronous Machine Using, “Constant Flux-Linkage Theorem” and “Theory of Small Perturbation” This particular section deals mainly with the development of an equivalent circuit of a three-phase salient pole synchronous generator in a transient state, in general. As transients in synchronous machines are significantly different than the transients in static electrical networks, due to the appearance of the term “speed” in the mathematical model of the machine, it needs much mathematical depth while formulating the problem. All such detailed mathematical works are presented in subsequent sections.

7.2.1 Concept and Mathematical Model of “Constant Flux-Linkage Theorem” The Constant Flux Linkage Theorem is of considerable importance in investigating the electrical transients of a three-phase synchronous generator. This concept is stated as—“The magnetic flux linkage after a sudden disturbance in an electrical closed circuit having finite inductance, zero resistance and zero capacitance remain constant at their values before the perturbation, if the period of disturbance is very small”. In a normal situation, there is no capacitance in the armature and the field windings of an alternator. Also, their resistances are negligibly small in comparison with their inductance. Thus, armature and field windings may be assumed to be purely inductive, and the flux linkages in the armature and field circuits cannot be changed immediately by the change of current in one winding must be accompanied by a change of current in the other to keep the flux linkages constant. Constant Flux Linkage Theorem The voltage balance equations for any circuit can be written in the form given below: ) { n ( n { dϕk qk N + ek = i k Rk + dt c k=1 k k=1 k=1 k=1

n {

n {

(7.68)

In the above said equation, the symbols are defined as: {n ek Sum of potentials of all emf sources, in volts. {k=1 n i R k=1 k k Sum of ohmic drops on all resistors (of resistance “R”, carrying current “i”), in volts. N Number of turns of each inductor coil.

162

t ϕ q C

7 Modelling Approach Using Generalized Theory of Electrical Machines

Time instant, in seconds. Magnetic flux, in weber. Electric charge. Capacitance.

Using the symbol Ψ for the magnetic flux linkage (which is the product of ‘magnetic flux’ and ‘number of turns’), the Eq. (7.68) may be written as follows: n {

n {

n n n { { d { dΨk qk = = Ψk ek − i k Rk − Ck dt dt k=1 k=1 k=1 k=1 k=1

(7.69)

Equation (7.69) may be written in the alternative form as given by: n d { Ψk = e1 dt k=1

(7.70)

where e1 is the resultant voltage, which will be the function of the time and q and C stand for electric charge and capacitance, respectively. Integrating the Eq. (7.70), the magnetic flux linkage for any time duration from t = 0 to t = t, may be expressed as n {

(t Ψk =

k=1

e1 dt

(7.71)

0

Here, in general, “t” may be a large time. If the time duration is in terms of small change “ (small interval of time), then Δ

n { k=1

Δ

(Δt Ψk =

e1 dt

(7.72)

0

As the term, “Δt” tends to zero, so will be integral in Eq. (7.72). Hence, { n k=1 Ψk = 0. Therefore, the instantaneous change of flux linkage is zero.

7.2.2 Theory of Small Perturbation in Terms of Taylor’s Series Expansion Taylor’s series expansion of a function, f (x) of a single variable, “x”, around an operating point (x = X 0 ) with an infinitely small increment or decrement of value “h”, may be expressed as,

7.2 Development of the Sub-Transient, Transient and Steady State …

163

} h2 { } { I I f '' (x)Ix=x0 + . . . . (7.73) f (x + h)|x=x0 = f (x)|x=x0 + h f ' (x)Ix=x0 + 2! As “h” is very small, the quantities like “h 2 ”, “h 3 ” …. etc., may be neglected. Based on this reasoning, the preceding equation may be modified as, Δy = m(Δx)

(7.74)

y = f (x)

(7.75)

where,

Δy = f (x + h)|x=x0 − f (x)|x=x0 =Small change in y at the point having abcissa, x = X 0

(7.76)

and I m = f ' (x)Ix=x0 .

(7.77)

In the Eq. (7.77), the term, “m” represents the slope of the function at the point having abscissa, x = X 0 . The equation, Δy = m(Δx), obviously indicates a straight line and also such a technique is called “Linearization Technique”. In general, in any engineering application, the independent variable may be subjected to certain small changes or perturbations from any external source which may not be controllable. Such cases are modelled by the above-said method using “Linearization Technique” or “Theory of Small Perturbation”. The interesting point is that there are so many engineering applications where “Steady State Stability” is assessed by applying the “Theory of Small Perturbation”. Some case studies are hereby presented: Case 1: Applications of Theory of Small Perturbation: Analysis of steady-state stability of a motor-load combination: Problem Definition: To determine the criterion for steady-state stability of any motor load combination, with frictional torque being neglected and the starting speed being not specified. Problem Formulation: The torque dynamic equation of any motor-load combination (without any friction or any frictional torque being considered as a part of load torque), may be expressed as, Te − TL = J where,

dω , dt

(7.78)

164

Te TL J dω dt

7 Modelling Approach Using Generalized Theory of Electrical Machines

Electromagnetic torque. Load torque. Moment of inertia of the motor-load combination. Time derivative of angular velocity (or, motor speed in radians/second). It is well known that by applying the Theory of Small Perturbation, one can write, Te = Te0 + ΔTe

(7.79a)

TL = TL0 + ΔTL

(7.79b)

ω = ω0 + Δω

(7.79c)

In the above-said equations, the variables with suffices “e0”, “L0” and “0” represent the quiescent values or fixed operating point values or equilibrium values or steady state values of electromagnetic torque, load torque, and motor speed (in radians/second), respectively. Similarly, in the same equations, ΔTe , ΔTL and Δω represent the small perturbations of Te , TL and ω, respectively. From the Eqs. (7.78) to (7.79c), we have, (Te0 + ΔTe ) − (TL0 + ΔTL ) = J

dω0 dΔω +J . dt dt

As at the steady state condition, that is, at t = 0, Te0 = TL0 , and ΔTe − ΔTL = J

dΔω dΔω dω0 +J =J . dt dt dt

(7.80) dω0 dt

= 0, we get, (7.81)

We can express ΔTe and ΔTL (using the Theory of Small Perturbation) as, (( ) ) )I ∂ Te II ΔTe Δω ≈ ΔTe = Δω Δω ∂ω Iω=ω0 (( ) ) )I ( ∂ TL II ΔTL Δω ≈ ΔTL = Δω Δω ∂ω Iω=ω0 (

(7.82)

(7.83)

As T e and T L are functions of “ω“ only, it may be written that, I )I ∂ Tθ II dTθ II = , ∂ω Iω=ω0 dω Iω=ω0 I )I ( ∂ TL II dTL II = . ∂ω Iω=ω0 dω Iω=ω0 (

(7.84) (7.85)

7.2 Development of the Sub-Transient, Transient and Steady State …

165

Therefore, from the preceding equations, it yields, dΔω dt dΔω(t) , =J dt

K (Δω) = J

(7.86)

where, ( K =

) ( ) I I dTe II dTL II − . dω Iω=ω0 dω Iω=ω0

(7.87)

In the preceding equation, the variables Δω and Δω(t) are the same symbol because fundamentally Δω is a time-varying function. The above-said differential Eq. (7.86), in “Δω”, , may be solved either by using the classical method (Method of Particular Integral and Complementary Function) or by applying Laplace Transform. For having a more generalized approach, the second method (Laplace Transform) is hereby adopted. Solution Methodology: Applying Laplace Transform to the differential Eq. (7.86), we get, ( ΔΩ(s) =

) ( ) J J sΩ(s) − Δω(0) K K

(7.88)

where, ΔΩ(s) = L{Δω(t)}

(7.89a)

Δω(0) = {Δω(t)}|t=0

(7.89b)

and

Hence, we get, ( ΔΩ(s) =

) 1 ( ) {Δω(0)} s − KJ

(7.90)

Therefore, applying Inverse Laplace Transform, we get, Δω(t) = L −1 {ΔΩ(s)} K = {Δω(0)}e( J )t .

(7.91)

166

7 Modelling Approach Using Generalized Theory of Electrical Machines

Determination of the Stability Criterion: If the said motor-load combination is to be stable in the sense of Steady State Stability, as per the established condition, as t → ∞, Δω(t) → 0. To satisfy this condition, with reference to the Eq. (7.91), we must have, K < 0 i.e., Therefore, the criterion for steady state stability should be expressed as (

) ( ) I I dTe II dTL II < . dω Iω=ω0 dω Iω=ω0

(7.92)

Case 2: Applications of Theory of Small Perturbation: Analysis of steady-state stability of a power system: Problem Definition: Let us analyze the steady-state stability aspects of a power system. The description of the system is given as follows. The system consists of a three-phase turbo-generator feeding the electrical power to an infinite bus through a transformer and transmission line. The excitation e.m.f, the terminal voltage and the load angle of the generator are E f volts/phase and Vt volts/phase and δ radians, respectively. The combined value of the reactance of the system involving the synchronous reactance of the generator and the reactance of the transformer and the transmission line is X ohms per phase. The total mechanical power input (Pm ) to the generator remains unchanged during the operation. The armature winding resistance of the generator and resistances of the transmission line and transformer are neglected, being very small. Problem Formulation: The established Swing Equation for a cylindrical rotor synchronous generator, as a power system component may be presented as, M

d 2δ = Pm − Pe , dt 2

(7.93)

where Pm Pe M δ t

Total mechanical power input to the generator in MW (or, Megawatts) unit. Total electrical power output from the generator in MW (or, Megawatts) unit. Equivalent Moment of Inertia in MW s2 /electrical radian unit. Load angle or, power angle of the generator, in electrical radians unit. Time instant in seconds. Using the theory of small perturbation, we may write, Pm = Pm0 + ΔPm ,

(7.94a)

7.2 Development of the Sub-Transient, Transient and Steady State …

167

Pe = Pe0 + ΔPe ,

(7.94b)

δ = δ0 + Δδ,

(7.94c)

where the quantities Pm0 , Pe0 and δ0 indicate the quiescent values (fixed) and the quantities ΔP m , ΔP e and Δδ indicate the small changes in values (perturbation), of the variables Pm , Pe and δ, respectively. Using those preceding equations, the Swing equation in linearized form may be expressed as, M

d 2 (δ0 + Δδ) = (Pm0 + ΔPm ) − (Pe0 + ΔPe ). dt 2

(7.95)

It is known that, at steady state or, equilibrium condition, Pm0 = Pe0 .

(7.96)

Also, it may be noted that as per the definition of the problem, ΔPm = 0.

(7.97)

d 2 (δ0 ) =0 dt 2

(7.98)

As δ0 is fixed, we can write,

Based on the Equations from (7.95) to (7.98), we get, M

d 2 (Δδ) = −(ΔPe ). dt 2

(7.99)

Now, on the lines similar to the treatment in the Case Studies (1) of this section, we can write, (( ) ) )I ( ∂ Pe II ΔPe Δδ ≈ ΔPe = Δδ. (7.100) Δδ ∂δ I δ=δ0

Now, let (

)I ∂ Pε II = K. ∂δ Iδ=δ0

Hence from the above said equations, we get

(7.101)

168

7 Modelling Approach Using Generalized Theory of Electrical Machines

M

d 2 (Δδ) = −(K Δδ). dt 2

(7.102)

It is well known that the power-angle relation of a lossless synchronous generator is expressed as, Pe = Pmax sin δ

(7.103)

where, Pmax =

3 E f Vt . x

(7.104)

Hence, from Eqs. (7.101) and (7.104), it yields, ( K =

)I ∂ Pε II ∂δ Iδ=δ0

= Pmax (cos δ0 ).

(7.105)

Solution Methodology: As such no solution is needed for the present problem rather only the stability (steady state stability) is to be assessed. For this, the Characteristic Equation, in the s-domain (“s” being the complex frequency, the Kernel of Laplace Transform) is to be formed. Applying Laplace Transform to the above said perturbed (small perturbation) Swing Eq. (7.102), the characteristic equation comes out to be, Ms 2 + K = 0.

(7.106)

Assessment about the Steady State Stability Aspects: The roots of the preceding characteristic equation are: / s = ±j

K . M

(7.107)

The roots of the characteristic equation lie completely on the imaginary axis of the s-plane and such nature of the roots indicates that the system response will be marginally stable.

7.2 Development of the Sub-Transient, Transient and Steady State …

169

7.2.3 Combined Effect of the Above-Said Two Ideas to Develop the Resultant Equivalent Circuit With reference to the primitive machine model of three-phase salient pole synchronous machine in Fig. 7.7, the equivalent circuits at the sub-transient and other states can be developed for d-axis and q-axis, separately. The analysis which is necessary for developing the equivalent circuits for direct and quadrature axes is being presented separately, as follows. Analysis for d-axis: With reference to the primitive machine model (Fig. 7.7) of the salient pole synchronous machine, all the variables and parameters are assumed as per unit quantities. The currents flowing the D-coil, F-coil, and KD-coils are i d , i f and i kd , respectively. ( ) The magnetic flux linkages with D-coil (ψd ), F-coil ψ f , and KD-coil (ψkd ) are expressed as follows: ψd = L d i d + L md i f + L md i kd

(7.108)

Captions: (1) Centre of the circle (2) Rotor (periphery of the circle) (3) Direct axis (4) Quadrature axis (5) D-coil (part of armature winding) (6) Field winding (F-coil) (7) Direct axis damper winding (KD-coil) (8) Quadrature axis damper winding (KQ-coil) (9)Q-coil (part of armature winding) Fig. 7.7 Primitive machine model of three-phase salient pole synchronous machines with d- and q-axes damper windings

170

7 Modelling Approach Using Generalized Theory of Electrical Machines

ψ f = L md i d + L ff i f + L md i kd

(7.109)

ψkd = L md i d + L md i f + L kkd i kd

(7.110)

As per the previous discussions (see Sect. 7.1.2.4), the self-inductances of the said coils may be decomposed as: L d = L md + la

(7.111a)

L ff = L md + l f

(7.111b)

L kkd = L md + lkd ,

(7.111c)

where L d , L ff and L kkd are the self-inductances and la , l f and lkd are the leakage inductances of the D-, F- and KD-coils, respectively and L md is the mutual inductance between each pair of coils or windings. Substituting Eqs. (7.111a)–(7.111c) in the Eqs. (7.108), (7.109) and (7.110), it yields, ψd = (L md + la )i d + L md i f + L md i kd

(7.112)

) ( ψ f = L md i d + L md + l f i f + L md i kd

(7.113)

ψkd = L md i d + L md i f + (L md + lkd )i kd

(7.114)

Applying the theory of small perturbations to the above said expressions for flux linkages with the D-, F- and KD-coils in Eqs. (7.112)–(7.114), we get Δψd = (L md + la )Δi d + L md Δi f + L md Δi kd

(7.115)

( ) Δψ f = L md Δi d + L md + l f Δi f + L md Δi kd

(7.116)

Δψkd = L md Δi d + L md Δi f + (L md + lkd )Δi kd

(7.117)

Whenever a sudden disturbance (e.g., sudden short circuit or, sudden load change) at the armature terminals of the synchronous machine is imposed, the other winding members on that particular axis (i.e., d-axis) must maintain the “Constant flux linkage theorem” as the particular windings have a finite amount of electrical inertia (i.e., inductance). Therefore, we can write, ) ( Δψ f = L md Δi d + L md + l f Δi f + L md Δi kd = 0

(7.118)

7.2 Development of the Sub-Transient, Transient and Steady State …

Δψkd = L md Δi d + L md Δi f + (L md + lkd )Δi kd = 0

171

(7.119)

From the expression, Δψ f = 0 in Eq. (7.118), we get (( Δi kd = −

L md + l f L md

)) Δi f − Δi d

(7.120)

Substituting this expression for Δi kd from the preceding equation in the expression, Δψkd = 0 in Eq. (7.119), we get, ((

) ) L md + l f (L md + lkd ) L md Δi d + L md Δi f − Δi f − (L md + lkd )Δi d = 0 L md (7.121) ( ) (L md )2 − (L md )2 − L mdl f − L mdlkd − l f lkd = lkd (Δi d ) Or, Δi f (7.122) L md ) ( L mdlkd Or, Δi f = − (7.123) (Δi d ), D1 where, D1 = L mdl f + L mdlkd + l f lkd .

(7.124)

Substituting this expression for Δi f from Eq. (7.123), in Eq. (7.120), it yields, ) ) L md + l f (L md + lkd ) − 1 Δi d = L md D1 ) ( L mdl f =− (Δi d ) D1 ((

Δi kd

(7.125)

Substituting this expression for Δi f from the Eq. (7.123), and for Δi kd from the Eq. (7.125), in the expression for Δi d in Eq. (7.115), it yields, ( Δψd = (L md + la )Δi d − L md

) ( ) L mdl f L mdlkd (Δi d ) − L md (Δi d ) D1 D1

(7.126)

Substituting the expression for D1 from the Eq. (7.124), in the expression for Δψ d in Eq. (7.126), we have

(7.127)

172

7 Modelling Approach Using Generalized Theory of Electrical Machines

Equation (7.127) can be modified as given by, ⎧ ⎨ Δψd = (la ) + ( ⎩ Δi d

⎫ ⎬ ) ( ) ( ) . 1 + l1f + lkd1 ⎭ lmd 1

(7.128)

The right-hand side term of the above-said Eq. (7.128) represents an expression for ( a'' )new type inductance which is known as “Direct axis sub-transient inductance L d ”, because fundamentally, any inductance is nothing but the ratio of magnetic flux linkage and the concerned current flowing through that winding. ( ) If the supply frequency is ωs electrical radians per second, then the reactance X d '' corresponding to the “Direct axis sub-transient inductance '' can be expressed as, X d '' = ωs L d '' ⎡

⎧ ⎨

⎫⎤ ⎬ ( ) ⎦ = ωs ⎣(la ) + ( ) ⎩ 1 + 1 + 1 ⎭ L md lf lkd ⎧ ⎫ ⎨ ⎬ 1 = (xa ) + ( ) ( ) ( ) ⎩ 1 + 1 + 1 ⎭ xmd

1 ( )

xf

(7.129)

xkd

In the above-said expression for X d '' , xa , X md , x f and xkd are the armature leakage reactance, direct axis mutual reactance, field winding leakage reactance and direct axis damper winding leakage reactance, respectively. X d '' is called as “Direct axis armature sub-transient synchronous reactance” of a three-phase salient pole synchronous machine. The above said expression for X d '' in Eq. (7.129), directly indicates that the reactance parameters, X md , x f and xkd are connected electrically in parallel and finally xa becomes connected electrically, in series with this parallel combination. Such a physical view of the expression for Δψ d leads to the development of a direct axis sub-transient equivalent circuit of the Δi d three-phase salient pole synchronous machine and it is shown in Fig. 7.8, as follows. In the d-axis sub-transient equivalent circuit of Fig. 7.8, when the induced current in the d-axis damper winding branch becomes zero due to circuit attenuation (or, corresponding L/R effect), the sub-transient component dies out and transient current starts. In other words, from the network viewpoint, when xkd → ∞, the transient condition starts. Thus the transient equivalent circuit along the direct axis is obtained and this is shown in Fig. 7.9 as follows. In Fig. 7.9, if x f → ∞, then it indicates that the transient condition dies out and the steady state starts. Thus the steady-state equivalent circuit of a three-phase salient pole synchronous machine along the direct axis (d-axis) is obtained and it is shown in Fig. 7.10.

