# Fundamental Numerical Methods for Electrical Engineering

Springer Science & Business Media, Jul 17, 2008 - Technology & Engineering - 284 pages
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.

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### Contents

 Methods for Numerical Solution of Linear Equations 1 112 The GaussJordan Elimination Method 9 113 The LU Matrix Decomposition Method 11 114 The Method of Inverse Matrix 14 12 Indirect or Iterative Methods 17 122 Jacobi and GaussSeidel Methods 18 13 Examples of Applications in Electrical Engineering 23 References 27
 532 The Simpson Cubature Formula 148 54 An Example of Applications 151 References 154 Numerical Differentiation of One and Two Variable Functions 155 61 Approximating the Derivatives of One Variable Functions 157 62 Calculating the Derivatives of One Variable Function by Differentiation of the Corresponding Interpolating Polynomial 163 63 Formulas for Numerical Differentiation of Two Variable Functions 168 64 An Example of the TwoDimensional Optimization Problem and its Solution by Using the Gradient Minimization Technique 172

 Methods for Numerical Solving the Single Nonlinear Equations 29 21 Determination of the Complex Roots of Polynomial Equations by Using the Lins and Bairstows Methods 30 212 Bairstows Method 32 213 Laguerre Method 35 22 Iterative Methods Used for Solving Transcendental Equations 36 221 Bisection Method of Bolzano 37 222 The Secant Method 38 223 Method of Tangents NewtonRaphson 40 23 Optimization Methods 42 24 Examples of Applications 44 References 47 Methods for Numerical Solution of Nonlinear Equations 49 32 The Iterative Parameter Perturbation Procedure 51 33 The Newton Iterative Method 52 34 The Equivalent Optimization Strategies 56 35 Examples of Applications in the Microwave Technique 58 References 68 Methods for the Interpolation and Approximation of One Variable Function 69 41 Fundamental Interpolation Methods 72 412 The Lagrange Interpolating Polynomial 73 413 The Aitken Interpolation Method 76 414 The NewtonGregory Interpolating Polynomial 77 415 Interpolation by Cubic Spline Functions 82 416 Interpolation by a Linear Combination of Chebyshev Polynomials of the First Kind 86 42 Fundamental Approximation Methods for One Variable Functions 89 422 The Maximally Flat Butterworth Approximation 94 423 Approximation Curve Fitting by the Method of Least Squares 97 424 Approximation of Periodical Functions by Fourier Series 102 43 Examples of the Application of Chebyshev Polynomials in Synthesis of Radiation Patterns of the InPhase Linear Array Antenna 111 References 120 Methods for Numerical Integration of One and Two Variable Functions 121 51 Integration of Deﬁnite Integrals by Expanding the Integrand Function in Finite Series of Analytically Integrable Functions 123 52 Fundamental Methods for Numerical Integration of One Variable Functions 125 522 The Romberg Integration Rule 130 523 The Simpson Method of Integration 132 524 The NewtonCotes Method of Integration 136 525 The Cubic Spline Function Quadrature 138 526 The Gauss and Chebyshev Quadratures 140 53 Methods for Numerical Integration of Two Variable Functions 147
 References 177 Methods for Numerical Integration of Ordinary Differential Equations 178 72 The OneStep Methods 180 722 The Heun Method 182 723 The RungeKutta Method RK 4 184 724 The RungeKuttaFehlberg Method RKF 45 186 73 The Multistep PredictorCorrector Methods 189 731 The AdamsBashforthMoulthon Method 193 732 The MilneSimpson Method 194 733 The Hamming Method 197 74 Examples of Using the RK 4 Method for Integration of Differential Equations Formulated for Some Electrical Rectiﬁer Devices 199 742 The FullWave Rectiﬁer Integrated with the ThreeElement LowPass Filter 204 743 The Quadruple Symmetrical Voltage Multiplier 208 75 An Example of Solution of Riccati Equation Formulated for a Nonhomogenous Transmission Line Segment 215 76 An Example of Application of the Finite Difference Method for Solving the Linear Boundary Value Problem 219 References 221 The Finite Difference Method Adopted for Solving Laplace Boundary Value Problems 223 81 The Interior and External Laplace Boundary Value Problems 226 82 The Algorithm for Numerical Solving of TwoDimensional Laplace Boundary Problems by Using the Finite Difference Method 228 821 The Liebmann Computational Procedure 231 822 The Successive OverRelaxation Method SOR 238 83 Difference Formulas for Numerical Calculation of a Normal Component of an Electric Field Vector at Good Conducting Planes 242 Impedance and Attenuation Coefﬁcient for Some TEM Transmission Lines 245 841 The Shielded Triplate Stripline 246 842 The Square Coaxial Line 249 843 The Triplate Stripline 251 844 The Shielded Inverted Microstrip Line 253 845 The Shielded Slab Line 258 846 Shielded Edge Coupled Triplate Striplines 263 References 268 Equation of a Plane in ThreeDimensional Space 269 The Inverse of the Given Nonsingular Square Matrix 271 The Fast Elimination Method 273 The Doolittle Formulas Making Possible Presentation of a Nonsingular Square Matrix in the form of the Product of Two Triangular Matrices 275 Difference Formula for Calculation of the Electric Potential at Points Lying on the Border Between two Looseless Dielectric Media Without Electrical... 277 Complete Elliptic Integrals of the First Kind 279 Subject Index 281 Copyright