Mathematical Foundations for Linear Circuits and Systems in Engineering

Extensive coverage of mathematical techniques used in engineering with an emphasis on applications in linear circuits and systems Mathematical Foundations for Linear Circuits and Systems in Engineering provides an integrated approach to learning the necessary mathematics specifically used to describe and analyze linear circuits and systems. The chapters develop and examine several mathematical models consisting of one or more equations used in engineering to represent various physical systems. The techniques are discussed in–depth so that the reader has a better understanding of how and why these methods work. Specific topics covered include complex variables, linear equations and matrices, various types of signals, solutions of differential equations, convolution, filter designs, and the widely used Laplace and Fourier transforms. The book also presents a discussion of some mechanical systems that mathematically exhibit the same dynamic properties as electrical circuits. Extensive summaries of important functions and their transforms, set theory, series expansions, various identities, and the Lambert W–function are provided in the appendices. The book has the following features: Compares linear circuits and mechanical systems that are modeled by similar ordinary differential equations, in order to provide an intuitive understanding of different types of linear time–invariant systems. Introduces the theory of generalized functions, which are defined by their behavior under an integral, and describes several properties including derivatives and their Laplace and Fourier transforms. Contains numerous tables and figures that summarize useful mathematical expressions and example results for specific circuits and systems, which reinforce the material and illustrate subtle points. Provides access to a companion website that includes a solutions manual with MATLAB code for the end–of–chapter problems. Mathematical Foundations for Linear Circuits and Systems in Engineering is written for upper undergraduate and first–year graduate students in the fields of electrical and mechanical engineering. This book is also a reference for electrical, mechanical, and computer engineers as well as applied mathematicians. John J. Shynk, PhD, is Professor of Electrical and Computer Engineering at the University of California, Santa Barbara. He was a Member of Technical Staff at Bell Laboratories, and received degrees in systems engineering, electrical engineering, and statistics from Boston University and Stanford University. INDICE: Preface .Notation and Bibliography .1 Overview and Background .1.1 Introduction .1.2 Mathematical Models .1.3 Frequency Content .1.4 Functions and Properties .1.5 Derivatives and Integrals .1.6 Sine, Cosine, and .1.7 Napier s Constant e and Logarithms .Part I Circuits, Matrices, and Complex Numbers .2 Circuits and Mechanical Systems .2.1 Introduction .2.2 Voltage, Current, and Power .2.3 Circuit Elements .2.4 Basic Circuit Laws .2.4.1 Mesh–Current and Node–Voltage Analysis .2.4.2 Equivalent Resistive Circuits .2.4.3 RC and RL Circuits .2.4.4 Series RLC Circuit .2.4.5 Diode Circuits .2.5 Mechanical Systems .2.5.1 Simple Pendulum .2.5.2 Mass on a Spring .2.5.3 Electrical and Mechanical Analogs .3 Linear Equations and Matrices .3.1 Introduction .3.2 Vector Spaces .3.3 System of Linear Equations .3.4 Matrix Properties and Special Matrices .3.5 Determinant .3.6 Matrix Subspaces .3.7 Gaussian Elimination .3.7.1 LU and LDU Decompositions .3.7.2 Basis Vectors .3.7.3 General Solution of Ay = x .3.8 Eigendecomposition .3.9 MATLAB Functions .4 Complex Numbers and Functions .4.1 Introduction .4.2 Imaginary Numbers .4.3 Complex Numbers .4.4 Two Coordinates .4.5 Polar Coordinates .4.6 Euler s Formula .4.7 Matrix Representation .4.8 Complex Exponential Rotation .4.9 Constant Angular Velocity .4.10 Quaternions .Part II Signals, Systems, and Transforms .5 Signals, Generalized Functions, and Fourier Series .5.1 Introduction .5.2 Energy and Power Signals .5.3 Step and Ramp Functions .5.4 Rectangle and Triangle