Introduction to Electrical Circuit Analysis PDF

Table of Contents Cover Title Page Copyright Dedication Important Units Conventions with Examples Preface About the Companion Website Chapter 1: Introduction 1.1 Circuits and Important Quantities 1.2 Resistance and Resistors 1.3 Independent Sources 1.4 Depe...

Table of Contents Cover Title Page Copyright Dedication Important Units Conventions with Examples Preface About the Companion Website Chapter 1: Introduction 1.1 Circuits and Important Quantities 1.2 Resistance and Resistors 1.3 Independent Sources 1.4 Dependent Sources 1.5 Basic Connections of Components 1.6 Limitations in Circuit Analysis 1.7 What You Need to Know before You Continue Chapter 2: Basic Tools: Kirchhoff's Laws 2.1 Kirchhoff's Current Law 2.2 Kirchhoff's Voltage Law 2.3 When Things Go Wrong with KCL and KVL 2.4 Series and Parallel Connections of Resistors 2.5 When Things Go Wrong with Series/Parallel Resistors 2.6 What You Need to Know before You Continue Chapter 3: Analysis of Resistive Networks: Nodal Analysis 3.1 Application of Nodal Analysis 3.2 Concept of Supernode 3.3 Circuits with Multiple Independent Voltage Sources 3.4 Solving Challenging Problems Using Nodal Analysis  3.5 When Things Go Wrong with Nodal Analysis 3.6 What You Need to Know before You Continue Chapter 4: Analysis of Resistive Networks: Mesh Analysis 4.1 Application of Mesh Analysis 4.2 Concept of Supermesh 4.3 Circuits with Multiple Independent Current Sources 4.4 Solving Challenging Problems Using the Mesh Analysis 4.5 When Things Go Wrong with Mesh Analysis 4.6 What You Need to Know before You Continue Chapter 5: Black-Box Concept 5.1 Thévenin and Norton Equivalent Circuits 5.2 Maximum Power Transfer 5.3 Shortcuts in Equivalent Circuits 5.4 When Things Go Wrong with Equivalent Circuits 5.5 What You Need to Know before You Continue Chapter 6: Transient Analysis 6.1 Capacitance and Capacitors 6.2 Inductance and Inductors 6.3 Time-Dependent Analysis of Circuits in Transient State 6.4 Switching and Fixed-Time Analysis 6.5 Parallel and Series Connections of Capacitors and Inductors 6.6 When Things Go Wrong in Transient Analysis 6.7 What You Need to Know before You Continue Chapter 7: Steady-State Analysis of Time-Harmonic Circuits 7.1 Steady-State Concept 7.2 Time-Harmonic Circuits with Sinusoidal Sources 7.3 Concept of Phasor Domain and Component Transformation 7.4 Special Circuits in Phasor Domain 7.5 Analysis of Complex Circuits at Fixed Frequencies 7.6 Power in Steady State  7.7 When Things Go Wrong in Steady-State Analysis 7.8 What You Need to Know before You Continue Chapter 8: Selected Components of Modern Circuits 8.1 When Connections Are via Magnetic Fields: Transformers 8.2 When Components Behave Differently from Two Sides: Diodes 8.3 When Components Involve Many Connections: OP-AMPs 8.4 When Circuits Become Modern: Transistors 8.5 When Components Generate Light: LEDs 8.6 Conclusion Chapter 9: Practical Technologies in Modern Circuits 9.1 Measurement Instruments 9.2 Three-Phase Power Delivery 9.3 AD and DA Converters 9.4 Logic Gates 9.5 Memory Units 9.6 Conclusion Chapter 10: Next Steps 10.1 Energy Is Conserved, Always! 10.2 Divide and Conquer Complex Circuits 10.3 Appreciate the Package 10.4 Consider Yourself as a Circuit Element 10.5 Safety First Chapter 11: Photographs of Some Circuit Elements Appendix A A.1 Basic Algebra Identities A.2 Trigonometry A.3 Complex Numbers Appendix B: Solutions to Exercises Chapter 1 Chapter 2  Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Index End User License Agreement List of Illustrations Chapter 1: Introduction Figure 1.1 A simple circuit involving a bulb connected to a battery. The connection between the bulb and battery is shown via simple lines. Figure 1.2 A circuit involving connections of four components labeled from to. From the circuit-analysis perspective, connection shapes are not important, and these three representations are equivalent. Figure 1.3 Electric field lines created individually by a positive charge and a negative charge. An electric field is assumed to be created whether there is a second test charge or not. If a test charge is located in the field, repulsive or attractive force is applied on it. Figure 1.4 Movement of a charge in an electric field created by external sources. The energy absorbed or released by the charge does not depend on the path but depends on the potentials at the start and end points. The electric potential (voltage) is always defined between two points, while selecting a reference point as a ground enables unique voltage definitions at all points. Figure 1.5 On a metal wire, the conventional current direction, which is defined as positive charge flow, is the opposite of the actual electron movements. In a circuit, voltages are defined at the nodes, as well as across components, using the sign convention. Figure 1.6 Power of a device for given current and voltage across it. Figure 1.7 Transient state and steady state in DC and AC signals.  Figure 1.8 Structure of a general coaxial cable and a representation of the drift velocity of an electron under an electric field. Figure 1.9 The resistance of a rod with conductivity is often approximated as , where and are the length and cross- section area, respectively, of the rod. In circuit analysis, a resistor is a two-terminal device that usually consumes energy. Figure 1.10 Short circuit and open circuit can be interpreted as special cases of resistors, with zero and infinite resistance values, respectively. Figure 1.11 There are alternative symbols to show voltage and current sources; circular representations are used in this book. For any source, the polarity should be clearly indicated. In addition to sources with constant values (DC sources), the source values and may depend on time (AC sources). Figure 1.12 Dependent sources have fixed voltage/current values, depending on some other voltage/current values in the circuit. Figure 1.13 Series and parallel connections, where the current and voltage are common values, respectively, for the components. Figure 1.14 Some possible and impossible configurations using ideal components. Figure 1.15 Some possible circuits involving only one or two components that are connected consistently. In the first and second circuits, where there is a current and a voltage source, the sources do not produce any power. However, they still provide the current and voltage values, in accordance with their definitions. In the third circuit, the voltage source absorbs power (100 W), while the current source delivers power (100 W). Figure 1.16 Two simple circuits that can be interpreted incorrectly as impossible. In fact, both two circuits are possible and they involve consistent voltage and current values. In the first circuit, a voltage drop (by an amount of V) exists across the resistor. In the second circuit, a nonzero current ( A) flows through the resistor. These values can easily be found by applying Kirchhoff's laws, as described in the next chapter. Chapter 5: Black-Box Concept  Figure 5.1 Thévenin and Norton equivalent circuits. Figure 5.2 Open-circuit and short-circuit cases to find the values in a Thévenin equivalent circuit. Figure 5.3 Open-circuit and short-circuit cases to find the values in a Norton equivalent circuit. Chapter 7: Steady-State Analysis of Time-Harmonic Circuits Figure 7.1 Steady-state equivalents of capacitors and inductors when only DC sources are involved. Figure 7.2 A representation of passage from transient to steady state in an AC circuit. Figure 7.3 Sinusoidal voltage and current sources. Figure 7.4 A resistor connected to a sinusoidal voltage source. Figure 7.5 A capacitor connected to a sinusoidal voltage source. Figure 7.6 A capacitor connected to a sinusoidal current source at different frequencies. Figure 7.7 An inductor connected to a sinusoidal current source. Introduction to Electrical Circuit Analysis Özgür Ergül Middle East Technical University, Ankara, Turkey This edition first published 2017 © 2017 John Wiley and Sons Ltd All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law.Advice on how to obtain permision to reuse material from this title is available at http://www.wiley.com/go/permissions. 