Essentials of Advanced Circuit Analysis E-Book

Djafar K. Mynbaev

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 816

Veröffentlichungsjahr: 2024

Beschreibung

ESSENTIALS OF ADVANCED CIRCUIT ANALYSIS

Comprehensive textbook answering questions regarding the Advanced Circuit Analysis subject, including its theory, experiment, and role in modern and future technology

Essentials of Advanced Circuit Analysis focuses on fundamentals with the balance of a systems theoretical approach and current technological issues. The book aims to achieve harmony between simplicity, engineering practicality, and perceptivity in the material presentation.

Each chapter presents its material on various levels of technological and mathematical difficulty, broadening the potential readership and making the book suitable for both engineering and engineering technology curricula.

Essentials of Advanced Circuit Analysis is an instrument that will introduce our readers to real-life engineering problems—why they crop up and how they are solved. The text explains the need for a specific task, shows the possible approaches to meeting the challenge, discusses the proper method to pursue, finds the solution to the problem, and reviews the solution's correctness, the options of its obtaining, and the limitations of the methods and the results.

Essentials of Advanced Circuit Analysis covers sample topics such as:

• Traditional circuit analysis's methods and techniques, concentrating on the advanced circuit analysis in the time domain and frequency domain
• Application of differential equations for finding circuits’ transient responses in the time domain, and classical solution (integration) of circuit’s differential equation, including the use of the convolution integral
• Laplace and Fourier transforms as the main modern methods of advanced circuit analysis in the frequency domain

Essentials of Advanced Circuit Analysis is an ideal textbook and can be assigned for electronics, signals and systems, control theory, and spectral analysis courses. It’s also valuable to industrial engineers who want to brush up on a specific advanced circuit analysis topic.

Leseprobe

Essentials of Advanced Circuit Analysis

A Systems Approach

Djafar K. Mynbaev

New York City College of Technologyof the City University of New York

This edition first published 2024

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Library of Congress Cataloging-in-Publication Data

Names: Mynbaev, Djafar K., author.

Title: Essentials of advanced circuit analysis : a systems approach / Djafar K. Mynbaev, New York City College of Technology of the City University of New York.

Description: Hoboken, NJ, USA : John Wiley & Sons, Inc., 2024. | Includes bibliographical references and index.

Identifiers: LCCN 2023024442 (print) | LCCN 2023024443 (ebook) | ISBN 9781119847229 (hardback) | ISBN 9781119847236 (ePDF) | ISBN 9781119847243 (epub)

Subjects: LCSH: Electric circuit analysis. | Electronic circuits. | Electrical engineering--Mathematics.

Classification: LCC TK454 .M96 2024 (print) | LCC TK454 (ebook) | DDC 621.3815--dc23/eng/20230718

LC record available at https://lccn.loc.gov/2023024442

LC ebook record available at https://lccn.loc.gov/2023024443

Cover Image: © John W Banagan/Getty Images

Cover Design: Wiley

Set in 9.5/12.5pt STIXTwoText by Integra Software Services Pvt. Ltd, Pondicherry, India

To Bronia

Brief Contents

Preface

Acknowledgments

About the Companion Website

Part 1 Background – Steady-State Analysis of Electrical Circuits

1 Components, Topologies, and Basic Laws of Electrical Circuits

2 Methods of DC Circuit Analysis

3 Basics of AC Steady-State Circuit Analysis

Part 2 Advanced Circuit Analysis in Time Domain

4 Advanced Circuit Analysis in Time Domain—I

5 Advanced Circuit Analysis in Time-Domain—II

6 Advanced Circuit Analysis in Time Domain–Convolution-Integral Technique

Part 3 Frequency-Domain Advanced Circuit Analysis

7 What and Why of the Laplace Transform

8 Laplace Transform Application to Advanced Circuit Analysis

9 Advanced Consideration of the Laplace Transform and its Application to Circuit Analysis

10 Fourier Transform in Advanced Circuit Analysis

Bibliography and References

Index

Preface

Rationale (Advanced Circuit Analysis)

