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

Preface
	The Place of Magnetic Levitation and Propulsion as a Major Application Area of Electromagnetic Field Theory
	Applications in Fluid Mechanics
	Mathematical Modelling Using the Generalized Theory of Electrical Machines
	Mathematical Modelling of a Non-linear System Using Circuit Theory Approach and Linearization Technique
	Applications of Signal Processing to Certain Classes of Electrical Power System Problems
	Organization
	Suggestions for Using This Book
	Notable Features
Acknowledgements
Contents
About the Editors
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
	1.2 Formulation for Fields and Currents
		1.2.1 Boundary Conditions and Determination of the Coefficients, C1,C2,C3 andC4
		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,C4andC6
		1.5.3 Calculation of Flux Densities and Current Density in the Rotor Sheet
		1.5.4 Calculation of Forces
	1.6 Conclusions
	References
2 Mathematical Modelling of Electromagnetic Forces Due to Finite Width Effects of a Single-Sided Linear Induction Motor
	2.1 Introduction
	2.2 Problem Formulation
		2.2.1 Formulation for the Field Due to Stator Current Sheet
	2.3 Solution for Stream Function uy
	2.4 Steps for Algorithm for Numerical Solution for uy, By2
	2.5 Calculation of Forces
	2.6 Results and Discussion
	2.7 Conclusions
3 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 uy, By2, Bx2andBz2
	3.4 Calculation of Forces
	3.5 Results and Discussion
	3.6 Conclusions
4 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
	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
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
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
	References
7 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
8 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
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
	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
Appendix A
A.1 Rotor Resistivity Correction Factor (Kr)
Appendix B
B.1 Evaluation of the Integral int - inftyinfty e - jkp ( p2 + b2  )12  dp
Appendix C Brief Description of the Fabrication of a Single-Sided Linear Induction Motor
C.1 Description of the Experimental Model
C.1.1 Stator Structure and Stator Winding
C.1.2 Stabilizing Coils for Lateral Stabilization
C.1.3 Rotor Structure