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Library of Congress Cataloging-in-Publication data applied for ISBN: 9781119284932 Cover design by Wiley Cover image: (Circuit Board) © ratmaner/Gettyimages; (Electronic Components) © DonNichols/Gettyimages; (Formulas) © Bim/Gettyimages For my wife Ayça, three cats (Boncuk, Pepe, and Misket), and the dog, who all suffered during the writing of this book Important Units Ampere (A) Coulomb (C) Farad (F): C/V Henry (H): weber/A Hertz (Hz): 1/s Joule (J): N m kg /s kilo (k ): meter (m) micro ( ): milli (m ): Newton (N): kg m/s Second (s) Siemens (S): A/V Volt (V): J/C Volt-ampere (V A): J/s Watt (W): J/s Conventions with Examples Fractions: A A A Irrational numbers: A A A Approximation: A A Scientific notation: and Multiplication without sign: Number ranges: Limit of a number from left and right: Preface Since the first known electricity experiments more than 25 centuries ago by Thales of Miletus, who believed that there should be better ways than mythology to explain physical phenomena, humankind has worked hard to understand and use electricity in many beneficial ways. The last three centuries have seen rapid developments in understanding electricity and related concepts, leading to constantly accelerating technology advancements in the last several decades. Today, most of us simply cannot live without electricity, and it is almost ubiquitous in daily life. We are so attached to and dependent on electricity that there are even post-apocalyptic fiction movies and film series based on sudden electrical power blackouts. And they are terrifying. Electricity is one of a few subjects with which we have a strange relationship. The more we use it, less we know about it. Electrical and electronic devices, where electricity is somehow used to produce beneficial outputs, are a closed book to most of us, until we open them (not a suggested activity!) and see that they contain incredibly small but highly intelligent parts. These parts, some of which once had huge dimensions and even filled entire rooms, are now so tiny that we are able to place literally billions of them (at the time of writing) in a smartphone microprocessor. One billion is a huge number; at a rate of one a second, it takes 31 years to count. And we are able to put these uncountable (OK, countable, but not feasibly so) numbers of components together and make them work in harmony for our enjoyment. Yet most of us know little about how they actually work. The topic of circuit analysis has naturally developed in parallel with electrical circuits and devices starting from centuries ago. To provide some intuition, Ohm's law has been known since 1827, while Kirchhoff's laws were described in 1845. Nodal and mesh analysis methods have been developed and used for systematically applying Kirchhoff's laws. Phasor notation is borrowed from mathematics to deal with time-harmonic circuits. These fundamental laws have not changed, and they will most probably remain the same in the coming years. In general, basic laws describe everything when they are wisely used. Hence, more and more sophisticated circuits in future technologies will also benefit from them, independent of their complexity. Circuit analysis is naturally linked to all other technologies involving electricity, including medical, automotive, computer, energy, and aerospace industries, as including medical, automotive, computer, energy, and aerospace industries, as well as all subcategories of electrical and electronic engineering. Interestingly, with the rapid development of technology, we tend to learn fundamental laws more superficially. One can identify two major factors, among many: As circuits become more complicated and specialized, we are attracted and guided to focus on higher-level representations, such as inputs and outputs of microchips with well-defined functions, without spending time on fundamental laws. Great advancements in circuit-solver software “eliminate” the need to fully understand fundamental laws and appreciate their importance in everyday life, reducing circuit analysis to numbers. Unfortunately, without absorbing fundamental laws, we tend to make major conceptual mistakes. Most instructors have had a student who offers infinite energy by rotating something (usually a car wheel if s/he is a mechanical engineering student), disregarding the conservation of energy. It is often a confusing issue for a biomedical student to appreciate the necessity of grounding for medical safety. And it is probably a computational mistake but not a new technology if a circuit analyzer program provides a negative resistor value. The aim of this book is to gradually construct the basics of circuit analysis, even though they are not new material, while accelerating our understanding of electrical circuits and all technologies using electricity. This is intended as an introductory book, mainly designed for college and university students who may have different backgrounds and, for whatever reason, need to learn about circuits for the first time. It mainly focuses on a few essential components of electrical components, namely, resistors, independent voltage and current sources, dependent sources (as closed components, not details), capacitors, and inductors. On the other hand, transistors, diodes, OP-AMPs, and similar popular and inevitable components of modern circuits, which are fixed topics (and even starting points) in many circuit books, are not detailed. The aim of this book is not to teach electrical circuits, but rather to teach how to analyze them. From this perspective, the components listed above provide the required combinations and possibilities to cover the fundamental techniques, namely, possibilities to cover the fundamental techniques, namely, Ohm's and Kirchhoff's laws, nodal analysis, mesh analysis, the black-box approach and Thévenin/Norton equivalent circuits. This book also covers the analysis methods for both DC and AC cases in transient and steady states. To sum up, the technology that is covered in this book is well established. The analysis methods and techniques, as well as components, listed above have been known for decades. However, the fundamental methods and components need to be known in sufficient depth in order to understand how electrical circuits work, including state-of-the-art devices and their ingredients. Many books in this area are dominated by an increasing number of new electrical and electronic components and their special working principles, while the fundamental techniques are squeezed into short descriptions and limited to a few examples. Therefore, the purpose of this book is to provide sufficient basic discussion and hands-on exercises (with solutions at the back of the book) before diving into modern circuits with higher-level properties. Enjoy! About the Companion Website This book is accompanied by a companion website: www.wiley.com/go/ergul4412 The website includes: Exercise sums and solutions Videos Chapter 1 Introduction We start with the iconic figure (Figure 1.1), which depicts a bulb connected to a battery. Whenever the loop is closed and a full connection is established, the bulb comes on and starts to consume energy provided by the battery. The process is often described as the conversion of the chemical energy stored in the battery into electrical energy that is further released as heat and light by the bulb. The connection between the bulb and battery consists of two wires between the positive and negative terminals of the bulb and battery. These wires are shown as simple straight lines, whereas in real life they are usually coaxial or paired cables that are isolated from the environment. Figure 1.1 A simple circuit involving a bulb connected to a battery. The connection between the bulb and battery is shown via simple lines. The purpose of this first chapter is to introduce basic concepts of electrical circuits. In order to understand circuits, such as the one above, we first need to understand electric charge, potential, and current. These concepts provide a basis for recognizing the interactions between electrical components. We further discuss electric energy and power as fundamental variables in circuit analysis. The time and frequency in circuits, as well as related limitations, are briefly considered. Finally, we study conductivity and resistance, as well as resistors, independent sources, and dependent sources as common components of basic circuits. 