Fundamental Numerical Methods for Electrical Engineering (eBook)

Stanislaw Rosloniec (Autor)

eBook Download: PDF

2008 | 2008
XIV, 284 Seiten
Springer Berlin (Verlag)
978-3-540-79519-3 (ISBN)

Infos

Stormy development of electronic computation techniques (computer systems and software), observed during the last decades, has made possible automation of data processing in many important human activity areas, such as science, technology, economics and labor organization. In a broadly understood technology area, this developmentledtoseparationofspecializedformsofusingcomputersforthedesign and manufacturing processes, that is: - computer-aided design (CAD) - computer-aided manufacture (CAM) In order to show the role of computer in the rst of the two applications m- tioned above, let us consider basic stages of the design process for a standard piece of electronic system, or equipment: - formulation of requirements concerning user properties (characteristics, para- ters) of the designed equipment, - elaboration of the initial, possibly general electric structure, - determination of mathematical model of the system on the basis of the adopted electric structure, - determination of basic responses (frequency- or time-domain) of the system, on the base of previously established mathematical model, - repeated modi cation of the adopted diagram (changing its structure or element values) in case, when it does not satisfy the adopted requirements, - preparation of design and technological documentation, - manufacturing of model (prototype) series, according to the prepared docum- tation, - testing the prototype under the aspect of its electric properties, mechanical du- bility and sensitivity to environment conditions, - modi cation of prototype documentation, if necessary, and handing over the documentation to series production. The most important stages of the process under discussion are illustrated in Fig. I. 1. xi xii Introduction Fig. I.