7.2 Development of the Sub-Transient, Transient and Steady State …

173

Fig. 7.8 Sub-transient equivalent circuit of a three-phase salient pole synchronous machine along the direct axis (d-axis) Fig. 7.9 Transient equivalent circuit of a three-phase salient pole synchronous machine along the direct axis (d-axis)

Fig. 7.10 Steady-state equivalent circuit of a three-phase salient pole synchronous machine along the direct axis (d-axis)

174

7 Modelling Approach Using Generalized Theory of Electrical Machines

Analysis for q-axis: With reference to the primitive machine model (Fig. 7.7) of the salient pole synchronous machine, all the variables and parameters are assumed as per unit quantities. The currents flowing the Q-coil,(and)KQ-coils are i(q , andi ) kq , respectively. The magnetic flux linkages with Q-coil ψq and KQ-coil ψkq are expressed as follows: ψq = L q i q + L mq i kq

(7.130)

ψkq = L mq i q + L kkq i kq

(7.131)

Furthermore, as per the previous discussions (see Sect. 7.1.2.4), the selfinductances of the said coils may be decomposed as: L q = L mq + la

(7.132)

L kkq = L mq + lkq

(7.133)

where L q and L kkq are the self-inductances and la and lkq are the leakage inductances of the Q- and KQ-coils, respectively and L mq is the mutual inductance between each pair of coils or windings. Substituting Eqs. (7.132) and (7.133) in the Eqs. (7.130) and (7.131) respectively, it yields, ( ) ψq = L mq + la i q + L mq i kq

(7.134)

) ( ψkq = L mq i q + L mq + lkq i kq

(7.135)

Applying the theory of small perturbations to the above said expressions for flux linkages with the Q-and KQ-coils in Eqs. (7.130) and (7.131), we get ( ) Δψq = L mq + la Δi q + L mq Δi kq

(7.136)

) ( Δψkq = L mq Δi q + L mq + lkq Δi kq

(7.137)

Whenever a sudden disturbance (e.g., sudden short circuit or, sudden load change) at the armature terminals of the synchronous machine is imposed, the other winding member on that particular axis (i.e., q-axis) must maintain the “Constant flux linkage theorem” as the particular winding has a finite amount of electrical inertia (i.e., inductance). Therefore, we can write, ) ( Δψkq = L mq Δi q + L mq + lkq Δi kq = 0

(7.138)

7.2 Development of the Sub-Transient, Transient and Steady State …

175

From the expression, Δψkq = 0 in Eq. (7.138), it yields, (

(

Δi kq = − (

L mq

)

)

L mq + lkq

) Δi q

(7.139)

Substituting this expression for Δi kq in Eq. (7.138), one may write that, ( ( ) ) ) L mq ) Δi q Δψq = L mq + la Δi q − L mq ( L mq + lkq (

(7.140)

The preceding equation, may be further simplified as, (7.141)

Hence, it yields ⎧ ⎨ Δψq = (la ) + ( ⎩ Δi q

1 L mq

)

⎫ ⎬

1 +

( ) . 1 ⎭

(7.142)

lkq

The right-hand side term of the above-said Eq. (7.142) represents a compact expression for a new type (of inductance which is known as “Quadrature axis armature ) sub-transient Inductance L q '' ”, because fundamentally, any inductance is nothing but the ratio of magnetic flux linkage and the concerned current flowing through that (winding. ) If the supply frequency is ωs electrical radians per second, then the reactance X q '' corresponding to the “Quadrature axis armature sub-transient inductance”, can be expressed as, X q '' = ωs L q '' = (xa ) +

⎧ ⎨ ⎩

(

1 xmq

)

1 +

(

⎫ ⎬

1 xkq

) . ⎭

(7.143)

In the above-said expression of X q '' , xa , X mq and xkq are the armature leakage reactance, quadrature axis mutual reactance and quadrature axis damper winding leakage reactance, respectively. X q '' is called as “Quadrature axis armature sub-transient synchronous reactance” of a three-phase salient pole synchronous machine. The above said expression for X q '' in Eq. (7.143), directly indicates that the reactance parameters, X mq and xkq are connected electrically in parallel and finally xa becomes connected electrically, in series with this parallel combination. Such physical view of the expression for Δψq leads to the development of a quadrature axis sub-transient equivalent circuit Δi q

176

7 Modelling Approach Using Generalized Theory of Electrical Machines

Fig. 7.11 Sub-transient equivalent circuit of a three-phase salient pole synchronous machine along quadrature axis (q-axis)

of a three-phase salient pole synchronous machine and it is shown in Fig. 7.11, as follows. In Fig. 7.11, if the current through the branch containing xkq tends to die out, then only the sub-transient state also tends to die out to give birth to the new state which is the steady state. This is possible only when, xkq → ∞. This concerned status is drawn in Fig. 7.12. So far, the importance of the research work involving the sub-transient and transient reactance of synchronous machines is concerned, recent literature [5] gives a nice mathematical treatment based on the superconductor simulation method (SSM). The special feature of this paper is that it uses electromagnetic field theory equations to give such analysis, a complete shape along with verifying the simulation results with the experimental results.

Fig. 7.12 Steady-state equivalent circuit of a three-phase salient pole synchronous machine along quadrature axis (q-axis)

References

177

References 1. Adkins B, Harley RG (1975) The general theory of alternating current machines: application to practical problems. Chapman and Hall, London, U.K 2. Frederic BM et al (2017) Dynamic models of electromechanical transducers: equations of generalised electrical machines. Res Rev: J Appl Sci Innov RRJASI 1(2):24–28 3. Tielens P, Henneaux P, Cole S (2018) Penetration of renewables and reduction of synchronous inertia in the European power system—Analysis and solutions. In: The ASSET project, funded by the European Commission. DG Energy, Unit C.2. [email protected], pp (11–12)/72 4. Bollen MH, Hassan F (2011) Integration of distributed generation in the power system, WileyIEEE Press 5. Zhanga S et al (2022) Numerical calculation and experimental verification of subtransient and transient reactances of a pumped storage machine. In: 2022 The 4th international conference on clean energy and electrical systems (CEES 2022), Tokyo, Japan, pp 1022–1029

Chapter 8

Digital Modeling Approach for Stability Analysis of Synchronous Motor Drive

8.1 Introduction Digital modelling of a three-phase synchronous motor drive system calls for the development of primarily, a suitable modelling package in the continuous domain for a general application area. However, it is expected that at the initial stage, the time differential equation of the drive system is converted into a mathematical model in the complex frequency domain(s-domain). Once this part is finished, then comes the responsibility of converting this model into a suitable digital model using the everefficient z-Transform method. But before going to the detailed steps of this methodology, it will appear to be very advantageous before the readers or the researchers, from the view point of grasping the formulation strategy if some relevant literature review of this particular topic is presented in a proper manner. Permanent magnet Synchronous Motor (PMSM) drive systems are becoming more popular due to their advantages such as utility of self-control, good efficiency and operation near to unity power factor, small inertia etc. Due to these advantages synchronous motors are also serious competitors to both dc motor drives and induction motor drives [1]. When compared to an AC induction motor, PMSM has superior advantage to achieve higher efficiency as it produces the rotor magnetic flux with permanent magnets. For this reason, PMSMs are used in high-end appliances and equipment that require high efficiency and reliability [2]. PMSMs are gaining increasing popularity and demand in various areas such as automobiles, robotics and aerospace engineering [3]. In addition, Z Source-Based Permanent Magnet Synchronous Motor Drive System [4] is also gaining importance. Design of controller for PMSM has been reported in [5]. PMSM has many attractive characteristics such as high-power density, hightorque-to inertia ratio, wide speed operation range, and free from maintenance, and it is suitable for many industrial applications [6, 7]. So far the literature review is concerned, digital implementation of an adaptive speed regulator for a PMSM has been reported in [8]. However, this work does not focus on the stability analysis from the digital domain point of view. To evaluate the field performance accurately © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. B. Chattopadhyay et al., Mathematical Modeling of Physical Systems, Advances in Intelligent Systems and Computing 1436, https://doi.org/10.1007/978-981-19-7558-5_8

179

180

8 Digital Modeling Approach for Stability Analysis of Synchronous …

we need accurate digital simulation tools. Hence, the design of digital controllers merges to be the main area of interest for such drive systems. One disadvantage with PMSM is that for accurate analysis of the performance of the drive systems, circuit theory loses its advantage, whereas in such problem, electromagnetic field theory becomes quite successful to analyse and then to come to a concrete conclusion.

8.2 Problem Formulation A primitive machine model of the Synchronous motor pertaining to the model shown in Fig. 8.1 has been developed and the detailed mathematical analysis is presented in [9].

8.2.1 Development of Transfer Function in Continuous Domain The torque dynamic equation of a synchronous motor can be written as, Te − TL = J

dω dt

(8.1)

where, Te TL ω J ΔTe (s)

is electromagnetic torque in N m is Load Torque in N m is Motor speed in mechanical rad./s is Polar moment of inertia of motor and load (combined) kg m2 is perturbed quantity (transformed) of electromagnetic torque and the expression for it is derived from [I] is given as

Fig. 8.1 Drive configuration for open loop current fed synchronous motor

8.2 Problem Formulation

181

] x1 s 3 + x2 s 2 + x3 s + x4 Δβ(s) ΔTe (s) = l1 s 3 + l2 s 2 + l3 s + l4 [

(8.2)

The small change in speed ‘ω’ equal to Δω can be related to small change in field angle, Δβ as given by, Δω = −

d(Δβ) dt

(8.3)

The negative sign in equation physically indicates a drop in speed (ω) due to increase in field angle (β). Based on Eq. (8.3), the following expression can be written, J

[ ] d d d2 d(Δω) =J − (Δβ) = −J 2 (Δβ) dt dt dt dt

(8.4)

The small-perturbation model of Eq. (8.4) can be written as, ΔTe − ΔTL = J

d(Δω) dt

(8.5)

d2 (Δβ) dt 2

(8.6)

Combining Eqs. (8.4) and (8.5), it yields, ΔTe − ΔTL = −J

The transformed version of Eq. (8.6), with initial condition relaxed, comes out to be ΔTe (s) − ΔTL (s) = −J s 2 Δβ(s)

(8.7)

Substituting the expression for ΔTe(s) from Eq. (8.2) in Eq. 8.7), it yields [

] x1 s 3 + x2 s 2 + x3 s + x4 Δβ(s) + J s 2 Δβ(s) = ΔTL (s) l1 s 3 + l2 s 2 + l3 s + l4

(8.8)

Equation (8.8) gives after manipulation, a Transfer Function, T (s) expressed as, T(s) =

l1 s 3 + l2 s 2 + l3 s + l4 Δβ(s) = 5 ΔTL (s) K1s + K2s4 + K3s3 + K4s2 + K5s + K6

where, K 1 =Jl1 K 2 =Jl2

(8.9)

182

8 Digital Modeling Approach for Stability Analysis of Synchronous …

K 3 =(Jl3 + x1 ) K 4 =(Jl4 + x2 ) K 5 =x3 K 6 =x4 Once the transfer function has been developed in s-domain, the z-transfer function has been obtained using MATLAB as explained in the next section.

8.2.2 Development of Transfer Function in Discrete Domain The transfer function in the continuous time domain was given as [8]: T(s) = T(s) =

Δβ(s) = ΔTL (s)

l1 s 3 + l2 s 2 + l3 s + l4 Δβ(s) = ΔTL (s) K1s5 + K2s4 + K3s3 + K4s2 + K5s + K6

(8.10)

where I.

The algebraic expressions of l1 , l 2 , l 3 , l 4 and K 1 , K 2 , K 3 , K 4 , K 5 , K 6 are available in [8]. II. Δβ(s) is the transformed field angle (small perturbation). III. ΔTL (s) is the transformed load torque (small perturbation). Here, the terminology “transformed” means that the mathematical tool Laplace Transform has been applied. The polynomial coefficients (K 1 –K 6 ) [8] were calculated using the machine data present in the appendix. The values of K 1 –K 6 are as follows. K 1 =2.2871 K 2 =0.3585 K 3 =0.5577 K 4 =0.0776 K 5 =0.0023 K 6 =2.419 × 10−6 The numerator polynomial coefficients were calculated using the equations and the machine data [I]. The values of l1 –l 4 are as follows: I1 = 0.2857, I2 = 0.0447, I3 = 1.615 × 10−3 , I4 = 1.755 × 10−6 .

8.3 Stability Analysis of Discrete Time Systems

183

The above equation in continuous form is converted to discrete form (i.e. converted to z domain). The discrete domain conversion was done using MATLAB software with sampling time as 0.1 s. Z transform was taken after substituting the coefficient values. The transfer function obtained is represented as T (z). Δβ(z) ΔTL (z) 0.0006245z 4 − 0.001239z 3 − 9.676 × 10−6 z 2 + 0.001239z − 0.0006148 = z 5 − 4.982z 4 + 9.931z 3 − 9.899z 2 + 4.935z − 0.9844 (8.11)

T (z) =

8.3 Stability Analysis of Discrete Time Systems 8.3.1 Stability Analysis Using Pole-Zero Mapping The stability analysis of the discrete time equation is done by plotting a pole zero map using MATLAB software. Figure 8.2 shows the pole zero map obtained for the z-transfer function. Since all the poles lie inside the unit circle, the steady state stability is assured for the system in discrete domain. Even though pole-zero locations give a basic information about the system stability, investigations on the coefficients of the characteristic equation of the same system give a more clear picture on the stability assessment. Such investigation is known as Jury’s test and it is taken up in the next section for a through result and discussion.

8.3.2 Stability Analysis Using Jury’s Test Jury’s stability criterion is a method to analyse the stability of the discrete time system using the coefficients of the characteristic equation derived from the transfer function. Using Jury’s test we can do stability analysis of a discrete time system without having to calculate the poles of the system. Detailed mathematical analysis of jury test on the system is given below. The characteristic equation is: f (z) = z 5 − 4.982z 4 + 9.931z 3 − 9.899z 2 + 4.935z − 0.9844

(8.12)

The necessary and sufficient conditions to satisfy for the system to be considered stable are:

184

8 Digital Modeling Approach for Stability Analysis of Synchronous …

Fig. 8.2 Pole zero map of the transfer function

Rule 1 If z is 1, the system output must be positive: f (1) > 0 Rule 2 If z is −1, then the following relationship must hold: (−1)n f (−1) > 0 where n is highest power of the characteristic equation. Rule 3 The absolute value of the constant term (a0 ) must be less than the value of the highest coefficient (an ): |a0 | < |an |, where the polynomial is given as f (z) = an + an−1 z 1 + an−2 z 2 + an−3 z 3 ...... + a0 z n If Rule 1 Rule 2 and Rule 3 are satisfied, construct the Jury Array. Rule 4 Once the Jury Array has been formed, all the following relationships must be satisfied until the end of the array: |b0 | > |bn-1 |

8.3 Stability Analysis of Discrete Time Systems

185

Table 8.1 Jury’s array pertaining to the stability assessment in discrete time domain Z0

Z1

Z2

Z3

Z4

Z5

−0.98 44

4.935

−9.899

9.931

−4.982

1

1

−4.982

9.931

−9.899

4.935

−0.9844

−0.03 09

0.1239

−0.186

0.1229

−0.0307

×

−0.03 07

0.1229

−0.186

0.1239

−0.0309

×

1.232 ×

1.594 ×

6.1 ×

×

×

6.1 × 10–6

0.0372 × 10–3

1.594 × 10–4

1.232 × 10–5

×

×

1.458 × 10–10

1.6814 × 10–9

−0.0514 × 10–8

×

×

×

10–5

10–4

0.0372 ×

10–3

10–6

|c0 | > |cn-2 | |d 0 | > |d n-3 | And so on until the last row of the array. If all these conditions are satisfied, the system is stable. Jury array for the system is shown in Table 8.1. The results are in Table 8.1, hereby assure the stability of the system. It is well known that if the coefficients of a characteristic equation are perturbed, the system may move to the unstable zone. This will give a rough indication of the degree of robustness of the system. The stability ranges for coefficients were found by changing the value of one coefficient while keeping the other coefficients constant. The stability analysis was done by plotting the pole zero maps using MATLAB for the transfer function. If the poles of the equation lie inside the unit circle then the system was concluded to be stable and if the poles lie outside the unit circle then the system was concluded to be unstable. Table 8.2 represents the range of the co-efficient. The original value of the coefficients is represented in the brackets. For example when K 1 is given value more than 2.4 the system becomes unstable. Similar analysis was done for rest of the coefficients as well. Figure 8.3 represents the pole zero map of the stable system when there was no change in the coefficient values. Figure 8.4 represents the pole zero map of unstable system. In this transfer function the K 1 value was taken as 2.5 which clearly exceeds the maximum range which is 2.4. Therefore, the system becomes unstable. Once the range of coefficients of the characteristic equations are known from the view point of stability, it becomes necessary to find the impulse response of the system as an important part of the whole digital modelling. The analysis involving the impulse response is presented below. Table 8.2 Range for the coefficients

Range

Original values (stable)

K 1 ≤ 2.4

(2.2871)

K 2 ≤ 15.9

(0.3585)

K 3 ≤ 52.75

(0.5577)

K 4 ≤ 0.081

(0.0776)

186

8 Digital Modeling Approach for Stability Analysis of Synchronous …

Fig. 8.3 Pole zero map for stable system

Fig. 8.4 Pole zero map for unstable system

8.4 Impulse Response Analysis in Continuous and Discrete Domain It is well known that an Impulse response of a system can be looked upon as Inverse Laplace Transform of the Transfer Function. Hence, it is a very important analysis of a control system. For example, if we consider an AVR control system that controls the output voltage of the generator, when designing a controller for this system, one has to consider worst case scenarios or conditions. For example, if we consider the worst case scenario can be a lightning for a very short period of time (in microseconds).

8.4 Impulse Response Analysis in Continuous and Discrete Domain

187

This causes a very big voltage to be impressed at the terminals of the generator. In this case the control system must be able to secure the generator from the worst possible scenario. In such case stability analysis using the impulse response method becomes a strong tool for analysis. In the following section, the stability analysis of the system with input as the unit impulse is performed and subsequently a detailed mathematical analysis is represented.

8.4.1 Impulse Response Analysis in S-domain With reference to Fig. 8.1, there may occur so many types of variations in the magnitude of change in load torque. Due to abnormal behaviour of the mechanical coupling system or any other reason there may appear a sudden change in load torque. In other words, change in load torque may be considered as a unit impulse function and our objective in this subsection is to investigate the stability assessment of the particular system with input as δ(t) through the method of Pole-Zero mapping. ΔT L (S) = Lδ(t) = 1

(8.13)

Δβ(t) = L −1 Δβ(S)

(8.14)

where L and L −1 are Laplace and inverse Laplace transform operator, respectively. Hence, Δβ(t) = e−(p1)t + Be−(p2)t + Ce−(p3)t + De−(p4)t + Ee−(p5)t

(8.15)

where p1, p2, p3, p4, p5 are the roots of the characteristic equation and A, B, C, D, E are the coefficient values. p1 = − 0.0071 + 0.4874i p2 = − 0.0071 − 0.4874i p3 = − 0.1011 p4 = − 0.0403 p5 = − 0.0011 A = − 0.0036 − 0.1284i B = − 0.0036 + 0.1284i C =0 D = 0.0072 E =0

188

8 Digital Modeling Approach for Stability Analysis of Synchronous …

Fig. 8.5 Impulse response of the transfer function in continuous time domain

Since all the poles of the characteristic equation have negative real parts assured stability is guaranteed in this context, plot of change in amplitude versus time has been presented in Fig. 8.5. This plot clearly indicates a decaying sinusoid having both the upper envelope and lower envelope. It is also a point of interest that such a plot is treated as a symmetrical wave form leading to the major physical conclusion, i.e., the DC component is absent. Such concept appears to be automatically correct because of the ordinate of Fig. 8.5 is a change in variable (not absolute value).