1.1 Circuits and Important Quantities An electrical circuit is a collection of components connected via metal wires. Electrical components include but are not limited to resistors, inductors, capacitors, generators (sources), transformers, diodes, and transistors. In circuit analysis, wire shapes and geometric arrangements are not important and they can be changed, provided that the connections between the components remain the same with fixed geometric topology. Wires often meet at intersection points; a connection of two or more wires at a point is called a node. Before discussing how circuits can be represented and analyzed, we first need to focus on important quantities, namely, electric charge, electric potential, and current, as well as energy and power. Figure 1.2 A circuit involving connections of four components labeled from to. From the circuit-analysis perspective, connection shapes are not important, and these three representations are equivalent. 1.1.1 Electric Charge Electric charge is a fundamental property of matter to describe force interactions among particles. According to Coulomb's law, there is an attractive (negative) force between a proton and an electron given by which is significantly larger than (around times) the gravity between these particles. In the above, is the distance between the proton and electron, given in meters (m). This law can be rewritten by using Coulomb's constant as where are the electrical charges of the proton and electron, respectively, in units of coulombs (C). Coulomb's constant enables the generalization of the electric force between any arbitrary charges and as where and are assumed to be point charges (theoretically squeezed into zero volumes), which are naturally formed of collections of protons and electrons. The definition of the electric force above requires at least two charges. On the other hand, it is common to extend the physical interpretation to a single charge. Specifically, a stationary charge is assumed to create an electric field (intensity) that can be represented as where is now the distance measured from the location of the charge. This electric field is in the radial direction, either outward (positive) or inward (negative), depending on the type (sign) of the charge. Therefore, we assume that an electric field is always formed whether there is a second test charge or not. If there is at a distance , the electric force is now measured as either as repulsive (if and have the same sign) or attractive (if and have different signs). Figure 1.3 Electric field lines created individually by a positive charge and a negative charge. An electric field is assumed to be created whether there is a second test charge or not. If a test charge is located in the field, repulsive or attractive force is applied on it. The definition of the electric field is so useful that, in many cases, even the sources of the field are discarded. Consider a test charge exposed to some electric field. The force on can be calculated as without even knowing the sources creating the field. This flexibility further allows us to define the electric potential concept, as discussed below. 1.1.2 Electric Potential (Voltage) Consider a charge in some electric field created by external sources. Moving the charge from a position to another position may require energy if the movement is opposite to the force due to the electric field. This energy can be considered to be absorbed by the charge. If the movement and force are aligned, however, energy is extracted from the charge. In general, the path from to may involve absorption and release of energy, depending on the alignment of the movement and electric force from position to position. In any case, the net energy absorbed/released depends on the start and end points, since the electric field is conservative and its line integral is path-independent. Figure 1.4 Movement of a charge in an electric field created by external sources. The energy absorbed or released by the charge does not depend on the path but depends on the potentials at the start and end points. The electric potential (voltage) is always defined between two points, while selecting a reference point as a ground enables unique voltage definitions at all points. Electric potential (voltage) is nothing but the energy considered for a unit charge (1 C) such that it is defined independent of the testing scheme. Specifically, the work done in moving a unit charge from a point to another point is called the voltage between and. Conventionally, we have as the voltage between and , corresponding to the work done in moving the charge from to. If , then work must be done to move the charge (the energy of the charge increases). On the other hand, if , then the work done is negative, indicating that energy is actually released due to the movement of the charge. The unit of voltage is the volt (V), and 1 volt is 1 joule per coulomb (J/C). A proper voltage definition always needs two locations and a polarity definition. Considering three separate points , , and , we have and The equality above is a result of the conservation of the electric energy (conservative electric field). On the other hand, , , and are not yet uniquely defined. In order to simplify the analysis in many cases, a location can be selected as a reference with zero potential. In circuit analysis, such a location that corresponds to a node is called ground, and it allows us to define voltages at all other points uniquely. For example, if in the above, we have. 1.1.3 Electric Current A continuous movement of electric charges is called electric current. Conventionally, the direction of a current flow is selected as the direction of movement of positive charges. The unit of current is the ampere (A), and 1 ampere is 1 coulomb per second (C/s). Formally, we have where and represent charge and time, respectively. The current itself may depend on time, as indicated in this equation. But, in some cases, we only have steady currents, , where does not depend on time. Different types of current exist, as discussed in Section 1.2.1. In circuit analysis, however, we are restricted to conduction currents, where free electrons of metals (e.g., wires) are responsible for current flows. Since electrons have negative charges and an electric current is conventionally defined as the flow of positive charges, electron movements and the current direction on a wire are opposite to each other. Indeed, when dealing with electrical circuits, using positive current directions is so common that the actual movement of charges (electrons) is often omitted. When charges move, they interact with each other differently such that they cannot be modeled only with an electric field. For example, two parallel wires carrying currents in opposite directions attract each other, even though they do not possess any net charges considering both electrons and protons. Similar to the interpretation that electric field leads to electric force, this attraction can be modeled as a magnetic field created by a current, which acts as a magnetic force on a test wire. Electric and magnetic fields, as well as their coupling as electromagnetic waves, are described completely by Maxwell's equations and are studied extensively in electromagnetics. Figure 1.5 On a metal wire, the conventional current direction, which is defined as positive charge flow, is the opposite of the actual electron movements. In a circuit, voltages are defined at the nodes, as well as across components, using the sign convention. 