Inhalt

Contents 5
About the Author 9
Introduction 11
Methods for Numerical Solution of Linear Equations 14
1.1 Direct Methods 18
1.1.1 The Gauss Elimination Method 18
1.1.2 The Gauss–Jordan Elimination Method 22
1.1.3 The LU Matrix Decomposition Method 24
1.1.4 The Method of Inverse Matrix 27
1.2 Indirect or Iterative Methods 30
1.2.1 The Direct Iteration Method 30
1.2.2 Jacobi and Gauss–Seidel Methods 31
1.3 Examples of Applications in Electrical Engineering 36
References 40
Methods for Numerical Solving the Single Nonlinear Equations 42
2.1 Determination of the Complex Roots of Polynomial Equations by Using the Lin’s and Bairstow’s Methods 43
2.1.1 Lin’s Method 43
2.1.2 Bairstow’s Method 45
2.1.3 Laguerre Method 48
2.2 Iterative Methods Used for Solving Transcendental Equations 49
2.2.1 Bisection Method of Bolzano 50
2.2.2 The Secant Method 51
2.2.3 Method of Tangents (Newton–Raphson) 53
2.3 Optimization Methods 55
2.4 Examples of Applications 57
References 60
Methods for Numerical Solution of Nonlinear Equations 62
3.1 The Method of Direct Iterations 62
3.2 The Iterative Parameter Perturbation Procedure 64
3.3 The Newton Iterative Method 65
3.4 The Equivalent Optimization Strategies 69
3.5 Examples of Applications in the Microwave Technique 71
References 81
Methods for the Interpolation and Approximation of One Variable Function 82
4.1 Fundamental Interpolation Methods 85
4.1.1 The Piecewise Linear Interpolation 85
4.1.2 The Lagrange Interpolating Polynomial 86
4.1.3 The Aitken Interpolation Method 89
4.1.4 The Newton–Gregory Interpolating Polynomial 90
4.1.5 Interpolation by Cubic Spline Functions 95
4.1.6 Interpolation by a Linear Combination of Chebyshev Polynomials of the First Kind 99
4.2 Fundamental Approximation Methods for One Variable Functions 102
4.2.1 The Equal Ripple (Chebyshev) Approximation 102
4.2.2 The Maximally Flat (Butterworth) Approximation 107
4.2.3 Approximation (Curve Fitting) by the Method of Least Squares 110
4.2.4 Approximation of Periodical Functions by Fourier Series 115
4.3 Examples of the Application of Chebyshev Polynomials in Synthesis of Radiation Patterns of the In- Phase Linear Array Antenna 124
References 133
Methods for Numerical Integration of One and Two Variable Functions 134
5.1 Integration of Definite Integrals by Expanding the Integrand Function in Finite Series of Analytically Integrable Functions 136
5.2 Fundamental Methods for Numerical Integration of One Variable Functions 138
5.2.1 Rectangular and Trapezoidal Methods of Integration 138
5.2.2 The Romberg Integration Rule 143
5.2.3 The Simpson Method of Integration 145
5.2.4 The Newton–Cotes Method of Integration 149
5.2.5 The Cubic Spline Function Quadrature 151
5.2.6 The Gauss and Chebyshev Quadratures 153
5.3 Methods for Numerical Integration of Two Variable Functions 160
5.3.1 The Method of Small (Elementary) Cells 160
5.3.2 The Simpson Cubature Formula 161
5.4 An Example of Applications 164
References 167
Numerical Differentiation of One and Two Variable Functions 168
6.1 Approximating the Derivatives of One Variable Functions 170
6.2 Calculating the Derivatives of One Variable Function by Differentiation of the Corresponding Interpolating Polynomial 176
6.2.1 Differentiation of the Newton–Gregory Polynomial and Cubic Spline Functions 176
6.3 Formulas for Numerical Differentiation of Two Variable Functions 181
6.4 An Example of the Two-Dimensional Optimization Problem and its Solution by Using the Gradient Minimization Technique 185
References 190
Methods for Numerical Integration of Ordinary Differential Equations 192
7.1 The Initial Value Problem and Related Solution Methods 192
7.2 The One-Step Methods 193
7.2.1 The Euler Method and its Modified Version 193
7.2.2 The Heun Method 195
7.2.3 The Runge–Kutta Method (RK 4) 197
7.2.4 The Runge–Kutta–Fehlberg Method (RKF 45) 199
7.3 The Multi-step Predictor –Corrector Methods 202
7.3.1 The Adams–Bashforth–Moulthon Method 206
7.3.2 The Milne–Simpson Method 207
7.3.3 The Hamming Method 210
7.4 Examples of Using the RK 4 Method for Integration of Differential Equations Formulated for Some Electrical Rectifier Devices 212
7.4.1 The Unsymmetrical Voltage Doubler 212
7.4.2 The Full-Wave Rectifier Integrated with the Three-Element Low- Pass Filter 217
7.4.3 The Quadruple Symmetrical Voltage Multiplier 221
7.5 An Example of Solution of Riccati Equation Formulated for a Nonhomogenous Transmission Line Segment 228
7.6 An Example of Application of the Finite Difference Method for Solving the Linear Boundary Value Problem 232
References 234
The Finite Difference Method Adopted for Solving Laplace Boundary Value Problems 236
8.1 The Interior and External Laplace Boundary Value Problems 239
8.2 The Algorithm for Numerical Solving of Two-Dimensional Laplace Boundary Problems by Using the Finite Difference Method 241
8.2.1 The Liebmann Computational Procedure 244
8.2.2 The Successive Over-Relaxation Method (SOR) 251
8.3 Difference Formulas for Numerical Calculation of a Normal Component of an Electric Field Vector at Good Conducting Planes 255
8.4 Examples of Computation of the Characteristic Impedance and Attenuation Coefficient for Some TEM Transmission Lines 258
8.4.1 The Shielded Triplate Stripline 259
8.4.2 The Square Coaxial Line 262
8.4.3 The Triplate Stripline 264
8.4.4 The Shielded Inverted Microstrip Line 266
8.4.5 The Shielded Slab Line 271
8.4.6 Shielded Edge Coupled Triplate Striplines 276
References 281
A Equation of a Plane in Three-Dimensional Space 282
B The Inverse of the Given Nonsingular Square Matrix 284
C The Fast Elimination Method 286
D The Doolittle Formulas Making Possible Presentation of a Nonsingular Square Matrix in the form of the Product of Two Triangular Matrices 288
E Difference Formula for Calculation of the Electric Potential at Points Lying on the Border Between two Looseless Dielectric Media Without Electrical Charges 290
F Complete Elliptic Integrals of the First Kind 292
Subject Index 294

Produkt-Details

• Erscheint lt. Verlag: 17.7.2008
• Reihe/Serie: Lecture Notes in Electrical Engineering
• Zusatzinfo: XIV, 284 p. 301 illus.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Informatik ► Theorie / Studium ► Künstliche Intelligenz / Robotik
• Themenwelt: Mathematik / Informatik ► Mathematik
• Themenwelt: Technik ► Elektrotechnik / Energietechnik
• Schlagworte: Automation • Computer • Computer-Aided Design (CAD) • Computer-Aided Manufacturing (CAM) • Development • differential equation • Electrical Engineering • Manufacturing • Model • Numerical Methods • ordinary differential equation • Production • Software • Standard
• ISBN-10: 3-540-79519-7 / 3540795197
• ISBN-13: 978-3-540-79519-3 / 9783540795193