8.4.2 Impulse Response Analysis in Z-domain It may be noted that the concept put in Sect. 8.2.1 was valid in continuous time domain and that is why Laplace Transform operator was used to investigate the satiability assessment phenomena. However, as the present paper deals with the digital modelling of the system, at this stage it becomes necessary to convert the above said philosophy in Sect. 8.2.2 to discrete time domain. As a natural responsibility the use of ‘Z’ transform must be involved in such case to investigate the stability phenomenon in the discrete time domain. Based on such philosophy the related mathematical equations are presented as follows: Since Z [δ(n)] = 1, Δβ(z) = T (z) ∗ 1

(8.16)

Δβ(n) = Z −1 [T (z)]

(8.17)

8.4 Impulse Response Analysis in Continuous and Discrete Domain

189

Hence, Δβ(n) = Aδ(n) + Bl1n + Cl2n + Dl3n + El4n + Fl5n

(8.18)

Here l 1 , l 2 , l 3 , l 4 , l 5 are the roots of the characteristic equation and A, B, C, D, E, F are the coefficients. l1 = 1.1875 + 0.1692i l2 = 1.1875 − 0.1692i l3 = 0.9014 + 0.1956i l4 = 0.9014 − 0.1956i l5 = 0.8041 A = 0.0006 B = 0.0008 − 0.0007i C = 0.0008 + 0.0007i D = −0.0005 − 0.001i E = −0.0005 + 0.001i F = −0.0012 The explanation behind the presentation of Fig. 8.5 can be reproduced in similar lines for simulations in discrete time domain. Such approach leads to the development of Fig. 8.6; where impulse response of transfer function is presented in discrete domain. It is natural to put a question on the fact that Δβ(t) or Δβ(n) are considered to be variables of importance, and why not β(t) or β(n). The answer lies in the fact that the very purpose of this research is to investigate the steady state stability of the synchronous motor drive system from various viewpoints. Moreover, the concept of the steady state stability is based on the theory of small perturbations. At this stage it is felt that even though necessary analysis has been presented involving the coefficients of the characteristic equation, still some more study from the view point of design aspects of the motor can be done. This study will be based on perturbing the real machine design parameters like Rf (Field winding resistance), Rkd (Direct axis damper windings) etc. Due to these changes, the coefficients like K 1 , K 2 etc. will be perturbed and hence steady state stability of the system may get affected. Such analysis with the relevant results are presented in the next section.

190

8 Digital Modeling Approach for Stability Analysis of Synchronous …

Fig. 8.6 Impulse response of transfer function in time discrete domain

8.5 Analysis of Parameter Perturbation on Stability Based on the stability range for the coefficients given in the Table 8.1, it reveals that changing the parameter affects the design from the view point of steady state stability assessment through digital modelling. To find out which machine parameter affects the stability of the system most a method is proposed with a detailed explanation in an example below. If coefficient K 6 is considered as K 6 = X 4 as in (8.2) x4 = n 4 i 5 cos Δβ − m 4 i s sin Δβ

(8.19)

n 4 = c2 f 2 b3

(8.20)

where,

and C1 =

[(

) ] L d − L q i q0 − L mq i kq0

(8.21)

8.5 Analysis of Parameter Perturbation on Stability

C2 =

[(

) ] L d − L q i d0 + L md i kd0 + L md i f 0

191

(8.22)

f 2 = Rkq

(8.23)

b3 = Rkd R f

(8.24)

Let, i s cos Δβ = P1 and i s sin Δβ = P2. Therefore, (8.16) can be rewritten as x4 = c2 f 2 b3 P1 − c1 b3 f 2 P2 ) ] ) L d − L q i d0 + L md i kd0 + L md i f 0 P1 [( ) ] − L d − L q i q0 − L mq i kq0 P2

(8.25)

( [( K 6 = X 4 = Rkd R f Rkq

(8.26)

Substituting the constant values given in appendix we get, K 6 = Rkd R f Rkq

{[(

) ] [( ) ] } L d − L q i d0 + L md i f 0 P1 − L d − L q i q0 P2

(8.27)

At this stage, it will be a matter of justice if the physical significances of variables with suffice “0” (i.e. βo, if 0 , id0 , iq0 , ikq0 , ikd0 ) are stated clearly. Basically, the theory of calculus states that Taylor’s series expression of any function becomes mathematically defined about an operating point (or quiescent point). id0 , iq0 etc. are the quiescent values. Practically also, one cannot perturb a system until and unless the equilibrium state is known. We can see that K 6 is a function of (Rkd , Rf , Rkq , L md ). Now from the Table 8.1 we know that the system is stable when K 6 ≤ 0.00027 the system is stable. The system becomes unstable at K 6 = 0.0003. Therefore, dK6 = 0.0003 − 0.00027 or, dK6 = 3 × 10−5 .

(8.28)

Now using the Euler’s differential formula we can write it as, d K6 =

∂f ∂f ∂f ∂f d RKd + d RKq + dRf + d L md ∂ RKd ∂ RKq ∂Rf ∂ L md

Case 1: When we consider RKd is changing,

(8.29)

192

8 Digital Modeling Approach for Stability Analysis of Synchronous …

d K6 =

∂f d RKd + 0 ∂ RKd

(8.30)

Since only RKd is changing and the other coefficients are constant, the rest of the equation becomes zero. Now Eqs. (8.28) and (8.30) we get 3 × 10−5 =

∂f d RKd + 0 ∂ RKd

∂ {RKd R f RKq P1[(L d − L q )i d0 + L md i f 0 ]} ∂ RKd ∂ − {RKd R f RKq P2(L d − L q )i q0 } ∂ RKd = [R f RKq P1[(L d − L q )i d0 + L md i f 0 − R f RKq P2(L d − L q )i q0 ]ΔRKd

=

(8.31)

By substituting the values for the parameters given in the appendix we get ΔRKd = 0.372

(8.32)

Case 2: When we consider Rf is changing, d K6 =

∂f dRf + 0 ∂Rf

(8.33)

Since only Rf is changing and the other coefficients are constant, the rest of the equation becomes zero. Now Eqs. (8.28) and (8.33) we get 3 × 10−5 =

∂f dRf + 0 ∂Rf

∂ {RKd R f RKq P1[(L d − L q )i d0 + L md i f 0 ]} ∂Rf ∂ {RKd R f RKq P2(L d − L q )i q0 } − ∂Rf

=

= [RKd RKq P1[(L d − L q )i d0 + L md i f 0 ]} − {RKd RKq P2(L d − L q )i q0 ]ΔR f (8.34) By substituting the values for the parameters given in the appendix we get. ΔR f = 1.86 × 10−2

(8.35)

8.5 Analysis of Parameter Perturbation on Stability

193

Case 3: Now considering RKq is changing, d K6 =

∂f d RKq + 0 ∂ RKq

(8.36)

Since only RKq is changing and the other coefficients are constant, the rest of the equation becomes zero. Now Eqs. (8.28) and (8.36) we get 3 × 10−5 =

∂f d RKq + 0 ∂ RKq

∂ {RKd R f RKq P1[(L d − L q )i d0 + L md i f 0 ]} ∂ RKq ∂ − {RKd R f RKq P2(L d − L q )i q0 } ∂ RKq

=

= [RKd R f P1[(L d − L q )i d0 + L md i f 0 ]} − {RKd R f P2(L d − L q )i q0 ]ΔRKq (8.37) By substituting the values for the parameters given in the appendix we get. ΔRKq = 0.527

(8.38)

Case 4: Similarly considering L md is changing, d K6 =

∂f d L md + 0 ∂ L md

(8.39)

Since only RKq is changing and the other coefficients are constant, the rest of the equation becomes zero. Now Eqs. (8.28) and (8.39) ∂f d L md + 0 ∂ L md ∂ = {RKd R f RKq P1[(L d − L q )i d0 + L md i f 0 ]} ∂ L md ∂ − {RKd R f RKq P2(L d − L q )i q0 } ∂L ] ] [[ md = RKd R f RKq P1L md i f o ΔL md

3 × 10−5 =

(8.40)

By substituting the values for the parameters given in the appendix we get.

194

8 Digital Modeling Approach for Stability Analysis of Synchronous …

ΔL md = 17.89

(8.41)

The physical interpretation of such change can be explained as follows, with reference to (8.41), the specified value of the change in L md basically indicates a change in direct-axis air-gap. If the said value of perturbation in L md is not acceptable, then the direct axis air-gap will have to be changed and it will in turn affect the input power factor of the drive system. If, the particular value of change in input power factor is not allowed in practice, then field winding current of the motor will have to be adjusted.

8.6 Conclusion This research concludes the following salient points: • Assessment of steady state stability has been performed in discrete time domain, by the method of “pole-zero” mapping and “Jury’s Test”. The result of both the methods do not conflict. • As changing the coefficients of the characteristic equation shows a direct impact on the stability of the system, a new view point based on machine design aspect is developed, which is written under the next point. • The machine design parameters (R f , RKd , L md etc.) can be perturbed. This aspects has been studied using necessary mathematical formulations based on “Partial Differentiation”. • It was observed that a minimum amount of perturbation on field winding resistance affects the stability most. This is due to the fact that a change in field winding resistance changes the field current. As a result the flux linkage changes and ultimately the electromagnetic torque is affected. If this effect happens in a negative direction, stability is affected and the motor may finally cease to take the load.

References 1. Pillay P, Krishnan R (1991) Application characteristics of permanent magnet synchronous and brushless dc motors for servo wives. IEEE Trans Ind Appl 27:986–996 2. Lee S, Song BM, Won TH (2010) Evaluation of software configurable digital controller for the permanent magnet synchronous motor using filed oriented control. In: 42nd South Eastern symposium on system theory University of Texas at Tyler, TX, USA 3. Samaramayake L, Chin YK (2003) Speed synchronized control ofpermanent magnet synchronous motors with field-weakening. In: International Conference on Power and Energy Systems, EuroPES2003, vol 3, pp 547–552 4. Mahmoudi H, Aleenejad M, Ahmadi R, Modulated model predictive control for a Z source based permanent magnet synchronous motor drive system. IEEE Trans Ind Electron PP(99):1–1. https://doi.org/10.1109/TIE.2017.2787566

References

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5. Zheng Y, Wang Z, Zhang J (2013) Research of harmonics and circulating current suppression in paralleled inverters fed permanent magnet synchronous motor drive system. In: 2013 international conference on electrical machines and systems (ICEMS), Busan, pp 1068–1073 6. Lin FJ, Liu YT, Yu WA (2018) Power perturbation based MTPA with an online tuning speed controller for an IPMSM drive system. IEEE Trans Ind Electron 65(5):3677–3687. https://doi. org/10.1109/TIE.2017.2762634 7. Lin FJ, Hung YC, Chen JM, Yeh CM (2014) Sensorless IPMSM drive system using saliency back-EMF-based intelligent torque observer with MTPA control. IEEE Trans. Ind. Informat. 10(2):1226–1241 8. Choi HH, Vu NTT, Jung JW (2011) Digital implementation of an adaptive speed regulator for a PMSM. IEEE Trans Power Electron 26(1):3–8 9. Chattopadhyay AB, Thomas S, Chatterjee R (2011) Analysis of steady state stability of a CSI fed synchronous motor drive system with damper windings included. Trends Appl Sci Res

Chapter 9

Unscented and Complex Unscented Kalman Filtering for Parameter Estimation of a Single and Multiple Sinusoids in the Area of Power and Communication Signals

9.1 General Introduction The problem of estimating the sinusoidal parameters in white Gaussian noise is a fundamental well studied research topic. Its numerous applications include radar [1], nuclear magnetic resonance [2], control [3], instrumentation [4], power networks [5], digital communication [6], biomedical engineering [7], etc. For example, in power networks, accurate estimation of parameters is required not only for control and protection, but also for proper operation of distributed generation systems in a microgrid environment. In biomedicine, where ultrasound signal, heart sound signal, heart rate of a foetus, etc. from a patient needs to be analyzed to give accurate parameters of interest for diagnosis. Online monitoring of Radar, Sonar and Speech signals are some other applications of parameter estimation. In most practical cases, we need to estimate the time varying or quasi-periodic parameters of the signal. Hence, the problem of estimating amplitude, phase and frequency, directly from the discrete measurement in the presence of noise both in stationary and nonstationary environments is presented in this chapter. The state-space approach is convenient for handling multivariate data and nonlinear or non-Gaussian processes. In the Bayesian approach, for the optimal solution to the problem, one requires to construct the description of the full posterior probability density function (PDF) [8], which includes all the available statistical information. This solution is extremely general and conceptual solution, and incorporates aspects such as multimodality, asymmetries, and discontinuities. However, because the form of the PDF is not restricted, it cannot, in general, be described using a finite number of parameters. Therefore, any practical estimator must use an approximation of some kind. Many different types of approximations have been developed, unfortunately, most are either computationally unmanageable or require special assumptions about the form of the process and observation models that cannot be satisfied in practice.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. B. Chattopadhyay et al., Mathematical Modeling of Physical Systems, Advances in Intelligent Systems and Computing 1436, https://doi.org/10.1007/978-981-19-7558-5_9

197

198

9 Unscented and Complex Unscented Kalman Filtering for Parameter …

For these and other reasons, Rudolf Emil Kalman proposed a recursive algorithm in 1960 [9], which was derived from Wiener Filter [10] with the change to state-space form, which enables the filter to be used as smoother, a filter or predictor. Further, the state-space representation also allows the implementation of the filter in a discrete domain, which is the reason for the widespread application of the Kalman Filter. The Kalman Filter only utilizes the first two moments of the state (mean and covariance) in its update rule. Kalman Filtering is a relatively simple state representation, still it offers a number of important practical benefits. The mean and covariance of an unknown distribution require the maintenance of only a small and constant amount of information, but that information is sufficient to support most kinds of operational activities. Thus, it is a successful compromise between computational complexity and representational flexibility. The mean and covariance are linearly transformable quantities. Hence, when the system is linear or quasi-linear, the mean and covariance estimates can be maintained effectively. In other words, Kalman Filter provides an exact solution for linear Gaussian prediction. Similar results do not hold for other nonzero moments of a distribution. To characterize additional features of distribution, many researchers use sets of mean and covariance estimates. The Extended Kalman Filter [11, 12] is probably the most widely used adaptive filter algorithm for estimating nonlinear signals. Extended Kalman filtering is a nonlinear sequential state estimation technique. It uses relatively simple state representation. It utilizes only the first two moments of the state in its update rule, they are mean and covariance. But the performance of the Extended Kalman Filter is limited due to the crude assumptions made for signal parameter estimation, instability due to linearization, costly calculation of the derivatives and the biased nature of the estimates. To address the deficiencies of EKF, Julier proposed a new unscented transform (UT), which is used to evaluate the statistics of a random variable undergoing nonlinear transformation. Further, the UT relies on the principle that a Gaussian distribution is relatively easy to approximate than a nonlinear function and provides higher-order information about the distribution of the system state, even using a small number of sample points. The Unscented transform paved the way for the realization of the novel filter known as the Unscented Kalman filter (UKF) [13, 14], which is shown to have an approximately second-order convergence for Taylor series expansion. The main advantage of unscented transformation used in UKF is that it does not use linearization for computing the state and error covariance matrices resulting in a more accurate estimation of the parameters of a nonstationary signal. However, its accuracy significantly decreases, if SNR is low and the noise covariances and some of the parameters used in unscented transformation are not chosen correctly. For most of the estimators, the performance is evaluated according to its accuracy in estimation, processing time, computational requirement and ability to handle multiple signals. It is found that most of the estimation techniques have a tradeoff between estimation accuracy and computational complexity. Some of the estimators provide the best performance with high computational complexity, limiting their use in practical implementation. Other existing estimators have a low computational

9.2 Outline of Basic Estimation Methods

199

requirement, but are unable to achieve good performance. The accuracy of the estimator is evaluated by comparing the minimum possible error of the estimator with the Cramer–Rao Lower bound (CRLB).

9.2 Outline of Basic Estimation Methods The most widely used signal parameter estimation methods are DFT and FFT. But these methods suffer from aliasing, leakage and picket fence effects, and hence need error compensation and adaptive window width. Many of the other parametric methods for signal parameter estimation use a mathematical model to estimate the signal parameters. Some of the existing well known conventional parametric estimation method include Maximum Likelihood (ML) estimator [15], Linear predictor (LP) technique [16], Pisarenko’s method [17] and the MUSIC algorithm [18] which exploit the orthogonal property of the signal and noise subspaces, but result in large computational overhead. The two signal processing algorithms like LMS [19] and RLS [20] are widely used due to their simple mathematical models and estimation accuracy and convergence. However, they are well suited for amplitude and phase estimation of fundamental and harmonic components, if frequency is known apriori. Similarly, some other well known techniques like artificial neural networks [21] and Adalines [22] have been used for time varying signal parameter estimation. Most of these algorithms require heavy computational outlay and suffer from inaccuracies in the presence of noise with low signal to noise ratio (SNR). This section describes a few well known estimation techniques like the Maximum likelihood estimation (MLE), the MUSIC algorithm, Linear predictor (LP), extended Kalman Filter (EKF), etc. available inliterature. Stationary Signal A signal is said to be stationary if the signal parameters does not change with time. In other words, the frequency content of the stationary signal does not change with time. In mathematical sense, signal is stationary so that infinite averages can be estimated from finite averages. Let the stationary signal is given as Sk = Ak cos(ωk kTs + φk )

(9.1)

where Ak , ωk and φk are amplitude, frequency and phase of the sinusoid, respectively, and Ts is the sampling time. A stationary signal is shown in Fig. 9.1. Nonstationary Signals A signal is said to be nonstationary if the signal parameters change with time. In other words, a nonstationary signal is one whose statistical characteristics like amplitude distribution and standard deviation change with time. Many applications where nonstationary signals occur are power networks, human speech, biomedical signals, etc. The nonstationary signal can also be represented in a similar way as shown in Eq. (9.1) with the parameter Ak , ωk and φk . For example, consider a nonstationary

200

9 Unscented and Complex Unscented Kalman Filtering for Parameter …

Fig. 9.1 Stationary signal

signal where for the first 100 samples, the signal frequency is 50 Hz, and which changes to 70 Hz and then to 100 Hz for rest of the samples. The generated signal is shown in Fig. 9.2. Then the nonstationary signal discussed above is corrupted with 20 dB additive white Gaussian noise and is shown in Fig. 9.3.

Fig. 9.2 Non stationary signal

9.2 Outline of Basic Estimation Methods

201

Fig. 9.3 Non stationary signal with 20 dB noise

Before reviewing the conventional methods, we will discuss an important measure of signal parameter estimation algorithm, the Cramer–Rao lower bound (CRLB).

