Mathematical Foundations for Linear Circuits and Systems in Engineering (eBook)

John J. Shynk (Autor)

eBook Download: EPUB

2016
John Wiley & Sons (Verlag)
978-1-119-07339-0 (ISBN)

Infos

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.

Textauszug

CHAPTER 1
OVERVIEW AND BACKGROUND

1.1 INTRODUCTION

In this book, we develop and examine several mathematical models consisting of one or more equations that are used in engineering to represent various physical systems. Usually, the goal is to solve these equations for the unknown dependent variables, and if that is not possible, the equations can be used to simulate the behavior of a system using computer software such as MATLAB.1 In most engineering courses, the equations are usually linear or can be linearized as an approximation, but sometimes they are nonlinear and may be difficult to solve. From such models, it is possible to design and analyze components of a proposed system in order to achieve required performance specifications before developing a prototype and actually implementing the physical system.

Definition: System

A system is a collection of interacting elements or devices that together result in a more complicated structure than the individual components alone, for the purpose of generating a specific type of signal or realizing a particular process.

The term system, as used in this book, also describes several interrelated equations called a system of equations, which are usually linear and can be represented by a matrix equation. The distinction between a physical system and a system of linear equations will be evident from the specific application.

Definition: Mathematical Model

A mathematical model is an equation or set of equations used to represent a physical system, from which it is possible to predict the properties of the system andits output response to an input, given known parameters, certain variables, and initial conditions.

Generally, we are interested in the dynamic behavior of a system over time as it responds to one or more time-varying input signals. A block diagram of a system with single input and single output (single-input single-output (SISO)) is shown in Figure 1.1(a), where is continuous time. The time variable can be defined for the entire real line : , but often we assume nonnegative : . In this scenario, a mathematical model provides the means to observe how varies with over , assuming known initial conditions (usually at ), so that we can predict the future behavior of the system. For the electric circuits described in Chapter 2, the inputs and outputs are currents through or voltages across the circuit components. For convenience, Table 1.1 summarizes the notation for different sets of numbers used in this book (though quaternions are only briefly discussed in Chapter 4).

Figure 1.1 Systems with a single input and a single output (SISO). (a) General system with input and output . (b) Linear system with sinusoidal input and output.

Table 1.1 Symbols for Sets of Numbers

Symbol

Domain

Set

Real numbers

Nonnegative real numbers

Integers

Nonnegative integers

Natural numbers

with and

Rational numbers

with and

Imaginary numbers

with and

Complex numbers

Quaternions

with and

Figure 1.1(b) shows a linear SISO system with sinusoidal input where is ordinary frequency in hertz (Hz). As discussed in Chapter 7, a sinusoidal signal is an eigenfunction of a linear system, which means that the output is also sinusoidal with the same frequency . For such a signal, the output differs from the input by having a different magnitude, which is in the figure, and possibly a phase shift . This is an important characteristic of linear systems that allows us to investigate them in the so-called frequency domain, which provides information about their properties beyond those observed in the time domain.

In order to more easily solve for the unknown variables of a mathematical model, the techniques usually require knowledge of matrices and complex numbers. The matrices covered in Chapter 3 are useful for describing a system of linear equations with constant coefficients. Chapter 4 provides the motivation for complex numbers and summarizes many of their properties. Chapter 5 introduces several different waveforms that are used to represent the signals of a system: inputs, outputs, as well as internal waveforms. These include the well-known sinusoidal and exponential signals, as well as the unit step function and the Dirac delta function. The theory of generalized functions and some of their properties are briefly introduced. Systems represented by linear ordinary differential equations (ODEs) are then covered in Chapter 6, where they are solved using conventional time-domain techniques. The reader will find that such techniques are straightforward for first- and second-order ODEs, especially for the linear circuits covered in this book, but are more difficult to use for higher order systems.

Chapter 7 describes methods based on the Laplace transform that are widely used in engineering to solve linear ODEs with constant coefficients. The Laplace transform converts an ODE into an algebraic equation that is more easily solved using matrix techniques. Finally, Chapter 8 introduces methods for analyzing a system in the frequency domain, which provides a characterization of its frequency response to different input waveforms. In particular, we can view linear circuits and systems as filters that modify the frequency content of their input signals.

We focus on continuous-time systems, which means are defined with support or where the functions are nonzero. Discrete-time systems and signals are defined for a countable set of time instants such as , , or . Different but related techniques are used to examine discrete-time systems, though these are beyond the scope of this book.

1.2 MATHEMATICAL MODELS

Consider again the system in Figure 1.1(a) and assume that we have access only to its input and output as implied by the block diagram. There is no direct information about the internal structure of the system, and the only way we can learn about its properties is by providing input signals and observing the output signals. Such an unknown system is called a “black box” (because we cannot see inside), and the procedure of examining its input/output characteristics is a type of reverse engineering. We mention this because the mathematical models used to represent physical devices and systems are typically verified and even derived from experiments with various types of input/output signals. Such an approach yields the transfer characteristic of the system, and for linear and time-invariant (LTI) systems, we can write a specific transfer function as described in Chapter 7.

Example 1.1

Suppose input of an unknown system is varied over and we observe the output shown in Figure 1.2. This characteristic does not change with time, and so we have suppressed the time argument for the input and output. The plot of is flat for three intervals: , , and , and it is linearly increasing for two intervals: and . For this piecewise linear function, the equation for each interval has the form where is the slope and is the ordinate, which is the point where the line crosses the -axis if it were extended to . For the first linearly increasing region, the slope is obviously . When , the extended line crosses the -axis at , which gives . Similarly, for the second linearly increasing region, and . The remaining three regions have zero slope but different ordinates (these equations are of the form ), and so the overall transfer characteristic for this system is

1.1

The values of match at the boundaries for each interval of as shown in the figure. The mapping in (1.1) is a mathematical model for a particular system that can be used to study its behavior even if it is included as part of a larger system. Note that this input/output characteristic does not provide any direct information about the individual components or the internal dynamics of the system. When the input is a function of time, the output is also time varying. For example, suppose that as illustrated in Figure 1.3 for one period of the sine function with frequency Hz. The output is computed using (1.1) at each time instant on the closed interval in seconds (s). Observe that is truncated relative to the input waveform due to this particular input/output mapping. Similar results for can be derived for any input function by using the model in (1.1).

Figure 1.2 Input/output characteristic for the nonlinear system in Example 1.1.

Figure 1.3 Output for the transfer characteristic in (1.1) in Example 1.1 with input for .

The output is not sinusoidal because...

Produkt-Details

• Erscheint lt. Verlag: 22.2.2016
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Technik ► Elektrotechnik / Energietechnik
• Schlagworte: Applied mathematics • Applied Mathematics in Engineering • Circuit Laws • Complex variables • Electrical & Electronics Engineering • Elektrotechnik u. Elektronik • Fourier Transforms • Frequency Response • Laplace Transforms • linear systems • Mathematics • Mathematik • Mathematik in den Ingenieurwissenschaften • Mechanical Systems • Numerical Methods & Algorithms • Numerische Methoden u. Algorithmen • system of linear equations
• ISBN-10: 1-119-07339-1 / 1119073391
• ISBN-13: 978-1-119-07339-0 / 9781119073390