A circuit analysis, a discipline of electrical engineering, finds the voltage across and current through all the circuit components provided that a circuit operates in a steady-state regime and the input signals are either dc or sinusoidal. An advanced circuit analysis does the same work but includes into consideration the transient process and any input signal, which complicates obtaining the answers drastically. A system’s approach, while performing the advanced circuit analysis, focuses on finding the input–output relationship in a circuit. Therefore, this book searches the complete (transient plus steady-state) output in response to the circuit’s excitation by an arbitrary input.1

There are two strategic objectives of this textbook: first, to focus on the fundamentals of advanced circuit analysis with the balance between a systems theoretical approach and the practical concerns in the current technological issues; second, to create a textbook that is not merely a source of information but is an instrument that will teach the readers why real-life engineering problems appear and what are the strategies and techniques for finding their solutions. Let’s discuss the issues outlined above.

Modern electrical engineering technology changes rapidly, whereas its theoretical foundation evolves slower. Thus, finding the right balance between theory and practice is the primary objective of academic courses. Our book offers such a balance by concentrating on the subject’s fundamentals and relating these fundamentals to professional responsibilities through text discussions and—mainly—examples. Such an approach is hardly innovative since many textbooks do this. What makes our book unique is that the examples are not merely the exercises in plugging given numbers into equations but the problems those students will meet in their professional careers. Many examples are woven into the text fabric; those offered in a formal problem–solution format are accompanied by thorough discussions pinpointing the solution’s advantages, drawbacks, limitations, and implications. Thus, our examples serve as essential teaching tools.

Most textbooks traditionally serve as sources of information by introducing physical laws, deriving equations, and explaining how devices and systems work. This function is still valuable, but in the Internet era, when all information is just a click away, its importance is diminishing. Our book, still providing necessary information, teaches the reader why real-life engineering problems surface and how they are solved. We explain the need for a specific task, show possible approaches to meet the challenge, discuss methods to pursue, and consider their possible implementation. In other words, we do not merely present ready-to-implement solutions but encourage our readers to participate in finding and applying them. The objective is to teach our readers the approaches to finding solutions, the skills that remain with professionals throughout their careers regardless of changes in technology.

An engineer’s head is not a warehouse whose shelves an educator must fill with laws, formulas, and instructions on how to do it. Today, all the needed factual information can be easily obtained online, so such an approach to education is a method from the past. What engineers can’t find online is the ability to analyze the situation, formulate the problem, find the optimal solution, and verify the solution’s validity. And they must do all these steps by considering not a single task at hand but the operation of an entire system whose part this task is.

Pedagogical Issues

Discussion of most topics includes three levels of difficulty: basic, introductory, and advanced. This method allows instructors to choose the proper tier for an individual student (personalization) and gives the student a chance to switch among the levels depending on their progress (adaptive learning). Such an approach also gives the instructors the latitude to cover all the material in a single semester if they work with advanced students or present it more leisurely over an entire year.

More often than not, students study new material without having a solid background, and therefore they need to compensate for this deficiency by memorizing facts, equations, and laws. Since we introduce each topic’s basics in the book’s Part 1, students can readily refresh their memory without additional sources.

The textbook shows how a problem arises from a real-life need, what approaches can be taken to tackle the problem, what reasoning and logical steps scientists and engineers take to solve it, and what they do to implement it. Historical notes and short biographies of the greatest scientists and engineers also show the students that the problems discussed in this text stemmed from real-life situations and required tremendous efforts by people who created the technology we enjoy today.

Industry surveys consistently show that one of the significant shortcomings of new college graduates is their inability to see a problem and ask questions. The newly minted professionals frequently take everything for granted and often believe that their responsibilities are to plug given numbers into given equations or follow the instructions when operating with actual circuitry. But, in solving a problem or designing a device or circuit, engineers must ask: What is the solution to the problem? Is this the best solution? To what limits does this approach (equation) work? What could be wrong? How does my solution affect other parts of a system to which my circuit or devices belong? Asking these questions, the engineers would consider all possible situations their device and circuit can meet. This ability is critical for an engineer or technologist, so nurturing this ability is one of our objectives. We typically start every new topic by asking questions about the need for its discussion. Furthermore, questions are included at the end of each subject. In short, we encourage the readers to learn what real-life problems are posed and how to solve them best.