9.2.1 Cramer–Rao Lower Bound (CRLB) The performance of the parameter estimator is evaluated by the unbiasness, that is, the estimator converges to the true unknown parameter of interest on an average basis. The Cramer–Rao lower bound (CRLB) [23] is a limit to the variance that can be attained by an unbiased estimator of parameter of interest. As we are provided with the noisy observations of the data, all the information is embodied in the data and in the underlying probability density function (PDF). CRLB determines the minimum variance that can be attained by any unbiased estimator. The CRLB for a real sinusoidal signal is obtained as follows: Assuming the PDF denoted as p(x, θ ) satisfies the regularity condition ] ∂ ln p(x, θ ) =0 E ∂θ [

(9.2)

then the variance of any unbiased estimator θˆ must satisfy ˆ ≥ var(θ)

−E

[

1 ∂ 2 ln p(x,θ ) ∂θ 2

]

(9.3)

202

9 Unscented and Complex Unscented Kalman Filtering for Parameter …

where the derivative is evaluated at the true value of θ . The denominator in Eq. (9.3) is referred to as Fisher information. A sinusoidal signal corrupted with white Gaussian noise is given as x(k) = S(k, θ ) + v(k).

(9.4)

And the variance is given as var(θˆ ) ≥

N∑ −1 ( k=0

σ2 ∂s(k,θ ) ∂θ

)2

(9.5)

With σ 2 is the variance of the noise signal. The next section provides a description of other well known signal parameter estimation paradigms.

9.2.2 Maximum Likelihood Estimation (MLE) The maximum likelihood estimate of an unknown parameter is that value, which maximizes the likelihood function of the parameter to be estimated for the given data. MLE [15] has many optimal properties like consistency, sufficiency and efficiency. Its performance is optimal for large enough data record. Let the signal model used for parameter estimation is given by yk = Sk + vk , k = 1, 2, ....N

(9.6)

where Sk =

M ∑

Aik cos(ωik tk + φik )

(9.7)

i=1

where Aik , ωik and φik are amplitude, frequency and phase of the ith sinusoid, respectively, tk = kTs , where Ts is the sampling time. The modelling error vk is a zero mean white Gaussian noise with variance σv2 , and k is number of samples of the signal. In maximum likelihood estimation, the maximization of the probability density function (PDF) parameterized by sinusoidal parameters is equivalent to minimizing the sum of the squares of the residue.

9.2 Outline of Basic Estimation Methods

203

The PDF is given by [ ∑ ] N 2 1 k=1 (yk − Sk ) p(y, A, w, φ) = exp − , (2π σ 2 ) N /2 2σ 2

(9.8)

where A = [A1 , A2 , A3 , . . . A M ]T , w = [ω1 , ω2 , ω3 , . . . ω M ]T , and φ = [φ1 , φ2 , φ3 , . . . φ M ]T

Taking logarithm both side of the PDF yields log( p(y, A, w, φ)) =

N 1 ∑ −N log (2π σ 2 ) − (yk − Sk )2 , 2 2σ 2 k=1

(9.9)

and according to the definition of MLE, the parameters that maximizes the PDF is equivalent to minimizing the least square function as θˆ = arg min

A, w, φ

N ∑

(yk − Sk )2

(9.10)

k=1

Although MLE is consistent and efficient, but its computational cost of minimizing the least square function is very high. And for a global optimization, a good initialization is required.

9.2.3 Linear Predictor (LP) In linear predictor (LP) [16], the problem of predicting the current sample value from the previous sample values is based on the fact that the current sample of the signal is a linear combination of its paste 2 M samples, and the predicted value is given as ˜ S(k) =

2M ∑

ar S(k − r ),

(9.11)

r =1

where ar = 1, ... 2M, are the linear prediction coefficients. The prediction error is given as ˜ e(k) = S(k) − S(k) = S(k) −

2M ∑ r =1

ar S(k − r ),

(9.12)

204

9 Unscented and Complex Unscented Kalman Filtering for Parameter …

Now minimizing the error square by finding the optimal set of values of ar , and setting the derivative to zero. By the linear predictor properties, the system can be set in matrix form as c = C a˜

(9.13)

where c = [y(N ) + y(N − 2M)y(N − 1) + y(N − 2M − 1)y(N − 2) + y(N − 2M − 2)..y(2M) + y(1)]T

(9.14)

⎤ y(N − 2) + y(N − 2M) . . . . .y(N − M − 1) ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ . C =⎢ ⎥ ⎥ ⎢ . ⎦ ⎣ y(2M − 1) + y(1) . . . y(M) ⎡

(9.15)

and

a˜ = [a1 , a2 , . . .

. a M ]T

(9.16)

Linear Predictor is computationally efficient than MLE but statistically inefficient.

9.2.4 Extended Kalman Filter (EKF) The Kalman filter is a set of mathematical equations that provides an efficient recursive computational means to estimate the state of a process, in a way that minimizes the mean of the squared error. Unlike the Wiener filters, which are designed under the assumption that the signal and noise are stationary, the Kalman filter has the ability to adapt itself to nonstationary environments. Hence, the Kalman filter can be viewed as a sequential minimum MSE estimator of a signal in additive noise. If the signal and noise are jointly Gaussian, the then Kalman filter is optimal in a minimum MSE sense. This means the belief state must be unimodal, which is inappropriate for many problems, especially those involving qualitative (discrete) variables. For example, some systems have multiple modes or regimes of behaviour, there Kalman filter will not provide optimal solution to the problem. Hence, the Kalman Filter addresses the problem of estimating the state of a linear stochastic discrete time system. But if the system state to be estimated or the measurement relationship to the process is nonlinear, then Extended Kalman Filter is used, which is the standard extension of the Kalman fiter to nonlinear models. The Extended

9.2 Outline of Basic Estimation Methods

205

Kalman Filter was proposed by Stanley F. Schmidt [24] earlier known as “KalmanSchmidt” filter. The Extended Kalman Filter generalizes the use of likelihood function to non-Gaussian processes in a rather straight forward manner. With the assumption that all transformations are quasi-linear, the EKF simply linearizes all nonlinear transformations and substitutes Jacobean matrices for the linear transformations in the basic KF equations. The time varying signal using nonlinear stochastic difference equations, and is modelled in state space as xk+1 = φ (xk , wk )

(9.17)

where xk is the state vector of the process at time k, φ is the nonlinear state transition matrix of the process which relates the state at k to the state at k + 1, wk is the process noise. Measurements on this variable can be modelled as yk = h( xk , vk )

(9.18)

where yk is the measurement of xk at time k, h is the nonlinear measurement transition matrix, which is a noiseless connection between the state vector and the measurement vector. vk is the associated measurement noise. The derivation begins with an equation that linearizes the estimate about Eqs. (9.17) and (9.18), xk+1 = x˜k+1 + A(xk − xˆk ) + W wk

(9.19)

yk = y˜k + H (xk − x˜k ) + V vk

(9.20)

where x˜k and y˜k are the approximate state and measurement vectors, xˆk to be the a posteriori state estimate at time step k, A is the Jacobian matrix of partial derivatives of φ with respect to x, A[i, j] =

∂φ[i] ∂ x[ j]

(9.21)

W is the Jacobian matrix of partial derivatives of φ with respect to w, W[i, j] =

∂φ[i] ∂w[ j]

(9.22)

H is the Jacobean matrix of partial derivatives of h with respect to x, H[i, j] =

∂h [i] ∂ x[ j]

V is the Jacobean matrix of partial derivatives of h with respect to v,

(9.23)

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9 Unscented and Complex Unscented Kalman Filtering for Parameter …

V[i, j] =

∂h [i] ∂v[ j]

(9.24)

Now the prediction error and measurement errors can be calculated as e˜xk = xk − x˜k and e˜ yk = yk − y˜k

(9.25)

( ) e˜xk = A xk−1 − xˆk−1 + εk and e˜ yk = H e˜xk + n k

(9.26)

or it can be written as

where εk and ηk are the random variables having zero mean and covariance matrices as W QW T and V RV T with Q and R. Now these equations can be used for time and measurement update as xˆk = xˆk + K k e˜ yk = x˜k + K k (yk − y˜k )

(9.27) '

The time update equations can be summarized by assuming x˜k = xˆk , the a priori state estimate as '

xˆk+1 = φ( xˆk , wk )

(9.28)

T T Pk+1 = Ak+1 Pk Ak+1 + Wk+1 Q k Wk+1

(9.29)

'

And the measurement update equations are given as '

K k = Pk HkT (Hk PkT HkT + Vk Rk VkT )−1 '

'

xˆk = xˆk + K k (yk − h(xˆk , 0)) '

Pk = (I − K k Hk )Pk

(9.30) (9.31) (9.32)

From these equations, it is clear that the linearization can be viewed as a truncation of the Taylor series expansion of the nonlinear function around the mean. The equations for EKF are summarized in the Fig. 9.4 [25]. Although the EKF is computationally efficient recursive update form of the KF, it suffers a number of serious limitations. Linearized transformations are only reliable if the error propagation can be well approximated by a linear function. If this condition does not hold, the linearized approximation can be extremely poor. At best, this undermines the performance of the filter. At worst, it causes its estimates to diverge altogether. Calculating Jacobean matrices can be a very difficult and error-prone process. The Jacobean equations frequently produce many pages of dense algebra that must be converted to code. This introduces numerous opportunities for human

9.2 Outline of Basic Estimation Methods

207

Measurement Update Time Update

(1) Compute Kalman gain (1) State projection ʹ

(2)Error covariance projection

(2)Update the estimate

(3)Update error covariance

Initial estimates for

,

Fig. 9.4 Summary of equations of EKF

coding errors that may undermine the performance of the final system in a manner that cannot be easily identified and debugged, especially given the fact that it is difficult to know what quality of performance is expected. Regardless of whether the obscure code associated with a linearized transformation is or is not correct, it presents a serious problem for subsequent users who must validate it for use in any high integrity system. Hence, Extended Kalman filter can be applied to nonlinear systems if a consistent set of a priori estimated quantities are available.

9.2.5 Unscented Kalman Filter The classical nonlinear Extended Kalman filter (EKF) [11, 12], and its equations are similar to the linear Kalman filter equations and are obtained by linearizing the nonlinear equations based on first-order Taylor series expansion at the predicted points. So the filter may diverge when the observability of the system is low, or system becomes unstable due to linearization and erroneous parameters, it includes costly calculations of derivatives. Hence, to address the deficiencies of EKF, Julier proposed a new and more direct method for transforming mean and covariance information. Unscented transformation (UT) [13, 14], is efficient and unbiased for the mean and variance calculation and it is based on the idea of approximating the probability distribution rather than to approximate an arbitrary nonlinear function. Based on this transform, the Unscented Kalman filter was proposed. As compared with EKF, which is accurate up to the first-order of the Taylor series expansion, UKF is accurate up to the third-order of the Gaussian data and at least second-order of non-Gaussian data.

208

9 Unscented and Complex Unscented Kalman Filtering for Parameter …

The main advantage of unscented transformation used in UKF is that it does not use linearization for computing the state and error covariance matrices resulting in a more accurate estimation of the parameters of a nonstationary signal. Instead of linearizing the Jacobean matrices, the UKF uses a deterministic sampling approach to capture mean and covariance estimates exactly. A fixed number of sigma points are generated (set of 2 × L + 1, sigma points, where L is the state dimension). These sigma points are propagated through the nonlinearity, without approximation, and a distribution is estimated from them. Then a weighted mean and covariance is found. Like the EKF, the UKF uses a recursive algorithm that uses the system model, measurements and known statistics of the noise mixed with the signal. The UKF was originally designed to estimate the states of a dynamic system and for nonlinear control applications. The most mentionable advantage of UKF over EKF is its derivative-free-nonlinear estimation (no need of calculation of Jacobians and Hessians), though its computational complexity is same as the EKF’s. A comparison of estimation accuracy between EKF and UT is shown in Fig. 9.5 [26] for a 2-D system, where a cloud of 5000 samples drawn from a Gaussian prior is propagated through an arbitrary highly nonlinear function and the true posterior sample mean and covariance are calculated by a Monte Carlo approach, as shown in left plot, which can be regarded as a ground truth of the two approaches, EKF and UT. The centre plot shows the performance of the UT (note only 5 sigma points are required); the right plot shows the results using a linearization approach as would be done in the EKF. The superior performance of the UT is clear. In spite of many advantages of UKF over EKF, its accuracy significantly decreases under high noise condition and the noise covariances and some of the parameters used in unscented transformation are not chosen correctly.

Fig. 9.5 Comparison of EKF and UT

9.3 Motivation Behind the Development of a New Model Taking Real …

209

Thus, for best signal tracking performance, this chapter introduces a novel adaptive Unscented Kalman Filtering technique, which recursively updates the model and measurement error covariances for better performance under high noise condition.

9.3 Motivation Behind the Development of a New Model Taking Real Sinusoid into Account Before applying any estimation algorithm, it is required to introduce the signal model used for parameter estimation of a single and multiple sinusoids. In real sinusoidal parameter estimation, the noisy measurement which is discrete time in nature can be modelled as yk =

M ∑

Aik cos(ωik tk + φik ) + vk , k = 1, 2, ......N

(9.33)

i=1

where Aik is amplitude,ωik is frequency and φik is phase of the ith sinusoid, respectively, tk = kTs , where Ts is the sampling time. Consider a signal consisting of M sinusoids. The modelling error vk is a zero mean Gaussian white noise with variance σv2 . In this chapter, first a method for estimating the amplitude and frequency of the fundamental components of a sinusoid buried in noise is presented. Thus, for the fundamental component estimation, signal can be model as yk = Ak cos(kωk Ts + φk ) + vk

(9.34)

The white noise vk is modelled as vk ∼ N (0, Rk ), and the covariance of the measured error is given by Rk = E(vk vkT )

(9.35)

The discrete signal can be represented in state space as xk+1 = G k xk + ηk

(9.36)

and the model error covariance is given as:ηk = N (0, Q k ) The state variables of the filter are expressed as xk (1) = A sin(xk (3)kTs + φ)xk (2) = A cos(xk (3).kTs + φ) xk (3) = ω The state transition matrix in this case becomes equal to

(9.37)

210

9 Unscented and Complex Unscented Kalman Filtering for Parameter …

⎡

⎤ cos(xk (3)Ts ) sin(xk (3)Ts ) 0 G k (xk ) = ⎣ −sin(xk (3)Ts ) cos(xk (3)Ts ) 0 ⎦ 0 0 1

(9.38)

The observation model is given by yk = h(xk ) + vk

(9.37)

[ ] where h(xk ) = 0 1 0 xk , and the observation matrix H is given by H = [0 1 0]

(9.38)

and the model noise covariance matrix is given by ⎡

⎤ q1 0 0 Q = ⎣ 0 q2 0 ⎦ 0 0 q3

(9.39)

In the presence of harmonics, the performance of the single sinusoid model might deteriorate, and therefore, the signal model should include harmonics also. The performance of the Unscented Kalman filter in tracking signal frequency, amplitude and phase is highly dependent on the correct choice of error covariances Q and R. To overcome the problems of UKF, many researchers have proposed an adaptive UKF, which uses the UKF algorithm with a modification, in which the measurement error covariance Rk and model error covariance Qk are updated in a recursive manner. One of the method to update Rk and Qk is discussed in the next section.

9.3.1 Algorithm Development for a Real Sinusoid The UKF uses a deterministic sampling approach [13, 14] to capture mean and covariance estimates with a minimal set of sample points and chooses 2 × L + 1, sigma points (L is the dimension of the state) based on a square-root decomposition of the prior covariance. These sigma points are propagated through the nonlinearity, without approximation, and a weighted mean and covariance is found. The UKF thus involves the recursive application of the sampling approach to the state-space equations of the signal. The UKF algorithm is summarized in the following steps: For the system state x, initialize with xˆ0 = E[x0 ] , P0 = E[(x0 − xˆ0 )(x0 − xˆ0 )T ]

(9.39)

9.3 Motivation Behind the Development of a New Model Taking Real …

211

For a given state vector at step k-1, sigma points can computed and stored in the columns of L · (2L + 1) sigma point matrix χ , where L = dimension of the state vector. For the present problem, L = 3 so χk−1 is a 3.7 matrix. The sigma points are computed as χ0,k−1 = xˆk−1 χi,k−1 = xˆk−1 + χi+L ,k−1 = xˆk−1 −

(√ (√

(l + λ)Pk−1

)

(L + λ)Pk−1

)i

, i = 1, 2 , . . . , L

i

, i = L + 1, . . . , 2L

(9.40)

The parameter ∟ is used to control the covariance matrix, and is given as λ = α 2 (L + K ) − L

(9.41)

where both ∟ and K are scaling parameters. The constant α determines the spread of the sigma points and its value is between 0.0001 < = α < = 1. After computing the sigma points, the time updates of state estimates are given by χ k|k−1 = Fk−1 (xk−1 ) xˆk− =

2L ∑

(9.42)

Wim χi, k|k−1

(9.43)

i=0

where the weights Wim are defined by W0(m) =

λ , L +λ

Wi(m) =

λ , 2(L + λ)

(m) Wi+L =

1 , 2(L + λ)

i = 1, 2 . . . , L (9.44)

the a priori error covariance is given by Pk =

2L ∑

[ ][ ]T Wi(c) χi,k|k−1 − xˆk− χi,k|k−1 − xˆk− + Q k

(9.45)

i=0

where the weights Wic are defined by λ + (1 − α 2 + β) (L + λ) 1 + (1 − α 2 + β) = 2(L + λ) 1 , i = 1, 2 . . . , L = 2(L + λ)

W0(c) = Wi(c) (c) Wi+L

(9.46)

212

9 Unscented and Complex Unscented Kalman Filtering for Parameter …

the estimated output is given as ) ( Yi,k|k−1 = H χi,k|k−1, yˆk− =

2L ∑

Wi(m) Yi,k|k−1

(9.47)

(9.48)

i−0

the a posterior state estimate is computed as ( ) Xˆ k = Xˆ k + K k yk − yˆk

(9.49)

where K k is the Kalman gain, and is given as K k = G k Sk−1

(9.50)

with Gk =

2L ∑

[ ][ ]T Wi(c) χi,k|k−1 − xˆk− Yi,k|k−1 − yˆk−

(9.51)

[ ][ ]T Wi(c) Yi,k|k−1 − yˆk− Yi,k|k−1 − yˆk− + Rk

(9.52)

i=0

and Sk =

2L ∑ i=0

Rk is measurement error covariance matrix. The a posterior estimate of the error covariance matrix is given by Pk = Pk−1 − K k Sk K kT

(9.53)

Like the EKF algorithm, the UKF parameters Qk , Rk , ⟨, ®, K are to be chosen by trial and error. The measurement error covariance Rk and model error covariance Qk are updated in the following manner that gives us the adaptive Unscented Kalman filter. The model error can be estimated at any instant k as follows: ( ) Z k = Xˆ k − Xˆ k = K k yk − yˆk ,

(9.54)

Note that, according to number of state which is three for the estimation of fundamental amplitude, frequency and phase components, in the above UKF model, Z k takes different value leading to different variance estimates. Hence, the model error covariance estimate is taken as the average of all the three terms as

9.3 Motivation Behind the Development of a New Model Taking Real …

) ( Q k = 1/2 abs(Z 1k )2 + abs(Z 2k )2 + abs(Z 3k )2 × I

213

Δ

(9.55)

and measurement error covariance is estimated as R = R0 + λ(abs|Z k |)(abs|Z k−1 |),

(9.56)

where λ is forgetting factor and 0 ≤ λ ≤ 1 In this way, at every instant, the model error covariance and measurement error covariance are updated. The new values of covariance matrix Q k ∧ and R are used to improve the estimate of the state through iterative procedure.

