Fundamental Numerical Methods for Electrical Engineering

by: Stanislaw Rosloniec

Springer-Verlag, 2008

ISBN: 9783540795193 , 284 Pages

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Fundamental Numerical Methods for Electrical Engineering


 

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