1.1.4 Electric Voltage and Current in Electrical Circuits In electrical circuit analysis, charges, fields, and forces are often neglected, while electric voltage and electric current are used to describe all phenomena. This is completely safe in the majority of circuits, where individual behaviors of electrons are insignificant (because the circuit dimensions are large enough with respect to particles), while the force interactions among wires and components are negligible (because the circuit is small enough with respect to signal wavelength). The behavior of components is also reduced to simple voltage– current relationships in order to facilitate the analysis of complex circuits. The limitations of circuit analysis using solely voltages and currents are discussed in Section 1.6. In an electrical circuit, voltages are commonly defined at nodes, while currents flow through wires and components. A wire is assumed to be perfectly conducting (see Section 1.2.4) such that no voltage difference occurs along it, that is, the voltage is the same on the entire wire. This is the reason why their shapes are not critical. On the other hand, a voltage difference may occur across a component, depending the type of the component and the overall circuit. For unique representation of a node voltage, a reference node should be selected as a ground. However, the voltage across a component can always be defined uniquely since it is based on two or more (if the component has multiple terminals) points. In circuit analysis, voltages and currents are usually unknowns to be found. Since they are not known, in most cases, their direction can be arbitrarily Since they are not known, in most cases, their direction can be arbitrarily selected. When the solution gives a negative value for a current or a voltage, it is understood that the initial assumption is incorrect. This is never a problem at all. For consistency, however, it is useful to follow a sign convention by fixing the voltage polarity and current direction for any given component. In the rest of this book, the current through a component is always selected to flow from the positive to the negative terminal of the voltage. 1.1.5 Electric Energy and Power of a Component Consider a component with a current and voltage , defined in accordance with the sign convention. If , one can assume that positive charges flow from the positive to the negative terminal of the component. In addition, if , these positive charges encounter a drop in their potential values, that is, they release energy. This energy must be somehow used (consumed or stored) by the component. Formally, we define the energy of the component as where the time integral is used to account for all charges passing during , assuming that the component is used from time. If , it is understood that the component consumes net energy during the time interval. On the other hand, if , the component produces net energy in the same time interval. We note that the unit of energy is the joule, as usual. Energy as defined above provides information in selected time intervals. In many cases, however, it is required to know the behavior (change of the energy) of the component at a particular time. For a device with a current and voltage , this corresponds to the time derivative of the energy, namely the power of the device, defined as Specifically, for a given component, its power is defined as the product of its voltage and current. The unit of power is the watt (W), where 1 watt is 1 volt ampere (V A) or 1 joule per second (J/s). If , the component absorbs energy at that specific time. Otherwise (i.e., if ), the component produces energy. Example 1 Electric power and energy are often underestimated. Consider an 80 W bulb, which is on for 24 hours. Using the energy spent by the bulb, how many meters can a 1000 kg object be lifted? Solution The energy spent by the bulb is Then, assuming m/s , and using for the potential energy, we have Example 2 There are approximately bulbs on earth. Assuming an average on period of 6 hours and 50 W average power, find the amount of coal required to produce the same amount of energy for 1 day. Assume that the thermal energy of coal is J/kg and the efficiency of the conversion of the energy is. Solution The required energy for the bulbs per day is The corresponding amount of coal can be found as Example 3 The voltage and current of a device are given by V and A, respectively, as functions of time. Find the maximum power of the device. Solution We have as the power of the device. We note that In order to find the maximum point for the power, we use leading to Then the maximum power is Figure 1.6 Power of a device for given current and voltage across it. Exercise 1 A device has a power of 60 W when it is active and 10 W when it is on standby. An engineer measures that it spends a total of 2664 kJ energy in 24 hours. How many hours was the device actively used? 1.1.6 DC and AC Signals Until now, we have considered the time concept in circuits for studying energy (which needs to be defined in time intervals) and power (which may depend on time). In fact, the time dependence of the power of a component corresponds to the time dependence of its voltage and/or current. This brings us to the definition of direct current (DC) and alternating current (AC), which are important terms in describing and categorizing circuits and their components. DC means a unidirectional flow of electric charges, leading to a current only in a single direction. However, the term ‘DC signal’ is commonly used to describe voltages and other quantities that do not change polarity. DC signals are produced by DC sources, whose voltages or currents are assumed to be fixed in terms of direction and amplitude. Examples of DC sources are batteries and dynamos. Voltage and current values of these sources may have very slight variations with respect to time, which are often neglected in circuit analysis. AC describes electric currents and voltages that periodically change direction and polarity. This periodicity is generally imposed by AC sources, which may provide voltage and currents in sinusoidal, triangular, square, or other periodic forms. AC is commonly used in all electricity networks, including homes. The reason for its common usage is its well-known advantage when transmitting AC signals over long distances. Specifically, the electric power can be transmitted with less ohmic losses in the AC form in comparison to the DC form. In addition, AC signals can be amplified or reduced easily via transformers, making it possible to use different voltage and current values in different lines of it possible to use different voltage and current values in different lines of electricity networks and electrical devices. In general, AC circuits have a fixed periodicity and frequency, which is set to 50–60 hertz (Hz = 1/s) in domestic usage. AC signals are also associated with electromagnetic waves (e.g., radiation from electrical components). AC and DC signals can be converted into each other. The conversion from AC to DC is achieved by rectifiers, while inverters are used to convert DC signals into AC signals. DC to DC and AC to AC converters are also common when the properties but not the types of the signals need to modified. Figure 1.7 Transient state and steady state in DC and AC signals. 1.1.7 Transient State and Steady State Whether DC or AC, any circuit in real life has a time dependency, at least when switching the circuit on and off. The short-time state, in which variations in voltage and current values are encountered due to outer effects (e.g., switching), is called the transient state. Whether it is a DC or AC circuit, any circuit can be in a transient state before it reaches an equilibrium. A transient state is usually an unwanted state, where voltage, current, and power values involve fluctuations that are not designed on purpose. In the long term, circuits that are not disturbed by outer effects enter into equilibrium, namely, the steady state. Theoretically, an infinite time is required to pass from transient state to equilibrium, while most circuits are assumed to reach steady state after a sufficient period (i.e., when fluctuations become negligible). For DC circuits in steady state, voltage and current values are assumed to be constant. In the first few chapters, a steady state is automatically assumed when only resistors and DC sources are considered. In fact, the time needed to pass from transient state to steady state depends on a time constant, needed to pass from transient state to steady state depends on a time constant, which is a contribution of both resistors and energy-storage elements (capacitors and inductors). Hence, circuits with only resistors and DC sources have zero transient time, that is, they can be assumed to be in steady state without any transient analysis. For AC circuits, voltages and currents in steady state oscillate with the time period dictated by the sources. Therefore, we emphasize that the steady state does not indicate constant properties for all circuits. 