9.3.2 Harmonic Estimation of Real Sinusoid Due to the increased use of nonlinear components in power networks, distortion occurs in current and voltage waveforms. Harmonics may cause extra power loss in some conventional loads such as motors, distribution transformers, feeders. To prevent the undesired effects of harmonics, the accurate estimation of the harmonic components became important. Many detection techniques have been proposed in literature for monitoring the system fundamental frequency and harmonics. Among the several methods for frequency, amplitude and phase estimation of nonstationary signals, discrete Fourier Transform (DFT), and Fast Fourier Transform (FFT) are widely used [27, 28]. However, both the above methods suffer from aliasing, leakage and picket fence effects, and hence need error compensation and adaptive window width [29]. Some of the known signal processing techniques like artificial neural networks [30, 31], adalines [32, 33], linear prediction technique [34, 35], adaptive filters [36], supervised Gauss–Newton algorithm [37, 38], least-error squares and its variants [39–41], Extended Kalman filters [42], have been used for time varying signal parameter estimation. Most of these algorithms require heavy computational outlay and suffer from inaccuracies in the presence of noise with low signal to noise ratio (SNR). As discussed in previous section, a novel unscented Kalman filter (UKF) [43] has been proposed to overcome the shortcoming of the of Extended Kalman filter (EKF) algorithm for its instability due to linearization, erroneous parameters, costly calculations derivatives, and the biased nature of the estimates. The main advantage of the unscented transformation used in UKF is that it does not use linearization for computing the state and error covariance matrices resulting in a more accurate estimation of the parameters of a nonstationary signal. However, its accuracy significantly reduces, if SNR is low and the noise covariances and some of the parameters used in UT are not chosen correctly. Thus, for best signal tracking performance, it is proposed in this chapter to use Adaptive Particle Swarm Optimization technique, for the optimal choice of AUKF parameters and error covariances.

214

9 Unscented and Complex Unscented Kalman Filtering for Parameter …

9.3.3 Rigorous Mathematical Modelling Multiple Sinusoids The signal model for estimating the time varying amplitude, phase, frequency and harmonics is represented as a discrete time equation ) ( f0 yk = an sin 2π n k + φn + ηk fs n=1 N ∑

(9.57)

where yk is the measured signal, an is the time varying peak amplitude of the nth harmonic component of the signal, N is the order of the harmonic, f 0 is frequency of the fundamental component and f s is the sampling frequency. ηk is the measurement noise with associated variance. The signal and observation models are derived in state-space form as follows: xk = f (xk−1 ) + vk

(9.58)

where the state vector xk is given by xk = [a1 sin φ1 a1 cos φ1 ........ an sin φn an cos φn ω]T and the angular frequency is given as ω = 2π f 0 . and f (xk ) = Ank xkT

(9.59)

The state transition matrix ⎡ Ank

A1 0 0 0

⎤

⎢ ⎥ ⎢ 0 ... 0 ⎥ ⎢ ⎥ =⎢ ⎥ ⎣ 0 An 0 ⎦ 0 0 1

(9.60)

where [ An =

cos(nωkTs ) sin(nωkTs ) − sin(nωkTs ) cos(nωkTs )

] (9.61)

The model error covariance matrix is given as ] [ Q = E vv T = q I,

(9.62)

where I is a N + 1 unit matrix The measurement model for the signal represented by Eq. (9.57) is obtained as

9.3 Motivation Behind the Development of a New Model Taking Real …

215

yk = Hk xk + ηk

(9.63)

Hk = [1 0 1 0. . . .. . . .1 0 0]

(9.64)

where

] [ and the measurement error covariance R = E ηη T

9.3.4 Algorithm Development for Harmonic Estimation As described in the previous section, the UKF is consider to overcome the disadvantages of EKF. For this signal model also, the adaptive UKF is considered to track the fundamental and harmonic components of a nonstationary signal that is found in power networks. Given a state vector at step k-1, sigma points are computed and stored in the columns of L · (2L + 1) sigma point matrix χk−1 where L = dimension of the state vector. For the estimation of fundamental signal frequency, amplitude and phase without modelling the harmonics, L = 3 and thus χk−1 is a 3.7 matrix. However, for the estimation of N number of harmonics and the fundamental frequency L = 2 N + 1, thus χk−1 is a (2 N + 1)(4 N + 3) matrix. Then the normal UKF algorithm is implemented. Usually, a small value for Q = q I (q = .0001, I = unit matrix of appropriate dimension) is adopted for most of the filter computations. Here a self-tuning procedure for adapting the covariances is presented for improving the performance of the filter during sudden changes in the amplitude, phase, or the frequency of the signal. AUKF uses the normal UKF equations with a modification in making error covariances iterative. The model error is estimated as '

'

vk = f (ˆxk ) − f (ˆxk ) = Kk (yk − yˆk )

(9.65)

The value of process covariance matrix is updated as the average of the variance square during the successive iterations. Thus, the incremental matrix Q k is obtained as { } 2L 2L ∑ ∑ (m) (m) 2 2 Qk = (1/2) (K k (yk − Wi Yi,k|k−1 )) +(Kk (yk − Wi Yi,k−1|k−2 )) i−0

i−0

(9.66) In a similar way, the measurement error covariance is updated as

216

9 Unscented and Complex Unscented Kalman Filtering for Parameter …

Rk = ρ Rk−1 + (1 − ρ)(yk −

2L ∑

Wi(m) Yi,k|k−1 )2

(9.67)

i−0

However, if the value of Q k obtained from Eq. (9.66) is sufficiently large, the adaptive UKF can tolerate high order error in the unscented transformation by enlarging the noise covariance matrix, and thereby improving the stability, but resulting in a large estimation error. Therefore, a threshold value of Q k is required to provide both, the accuracy and stability, in tracking harmonic signals. Through a proper choice of the sigma points, that is the parameters α,λ,β and the initial values of the covariances Q and R, the UKF assures a better performance than the EKF in estimating fundamental frequency, and amplitude and phase of the fundamental and harmonic components of a signal buried in noise. Thus, to improve the performance of AUKF, researchers have proposed stochastic optimization technique like genetic Algorithm, PSO [44–48] and its variants are used to obtain the parameters α,λ,β,Q and R instead of trial and error approach. These approaches are out of the scope of this chapter.

9.3.5 Outline of Experimental Setup Experimental test data is generated using the laboratory set up, as shown in Fig. 9.6. The load is fed from a 3 kVA, 230:230 V single-phase transformer, and the Data Acquisition subsystem (DAS) is activated by using another 230 V:12 V transformer. The DAS comprises a PCL-208 data acquisition card which has an inbuilt sample and hold device, and an A/D converter. The voltage across the load after transformation to 12v is sampled at a rate of 2.3 kHz and is digitized by the A/D system and the digital data is sent to PC using a programme written in “C” language. Signals with time varying amplitude and frequency are obtained using the waveform simulator by switching on and switching off the load, respectively.

9.3.6 A Comparative Discussion on Simulation and Experimental Result 9.3.6.1

Examples of Parameter Estimation for Single Sinusoid

For the estimation of amplitude, phase and frequency of the fundamental component of the sinusoid, the following test signal is used: yk = Ak cos (kωk Ts + φk ) + vk ,

(9.67)

UKF parameters Qk , R, ⟨, β and K need to be initialized, Here β and K are chosen as β = 2, and K = 0, and ⟨,Qk , R are initially chosen as ⟨ = 0.5, Qk = q.I3x3 and

9.3 Motivation Behind the Development of a New Model Taking Real …

217

CRO PC 0.35 0.3 0.25 0.2

0

0.15 0.1 0.05

DIGITAL DATA

0

ANALOG DATA

-0.05 0

100

200

300

400

230 : 230

500

600

PCL-208 DATA ACQUISITION CARD A/D

SAMPLE

ELECTRICAL

CONVERTER

HOLD

NETROK

230 : 12

150Ω

~

D A

500Ω

PC

C

Supply Subsystem

Data Acquisition Subsystem

Waveform simulator subsystem Fig. 9.6 Laboratory setup for simulation of time varying signals

R = 0.05, where I is a 3rd order unit matrix,. Since it is required to estimate three parameters of the signal, the value of L is set equal to 3, and thus the number of sigma points for this estimation is 2L + 1, i.e. 7. The angular frequency of the signal ω is varied linearly as For 0 ≤ k ≤ 300, w(k) = w0 for 300 ≤ k ≤ 700, w(k) = w0 +

w1 − w0 (k − 300) 400

(9.68)

and for k ≥ 700w(k) = w1 Case Study1 Large ramping in frequency is made from f0 = 45 Hz to f1 = 65 Hz in the presence of white gaussian noise. The time varying frequency, amplitude and phase of the signal are obtained for different SNR (signal to noise ratio) of the white noise varying from 30 to 10 dB. The errors between the tracked parameters and the actual parameters are shown in Figs. 9.7, 9.8 and 9.9 for different 30 dB noise for different estimation methods. From the figures, it is observed that the standard EKF with constant Q and R in Fig. 9.7, tracked frequency is found to contain noise and fluctuations during the initial samples and when the frequency is ramped. Same signal is analyzed using UKF in Fig. 9.8 and adaptive UKF in Fig. 9.9 shows the improvement in estimation of AUKF which uses tuning of measurement and model error covariance matrices.

218

9 Unscented and Complex Unscented Kalman Filtering for Parameter …

Fig. 9.7 Parameter estimation using EKF

Case Study 2 A similar situation is considered where the frequency is ramped and decreases from initial value of 55 Hz to 45 Hz. The performance of EKF and UKF is compared at different noise level as shown in Figs. 9.10 and 9.11.

9.3.6.2

Examples of Parameter Estimation of Harmonic Components

Case Study 3. The test power signal is assumed to comprise a fundamental and several harmonics and is given by y(k) = 1.2 ∗ sin(ωkTs + φ1 ) + 0.25 ∗ sin(3ωkTs + φ3 ) + 0.15 ∗ sin(5ωkTs + φ5 ) + 0.1 ∗ sin(11ωkTs + φ11 ) + η(k)

(9.69)

The fundamental frequency of the signal is varied linearly from the initial value of 50–51 Hz within a span of 2 cycles and a similar frequency variation is generated in the harmonic components with a white Gaussian noise of SNR = 20 dB added

9.3 Motivation Behind the Development of a New Model Taking Real …

219

Fig. 9.8 Parameter estimation using UKF

to the signal given in Eq. (9.69). Further, its angular frequency is varied in a similar way as in case Study1. The absolute estimation error for the fundamental frequency and amplitude of different harmonic components is presented in Table 9.1. From this table, it is quite evident that the UKF and its optimized variants exhibit less estimation error in comparison to the EKF filter. In a similar way, Fig. 9.12a–c depict the ramp frequency change and absolute error in amplitude tracking of 3rd and 11th harmonic components using EKF, UKF and AUKF, respectively. Similarly, another test is done by generating frequency ramp of 50–65 Hz with 20 dB and the estimation result is presented in Table 9.2.

9.3.6.3

Discussion

In this section, different adaptive filtering techniques have been discussed for the estimation of amplitude, phase and frequency of the stationary as well as time varying single sinusoid and also the harmonics. The signal model for both the single sinusoid and sinusoid with harmonic components has been discussed.

220

9 Unscented and Complex Unscented Kalman Filtering for Parameter …

Fig. 9.9 Parameter estimation using Adaptive UKF

9.4 Complex Unscented Kalman Filter for Signal Parameter Estimation In applications such as communication, radar and sonar, the information bearing signal component is modulated onto the carrier wave. The bandwidth of the information bearing signal is very small compared with carrier frequency. To obtain the information bearing signal, the input signal is translated down in frequency. In general, the information bearing signal so obtained is complex in nature. Although the real versions of both EKF [11, 12, 42] and UKF [13, 14] are used in many applications for tracking signal frequencies, the complex version of EKF is much simpler and direct as far as the frequency measurement is concerned and outperforms the real extended Kalman filter in terms of speed of convergence, accuracy and stability [49, 50]. Moreover, complex representations can make the measurement function the likelihood of linear formulation to reduce the influence of high order terms. Complex adaptive filters include real adaptive filter as special case. But the complex extended Kalman filter is still unable to track accurately the frequency in presence of harmonics and abrupt large transient disturbances. Hence, to overcome the above mentioned problems, a new complex unscented Kalman filter

9.4 Complex Unscented Kalman Filter for Signal Parameter Estimation

221

Fig. 9.10 Parameter estimation using UKF and EKF at 30 dB noise

(CUKF) algorithm is proposed in this chapter. The CUKF uses an unscented transformation to a complex state-space model of the sinusoid and computes the covariances and the Kalman gain from the measurements corrupted with white noise. Further to improve its performance during large frequency changes, a self-tuning technique is used to update the model and measurement error covariances and this filter is named as adaptive complex unscented Kalman filter (ACUKF).

9.4.1 Motivation Behind the Development of a New Model Taking the Complex Nature of the Signal into Account Consider a time varying sinusoidal signal represented in discrete form as Sk = Ak cos(kωk Ts + φk ) + vk ,

(9.70)

where Ak ,ωk ,φk and Ts are the amplitude, angular frequency, phase and sampling interval of the sinusoid, respectively; Ts is expressed as Ts = 1/ f s ,ωk = 2π f k , f s and f k are the sampling and fundamental frequency of the signal.

222

9 Unscented and Complex Unscented Kalman Filtering for Parameter …

Fig. 9.11 Parameter estimation using UKF and EKF at 10 dB noise

Table. 9.1 50–51 HzFrequency-RAMP-variation with 20 dB Noise

Order of harmonic

EKF

UKF

AUKF

Fundamental frequency error

0.09 Hz

0.0307 Hz

0.0067 Hz

Fundamental amp error

0.0107 V

0.0063 V

0.0049 V

3rd harmonic Amp error 0.0234 V

0.0137 V

0.0002 V

5th harmonic Amp error 0.0204 V

0.0107 V

0.0016 V

11th harmonic Amp error

0.0514 V

0.0075 V

0.0815 V

Since the observed signal is real, the development of a complex model requires a complex signal, and thus we use Hilbert transform [42] to convert the real signal sk to an analytic signal of Gabor yk as yk = Ae j (kωTs +φ) + νk

(9.71)

The analytical signal y(k) is modelled in the state space using the state variables x1k and x2k as

9.4 Complex Unscented Kalman Filter for Signal Parameter Estimation

223

Fig. 9.12 a Fundamental and Harmonic frequency estimation using EKF at 20 dB noise. b Fundamental and Harmonic frequency estimation using UKF at 20 dB noise. c Fundamental and Harmonic frequency estimation using Adaptive UKF at 20 dB noise

224

9 Unscented and Complex Unscented Kalman Filtering for Parameter …

Fig. 9.12 (continued)

Table. 9.2 50–70 HzFrequency-RAMP-variation with 20 dB noise

Order of Harmonic

EKF

UKF

AUKF

Fundamental freq error (Hz)

0.3031

0.2953

0.0063

Fundamental amp error (pu)

0.089

0.73

0.0131

3rd harmonic amp error (pu)

0.22

0.179

0.0119

5th harmonic amp error (pu)

0.133

0.0304

0.0031

11th harmonic amp error (pu)

0.0675

0.0479

0.0179

x1k = e j(ωTs ) ,

(9.72)

x2k = Ak e j (kωTs +φ) ,

(9.73)

Note that the model can easily be extended to represent signals containing harmonics. The signal model can be put in general form as

9.4 Complex Unscented Kalman Filter for Signal Parameter Estimation

225

xk+1 = f (xk ) + ηk ,

(9.74)

yk = h(xk ) + vk ,

(9.75)

f (xk ) = [x1k x1k x2k ]T

(9.76)

where

and [ h(xk ) = [0

1]

x1k x2k

] ,

(9.77)

The complex signal model used in Eqs. (9.72)–(9.77) will be used in estimating the frequency of a time varying sinusoid embedded in noise with low SNR. The next section describes the unscented transformation and filtering algorithm.

9.4.2 Algorithm Development UKF is a powerful nonlinear estimation technique and operates on the premise that it is easier to approximate a Gaussian distribution than it is to approximate an arbitrary nonlinear function. Instead of linearizing using Jacobean matrices, the UKF evaluates the nonlinear function with a minimal set of carefully chosen sampling points of 2 × L + 1, sigma points (L is the state dimension) based on a square-root decomposition of the prior covariance [13]. These sigma points are propagated through the nonlinearity, without approximation, and a weighted mean and covariance is found. Like the EKF, the UKF uses a recursive algorithm that uses the system model, measurements and known statistics of the noise mixed with the signal. The posterior mean and covariances estimated from these sample points are accurate to the second order for any nonlinearity. Consider the nonlinear system modelled by the discrete time state as in Eqs. (9.74) and (9.75), where xk ∈ R L , and yk ∈ R P are the signal state and measurement, respectively. The nonlinear mapping f () is assumed to be continuously differentiable with respect to xk . Moreover ηk and vk are uncorrelated zero mean Gaussian noise sequences with E

[

ηk η∗T k

]

[ = Qk =

q1

0

0

q2

]

] ] [ [ , E vk vk∗T = Rk , E ηk vk∗T = 0,

(9.78)

and the * sign indicates the complex conjugate of the quantity. The procedure for implementation of CUKF is similar to UKF with complex conjugate terms as follows:

226

9 Unscented and Complex Unscented Kalman Filtering for Parameter …

Step 1: Selection of Sigma Points Given a L × 1 state vector xˆk−1 at time step k − 1 and state error covariance matrix Pˆk−1 , compute a set of 2L + 1 sigma points as / / χk−1 = [ˆxk−1 xˆ k−1 + ζ Pˆ k−1 xˆk−1 − ζ Pˆ k−1 ]

(9.79)

√ where ζ = L + λ and λ = α 2 (L + κ) − L , the parameter λ decides the spread of ith sigma point around xˆk−1 . For λ > 0, the points are scaled further from xˆk−1 and when λ < 0, the points are scaled towards xˆk−1 . Further, λ can be defined as a function of two parameters α and κ, where the constant α is a small constant lying between 0.0001 and 1 and can be used to control the amount of the higher-order nonlinearities around xˆk−1 which can be taken into account. The parameter κ is a secondary scaling parameter which is usually set to 0 or 3 − L to ensure that the kurtosis of the sigma point distribution agrees with the kurtosis of a Gaussian distribution. The matrix ζ Pˆk−1 is assumed positive definite and its square root can, therefore, be computed by using the Cholesky decomposition. Step 2: Transformation of Sigma points through system function (time update) Each column of the sigma point matrix is propagated one step ahead through the dynamic function f () of Eq. (9.74) to get the “transformed sigma points” at time k ) ( χi,k = f χi,k−1 ,

i = 1, 2, ...2L + 1

(9.80)

Step 3: Computation of Prior State Estimates The prior state estimate xˆk− and its corresponding covariance matrix Pˆk/k−1 are approximated by the weighted mean and covariance of the transformed sigma points as follows: xˆ k− =

