**PREFACE **

The main goal of this book is to provide the mathematical background needed for the study of linear circuits and systems in engineering. It is more rigorous than the material found in most circuit theory books, and it is appropriate for upper-division undergraduate students and first-year graduate students. The book has the following features:

• A comparison of linear circuits and mechanical systems that are modeled by similar ordinary differential equations. This provides a greater understanding of the behavior of different types of linear time-invariant circuits and systems.

• Numerous tables and figures summarize several mathematical techniques and provide example results. Although the focus of the book is on the equations used in engineering models, it includes over 250 figures and plots generated using MATLAB that reinforce the material and illustrate subtle points.

• Several appendices provide background material on set theory, series expansions, various identities, and the Lambert W-function. An extensive summary of important functions and their transforms encountered in the study of linear systems is included in Appendix A.

• A brief introduction to the theory of generalized functions, which are defined by their properties under an integral. This theory is connected to the Laplace and Fourier transforms covered later, which are specific types of integral transforms of time-domain functions.

After the overview in Chapter 1, which includes a brief review of functions and calculus, the book is divided into two parts:

• Part I: Circuits and Mechanical Systems; Linear Equations and Matrices; Complex Numbers and Functions (Chapters 2–4).

• Part II: Signals, Generalized Functions, and Fourier Series; Differential Equation Models for Linear Systems; Laplace Transforms and Linear Systems; Fourier Transforms and Frequency Responses (Chapters 5–8).

Chapter 2 describes circuits consisting of resistors (R), capacitors (C), and inductors (L), as well as Kirchoff’s circuit laws and mesh and nodal analysis techniques. There is a brief study of nonlinear diode circuits and then a discussion of some mechanical systems that have the same time-domain properties as RL, RC, and RLC circuits. Linear algebra and systems of linear equations are covered in Chapter 3, along with the matrix determinant, matrix subspaces, LU and LDU decompositions, and eigendecompositions. Equations that model the voltages and currents in a resistive circuit are represented using matrices, and the solutions are derived using either Cramer’s rule or Gaussian elimination.

Chapter 4 contains a thorough discussion of complex numbers, with material not covered in most books on linear circuits and systems. It includes matrix representations of complex quantities, exponential rotations on the complex plane, the constant angular velocity of time-varying complex functions, and a brief discussion of quaternions. Chapter 5 gives definitions of several signals that describe the dynamic behavior of linear circuits and systems, including ordinary functions such as the exponential function and singular functions like the Dirac delta function. A brief introduction to the theory of generalized functions is provided, which illustrates several of their properties and in particular how their derivatives are found.

This chapter also includes Fourier series representations of periodic signals and a view of their coefficients as cross-correlations between the original signal and sinusoidal signals with increasing frequency. First- and second-order ordinary differential equations used to model RL, RC, and RLC circuits are then covered in Chapter 6. The solutions are derived entirely in the time domain, and it is demonstrated that second-order linear systems can have three types of responses depending on their parameter values. Phasor notation and impedance for circuits with sinusoidal source signals are also discussed.