Organization of Our Textbook and Instructional Concerns

The book consists of three parts that are subdivided into ten chapters. Each part covers a specific area—background, time-domain, and frequency-domain—of the advanced circuit analyses. The part comprises several chapters that deepen and broaden the topic, as the table of contents attests. This structure allows the students to start at the primary level and strive for higher and higher levels.

Since almost every chapter is self-sufficient, an instructor can choose the branches to suit an individual class’s needs. The chapters covering the basics of various topics can be merged to create a primary course. This course can be used at a sophomore level at any college, including a community college. Consequently, junior and senior courses can be devised from the introductory and advanced chapters.

The book utilizes several pedagogical devices, such as extensive discussions for all the examples, questions in the text that encourage students to think outside the box, chapter-opening notes, section and chapter summaries, and historical references. Sidebars and Appendixes present auxiliary but essential information. A reader can skip them without breaking the flow of the text mainstream; however, they enhance the book contents and deepen understanding of the material.

Homework problems are based on real-life questions that students will encounter in the workplace. These problems will also require students to comprehend an entire concept, not just solve an equation or understand how a specific circuit works. Such knowledge will help students develop a professional approach to solving practical problems. The assignments also include questions that require essay answers, which will help readers learn how to present their results in writing, a long-lost skill that is in high demand in the industry these days. Notably, the problems and questions not only test the student’s ability to plug numbers into memorized formulas but also gauge their knowledge of theory through its applications to real-world practice. Of course, we include design-oriented problems because we think the design-oriented approach must be the engineer’s hallmark. In addition, many problems in the Questions and Problems sections are, in essence, the assignments for mini-projects.

This text extensively employs MATLAB and Multisim. MATLAB is used to automate the calculations, solve the algebraic, matrix, and differential equations, and plot the graphs. But we have constantly reminded our readers that MATLAB is only a mathematical processing machine, and its outputs (results) are as good as its inputs (our manually derived formulas or calculations). In other words, MATLAB results cannot verify our answers. An independent tool must be exploited to validate the results, and Multisim is the such tool. Hence, Multisim simulations are widely utilized throughout the book. Not only do they verify the results of derivations and calculations, but—even more importantly—they lead to developing laboratory exercises.

The MATLAB and Multisim applications for the laboratory exercises have accompanied my Advanced Circuit Analysis course that I taught for many years, and I transferred my experience into the text, including its examples. As a result, each example can serve as a basis for a laboratory exercise because it contains all the necessary information for the lab. For instance, the examples provide instructions on performing transient circuit analysis with Multisim, and they contain detailed demonstrations of how to make measurements with MATLAB and Multisim graphs. Another example is the demonstration of how to configure the circuit’s initial conditions with Multisim in Sidebar 5S.1. We think that the ability to anticipate the expected results of an experiment is a vital engineering skill; appropriately, many examples contain prediction parts. All in all, to develop a lab manual, an instructor simply needs to use the example description. What’s more, even assignments to many problems can serve as laboratory exercises too.

We focus on the analysis of the passive circuits. They are still an invaluable part of contemporary electronics. Thanks to advances in research, development, and manufacturing, the passive components reduce their sizes to the micro and nano levels, significantly improve the accuracy and stability of their parameters, and increase their long-term steadiness and resistance to harsh ambient conditions. Thus, deep literacy in the operation of passive circuits and their components is a must feature of modern electrical and electronics engineers. Consider the following statement from the publication discussing the future in the transistor design: This approach “… also increases the transistor’s capacitance, thereby sapping some of its switching speed.2 ” A mere understanding of this statement, let alone the capability to provide its quantitative analysis, requires fundamental knowledge of the passive RC circuit operation.

I trust the book will be appealing to professionals who want to refresh their memory on the subject matter and take a new look at their everyday work.

By default, the book relies on extensive application of modern technology in a classroom, for the laboratory experiments, and for the personal use.