2L ∑

Wi(m) χi, k

(9.81)

i=0

Pˆ k/k−1 =

2L ∑

[ ] [ ]∗T χi,k − xˆ k− Wi(c) χi,k − xˆ k− + Q k−1

(9.82)

i=0

where Q k−1 is the process noise covariance matrix. The weights Wi(m) and Wi(c) are defined as W0(m) =

λ , L +λ

Wi(m) =

λ , 2(L + λ)

(m) Wi+L =

1 2(L + λ)

(9.83)

9.4 Complex Unscented Kalman Filter for Signal Parameter Estimation

λ + (1 − α 2 + ρ) (L + λ) 1 = + (1 − α 2 + ρ) 2(L + λ) 1 = i = 1, . . . , L 2(L + λ)

227

W0(c) = Wi(c) (c) Wi+L

(9.84)

where ρ is another parameter used to incorporate prior knowledge of the higher-order moments of the state distribution. The optimal choice of this parameter for Gaussian distribution is 2. Step 4: Computation of Predicted Observation The predicted values for the manifest observations at time step k can be obtained as the weighted sum of the projection of transformed sigma points through measurement function h: ) ( Yi,k = h χi,k , yˆk− =

2L ∑

Wi(m) Yi,k

(9.85)

(9.86)

i=0

The a posterior state estimate is computed as ( ) xˆk = xˆk− + K k yk − yˆk−

(9.87)

where K k is the Kalman gain given by Kk = Gk Sk−1

(9.88)

where Gk =

2L ∑

[ ][ ]∗T Wi(c) χi,k − xˆ k− Yi,k − yˆk−

(9.89)

[ ][ ]∗T Wi(c) yi,k − yˆk− Yi,k − yˆk− + Rk−1

(9.90)

i=0

Sk =

2L ∑ i=0

Rk−1 is the measurement noise covariance. The a posterior estimate of the error covariance matrix is given by Pˆ k = Pˆ k/k−1 − Kk Sk Kk∗T

(9.91)

228

9 Unscented and Complex Unscented Kalman Filtering for Parameter …

Like the UKF algorithm, the CUKF parameters Q k , Rk ,α,ρ,κ are to be chosen by trial and error basis. The above implementation of CUKF has considered the model error covariance Q k and measurement error covariance Rk are constant determined a priori. This approach presents a self-tuning update procedure for model error covariance Q k and the measurement error covariance Rk , in order to improve the filter adaptive capability and speed of the response. The model error can be estimated at any instant k from Eq. (9.87) as Zk = xˆ k − xˆ k− = Kk (yk − yˆk− ) = [ψ1k ψ2k ]T ,

(9.92)

As the model error is contributed by the white Gaussian noise, the calculated model error covariance matrix can be estimated from the above Eq. (9.92). Note that according to the number of states, which is two in the above CUKF model Z k takes different values, leading to different variance estimates given as qˆ1k = |ψ1k |2 ,

(9.93)

qˆ2k = |ψ2k |2 ,

(9.94)

and

The model error covariance estimate can be taken as the average of both the terms and is given as ) ( Qˆ k = 1/2 qˆ1k + qˆ2k × I,

(9.95)

By taking the average any large value of either qˆ1k or qˆ2k , which may be interpreted as a lack of accuracy of the whole model can be made adaptable. Similarly, the measurement error covariance is estimated using innovation error as ek = (yk − yˆk− ) Rk = λRk−1 + (1 − λ)|ek ||ek−1 |, and

(9.96)

where λ is forgetting factor and 0 ≤ λ ≤ 1. In this way, at every instant of time, the model and the measurement error covariances are updated. The new values of model and measurement error covariances Qˆ k and Rk are used to improve estimate of the state through iterative procedure. The initial setting of Qˆ 0 = 0 × I and R0 = 0, and Eqs. (9.95) and (9.96) are incorporated to the normal CUKF algorithm to make it adaptive ACUKF.

9.4 Complex Unscented Kalman Filter for Signal Parameter Estimation

229

9.4.3 Stability Analysis Thus to verify the performance of the proposed filter, the internal behaviour of the filter like, Kalman gain, model and measurement error covariances, of the filter is studied for tuned and untuned CUKF for a signal having abrupt change in frequency. The gain of the untuned filter settles to a fixed value even for change in signal parameter, where as the gain of the tuned filter adapts to the change in signal parameter, increasing the adaptive capability of the filter as shown in Fig. 9.13a. Similarly, Fig. 9.13b and c show the variation of the model and measurement error covariances of the filter. From the figures, it is clear that the error covariances change with change in signal parameter to reduce the estimation error. It also shows that the covariance matrix of estimation error converges to zero as time tends to infinity confirming the stability of the proposed filter.

9.4.4 A Comparative Discussion on Simulation and Experimental Results Different cases of time varying frequency change are tested using the CUKF and with different filters. Test 1 analyzes different types of major power signal variation problems, such as sudden frequency change using simulated waveforms and MATLAB software package. Test 2 analyzes distorted signals generated using an experimental setup as described in Sect. 9.3.5. The chosen sampling rate is 1 kHz, for Test 1 and 2.3 kHz for Test 2, and the frequency is normalized with respect to a base frequency. Test 1: The effectiveness of the Adaptive CUKF with complex-EKF, and UKF were demonstrated considering the nonstationary signal under different levels of noise, the signal to noise ratio (SNR) is varied from 60 to 10 dB. The nonlinear signal considered is given by yk = Ak cos(kωk Ts + φk ) + vk ,

(9.97)

where Ak ,ωk and φk are the amplitude, frequency and phase of the signal, respectively, and vk is Gaussian noise with zero mean. For tracking a time varying power signal of 50 Hz frequency, the sampling frequency is chosen as 1 kHz and the CUKF parameters are chosen as: Q k , Rk ,α, β and κ need to be initialized, Here β and κ are chosen as β = 2, and κ = 0, α = 0.5, Q k = qk I2×2 , where I is unit matrix, and qk = 0.5 and Rk = 0.05. Since the nonlinear model used in the proposed algorithm uses two states for modelling, the value of L is set equal to 2, and thus the number of sigma points for this estimation is 2L + 1, i.e. 5 and the augmented state vector χk−1 is a 2 × 5 matrix. Case 1: The frequency of the time varying signal is given by For

9 Unscented and Complex Unscented Kalman Filtering for Parameter …

a

b

ACUKF CUKF

1

Model error covariance(q3)

Fig. 9.13 a Comparison of Kalman gain of the filter. b Comparison of model error covariance. c Comparison of measurement error covariance

0.8 0.6 0.4 0.2 0 500 samples

450

550

-4

x 10

c Measurement error covariance(R)

230

ACUKF CUKF

10

8

6

4

2

0 450

500

550

600 samples

650

700

9.4 Complex Unscented Kalman Filter for Signal Parameter Estimation

231

0 ≤ k ≤ 300 f k = 50Hz, k ≥ 300 f k = 51 Hz,

(9.98)

The SNR of the signal is varied from 60 dB, 30 dB, 20 dB and 10 dB, respectively. The experiment was repeated 100 times. The frequency tracking ability of different algorithm is summarized in Table 9.3 which shows a comparison between the UKF, CEKF, CUKF and Adaptive CUKF for the signal in (9.97). Figure 9.14a shows the frequency tracking ability for different algorithms and Fig. 9.14b shows the frequency error for different filters. It is clear from the figure that the performance of ACUKF is much better than the UKF. Case 2: Similarly, a complex frequency variation like parabola is estimated using CEKF, CUKF and ACUKF at 30 dB noise. The signal is represented as in Eq. (9.97) and the signal frequency is changed in parabolic form as in Eq. (9.99). ( ( )∧ ) w(k) = 2∗ pi∗ 50 + 25∗ k∗ dt − 25∗ k∗ dt 2

(9.99)

From the Fig. 9.15, it is clear that the performance of ACUKF is much better than the CEKF. Test 2: Experimental test data is generated using practical setup, as explained in 9.3.5, and then different methods are applied to the acquired signals. The signal generated is a time varying signal where frequency of the signal changes from 10

Fig. 9.14 a Frequency tracked with different algorithms at SNR 30 dB, b Error in frequency with different algorithms at SNR 30 dB

Table. 9.3 The mean of MSE over 100 independent runs Error in dB (dB)

UKF

CEKF

CUKF

ACUKF

60

0.04

0.034

0.021

0.01

30

0.071

0.066

0.053

0.031

20

0.19

0.16

0.14

0.11

10

0.552

0.501

0.29

0.17

232

9 Unscented and Complex Unscented Kalman Filtering for Parameter …

Fig. 9.15 a Estimated frequency and error using CEKF at SNR 30Db. b Estimated frequency and error using CUKF at SNR 30 dB. c Estimated frequency and error using ACUKF at SNR 30 dB

to 6 Hz then comes back to 10 Hz as shown by the real time signal in Fig. 9.16a. The frequency tracking ability is summarized in Table 9.4 for different algorithms. Figure 9.16b and c show the frequency tracking ability of ACUKF and the inability of UKF to track the signal respectively. Figure 9.16d shows comparison of frequency tracking ability of all the three algorithms.

9.5 Conclusions We know that the use of EKF and its variants in the area of control, instrumentation, biomedical engineering, power networks, machine learning is wide spread. But the algorithm is limited by the crude assumptions made for signal parameter estimation, instability due to linearization, costly calculation of the derivatives and the biased nature of the estimates. In this chapter, looking in to the advantages of UKF an adaptive Unscented Kalman filter technique has been discussed in detail for the estimation of amplitude, phase and frequency of the stationary as well as time varying single sinusoid and also the harmonic estimation. The signal frequency is varied in different manner. The signal

9.5 Conclusions

233

Fig. 9.16 a Real time signal. b Signal tracked using ACUKF algorithmActual (Dotted), Estimated (Solid). c Signal tracked using UKF algorithm Actual (Dotted), Estimated (solid). d Real time signal Frequency tracked with different algorithms

234

Fig. 9.16 (continued)

9 Unscented and Complex Unscented Kalman Filtering for Parameter …

References

235

Table. 9.4 The mean of MSE over 100 independent runs Algorithm

60 dB

30 dB

20 dB

10 dB

UKF

0.082

0.093

0.127

0.242

CUKF

0.0054

0.045

0.108

0.21

ACUKF

0.0026

0.031

0.095

0.201

model for both the single sinusoid and sinusoid with harmonic components has been discussed. The model and the measurement error covariances of the unscented Kalman filter (UKF) are tuned iteratively so that the filter can accurately track small or large variations in the paramter of the signals in the presence of significant noise. From the computer simulation results, it is observed that the frequency and amplitude errors are found to be much smaller with an adaptively tuned unscented Kalman filter in comparison to the un tuned unscented Kalman filter in the estimation of fundamental frequency component and harmonic magnitudes. Several simulation results presented shows the efficiency of the tracking performance of this adaptively tuned optimized unscented Kalman filter. A nonlinear filter based on unscented transformation and a complex state-space signal model (CUKF) has been discussed for the estimation of frequency of a time varying sinusoid in the presence of high noise condition. The error performance of the different algorithms are analyzed. Further, the stability analysis of the proposed nonlinear filter has also been presented to prove its convergence property.

References 1. Evans DH, McDicken WN (2000) Doppler ultrasound-physics, instrumentation and signal processing, 2nd edn. Wiley, New York 2. Rife DC, Boorstyn RR (1974) Single-tone parameter estimation from discrete time observations. IEEE Trans Infor Theory IT 20:591–598 3. Ungoren AY, Peng H, Tseng HE (2002) Experimental verification of lateral speed estimation methods. In: Proceedings of AVEC 2002 6th international symposium on advanced vehicle control, Hiroshima, Japan, pp 361–366 4. Routray A, Pradhan AK, Rao KP (2002) A novel Kalman filter for frequency estimation of distorted signals in power system. IEEE Trans Instrum Meas 51(3):469–479 5. O’shea P, Farquharson M, Ledwich G (2003) Estimation of time varying mains frequencies. In: Proceedings of Australasian universities power engineering conference, 2003 6. Mitra SK (2001) Digital signal processing: a computer based approach. McGraw Hill 7. Thayer JF, Sollers JJI, Padial ER, Vila J (2002) Estimating respirotary frequency from autoregressive spectral analysis of heart period. IEEE Eng Med Biol Mag 21(4):41–45 8. Kushner HJ (1967) Dynamical equations for optimum nonlinear filtering. J Differ Equ 3:179– 190 9. Kalman RE (1960) A new approach to linear filtering and prediction problems. Trans ASME J Basic Eng 35–45 10. Weiner N (1949) Extrapolation, Interpolation and smoothing of stationary time series. Wiley, New York 11. Dash PK, Jena RK, Panda G, Routray A (2000) An extended complex Kalman filter for frequency measurement of distorted signals. IEEE Trans Instrum Meas 49:746–753

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12. Routray A, Pradhan AK, Rao KP (2002) A novel Kalman filter for frequency estimation of distorted signals in power system. IEEE Trans Instrum Meas 51(3):469–479 13. Julier SJ, Uhlmann JK (2004) Unscented filtering and nonlinear estimation. In: Proceedings of the IEEE, vol 92. No. 3 14. Xiong K, Zhang HY, Cham CW (2006) Performance evaluation of UKF-based nonlinear filtering. Automatica 22:261–270 15. Kenefic RJ, Nuttall AH (1987) Maximum likelihood estimation of parameters of tone using real discrete data. IEEE J Ocean Eng 1:279–280 16. So HC, Chan KW, Chan YT, Ho KC (2005) Linear prediction approach for efficient frequency estimation of multiple real sinusoids: algorithm and analysis. IEEE Trans Signal Process 53(7):2290–2305 17. Pisarenko VF (1973) The retrieval of harmonics from a covariance function. Geophys J Royal Astron Soc 33:511–531 18. Schmidt RO (1981) A signal subspace approach to multiple emitter location and spectral estimation. Ph.D. dissertation, Stanford University, Stanford 19. Pradhan AK, Routray A, Basak A (2005) Power system frequency estimation using least mean square technique. IEEE Trans Power Deliv 20(3):1812–1816 20. Mishra S (2005) Hybrid least squares adaptive bacterial foraging strategy for harmonic estimation. IEE Proc:-Gener Transm Distrib 152(3):379–389 21. Lin HC (2007) Intelligent neural network-based fast power system harmonic detection. IEEE Trans Industr Electron 54(1):43–53 22. wang YN, Gu JC, Cheu CM (2003) An improved adaline algorithm for on-line tracking of harmonic components. Int J Power Energy Syst 23(2):117–125 23. Kay SM (1993) Fundamentals of statistical signal processing, volume I: estimation theory. Prentice-Hall, Englewood Cliffs, NJ 24. Kalman RE (1960) A new approach to linear filtering and prediction problem. Trans ASME Ser D J Basic Eng 82:34–45 25. Costa FF, Cardoso AJM, Fernandes DA (2007) Harmonic analysis based on Kalman filtering and prony’s method. In: Proceedings of international conferences on power engineering, energy electrical drives, Setúbal, Portugal, pp 696–701 26. Rudolph van der Merwe and Eric A. Wan (2001) Efficient derivative-free Kalman filters for online learning. In: European symposium on artificial neural networks Bruges (Belgium), D-Facto public. ISBN 2-930307-01-3, pp 205–210. Accessed 25–27 April 2001 27. Thomas DWP, Woolfson MS (2001) Evaluation of frequency tracking methods. IEEE Trans Power Deliv 16(3):367–372 28. Girgis AA, Hann FM (1998). A quantitative study of pitfalls in FFT. IEEE Trans Aerosp Electron Syst 44(1):107–115 29. Zhu TX (2007) Exact harmonics/interharmonics calculation using adaptive window width. IEEE Trans Power Deliv 22(4):2279–2288 30. Lai LL et al (1999) Real-time frequency and harmonic evaluation using artificial neural networks. IEEE Trans Power Deliv 14(1):52–59 31. Lin HC (2007) Intelligent neural network-based fast power system harmonic detection. IEEE Trans Industr Electron 54(1):43–53 32. Jing WJ, Jindapetch N, Sengchuai K (2015) Adaptive on-line tracking of power system harmonics using ADALINE. In: 2015 12th International conference on electrical engineering/electronics, computer, telecommunications and information technology (ECTI-CON), 2015, pp 1–6. https://doi.org/10.1109/ECTICon.2015.7207024.. 33. wang YN, Gu JC, Cheu CM (2003) An improved adaline algorithm for on-line tracking of harmonic components. Int J Power Energy Syst 23(2):117–125 34. So HC, Ching PC (2004) Adaptive algorithm for direct frequency estimation. Proc Inst Elect Eng Radar Sonar Navig 151(6):359–364 35. So HC, Chan KW, Chan YT, Ho KC (2005) Linear prediction approach for efficient frequency estimation of multiple real sinusoids: algorithm and analysis. IEEE Trans Signal Process 53(7):2290–2305

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Chapter 10

Mathematical Modeling for Parameter Estimation of a Non Stationary Sinusoids in the Area of Power and Communication Signals

10.1 Introduction In the case of power networks, simultaneous estimation of the power signal parameters like the amplitude, phase and frequency are required not only for the control and protection, but also for the efficient operation of distributed generation systems in a microgrid environment using microcontrollers [1]. Further, the recent introduction of PMUs (Phasor measurement units) for wide area control and protection of power networks will require efficient and accurate estimation of fundamental signal parameters. The electrical parameter measurement of a power signal is relatively a straightforward matter, as long as the frequency of the signal is not time varying. However, if the frequency varies, the parameters of the power system network cannot be estimated accurately and this is further complicated due to the presence of harmonics, interharmonics, and noise in the signal. A similar scenario also prevails in the case of low frequency signals that occur in sonar, radar, communication and mechanical systems [2]. Thus there is a need for an algorithm that should perform consistently for a signal with time varying amplitude, phase and frequency harmonics in the presence of noise. In literature, there are few algorithms that can achieve accurate results in all these aspects. Amongst the three signal parameters, frequency is one of the most important and once it is measured accurately, estimation of other parameters like amplitude and phase are relatively easy. The most widely used methods for frequency estimation include: Zero crossing technique [3], discrete and Fast Fourier transform (FFT) [4], Adaline using the least mean squares technique (LMS) [5], and recursive least square (RLS) [6], adaptive notch filter [7], Newton’s type algorithms [8], Supervised Gauss–Newton algorithms [9], Kalman filters [10, 11], Neural Methods [12, 13], wavelet transform [14], linear prediction (LP) methods [15, 16], maximum likelihood estimates [17]. Although UKF-based filters discussed in chapter number nine are powerful, the order of the harmonic is very high like 64th the matrix size becomes very large, and the inverse of the matrix requires very high computation. Hence, we need to decouple the components. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. B. Chattopadhyay et al., Mathematical Modeling of Physical Systems, Advances in Intelligent Systems and Computing 1436, https://doi.org/10.1007/978-981-19-7558-5_10

239

240

10 Mathematical Modeling for Parameter Estimation of a Non Stationary …

In the modified Newton method [18], the frequency of single or multiple sinusoids are computed using a linear predictor formulation and subsequent optimization. This method, however, is only applicable to frequency estimation, but not for amplitude and phase estimation simultaneously. On the other hand, the recursive Gauss–Newton algorithm [19] assumes frequency of the sinusoidal signal a priori and computes the amplitude and phase of the signal only. Other methods include differentiation filter, iterative-loop approaching method, simple recursive methods, etc. [20–25]. Thus keeping the objective as stated in the beginning, there is a need of an approach for the estimation of the frequency, amplitude and phase of a sinusoid simultaneously. This chapter presents a multi objective Gauss–Newton (MGN) algorithm to simultaneously estimate the amplitude, phase, and frequency of a sinusoid.