1.1.8 Frequency in Circuits When AC sources are involved in a circuit, voltage and current values oscillate with respect to time. In most cases, the periodicity and frequency are fixed, that is, all voltages and currents change at the same rate, while there can be phase differences (delays) between them. The behavior of some components does not rely on the frequency, unless they are exposed to extreme conditions. As an example, resistors behave almost the same in a wide range of frequencies. On the other hand, many components, such as capacitors and inductors, strongly depend on the frequency. With DC sources, corresponding to zero frequency, capacitors/inductors act like open/short circuits, while they become almost the opposite at very high frequencies. Therefore, the behavior of an AC circuit directly depends on the frequency, as discussed extensively in time-harmonic analysis. 1.2 Resistance and Resistors Resistors (Figure 11.1) are fundamental components in electrical circuits. They are basically energy-consuming elements that are used to control voltage and current values in circuits. In addition, the energy conversion ability of resistors can be useful in various applications, where these elements are directly used for heating and lighting (conventional bulbs). Specifically, the energy consumed by a resistor is usually released as heat, and sometimes as useful light. Resistance is a common property of all metals, and even very conductive wires have resistances, which may need to be included in circuit analysis. 1.2.1 Current Types, Conductance, and Ohm's Law In order to understand resistance and resistors, first we need to define the conduction current. As described in Section 1.1.3, current is a continuous flow of charges. In electrolytes, gases, and plasmas, currents may be formed by ions, and even by moving protons. In some applications, electrons can be injected from special devices, leading to a current flow in a vacuum. In circuits, however, currents are mostly formed by the conduction of metals. In good conductors, one or more electrons from each atom is weakly bound to the atom. These electrons can move freely in the metal (especially on the surfaces), while these movements are random if the metal is not exposed to an electric field and potential. Therefore, without any excitation, there is no net flow of charges. When an electric field is applied, however, electrons collectively drift in the opposite direction, leading to a net measurable current. We note that the conventional current direction is also opposite to the movement of electrons, aligning it with the electric field. A simple relation between the current density and electric field intensity can be written by using Ohm's law as where is defined as the conductivity, given in siemens per meter (S/m). In the above equation, represents the current density, whose surface integral (on the cross-section) gives the overall current flowing through the metal. All materials can be categorized in terms of their conductivity values, as discussed below. Figure 1.8 Structure of a general coaxial cable and a representation of the drift velocity of an electron under an electric field. 1.2.2 Good Conductors and Insulators Most metals are good conductors, with conductivity values in the – S/m range for a wide band of frequencies. For example, copper has a conductivity of approximately S/m at room temperature. For all materials, conductivity values depend on temperature and other environmental conditions, as well as the frequency. Sea water is known to be conductive (with around 4–5 S/m conductivity), while its conduction mechanism is based on ions, not free electrons as in metals. Carbon has interesting properties, demonstrating extremely varying conductivity characteristics depending on the arrangement of its atoms. For example, diamond has a very low conductivity (around S/m), while graphite is as conductive as some metals (greater than S/m). A recently popular form of carbon called graphene may have conductivity values as large as S/m. There is often confusion between the velocity of electricity, velocity of electrons, and the drift velocity of electrons. In a typical metal without any excitation, electrons move randomly with a high (Fermi) velocity. These movements are of high speed (e.g., m/s for copper). However, due to their random nature, no net current flows along the metal. When the metal is exposed to a voltage difference, leading to an electric field, electrons continue their random movements, while they tend to drift in the opposite direction to the electric field. The corresponding drift velocity is usually very low (e.g., only m/s for a typical copper wire). On the other hand, the current measured along a wire is due to this drift velocity. Obviously, when AC sources are involved, electrons do not drift only in a single direction, but oscillate back and forth (in addition to high-velocity random movements) with the frequency of the signal. Since circuits are usually small with respect to wavelength, drift movements of electrons are almost synchronized through the entire circuit. Finally, the velocity of the electricity along a wire is not related to any actual movement of electrons. It is related to the speed of the electromagnetic wave through the wire (similar to sea waves that are not movements of water molecules). This speed is comparable to the speed of light in a vacuum, but it is reduced by a velocity factor depending on the properties of the material. In general, materials with low conductivity values are called insulators. Wood, glass, rubber, air, and Teflon are well-known insulators in real-life applications. Insulators are also natural parts of all circuits, for example for isolating components and wires from each other, as well as the parts of electrical components. Since they are not electrically active, however, they are not considered directly in circuit analysis. For example, when considering wires in circuits, we assume perfectly conducting metals without any insulator, while in real life, electrical wires have shielded or coaxial structures with layers of conducting metals and insulating materials separating them. 1.2.3 Semiconductors As their name suggests, semiconductors conduct electricity better than insulators and worse than good conductors. In addition, the conductivity of semiconductors can be altered by externally modifying their material properties permanently (via chemical processes) and temporarily (via electrical bias), making them suitable for controlling electricity. Silicon is the best-known semiconductor, and has been used in producing diverse components of integrated circuits. The key chemical operation is called doping, that is, modifying the conductivity of semiconductors by introducing impurities into their crystal lattice structures. This way, different (e.g., n-type, p-type) kinds of semiconductors can be produced and used to form junctions that enable control over electric current and voltage. Engineers use many different types of semiconducting devices, such as diodes and transistors, to construct modern circuits. These special components are discussed in Chapter 8. 1.2.4 Superconductivity and Perfect Conductivity Perfect conductivity is a theoretical limit when the conductivity of a metal becomes infinite, that is,. In this case, if a current exists along the metal, and there is no potential difference over it. Therefore, a perfect conductor does not dissipate power while conducting electricity. Perfect conductivity is an idealized property as all metals actually have finite conductivity, while some metals can be assumed to be perfect conductors to simplify their modeling. In circuit analysis, all wires are assumed to be perfect conductors (with no voltage drop across them), while any resistance due to imperfect conductivity can be modeled as a resistor component. Under the perfect conductivity assumption, the electric field is zero anywhere on a metal. This also means that all charges are distributed on the surface of the metal. For electromagnetic fields, where electric and magnetic currents are coupled, a zero electric field leads to a zero magnetic field. On the other hand, perfect conductivity does not enforce any assumption on a static magnetic field. Specifically, a static magnetic field inside a perfect conductor does not violate Maxwell's equations. Recently, superconductors have become popular due to their potential applications. Similar to perfect conductivity, superconductivity can be described as a limit case when the electrical conductivity goes to infinity. On the other hand, this infinite conductivity cannot be explained simply by electron movements, and quantum effects need to be considered to understand how a metal can become a superconductor. In a superconductor, magnetic fields are expelled toward its surface, making it different from theoretical perfect conductivity. Superconductivity is achieved in real life by cooling down special materials below critical temperatures. materials below critical temperatures. Figure 1.9 The resistance of a rod with conductivity is often approximated as , where and are the length and cross-section area, respectively, of the rod. In circuit analysis, a resistor is a two-terminal device that usually consumes energy. 