Djafar K. Mynbaev

New Jersey

April 2023

Notes

1

See

Section 4.1

for an in-depth discussion of this statement.

2

Marko Radosavljevic and Jack Kavalieros, “Taking Moore’s Law to New Heights,”

IEEE Spectrum

, December 2022, pp. 32–37.

Acknowledgments

This book could not exist without the help and support of many people. I am forever grateful for all the assistance I have received—this is as much your success as mine.

The first group that must be mentioned is that of my professional colleagues. The list starts with the President of New City College of Technology, Dr. Russel K. Hotzler, and his administration. Their gracious support extended well beyond the preparation of this book. Professional discussions and friendly conversations with my colleagues in the Department of Electrical and Telecommunications Engineering Technology inspired me to delve deeper into our vocation’s engineering and academic areas.

In addition to many people who directly or indirectly contributed to this book’s completion, several colleagues gave me a hand when I was working on specific topics or applications. Mr. Chi Jau Yuan, my long-term associate at our department, helped me with several issues in Multisim circuit simulations. Mr. Alex Ovrutsky, a friend and a computer guru for many years, resolved numerous problems that are inevitably encountered when working with computer applications. Dr. Jacob Sloujitel generously shared his knowledge in teaching matrices in his mathematics courses. Dr. Muhammad Ali Ummy taught the other Advanced Circuit Analysis class offered by our department for many years. Naturally, we discussed the teaching material, pedagogical problems, and their solutions. These stimulating conversations certainly affected this book, especially Chapters 7 and 8.

There are two more people whose help in my work cannot be overstated. Both were my students and grew to become high-level professionals capable of helping me with MATLAB applications. Dr. Vitaly Sukharenko, after graduation, became my research associate in the application of plasmonics in telecommunications. We published several scientific papers that made a helpful contribution to his doctoral thesis. Along the way, Vitaly helped me with MATLAB applications when I was writing my preceding and current books. Ms. Ina Tsikhanava, while studying at our department, participated in my research too. Her academic credentials won her a NASA internship. After graduation, she built a prosperous professional career. She wrote numerous MATLAB scripts for both my recently published and present book. I am happy to publicly express my deep appreciation to them both for many years of beneficial collaboration.

The above appreciative listing includes, of course, my students whose curiosity about the subject, desire for more in-depth learning, and general reaction to my teaching inspired me to write this book.

Mr. Brett Kurzman, Commissioning Editor at Wiley, has kept faith in me during many years of our collaboration, which included a long journey with my previous book, Essentials of Modern Communications, published in 2020, and slower than desired preparation of the current book. Others at Wiley also provided friendly support. To put it succinctly, my experience working with Wiley was nothing but pleasant.

Finally, this book is dedicated to my wife, Bronia (Bronislava). During the time spent writing this book (and the years spent writing the others) my family patiently endured not having my full attention and involvement in many events and activities. But no one experienced this more than Bronia, who had to deal with this on a day-to-day basis. Thus, this dedication is my way of asking forgiveness from Bronia and, through her, from the whole family. I’m all yours now, but you might miss the days when I had my head in my manuscript.

About the Companion Website

This book is accompanied by a companion website:

www.wiley.com/go/Mynbaev/AdvancedCircuitAnalysis

This website includes the answers to the questions and solutions to the exercises given in the text. It also contains the solutions to some of the most challenging problems. In addition, the qualified instructors can find here the solution manual.

Part 1 Background – Steady-State Analysis of Electrical Circuits

1 Components, Topologies, and Basic Laws of Electrical Circuits

1.1 Introduction

The textbook is written for those who are familiar with primary circuit analysis. Nonetheless, in Part 1, we provide a brief review of the basics of this topic to release the reader from the necessity to seek any additional sources for understanding the central part of the book. Since the transient processes in the electrical circuit are the main focus of this manuscript—this is what an advanced circuit analysis is all about—this review is done at the angle of the circuit transitions from one state to the other.

To facilitate reading the text that follows, Table 1.1 reminds the powers of 10 and their designations.