Functions .5.5 Exponential Function .5.6 Sinusoidal Functions .5.7 Dirac Delta Function .5.8 Generalized Functions .5.9 Unit Doublet .5.10 Complex Functions and Singularities .5.11 Cauchy Principal Value .5.12 Even and Odd Functions .5.13 Correlation Functions .5.14 Fourier Series .5.15 Phasor Representation .5.16 Phasors and Linear Circuits .6 Differential Equation Models for Linear Systems .6.1 Introduction .6.2 Differential Equations .6.3 General Forms of the Solution .6.4 First–Order Linear ODE .6.4.1 Homogeneous Solution .6.4.2 Nonhomogeneous Solution .6.4.3 Step Response .6.4.4 Exponential Input .6.4.5 Sinusoidal Input .6.4.6 Impulse Response .6.5 Second–Order Linear ODE .6.5.1 Homogeneous Solution .6.5.2 Damping Ratio .6.5.3 Initial Conditions .6.5.4 Nonhomogeneous Solution .6.6 Second–Order ODE Responses .6.6.1 Step Response .6.6.2 Step Response (Alternative Method) .6.6.3 Impulse Response .6.7 Convolution .6.8 System of ODEs .7 Laplace Transforms and Linear Systems .7.1 Introduction .7.2 Solving ODEs Using Phasors .7.3 Eigenfunctions .7.4 Laplace Transform .7.5 Laplace Transforms and Generalized Functions .7.6 Laplace Transform Properties .7.7 Initial– and Final–Value Theorems .7.8 Poles and Zeros .7.9 Laplace Transform Pairs .7.9.1 Constant Function .7.9.2 Rectangle Function .7.9.3 Triangle Function .7.9.4 Ramped Exponential Function .7.9.5 Sinusoidal Functions .7.10 Transforms and Polynomials .7.11 Solving Linear ODEs .7.12 Impulse Response and Transfer Function .7.13 Partial Fraction Expansion .7.13.1 Distinct Real Poles .7.13.2 Distinct Complex Poles .7.13.3 Repeated Real Poles .7.13.4 Repeated Complex Poles .7.14 Laplace Transforms and Linear Circuits .8 Fourier Transforms and Frequency Response .8.1 Introduction .8.2 Fourier Transform .8.3 Magnitude and Phase .8.4 Fourier Transforms and Generalized Functions .8.5 Fourier Transform Properties .8.6 Amplitude Modulation .8.7 Frequency Response .8.7.1 First–Order Lowpass Filter .8.7.2 First–Order Highpass Filter .8.7.3 Second–Order Bandpass Filter .8.7.4 Second–Order Bandreject Filter .8.8 Frequency Response of Second–Order Filters .8.9 Frequency Response of Series RLC Circuit .8.10 Butterworth Filters .8.10.1 Lowpass Filter .8.10.2 Highpass Filter .8.10.3 Bandpass Filter .8.10.4 Bandreject Filter .Appendices .Introduction .A Extended Summaries of Functions and Transforms .A.1 Functions and Notation .A.2 Laplace Transform .A.3 Fourier Transform .A.4 Magnitude and Phase .A.5 Impulsive Functions .A.5.1 Dirac Delta Function (Shifted) .A.5.2 Unit Doublet (Shifted) .A.6 Piecewise Linear Functions .A.6.1 Unit Step Function .A.6.2 Signum Function .A.6.3 Constant Function (Two–Sided) .A.6.4 Ramp Function .A.6.5 Absolute Value Function (Two–Sided Ramp) .A.6.6 Rectangle Function .A.6.7 Triangle Function .A.7 Exponential Functions .A.7.1 Exponential Function (Right–Sided) .A.7.2 Exponential Function (Ramped) .A.7.3 Exponential Function (Two–Sided) .A.7.4 Gaussian Function .A.8 Sinusoidal Functions .A.8.1 Cosine Function (Two–Sided) .A.8.2 Cosine Function (Right–Sided) .A.8.3 Cosine Function (Exponentially Weighted) .A.8.4 Cosine Function (Exponentially Weighted and Ramped) .A.8.5 Sine Function (Two–Sided) .A.8.6 Sine Function (Right–Sided) .A.8.7 Sine Function (Exponentially Weighted) .A.8.8 Sine Function (Exponentially Weighted and Ramped) .B Inverse Laplace Transforms .B.1 Improper Rational Function .B.2 Unbounded System .B.3 Double Integrator and Feedback .C Identities, Derivatives, and Integrals .C.1 Trigonometric Identities .C.2 Summations .C.3 Miscellaneous .C.4 Completing the Square .C.5 Quadratic and Cubic Formulas .C.6 Derivatives .C.7 Indefinite Integrals .C.8 Definite Integrals .D Set Theory 480 .D.1 Sets and Subsets .D.2 Set Operations .E Series Expansions .E.1 Taylor Series .E.2 Maclaurin Series .E.3 Laurent Series .F Lambert W–Function .F.1 Lambert W–Function .F.2 Nonlinear Diode Circuit .F.3 Nonlinear System of Equations .Glossary .Bibliography . 

• ISBN: 978-1-119-07347-5
• Editorial: Wiley–Blackwell
• Encuadernacion: Rústica
• Páginas: 656
• Fecha Publicación: 19/02/2016
• Nº Volúmenes: 1
• Idioma: Inglés