1.2.5 Resistors as Circuit Components As mentioned above, resistors are fundamental components of circuits. Given a resistor, the voltage–current relationship (obeying the sign convention) can be written as where is called the resistance. In general, the resistance of a structure depends on its dimensions and is inversely proportional to the conductivity of the material. The simple relationship above for the definition of the resistance is also commonly called Ohm's law. The unit of resistance is the ohm ( ), and 1 ohm is 1 volt per ampere (V/A). The power of a resistor can be found from which is always nonnegative. Therefore, resistors cannot produce energy themselves. In some cases, it is useful to use conductance, defined as The unit of the conductance is the siemens (S), and 1 siemens is 1 ampere per volt (A/V). Resistors in real life are made of different materials, including carbon. In addition to standard resistors with fixed resistance values, there are also adjustable resistors, such as rheostats, which can be useful in different applications. The resistance of a fixed resistor also demonstrates nonlinear behaviors, that is, it may change with temperature, which may rise during the use of the resistor, leading to a complicated relationship between the voltage and current. In circuit analysis, however, these nonlinear behaviors are often discarded, and a fixed resistor has always the same resistance value. Figure 1.10 Short circuit and open circuit can be interpreted as special cases of resistors, with zero and infinite resistance values, respectively. Two limit cases of resistors are of particular interest in circuit analysis. When , indicating a lack of resistance, we have a short circuit. Specifically, in a short circuit, we have while can be anything (may not be zero). While all wires with zero resistances can be categorized as short circuits, this definition is often used to indicate a direct connection between two points that are not supposed to be connected. For many components and devices, having a short circuit means a failure or breakdown. At the other extreme case, two points without a direct connection between them can be interpreted as a resistor with infinite resistance. Such a case is called an open circuit, which can be defined as while can be anything (may not be zero). Any two points without a direct connection in a circuit can be interpreted as an open circuit, while this definition is again used to indicate a special case, particularly a breakdown of a connection. 1.3 Independent Sources All circuits are excited with AC and DC sources. Among these, independent sources are defined as energy-delivering devices whose voltage or current values are fixed at a given value, independent of the rest of the circuit. Two types of independent sources are used in circuit analysis: voltage and current sources. An ideal voltage source is defined as where is given and independent of other parts of the circuit. If the voltage source is DC, we further have as a constant. We note that the current through a voltage source, , can be anything (not necessarily zero). In fact, if a voltage source is delivering energy, must be nonzero. An ideal current source is defined as where is given and independent of other parts of the circuit. Once again, if the current source is DC, we further have as a constant. We note that the voltage across a current source, , can be anything (not necessarily zero). Figure 1.11 There are alternative symbols to show voltage and current sources; circular representations are used in this book. For any source, the polarity should be clearly indicated. In addition to sources with constant values (DC sources), the source values and may depend on time (AC sources). In real life, batteries and dynamos can be considered as independent voltage sources. On the other hand, an independent current source, which has a predetermined current value no matter what the rest of the circuit does, is usually designed using a voltage source and some other components (e.g., diodes, transistors, and/or OP-AMPs). In this book, we always show an independent source as a single and ideal device, without detailed structures and any internal resistances. If a source has a nonideal resistance (e.g., nonzero resistance for a voltage source or finite resistance for a current source) it can be shown as a separate component in addition to the ideal part of the source. Under normal circumstances, voltage and current sources provide energy to their circuits. However, depending on the rest of the circuit, a voltage or current source may consume energy, which is a perfectly valid scenario. A source that consumes energy indicates that there is at least one other source that delivers energy. For a given isolated circuit, the sum of powers of all components must be zero due to the conservation of energy. 1.4 Dependent Sources Dependent sources are also energy-delivering devices, where, unlike independent sources, the voltage or current provided depends on another voltage or current in the circuit. While dependent sources are not frequently used practice, they are very common in circuit analysis for modeling a component, (e.g., transistors and OP-AMPs). Therefore, we assume that dependent sources exist as individual components of circuits, while the actual circuit structure may not be exactly the same. There are four types of dependent sources, which can be listed as follows. Figure 1.12 Dependent sources have fixed voltage/current values, depending on some other voltage/current values in the circuit. Voltage-controlled voltage source (VCVS): A voltage source whose voltage depends on another voltage in the circuit, i.e., , where is a  unitless quantity. Voltage-controlled current source (VCCS): A current source whose current depends on a voltage in the circuit, i.e., , where is measured in siemens. Current-controlled voltage source (CCVS): A voltage source whose voltage depends on a current in the circuit, i.e., , where is measured in ohms. Current-controlled current source (CCCS): A current source whose current depends on another current in the circuit, i.e., , where is unitless. The polarization of the voltage/current of a dependent source, as well as the reference voltage/current and the linkage constant ( , , , ), are given with the definition of the source. Similarly to independent sources, dependent sources can be DC or AC, depending on the reference voltage/current, and. 1.5 Basic Connections of Components In any given circuit, components are connected via wires that intersect at nodes. Considering multiple components, two basic types of connection may occur: series and parallel. Figure 1.13 Series and parallel connections, where the current and voltage are common values, respectively, for the components. If a common current flows through the components, they are connected in series. Hence, such components share the same current. If a common voltage is applied on the components, they are connected in parallel. Hence, such components share the same voltage. In general, series and parallel connections occur together, also with other types of connections, leading to a complex network. Figure 1.14 Some possible and impossible configurations using ideal components. Considering ideal components, some of the connections are impossible. Some basic examples of possible and impossible scenarios are as follows. A 10 A current source and a 20 A current source cannot be connected in series. A 10 A current source and an open circuit cannot be connected in series. A 10 A current source and a short circuit can be connected in series. If these are the only components of the circuit (with a full connection on both terminals), no voltage occurs across the current source; hence, it does not produce any power. A 10 V voltage source and a 20 V voltage source cannot be connected in parallel. A 10 V voltage source and an open circuit