Table 1.1 Powers of 10 and their designations.

Name

Math notation

Number

SI symbol

SI prefix

Quintillion

1,000,000,000,000,000,000

Exa

Quadrillion

1,000,000,000,000,000

Peta

Trillion

1,000,000,000,000

Tera

Billion

1,000,000,000

Giga

Million

1,000,000

Mega

Thousand

1,000

Kilo

one

1

Thousandth

0.001

Milli

Millionth

0.000 001

Micro

Billionth

0.000 000 001

Nano

Trillionth

0.000 000 000 001

Pico

Quadrillionth

0.000 000 000 000 001

Femto

Quintillionth

0.000 000 000 000 000 001

Atto

1.2 Main Electrical Parameters: Current and Voltage; Power and Energy

Definitions and Explanations

1.2.1 Electrical Charge and Electrical Current

The fundamental entity of electricity is an electrical charge. It is a natural property of matter responsible for all electric phenomena. Everyone is familiar with such phenomena produced by the electrical charges as lightning or sparks we encounter when wearing wool clothing. The smallest known unit charge is carried by an electron, an elementary particle. The unit of electrical charge is the charge of one electron, which is equal to , where C stands for coulomb,1 the International System of Units, SI, electrical charge unit. We agree to consider this electrical charge negative. There are also positive electrical charges. The like charges repel one another; the charges of the opposite polarities attract each other. The Coulomb law defines the force at which the electrical charges interact as

(1.1)

where is the constant, are the signed magnitudes of the charges, and is the distance between charges. A negative force sign means that the charges are attracted, and the positive sign indicates the repulsive force. The total electrical charge in a given system (and generally, in the universe) is conserved. This statement is known as the law of conservation of electrical charge.

Electrical charge lies in the foundations of electrical current, electrical voltage, and electromagnetic field. We start with electrical current.

Exercise Two electric charges have and . They are placed at the distance of 2.54 cm. At what force do they attract each other? Answer: . (Is this large or small force?)

Mini sidebar—Coulomb barrier

We know from our school physics courses that charged particles of the same sign repulse one another, and this phenomenon is attributed to Coulomb. However, as almost every law of nature, this law has its limitations. If these similarly charged particles collide, having extremely high energy levels, they can overcome the repulsion law and come close enough to start a nuclear reaction. This phenomenon is fundamental for developing nuclear fusion, a potential source of clean energy. After all, it is a fusion reaction that powers the sun. Nonetheless, the dream of creating a controlled nuclear fusion has remained elusive despite the tremendous efforts of the world’s most advanced research and development institutions. However, most recently American scientists achieved the controlled fusion reaction, in which generated energy exceeds the energy spent on the reaction ignition! It was the major breakthrough in solving this fundamental problem, though the reaction lasted a fraction of a nanosecond (a billionth of a second).

Electrical current is a stream of unit charges. In electrical circuits, which are the subject of this book, these charges are carried by the negatively charged electrons. As any stream, the current “flows” through an enclosure. In this case, the enclosure is a conductor in various forms. Everyone is familiar with an electrical wire, the most popular solid-state current conductor. In this book, we will imply this type of conductor when considering the current flow.

How can we measure the strength of a current? Since it is a stream of electrons, each carrying a unit negative charge , it’s reasonable to think that the greater the number of the charges, the stronger the current. Using , a standard designation for a current, we can write that the current (its strength, in fact) is proportional to the number of charges, i.e.,

(1.2)

Thus, the total electrical charge, , collectively carried by all N electrons involved, constitutes the total amount of electricity that exists in a given conductor’s locality, i.e.,

(1.3)

But an electrical current is a stream; therefore, its strength must be measured by the amount of electricity passing a specific point in one second, which means

(1.4a)

where stands for ampere.2 Equation 1.4a shows the average current measured over the interval . The instantaneous current value can be calculated as

(1.4b)

Therefore,

the strength of the electric current is measured in ampere, A, which is a flow of one coulomb of electricity per one second through the cross-section of a wire.

This definition is based on the value of the elementary charge given above and included in the International System of Units (SI) since 2019.