10.2 Signal Model in Complex Domain 10.2.1 Multi Objective Gauss–Newton Algorithm The major contribution of this algorithm discussed is to simplify the recursive Gauss– Newton method to reduce the computational complexity and produce a better accuracy in estimation under non-stationary conditions using two objective functions for error minimization. Here, the term multi-objective means that for amplitude and phase, one objective function is chosen for error minimization, while for frequency estimation another objective function comprising of three consecutive samples and a weighted error cost function is used. Further the forgetting factors of the algorithm are tuned iteratively for better estimation. The problem of sinusoidal parameter estimation is formulated for discrete-time noisy measurements as: y(k) = s(k) + v(k), k = 0, 1, 2, . . . ., N − 1

(10.1)

s(k) = A(k) sin(w(k)k + φ(k))

(10.2)

where,

with, A, w and φ being unknown values that denote the amplitude, frequency and phase of a real valued sinusoid, respectively, while v(k) is an additive white noise with unknown variance σv2 .

10.2 Signal Model in Complex Domain

241

10.2.2 Frequency Estimation An error objective function is formulated using consecutive 2R samples (R is the number of sinusoids in the signal) for estimating time varying frequency of the signal in Eq. (10.2). In this method the frequency of the signal is estimated using the properties of a linear predictor, and then the amplitude and phase of the signal are estimated using the recursive Gauss–Newton approach. It has been shown in [18] that s(k) can be uniquely expressed as a linear combination of its previous 2R samples as s(k) = −

2R ∑

an s(k − n)

(10.3)

n=1

where an are referred to as linear prediction coefficients. The relationship between w and an is given by [13] as: 2R ∑

an exp(− j wn) = 0, and an = a2R−n , n = 0, 1, 2, ..R

(10.4)

n=0

In the above equation a0 is chosen as 1, and the signal estimation error is defined as ew (k) =

R−1 ∑

a˜ n (y(k − n) + y(k − 2R + n)) + a˜ R y(k − R)

(10.5)

n=0

with a˜ n denotes the optimized value of an , here a˜ 0 may not be fixed to unity. For error minimization, an exponentially weighted cost function is used as follows: ε1 (k) =

k ∑

2 λk−n 1 ew (n)

(10.6)

n=0

where 0 < λ1 ≤ 1 is the forgetting factor. For the estimation of frequency of a single sinusoid (R = 1) according to Eq. (10.4), the linear predictor coefficients required are a0 , a1 and a2 ,(a2 = a0 ). In this case, Eq. (10.5) is rewritten with R = 1 as: ew (k) = a˜ 0 (y(k) + y(k − 2)) + a˜ 1 y(k − 1)

(10.7)

Hence the parameter vector to be estimated is given by: θ˜ (k) = [a˜ 0 a˜ 1 ]T . Since for time varying frequency, ew is not linear in a˜ 0 and a˜ 1 due to the time varying nature of the signal, conventional RLS [6] algorithm cannot be applied to minimize (10.6). Thus a recursive Gauss–Newton method [19] is used to minimize (10.6) and the updating equation for sinusoidal parameter estimation under noisy condition is as follows:

242

10 Mathematical Modeling for Parameter Estimation of a Non Stationary …

˜ − 1) − H −1 (k)ψ(k)ew (k) θ˜ (k) = θ(k H (k) =

k ∑

T λk−n 1 ψ(n)ψ (n)

(10.8)

(10.9)

n=0

where the gradient vector ψ, and the Hessian matrix H (k) [19] are obtained after some simple manipulation as: ψ(k) = H (k) =

k ∑ n=0

[ χ1k−n

∂ew (k) = ∂ θ˜

[

y(k) + y(k − 2) y(k − 1)

] (10.10)

(y(k) + y(k − 2))(y(k − 1)) (y(k) + y(k − 2))2 (y(k) + y(k − 2))(y(k − 1)) (y(k − 1))2

]

(10.11) To compute inverse Hessian matrix one can directly use the matrix inverse lemma at the cost of high computational complexity and thus an approximation is performed for fast computation of the frequency iteratively. Assuming that w is not near to 0 or π, H (k) can be approximated as: 1 − λk+1 1 H (k) = 1 − λ1 [ 2 ] 0 4 A cos2 (w) sin2 (w(k − 1) + φ(k − 1)) 0 A2 sin2 (w(k − 1) + φ(k − 1)) (10.12) The inverse of the Hessian matrix can easily be calculated as: H −1 (k) = [ ] 1/8c(k) A2 cos2 (w) sin2 (w(k − 1) + φ(k − 1)) 0 0 1/2c(k)A2 sin2 (w(k − 1) + φ(k − 1))

(10.13)

where c(k) =

1 − λk+1 1 2(1 − λ1 )

(10.14)

We can observe that c(k) can be computed recursively as: c(k) = λ1 c(k − 1) + 1/2

(10.15)

10.2 Signal Model in Complex Domain

243

Further by putting Eqs. (10.13) and (10.14) into Eq. (10.8) the following equations are obtained: a˜ 0 (k) = a˜ 0 (k − 1) − ew (k)/4c(k)A cos(w) sin(w(k − 1) + φ(k − 1))

(10.16)

a˜ 1 (k) = a˜ 1 (k − 1) − ew (k)/2c(k)A sin(w(k − 1) + φ(k − 1))

(10.17)

After estimating a˜ 0 , and a˜ 1 , the frequency of the sinusoid is computed as cos−1 (−a˜ 1 /2).

10.2.3 Amplitude and Phase Estimation Once the frequency is estimated, the amplitude and phase of the signal are calculated using the recursive Gauss–Newton method in the same iteration. For calculating amplitude and phase of a sinusoid, let the parameter vector be θ (k) = [A(k) φ(k)]T T ˆ ˆ φ(k)] and its estimate be θˆ (k) = [ A(k) Using θˆ (k − 1), the estimate of y(k) at time k is computed as: ˆ − 1) sin(wk + φ(k ˆ − 1)) yˆ (k) = A(k

(10.18)

The a priori estimation error at time k is given as ˆ − 1) sin(wk + φ(k ˆ − 1)) eθ (k) = y(k) − A(k

(10.19)

and a similar kind of cost function is taken as in Eq. (4.6) ε2 (k) =

k ∑

2 λk−n 2 eθ (n)

(10.20)

n=0

where 0 < λ2 ≤ 1 is the forgetting factor. In this case, recursive Gauss–Newton method is also used to minimize (10.20) in a similar way as mentioned above. The gradient vector is given by ] [ ˆ − 1)) ∂eθ (k) − sin(wk + φ(k ψ(k) = = ˆ − 1) cos(wk + φ(k ˆ − 1)) − A(k ∂ θˆ and the Hessian matrix is obtained as:

(10.21)

244

10 Mathematical Modeling for Parameter Estimation of a Non Stationary …

H (k) = [

k ∑

χ2k−n

n=0

] ˆ − 1) sin(2(wn + φ(k ˆ − 1)) ˆ − 1)))/2 A(k sin2 (w + φ(k ˆ − 1) sin(2(wn + φ(k ˆ − 1)))/2 Aˆ 2 (k − 1) cos2 (wn + φ(k ˆ − 1)) A(k (10.22)

In a similar manner for the frequency case, the Hessian matrix in Eq. (10.22) is approximated as H (k) =

k ∑ n=0

[ λk−n 2

] ] [ 1 − λk+1 1/2 0 1 0 2 = 0 Aˆ 2 (k − 1)/2 2(1 − λ2 ) 0 Aˆ 2 (k − 1)

(10.23)

and inverse Hessian matrix can easily be computed as H −1 (k) =

[

1/c(k) 0 0 1/ Aˆ 2 (k − 1)c(k)

] (10.24)

where c(k) is the same as in Eq. (10.14) with forgetting factor λ2 . Further by putting Eqs. (10.24) and (10.14) into Eq. (10.8), the amplitude and phase are calculated as: ˆ ˆ − 1) + sin(w(k ˆ − 1))eθ (k)/c(k) A(k) = A(k ˆ − 1)k + φ(k

(10.25)

ˆ − 1)c(k)) ˆ ˆ − 1) + cos(w(k ˆ − 1))eθ (k)/( A(k φ(k) = φ(k ˆ − 1)k + φ(k

(10.26)

10.2.4 Performance Analysis of Single Sinusoid From the above equations, we can observe that the forgetting factors λ1 and λ2 influence the estimation process. The initial values can be any arbitrary value within the range 0 < λ < 1, When the signal is contaminated with high random noise, forgetting factor close to 0.5 results in faster convergence, but increased sensitivity to noise. However, using the forgetting factor close to 1 (e.g., λ = 0.99) results in slow convergence, but better noise rejection property. Motivated by these aspects, a number of variable forgetting factor RLS algorithms have been developed [26]. The a priori estimation error is given by Eq. (10.19), and the a posteriori estimation function is given as: ˆ sin(wk + φ(k)) ˆ ς (k) = y(k) − A(k)

(10.27)

10.3 Algorithm Development for Multiple Sinusoids

245

To make the update equation stable assume: ( ) E ς 2 (k) = σv2

(10.28)

where σv2 is the power of the system noise. Solving for the above condition, a time-dependent variable forgetting factor can be generated as: λk =

σb (k)σv σe (k) − σv

(10.29)

where E(b2 (k)) = σb2 (k), and E(eθ (k)) = σe2 (k) is the power of error signal, and b(k) = ψ(k)T H (k)ψ(k) The power estimates are computed iteratively as: σˆ e2 (k) = τ σˆ e2 (k − 1) + (1 − τ )eθ2 (k)

(10.30)

σˆ b2 (k) = τ σˆ b2 (k − 1) + (1 − τ )b2 (k)

(10.31)

where τ = 1− K τ1D , is a weight factor, with K τ ≥ 2, and D is the number of elements in gradient vector, and the power of the system noise is estimated using longer data as: σˆ v2 (k) = β σˆ v2 (k − 1) + (1 − β)eθ2 (k)

(10.32)

with β = 1 − K β1 D K β ≻ K τ . Now with λk ∼ = σˆ v (k). Generally, = 1, leads to σˆ e (k) ∼ better estimates are expected with this tuning particularly in signal tracking with step changes of parameters and sudden changes of the network topology and the states. To implement the multi-objective Gauss–Newton Algorithm, the variables are chosen as: λ1 = 0.55, λ2 = 0.55, a(0) ˜ = [1 , 0]T for frequency, amplitude and phase estimation. The initial amplitude and phase are chosen from some prior knowledge about the signal parameters.

10.3 Algorithm Development for Multiple Sinusoids The algorithm discussed in Sect. 10.2 can be generalized for multiple harmonic and inter-harmonic component estimation. For multiple sinusoidal parameter estimation, the discrete time noisy measurement is given as:

246

10 Mathematical Modeling for Parameter Estimation of a Non Stationary …

s(k) =

R ∑

Ar (k) sin(wr (k)k + φr (k)), r = 1, 2, . . . R

(10.33)

r =1

where Ar ,wr and φr are unknown values that denote the amplitude, frequency and phase of the rth real valued sinusoid, respectively.

10.3.1 Frequency Estimation For R sinusoids according to Eq. (10.4), the linear predictor coefficients required are a0 ,a1 ,a2 ,.. a2R , with estimation error given in Eq. (10.5). Hence, the parameter vector to be estimated is given by: θ˜ (k) = [a˜ 0 a˜ 1 .......a˜ R ]T , and the updating equation for sinusoidal parameter estimation under noisy condition is as given in Eq. (10.8) and (10.9), where the gradient vector ψ, and the Hessian matrix H (k)[19] are obtained as in Eqs. (10.34) and (10.35). ⎤ y(k) + y(k − 2R) ⎢ y(k − 1) + y(k − 2R + 1) ⎥ ⎥ ⎢ ⎢ y(k − 2) + y(k − 2R + 2) ⎥ ⎥ ⎢ ⎥ ∂ew (k) ⎢ ······ ⎥ ⎢ ψ(k) = =⎢ ⎥ ˜ ⎥ ⎢ ······ . ∂θ ⎥ ⎢ ⎥ ⎢ ········· ⎥ ⎢ ⎣ y(k − R + 1) + y(k − R − 1) ⎦ y(k − R) ⎡

H (k) = ⎡

k ∑

(10.34)

λk−i 1

i=0

⎤ (y(k) + y(k − 2R)) (y(k − 1) + y(k − 2R + 1)) ..... (y(k) + y(k − 2R)) (y(k − R)) (y(k) + y(k − 2R))2 ⎢ ⎥ ⎢ (y(k) + y(k − 2R)) (y(k − 1) + y(k − 2R + 1)) (y(k − 1) + y(k − 2R))2 ..... (y(k − 1) + y(k − 2R)) (y(k − R))⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ....... ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ .......... ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ .......... ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ .......... ⎣ ⎦ 2 (y(k) + y(k − 2R)) (y(k − R)) (y(k − 1) + y(k − 2R)) (y(k − R)) ...... (y(k − R))

(10.35) To compute inverse Hessian matrix, a similar approximation is made to reduce the computational complexity assuming w is not near to 0 or π, where H (k) can be approximated as in Eqs. (10.36) and (10.37).

10.3 Algorithm Development for Multiple Sinusoids H (k) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

k ∑

247

λ1k−n

n=0

(y(k) + y(k − 2R))2 0 0

....

0

(y(k − 1) + y(k − 2R + 1))2 0

0 0

(y(k − 2) + y(k − 2R + 2))2

........ ........ .......... 0

0

.......

⎤ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(10.36)

(y(k − R))2

1 − λ1k+1 H (k) = 2(1 − λ1 ) ⎡ 0 ... 0 2R A21 cos 2 (Rw1 ) sin2 (w1 (k − R) + φ1 (k − R)) + 8A22 cos2 (Rw2 ) sin2 (w2 (k − R) ⎢ 2 2 2 ⎢ +φ (k − R) + ... + 8A cos (Rw ) sin (w (k − R) + φ (k − R)) 2 R R R R ⎢ ⎢ ⎢ .......... ⎢ ⎢ ⎢ .......... ⎢ ⎢ ⎢ .......... ⎢ ⎢ ⎢ 0 ...... 0 2 A21 sin2 (w1 (k − R) + φ1 (k − R)) + A22 sin2 (w2 (k − R) ⎣

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

+φ2 (k − R) + .............. + A2R sin2 (w R (k − R) + φ R (k − R))

(10.37) Then inverse Hessian matrix is easily calculated as done in Sect. 10.2. Then the parameters are obtained as: a˜ 0 (k) = a˜ 0 (k − 1) − ew (k)(A1 cos ( Rw1 ) sin(w1 (k − R) + φ1 (k − R)) + A2 cos ( Rw2 ) sin(w2 (k − R)+ φ2 (k − R)) + ... + A R cos(Rw R ) sin(w R (k − R)+ φ R (k − R)))/2Rc(k)X 1

(10.38)

a˜ 1 (k) = a˜ 1 (k − 1) − ew (k)(A1 cos( (R − 1)w1 ) sin (w1 (k − R) + φ1 (k − R)) A2 cos( (R − 1)w2 ) sin( w2 (k − R) + φ2 (k − R)) + ...+ A R cos( (R − 1)w R ) sin (w R (k − R) + φ R (k − R))) /2Rc (k)X 2 a˜ R (k) = a˜ R (k − 1) − ew (k)(A1 sin(w 1 (k − R) + φ1 (k − R))+ A2 sin(w2 (k − R) + φ2 (k − R)) + .. .. + A R sin(w R (k − R) + φ R (k − R)))/2c(k)X (R + 1)

(10.39)

(10.40)

where, the terms (1,1), (2,2), (3,3),…(R + 1,R + 1) of Eq. (10.37) are denoted as X1, X2,… X(R + 1) respectively.

248

10 Mathematical Modeling for Parameter Estimation of a Non Stationary …

10.3.2 Amplitude and Phase Once the frequencies are estimated using Eq. (10.4), amplitude and phase can be estimated as shown in Sect. 10.2, and the amplitude and phase of the harmonics and interharmonics are estimated as, ( ) (10.41) Aˆ r (k) = Aˆ r (k − 1) + sin wˆ r (k − 1)k + φˆr (k − 1) eθ (k)/c(k) ( ) ( ) φˆr (k) = φˆr (k − 1) + cos wˆ r (k − 1)k + φˆr (k − 1) eθ (k)/ Aˆ r (k − 1)c(k) (10.42)

10.3.3 Performance Analysis of Multiple Sinusoids Here, we will analyse the mean- square estimation errors of the parameters under stationary condition. Thus let us consider θ = [a0 , a1 , A , φ]T , and the covariance matrix in the MGN method denoted as cov(θˆ (k)), is calculated for two different objective functions as, cov(θˆ (k)) = E

=σ

2

[ k ∑ n=0

{[

∂ε12 (k) ∂ θˆ 2

]−1 [

]−1 T λk−n 1 ψ(n)ψ (n) n=0

k ∑

∂ε1 (k) ∂ θˆ

][

∂ε1 (k) ∂ θˆ

]T [

λ2(k−n) ψ(n)ψ T (n) 1

n

∂ε12 (k) ∂ θˆ 2

[ k ∑

]−1 } θˆ (k)=θ

]−1 T λk−n 1 ψ(n)ψ (n)

n=0

(10.43) where E denotes the expectation operator. Further when k is sufficiently large, the covariance reduces to [ ] 0 1/2c(k) A2 ˆ (10.44) cov(θ(k)) ≈ σ2 0 1/c(k)A2 Hence, the variance of the linear predictor coefficients is var(a˜ 0 (k)) = and

σ 2 (1 − λ1 ) ( ) A2 1 − λk+1 1

(10.45)

10.4 A Comparative Discussion on Simulation and Experimental Results

var(a˜ 1 (k)) =

2σ 2 (1 − λ1 ) ( ) A2 1 − λk+1 1

249

(10.46)

similarly analysing covariance matrix for the second cost function, the variance of the amplitude and phase are found to be 2σ 2 (1 − λ2 ) ˆ var( A(k)) = ( ) 1 − λk+1 2

(10.47)

2σ 2 (1 − λ2 ) ( ) A2 1 − λk+1 2

(10.48)

and, ˆ var(φ(k)) =

If all the forgetting factors are made equal to unity, then the variances will attain Cramer–Rao lower bound for sufficiently large value of k assuming the noise v(k) is Gaussian distributed. Hence, it is proved that the MGN algorithm attains optimal performance for stationary amplitude, phase and frequency in an asymptotic sense. The next section presents computational results for power sinusoids and low frequency signals.