can be connected in parallel. If these are the only components of the circuit, no current flows through the voltage source; hence, it does not produce any power. A 10 V voltage source and a short circuit cannot be connected in parallel. In order to understand why a connection may not be possible, one can directly use the definition of the components. For example, consider a series connection of 10 A and 20 A current sources. The 10 A source indicates that 10 A is passing through the line. On the other hand, the 20 A source, by definition, needs 20 A current to flow in the same line. Therefore, there is an inconsistency, since a wire cannot have different current values at the same time. Similar inconsistencies can be found for all impossible cases. Such impossible configurations are not due to a modeling incapability in circuit analysis; they actually correspond to physically impossible practices in real life. Consider actually correspond to physically impossible practices in real life. Consider another example involving a parallel connection of two voltage sources with different values. In real life, this configuration never exists since voltage sources have internal resistances, while the wires between them are also not perfectly conducting, leading to a voltage drop. Therefore, a more realistic model of the physical scenario would require a resistor between the voltage sources, leading to a perfectly valid circuit that can be analyzed. All impossible configurations described above have similar missing components, which can be added to convert them into possible scenarios. Figure 1.15 Some possible circuits involving only one or two components that are connected consistently. In the first and second circuits, where there is a current and a voltage source, the sources do not produce any power. However, they still provide the current and voltage values, in accordance with their definitions. In the third circuit, the voltage source absorbs power (100 W), while the current source delivers power (100 W). Impossible scenarios rarely occur, even when we consider ideal components in circuit analysis. In general, many circuits have multiple components and connections, where the voltage and current values become consistent. In order to find relations between voltage and current values, we use basic rules, namely Kirchhoff's laws, as described in the next chapter. These rules, which are based on the conservation of energy and charge, provide the necessary equations to relate different voltage and current values. Figure 1.16 Two simple circuits that can be interpreted incorrectly as impossible. In fact, both two circuits are possible and they involve consistent voltage and current values. In the first circuit, a voltage drop (by an amount of V) exists across the resistor. In the second circuit, a nonzero current ( A) flows through the resistor. These values can easily be found by applying Kirchhoff's laws, as described in the next chapter. At this stage, we can start analyzing some simple circuits, just by considering the definitions of the components. Example 4 Consider a circuit involving a 10 V voltage source connected to a resistor. Note that voltage and current directions are defined arbitrarily, but they must obey the sign convention. We can analyze the circuit as follows. Using the definition of the voltage source: V. Using the definition of the voltage between two points and considering that the voltage is fixed along a wire: V. Using the definition of the resistor (Ohm's law): V. Using the definition of the current: A. Using the definition of the power: W. Using the definition of the power: W. The signs of power values indicate that the voltage source delivers energy, while the resistor consumes the same amount of energy. As expected, we have due to the conservation of energy. Example 5 Consider a circuit involving a 10 A current source connected to a resistor. In this case, the current through the circuit is determined via the current source, while the voltage value is found by applying Ohm's law. We can analyze the circuit as follows. Using the definition of the current source: A. Using the definition of the current: A. Using the definition of the resistor (Ohm's law): V. Using the definition of the voltage between two points and considering that the voltage is fixed along a wire: V. Using the definition of the power: W. Using the definition of the power: W. Similarly to the previous example, the current source delivers energy, while the resistor consumes the same amount of energy. Example 6 Consider a circuit involving a 10 V voltage source connected to a 10 A current source. We can analyze the circuit as follows.  Using the definition of the voltage source and voltage: V. Using the definition of the current source and current: A. Using the definition of the power: W and W. In this circuit, the voltage source produces energy, while the current source (despite also being a source) consumes energy. Once again the total power is zero due to the conservation of energy. Example 7 Consider the following circuit involving a 10 V voltage source, a current- dependent voltage source, and two resistors. We can analyze the circuit as follows. First, using the definition of voltage, we have Then, using Ohm's law, we get and The definition of the current-dependent voltage source leads to  The definition of the current-dependent voltage source leads to Therefore, using the definition of voltage again, we derive Using Ohm's law once again, we have and Finally, the powers of all components can be found: We note that both the independent and dependent source provides energy to the circuit, while both resistors consume. 1.6 Limitations in Circuit Analysis A type of circuit analysis, which is used throughout this book, is based on lumped-element models. Specifically, we assume that the behavior of components and their interactions with each other can be described by voltage– current relations given by the descriptions of the components. In addition, we assume that all elements demonstrate their ideal characteristics, being independent of outer conditions. All these assumptions rely on two constraints on the sizes of the components and circuits. The circuit components are large enough to omit individual behaviors of protons and electrons. Hence, without dealing with individual particles, their bulk behaviors (i.e., voltage and current) are used directly to model the components.  The circuit components and circuits are much smaller than the wavelength of the signals. For example, oscillations in the current and voltage values through a wire are synchronized and no phase accumulation occurs. In addition, voltage/current phase differences are well defined in all components. Obviously, lumped-element models fail when the size constraints are not satisfied. For example, in circuits larger than the wavelength, connections may need to be modeled as transmission lines, where wave equations are solved. Some circuits may need the full application of Maxwell's equations to precisely describe the electromagnetic interactions of components with each other. Depending on the complexity of the circuit model, further assumptions are often made to simplify the analysis of circuits. In this book, we accept the following assumptions that are also common in the circuit analysis literature. The voltage–current relationship defined for a component does not depend on outer conditions (temperature, pressure, light, etc.). This also means that the circuit behaves always the same (e.g., change in resistance due to rising temperature as the circuit is used is omitted). The voltage–current relationship defined for a component does not depend on other components. For example, a resistor of always satisfies Ohm's law as , independent of other elements used in the same circuit. We also ignore cross-talk of circuits and their parts, other than the linkage through well-defined dependent sources. All components are ideal and we omit secondary effects, such as the resistance of a voltage source, inductance of a capacitor, or capacitance of a resistor. If these effects cannot be neglected, they can be represented as individual components. For example, the leakage of a capacitor can be represented by a resistor connected in parallel to the capacitor. Despite all these limitations and assumptions, circuit analysis methods presented in this book are widely accepted and used to analyze diverse circuits and electrical devices. In many cases, lumped elements are used as starting models before more complicated analysis techniques are applied. 