Note that the above consideration means that the term electrical current implies a physical phenomenon (the flow of charges) and a measure of this phenomenon (the strength or time rate of this flow).

Electrical current and charge are mutually dependent, which can be formulated as follows: One coulomb is the amount of electrical charge that flows per one second through a conductor’s cross-section when the current’s strength is one ampere. (Alternatively, one ampere is the electrical current created by one coulomb streaming through a conductor’s cross-section per one second.) One coulomb consists of 6.24 × 1018 elementary charges ; this number is inverse to the absolute value of the electron charge. As we can see, a coulomb is a tremendous amount of electricity.

Is a 1-ampere current large or small amount? It depends on applications: Commercial power lines deliver about 100 A to a typical house at 220-volts ac (alternating current), one-room individual air conditioner requires 15 A ac, a cell phone charger can deliver 1.7 A at 9-volts dc (direct current), a 60-watt equivalent LED lightbulb needs 0.2 A, whereas its incandescing counterpart typically required 0.5A at 110-volts ac. The integrated circuits (IC) that constitute majority of modern electronics operate at milliamperes, , or microamperes, , and individual transistor in these ICs needs a few nanoamperes, , to act. Later, we will introduce the unique machinery, Large Hadron Collider (LHC), that consumes thousands of amperes.

Exercise How many elementary (unit) electrical charges flow through a conductor’s cross-section area if the measured current is ? Answer: 6.24 × 1018 or one coulomb.

We’ve assumed that an electrical current is a stream of electrons, carriers of negative charges. But about 200 years ago, when scientists started investigating this new phenomenon, electricity, they hypothesized that the electrical particles are positive. Since then, we show that current flows in an electrical circuit from a positive terminal of a source to the negative one. This designation is called conventional flow because it reflects not a reality but convention among electrical engineers. The actual direction of the flow of the negative charge carriers is called electron flow. Figure 1.1 illustrates the concept of two flows in an electrical circuit. It is critical to know this distinction to avoid any confusion; it’s imperative for the circuits that include semiconductors, where the current is delivered by both negative (electrons) and positive (holes) charge carriers.

Figure 1.1 Conventional (clockwise) and electron (counterclockwise) flows of electrical current in an electrical circuit. Designations: is a battery (dc source) and is a resistor.

Considering electrical current as a stream of electrons is a convenient and easy-to-visualize presentation, but reality is far from this rudimentary model. First, it’s necessary to highlight again that electrical current is the process of passing an electrical charge from one place to the other within a conductor. Secondly, we need to know that there are two mechanisms for this passing.

One mechanism is a drift electrical current, , which indeed is a flow of electrons inside a metallic wire. This current can be calculated as

(1.5)

where is a unit charge, is charge carriers volume density, is conductor’s cross-section, is the wire length, and the drift velocity is given by . Thus, in (1.5) is the number of charge carriers in the given cross-section of a wire, is the total electrical charge that passes the cross-section at a given instant, and is the average drift velocity of electrons. Under typical electrical laboratory conditions, is about . (How many centimeters per hour?) This means that students can complete their laboratory exercise for three hours, but the electrons pushed into a switch at the beginning of this experiment would still sit in the switch. These numbers tell us that the drift current is not a significant factor in the whole operation of a regular electrical circuit. (However, a drift current might play an essential role in various electrical applications that are outside of our interest.)

How then, you might correctly ask, a character on a computer screen can immediately appear after clicking the keyboard button? How can light shine as soon as we turn the switch on? How can a telephone line support our live conversation without noticeable delays? We know that all these processes are provided by electrical circuits. The answer lies in existence of the other, main mechanism of delivering electrical current (signal in this case) called electrical charge passing. Consider an electrical wire completely filled with free charges. When we push into this wire an additional charge, this charge will be transferred from one electron to the other almost immediately, at the velocity close to the speed of light. The new charge at the “entrance” will replace one charge at the “exit,” thus saving the total amount of charges within the conductor. (Recall the law of conservation of charge.) Figure 1.2, which visualizes these explanations, shows the kind of material particles within a tube. Still, in reality, there are only the electrical charges inside a conductor, and no mechanical motion is involved in this process. In truth, the process consists of transferring a charge from one locality to the next. And even more precise, the passing-charge process is the change in the electrical field. This is how the electrical current propagates within a metallic wire. However, in our model, an electrical current is a flow of charges, which Figure 1.2 demonstrates.