10.4 A Comparative Discussion on Simulation and Experimental Results 10.4.1 Outline of Experimental Setup Experimental test has been carried out in a laboratory to establish the feasibility of the proposed algorithm in a real time environment. The proposed algorithm is simulated with a 200 V, 3-phase, 50 Hz voltage source supplying a 3- phase nonlinear load (shown in Fig. 10.1) using PSCAD software. The nonlinear load comprises a six pulse converter feeding a R-L load of 2.5 ohms and 0.1 mH. A capacitor of 100 μF across the R-L load is switched in at 0.3 s and taken out at 0.4 s. Phase a line outage is simulated at 0.5 s to obtain high harmonic content and abrupt signal variations. The signals are sampled with a step size of 200 μs, which effectively produces a sampling rate of 5 kHz. White Gaussian noise of 30 dB SNR (signal to noise ratio) is added to both voltage and current signals to account for the possible noise during measurement.

250

10 Mathematical Modeling for Parameter Estimation of a Non Stationary …

Fig. 10.1 System under study: measurements are done near to the voltage source

10.4.2 Single Sinusoid Computer simulations have been carried out to evaluate the performance of the proposed algorithm. The sampling frequency f s is chosen as 1.6 kHz based on 50 Hz fundamental frequency for the following case study. This experiment is done for a 50 Hz power signal with change in signal parameters. The signal is represented as: y(k) = A(k) sin(w(k)k + φ(k)) + vk

(10.49)

For the first 70 samples, freq = 50 Hz, A = 1pu, φ = π/4, for 70 to 150 sample, parameters changes to freq = 47 Hz, A = 1.2pu, φ = π/6, and then come back to initial values. The angular frequency of the signal is varied as: w(k) = w0 + (w1 − w0 )/80(k − 70)

(10.50)

where, w0 = 2 ∗ pi ∗ 50/ f s , w1 = 2 ∗ pi∗ 47/ f s , where f s is the sampling frequency. The signal parameters are estimated using Adaline [5], EKF [10], ANN [12], LP [15], RGN [18], HMA [19], GCMA [27] and the proposed method for comparison. Out of these some of the algorithms are used to compute frequency only, and some for amplitude and phase. The signal is corrupted with 30 dB noise. The estimated signal parameters using different algorithms are given in Fig. 10.2. The signal is then tested under different noise levels and the estimated frequency, amplitude and phase MSE in dB obtained in different algorithms are shown in Fig. 10.3. Absolute frequency error, amplitude error and phase error at different noise levels of different algorithms and their execution times are listed in Table 10.1. Although the ANN algorithm is capable of estimating the

10.4 A Comparative Discussion on Simulation and Experimental Results

251

Frequency

52 50 48 46

Amplitude

0

20

40

60

80

100

120

140

160

180

200

0

20

40

60

80

100

120

140

160

180

200

0

20

40

60

80

100

120

140

160

180

200

1.2 1 0.8

Phase

1

0.5

Samples

Fig. 10.2 Estimated parameters at 30 dB noise, EKF (dotted), RGN (dashed), MGN (solid)

harmonics present in the signal, its performance in estimating frequency component deteriorates under high noise condition. Then the computational complexity of the proposed algorithm is compared with some of the methods like RGN [18] and MFF [28]. Considering that the cosine and sine functions are determined from look-up tables, the computational requirements for the above mentioned methods are listed in Table 10.2.

10.4.3 Multiple Sinusoids 10.4.3.1

Case Study:1

This experiment is done for the analysis of the proposed algorithm on a highly distorted sinusoid signal with harmonics corrupted with 30 dB noise. In this case the sampling frequency is chosen as 7.2 kHz with a view to compute the highest harmonic component (50th harmonic) if any. The fundamental frequency for these cases is taken to be 60 Hz, as per ANSIC84.1- 1995 Electric Power Systems and Equipment Voltage Ratings. The damped sine wave with harmonics is modelled as: y(k) =

9 ∑ n=1

An e−(kTs α) sin(nw0 kTs + φn (k))

(10.51)

10 Mathematical Modeling for Parameter Estimation of a Non Stationary …

a -50

MS frequency error in dB

-100

-150

-200

-250

113 -300 5

10

15

20

25

30

20

25

30

20

25

30

SNR in dB

b

-60 -80 -100

MS amplitude error in dB

Fig. 10.3 a Mean Square frequency error in dB at different SNR EKF (dashed), RGN (dotted), MGN(solid). b Mean Square amplitude error in dB at different SNR. EKF (dashed), RGN (dotted), MGN(solid). c Mean Square phase error in dB at different SNR EKF (dashed), RGN (dotted), MGN (solid)

-120 -140 -160 -180 -200 -220 -240 5

10

15 SNR in dB

c -142.5 -143 -143.5 MS phase error in dB

252

-144 -144.5 -145 -145.5 -146

114

-146.5 5

10

15 SNR in dB

_

0.036

0.001

GCM

ANN

MGN(Proposed method)

_

HMA

0.032

0.087

_

_

0.071

0.091

0.050

0.021

LP

RGN

0.713

0.202

0.531

0.151

Adaline

20 dB

0.101

0.169

_

_

0.103

0.201

1.053

1.273

10 dB

0.007

0.007

0.010

0.008

0.009

_

0.021

0.072

30 dB

0.002

0.004

0.006

0.005

0.005

_

0.039

0.915

20 dB

0.019

0.038

0.39

0.30

0.034

_

0.609

0.717

10 dB

0.0005

0.0097

0.012

0.001

0.0009

_

0.0063

0.0097

30 dB

0.0001

0.0052

0.0041

0.0033

0.003

_

0.032

0.054

20 dB

Frequency estimation error (HZ) Amplitude estimation error (p.u) Phase estimation error

30 dB

EKF

Algorithm

Table 10.1 Performance comparison of different algorithms

0.004

0.0086

0.0069

0.0065

0.006

_

0.20

0.31

10 dB

0.59

_

0.61

0.58

0.86

_

1.53

_

Execution time (ms)

10.4 A Comparative Discussion on Simulation and Experimental Results 253

254

10 Mathematical Modeling for Parameter Estimation of a Non Stationary …

Table 10.2 Computational complexity comparison of different algorithms

Algorithm

×/÷

±

RGN

97

57

MFF (same λ)

21

8

MFF (different λ)

27

14

Proposed MGN (same λ)

19

8

Proposed MGN (variable λ)

26

12

and the parameters that are set as the fundamental frequency are 60 Hz, A1 = 1pu, A3 = 0.7, A5 = 0.3, A7 = 0.15, A9 = 0.05, α = 1.5, α3 = α5 = α7 = α9 = 1.5, φ1 = 0.8, φ3 = 0.4, φ5 = 0.7, φ7 = 0.6, φ9 = 0.5. Figure 10.4a, shows the damped sine wave with harmonics. Figure 10.4b shows the harmonic amplitude and 10.4c shows phase components of the estimated signal respectively. From the figures it is clear that the proposed algorithm outperforms even for estimation of such complex signal.

10.4.3.2

Experimental Validation

Experimental test has been carried out in a laboratory to establish the feasibility of the proposed algorithm in a real time environment as explained in Sect. 10.4.1 The figures given below represent the instantaneous estimations done on the acquired 3- phase current signals (inclusive of all harmonics due to the converter, capacitor switching and line outage). The tracking capabilities of the proposed algorithm are excellent as compared to the Running Discrete Fourier Transform (RDFT) algorithm. The scale of the graphs plotted shows how closely both the algorithms match for a real system signal analysis. Figure 10.5a shows the estimated phase a current signal using the proposed MGN algorithm and RDFT. As shown in Fig. 10.5b,

a

Fig. 10.4 a Damped sinusoid with harmonics. b Fundamental harmonic amplitude components. c Fundamental harmonic phase components

2

1.5 1

signal

0.5 0 -0.5 -1 -1.5 -2

0

50

100

150

200

250 300 samples

350

400

450

500

10.4 A Comparative Discussion on Simulation and Experimental Results

b

1

255

0.16

0.9

0.14 0.8

0.12

0.7

0.1

amplitude

amplitude

0.6 0.5 0.4

0.08

0.06 0.3 0.2

0.04

0.1

0.02

0

0

100

200

300

400

600 500 samples

700

800

900

1000

0

0

100

200

300

400

500 600 samples

700

200

300

400

500 600 samples

700

800

900

1000

0.7 0.06

0.6 0.05

0.5

amplitude

amplitude

0.04

0.4

0.3

0.02

0.2

0.1

0

0.03

0.01

0

100

200

300

400

500 600 samples

700

800

900

1000

0

0

100

phase

c

1

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2

0

Fig. 10.4 (continued)

100

200

300

400

500

600

700

800

900

1000

800

900

1000

256

10 Mathematical Modeling for Parameter Estimation of a Non Stationary …

a 0.15 MGN RDFT

Amplitude (in KA)

0.1 0.05 0 -0.05 -0.1 -0.15 -0.2

b

0

0.1

0.2

0.3

0.6 0.5 0.4 Time (in sec)

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4 0.5 0.6 Time (in sec)

0.7

0.8

0.9

1

0.1

0.2

0.3

0.4 0.5 0.6 Time (in sec)

0.7

0.8

0.9

1

amplitude

0.2 0.1 0 0.5

phase

200

0

c

0.025

3rd harmonic amplitude

-200

0.02 0.015 0.01 0.005 0 0

Fig. 10.5 a Phase a current signal. b Fundamental components of phase a current signal. c Third harmonic amplitude of phase a current signal. d Fifth harmonic amplitude of phase a current signal. e Seventh harmonic amplitude of phase a current signal. f Eleventh harmonic amplitude of phase a current signal

10.4 A Comparative Discussion on Simulation and Experimental Results

5th harmonic amplitude

d

257

0.035 0.03 0.025 0.02 0.015

121

0.01 0.005 0

e

0.03

7th harmonic amplitude

0

0.025

0.1

0.2

0.3

0.4 0.5 0.6 Time (in sec)

0.7

0.8

0.9

1

0.02 0.015 0.01 0.005 0 0 x 10

11th harmonic amplitude

f

0.1

0.2

0.3

0.4 0.5 0.6 Time (in sec)

0.7

0.8

0.9

1

0.1

0.2

0.3

0.6 0.5 0.4 Time (in sec)

0.7

0.8

0.9

1

-3

6

4

2

0 0

Fig. 10.5 (continued)

258

10 Mathematical Modeling for Parameter Estimation of a Non Stationary …

the proposed MGN algorithm responds faster to abrupt changes in the fundamental amplitude and phase component of the signal and exhibits superior performance. The presence of the harmonics during the load switching and line outage can clearly be observed from Fig. 10.5c–f. Similarly the signal components obtained for phase b current signal are shown below. Figure 10.6a shows phase b current signal obtained using both the algorithms. Figure 10.6b clearly shows the superior performance of the proposed algorithm over RDFT. Figure 10.6c–f show the harmonic components present in the signal. Similarly the signals obtained for phase c current signal are shown in Fig. 10.7a–e. A similar results are obtained for phase c current signal as observed in the case of phase b current signal.

10.5 Conclusion In this chapter, a multi-objective Gauss–Newton algorithm is presented to reduce the computational complexity and produce better accuracy in estimation even under nonstationary conditions. The algorithm uses two weighted error cost functions, one of which is used for minimizing the error obtained in frequency estimation based on the linear predictor technique, and the other for minimizing the error between the estimated value and the measured value of the sinusoids buried in noise with the help of the recursive Gauss–Newton algorithm. A forgetting factor has been used to consider the more recent signal samples to provide better tracking accuracy and speed. The performance of the proposed algorithm is found to be dependent on the choice of the forgetting factor used in the algorithm. Thus when the signal is contaminated with high random noise, forgetting factor close to 0.5 results in faster convergence, but increased sensitivity to noise. However, using a forgetting factor close to unity results in slow convergence, but better noise rejection property. Motivated by these aspects, a variable forgetting factor method is proposed in the thesis. Further to provide a simple adaptive filter for single or multi-frequency component signal, the resulting Hessian matrix of the Gauss–Newton algorithm is simplified. This simplification results in a decoupled estimation model for fundamental, harmonic, or any decaying dc component that might be present in the signal. The performance of the proposed algorithm is then analysed by comparing with some of the existing algorithms like Adaline, EKF, LP, RGN, HMA, GCMA, etc.. The absolute estimation error of different algorithms under different noise conditions is presented in this chapter. The proposed algorithm is found to estimate time varying parameters with a significant decrease in the estimation error. Also, the proposed filter is able to track the damped sinusoids with better accuracy, and less computational burden.

a

0.2

Amplitude (in KA)

10.5 Conclusion

0.1

259

MGN RDFT

0

-0.1

-0.2

0.1

0.2

0.3

0.6 0.5 0.4 Time (in sec)

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4 0.5 0.6 Time (in sec)

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4 0.5 0.6 Time (in sec)

0.7

0.8

0.9

1

b

0.2

amplitude

0

0.1

0

0.5

phase

200

0

-200

c 3rd harmonic amplitude

0.04 0.03 0.02 0.01 0 -0.01

Fig. 10.6 a: Phase b current signal. b Fundamental components of phase b current signal. c Third harmonic amplitude of phase b current signal. d Fifth harmonic amplitude of phase b current signal. e Seventh harmonic amplitude of phase b current signal. f: Eleventh harmonic amplitude of phase b current signal

260

10 Mathematical Modeling for Parameter Estimation of a Non Stationary …

d

0.035

5th harmonic amplitude

0.03 0.025 0.02 0.015 0.01 0.005 0 -0.005

e

0

0.1

0.2

0.3

0.4

0.6 0.5 Time(in sec)

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4 0.5 0.6 Time (in sec)

0.7

0.8

0.9

1

0.1

0.2

0.3

0.4 0.5 0.6 Time (in sec)

0.7

0.8

0.9

1

0.025

7th harmonic amplitude

0.02 0.015 0.01 0.005 0 -0.005

e

8

x 10

-3

11th harmonic amplitude

7 6 5 4 3 2 1 0 -1

0

Fig. 10.6 (continued)

10.5 Conclusion

261

a 0.2 MGN RDFT

Amplitude (in KA)

0.15 0.1 0.05 0 -0.05 -0.1 -0.15

amplitude

b

0

0.1

0.2

0.3

0.6 0.5 0.4 Time (in sec)

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4 0.5 0.6 Time (in sec)

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.5 0.4 0.6 Time (in sec)

0.7

0.8

0.9

1

0.2 0.1

0 0.5

phase

200

100

0

3rd harmonic amplitude

c

0.04 0.03 0.02 0.01 0

Fig. 10.7 a Phase c current signal. b Fundamental components of phase c current signal. c Third harmonic amplitude of phase c current signal. d Fifth harmonic amplitude of phase c current signal. e Seventh harmonic amplitude of phase c current signal

262

10 Mathematical Modeling for Parameter Estimation of a Non Stationary …

a 5th harmonic amplitude

0.03

0.02

0.01

0

7th harmonic amplitude

b

0

0.1

0.2

0.3

0.6 0.5 0.4 Time (in sec)

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.6 0.5 0.4 Time (in sec)

0.7

0.8

0.9

1

0.025 0.02 0.015 0.01 0.005 0 -0.005

Fig. 10.7 (continued)

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Appendix A

A.1 Rotor Resistivity Correction Factor (Kr ) The normal component of the stream function, u y in the rotor sheet satisfies the differential equation given by ∂2 ( ) ∂2 ( ) u + u y = j σ sωs B y(total) y ∂x2 ∂z 2

(A.1)

In Eq. (A.1), B y(total) can be expressed as B y(total) = B y1 + B y2

(A.2)

With reference to the above-said Eq. (A.2), B y1 and B y2 are the normal components of flux density due to the stator and rotor currents, respectively. As seen from the results of the distribution of a normal component of flux density against the width dimension (z) for a rotor model of finite width in Chap. 2, both B y1 and B y2 have approximately trapezoidal distributions. Therefore, the distribution of B y(total) can also be considered trapezoidal and it is shown in Fig. A.1. With reference to Fig. A.1, B y(total) distribution, by Fourier series analysis, can be expressed as ) ) ) ( ( 3π z nπ z πz + a3 cos + . . . + an cos =a1 cos wr wr wr [ )] ( Σ nπ z an cos = wr n=1,3,5... (

B y(total)

(A.3)

With reference to Eq. (A.3),} an for the present, is unknown and Wr is the rotor width. As the rotor is infinitely long, u y can be expressed as © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. B. Chattopadhyay et al., Mathematical Modeling of Physical Systems, Advances in Intelligent Systems and Computing 1436, https://doi.org/10.1007/978-981-19-7558-5

265

266

Appendix A

Fig. A.1 Approximate distribution of B y(total) against ‘z’

u y (x, z) = u y (z)e− jkx

(A.4)

Substituting B y(total) and u y (x, z) from Eqs. (A.3) and (A.4), respectively, in Eq. (A.1) yields −k 2 u y (z) +

( )] Σ [ ] nπ z d2 [ u (z) = j σ sω cos a y s n dz 2 Wr n=1,3,5.....

(A.5)

As current cannot escape from the sheet, u y (z) must become zero at the extremities (width wise) of the rotor. Therefore, the trial solution for u y (z) in Eq. (A.5) can be assumed as )] ( Σ [ nπ z bn cos (A.6) u y (z) = Wr n=1,3,5... In Eq. (A.6), “bn ” for the present, is an arbitrary constant. From Eqs. (A.5) and (A.6) we get, ⎡

⎤ − j sω (σ ) s ( 2 2 )} ⎦an bn = ⎣ { n π k 2 + (W )2 r

(A.7)

Appendix A

267

The time average of the propulsion force per unit length, Fx can be expressed as W {r /2

Fx =d −

( )}] 1 [{ ∗ Real −Jzr B yr dz 2

( ) W {r /2 ] [ 1 =−d Re/2al jku y (z)B y∗ dz 2

(A.8)

−

Substituting for bn from Eq. (A.7) in Eq. (A.6), we obtain the expression for u y (z) and then substituting it in Eq. (A.8), Fx , can be finally simplified to ( Fx =

⎡ ⎤ ) an2 dWr { σ sωs τ } Σ ⎣ { ( 2 2 )} ⎦ n τ 4 π 1 + n=1,3,5....

(A.9)

(Wr )

2

If the conductivity of the rotor material is reduced to the value, σeffective , and at the ( )2 same time if the rotor width “Wr ” is made very large, Wτ r tends to be zero and Fx approaches Fx(∞) given by ( Fx(∞) =

) { sω τ } Σ [ 2] dWr s {σeffective } an n=1,3,5..... 4 π

(A.10)

To have the value of force Fx(∞) given in Eq. (A.10) same as the value of Fx given in Eq. (A.9), we should have ⎡ σeffective

⎢ ⎢ = σ⎢ ⎣

[

Σ

{ ( 1+

n=1,3,5......

]⎤

an2 n2 τ 2 2

)}

[ (Wr]) Σ 2 n=1,3,5...... an

⎥ ⎥ ⎥ ⎦

(A.11)

The resistivity correction factor due to finite width effect, K r is defined as 1 σeffective = Kr σ

(A.12)

From Eqs. (A.11) and (A.12), K r is given by ⎡

Σ

[

⎢ 1 ⎢ n=1,3,5...... =⎢ Σ ⎣ Kr

{ ( 1+

n=1,3,5......

an2 n2 τ 2 (W )2

[ r] an2

]⎤ )}

⎥ ⎥ ⎥ ⎦

(A.13)

268

Appendix A

With reference to the distribution of B y(total) in Fig. A.1, we can express B y(total) as [ B y(total) = B0

(

z + W2s 1+ c

)]

( . . . . . . for

−

Ws +C 2

Ws ws