Figure 1.2 Charge passing in an electrical wire.

André-Marie Ampère discovered that electrical current flowing through a wire creates a magnetic field surrounding this wire. Given that both—electrical and magnetic—components of the electromagnetic field are produced by electrical charge, Ampere’s discovery is the other confirmation of the above statement.

Which mechanism—drift current or passing charge—plays the main role in electrical current? We hope you have a clear answer to this question now. Yet, despite all its limitations, the model of electrical current as a stream of electrons helps explain many details describing the electrical current behavior in electrical circuits. Bearing in mind all the constraints of this model, we will continue to use it in future discussions.

Equation 1.5 describing the drift current can be partly applied to the passing charge model because it highlights the role of a wire’s cross-section in transferring a current. Though it’s intuitively clear that the greater the wire cross-section, the more electricity can flow through this wire per second; (1.5) puts this statement on a solid mathematical footing. The wire sizes (cross-sections) are measured by their diameters, and the instruments used for their measurements are called gauges. Obviously, for a power line delivering 100 A of the current, the wire diameter must be much greater than that of a laptop charger carrying less than 2 A. Today, wire gauge also denotes the standard wire diameters collected in the American Wire Gauge (AWG) system. Similar system is also maintained by the International Electrotechnical Commission (IEC), and both systems are interrelated.

To continue using the electron-flow model for an electrical current, we must explain from where these free electrons appear. Metallic wires (solid-state conductors), which we consider in this book, consist of atoms whose nuclei are orbiting by the electrons. We refer to the well-known Bohr’s3atom planetary model of an atom shown in Figure 1.3a. An atom’s nucleus is composed of tightly packed protons and neutrons. The protons and neutrons have approximately the same mass, kilograms, which is extremely small. However, the electron’s mass is much smaller and can be considered negligible compared to this value. Hence, the mass of the whole atom is determined by the mass of its nucleus. In contrast, the atom’s size is defined by the electron orbits because the typical nucleus’s size is about times smaller than the orbit of the outmost electron. Protons carry a unit positive charge equal to that of an electron, . Since the neutrons have no charge, and the number of protons equals the number of electrons, the atom is electrically neutral. Coulomb forces attract electrons to the nucleus, thus forming the atom as a unit of matter. In metals (conductors), the farthest electrons can easily leave an atom; this is where free electrons in conductors come from. The modern view on the atom is more sophisticated and will be discussed shortly.

Figure 1.3 Atom models: a) Bohr planetary model; b) quantum-mechanical model of an atom. Not to scale. (Open source: UCDavisChemwiki, CC BY-NC-SA 3.0 US.)

Question Why can only the farthest but not all electrons easily leave an atom?

In metals, the nuclei are in fixed positions, making the metallic lattice, but the electrons can freely move due to conductor’s natural properties. Applying an external electric force to the wire, we can make these electrons flow in one direction, thus creating an electrical current. (In insulators, the electrons are strongly bounded, and we cannot make them flow.) This elemental model explains why only metals can conduct electricity. The best conductors are silver, copper, and gold. The specific applications in contacts and other circuit connections that require long-time stability and work well in harsh environments use gold. Interestingly, the organizers of the 2020 Olympic games in Japan used gold extracted from recycled smartphones and laptops to produce the Olympic medals. This fact highlights the scale of using gold in modern electronics.

The stream of free electrons—electrical current, that is—experiences resistance of the metal through which it flows. The resistance value depends on two factors. First, it is the natural property of the material. Silver, copper, and gold have the smallest resistivity measured in , though aluminum’s resistivity is just slightly greater than that of gold. Secondly, the resistivity of the same material increases with temperature rise. Consider a simplistic model of electrical current as of the flow of