1.7 What You Need to Know before You Continue Before proceeding to the next chapter, we summarize a few key points that need Before proceeding to the next chapter, we summarize a few key points that need to be known to understand the higher-level topics. Sign convention: In this book, the current through a component is selected to flow from the positive to the negative side of the voltage. Steady state: For DC circuits in steady state, voltage and current values are assumed to be constant. In the first few chapters, steady state is automatically assumed when only resistors and DC sources are considered. Short circuit and open circuit: Short circuit and open circuit can be interpreted as special cases of resistors, with zero and infinite resistance values, respectively. Sources: There are alternative symbols to show voltage and current sources; circular representations are used in this book. DC/AC types are indicated in the context of source values. Energy conservation: For a given isolated circuit, the sum of powers of all components must be zero due to the conservation of energy. Series connection: Components that share the same current are connected in series. Parallel connection: Components that share the same voltage are connected in parallel. Impossible configurations: Some connections of ideal components are not allowed due to inconsistency of voltage and current values enforced by the definitions of components. In the next chapter, we start with the most basic tools, namely Kirchhoff's laws, to analyze circuits. Chapter 2 Basic Tools: Kirchhoff's Laws Circuits with a few components can be analyzed by using only the definitions of the components. On the other hand, as the circuits become more complicated, involving connections of many components, well-defined solution tools are required to derive the necessary equations. This chapter presents the most basic laws of circuit analysis based on the conservation of charges (Kirchhoff's current law) and conservation of energy (Kirchhoff's voltage law). These fundamental rules, collectively named Kirchhoff's laws, can be used to derive useful equations at nodes and in loops, in order to relate the voltages and currents of components to each other. Solutions of the resulting equations lead to numerical values of these variables, hence the analysis of the given circuit. In general, Kirchhoff's laws should be sufficient to solve any type of circuit. On the other hand, when a circuit involves many resistors, a direct application of Kirchhoff's laws may lead to too many equations that can be difficult to solve. For scenarios of this kind, where multiple resistors are connected in series and parallel, one can derive shortcuts (again using Kirchhoff's laws) to simplify and represent the overall circuit using a few resistors. This chapter also presents such shortcuts for the analysis of large resistive circuits. 2.1 Kirchhoff's Current Law According to Kirchhoff's current law (KCL), the sum of currents entering ( ) and leaving ( ) a node should be zero, that is, where is the number of wires connected at the node. In the examples above, we have The conservation of charge is the background to KCL, which can be generalized to currents entering and leaving closed surfaces. While the selection could be arbitrary, we always select the entering and leaving currents as positive and negative, respectively. Obviously, KCL assumes that no charge accumulation occurs at the node or closed surface considered. In the analysis of circuits, we usually apply KCL at nodes. The format that we adopt is KCL( ): equation derived by applying KCL, where represents the node. 2.2 Kirchhoff's Voltage Law Kirchhoff's voltage law (KVL) states that the sum of voltages in a closed loop is zero, that is, where is the number of segments (e.g., components) along the loop. In the examples above, we have KVL is based on the conservation of energy. When applying KVL, an added voltage can be a voltage across a component or any voltage between two consecutive nodes in the loop. While the direction can be selected arbitrarily, we always use the clockwise direction when adding the voltages. A plus or minus sign is used depending on the polarization of the added voltage. In the analysis of circuits, we usually apply KVL in small closed loops, each of which is called a mesh. Similar to KCL, we use a format such as KVL( ): equation derived by applying KVL, where , , and represent the nodes forming the mesh. Example 8: Consider the following circuit involving a voltage source, a current source, and two resistors.  Find the voltage across the resistor. Solution With the given definitions of the voltages and currents on the circuit, we need to apply KCL and KVL systematically in order to solve the problem. One obtains KVL( ): V, KCL(2): A. Then, using Ohm's law, and , leading to. Finally, solving for , we obtain A, A, V, and as the voltage across the resistor. As one verification of the solution, one can calculate the powers of all components: and check that  As far as the power values are concerned, we can conclude that the current source delivers power, while the resistors, as well as the voltage source, consume power. Example 9: Consider the following circuit involving a voltage source, which is connected to two resistors and a bulb. Find the power of the bulb. Solution In most circuits, the voltage and current directions are not defined a priori. Therefore, when analyzing such a circuit, we define the directions arbitrarily, while enforcing the sign convention for all components. When a current/voltage value is found to be negative, we understand that the initial assumption is not correct. However, this is not a problem at all, provided that we are consistent with the directions throughout the solution. For the circuit above, we label the nodes, define the directions of the currents, and define the voltages in accordance with the sign convention, as follows.  Using KVL, one obtains KVL( ): V. Then, using Ohm's law, A, and we further have KCL(2): A. Finally, the power of the device is found to be Example 10: Consider the following circuit involving two voltage sources connected to three resistors.  Find the value of that passes through the 10 V source. Solution We again start with KVL to obtain KVL( ): V. We note that , , and , leading to Furthermore, V and A. Finally, applying KCL at node 2, we get KCL(2): A. Example 11: Consider the following circuit, where a voltage source and a current source are connected to three resistors. Find the value , that is, the current across the resistor. Solution First, we label the nodes, define the directions of the currents, and define the voltages in accordance with the sign convention, as follows. Then, using KVL and Ohm's law, we have , , and KVL( ): V. Therefore, we have Furthermore, using KCL (see below for some details), we derive KCL(2): A. Using and , we obtain Finally, solving two equations with two unknowns, we get In the above, we note that node 2 (as well as node 3) is defined simultaneously at two intersections, and KCL is written accordingly by considering all entering and leaving currents, as follows.  This is a common practice in circuit analysis in order to reduce the number of equations. Specifically, intersections without a component between them can be considered as a single node to avoid writing redundant equations with redundant unknowns. On the other hand, this not mandatory. For example, one can consider each intersection as a node, as follows. In this case, we need to define a current between nodes 2 and 4. Writing KCL at the nodes, we now have KCL(2): , KCL(4):. Obviously, when these equations are combined (directly added), we arrive at the same equation in the original solution, Example 12: Consider the following circuit.  Find the value of. Solution First, we label the nodes, define the directions of the currents, and define the voltages in accordance with the sign convention, as follows. Note that we again define node 2 as the combination of two intersection points. Besides, in order to simplify the solution, we do not define voltage variables separately as they are already related to the currents via Ohm's law. Using KVL, we derive  KVL( ): , KVL( ):. Then, using KCL, we obtain KCL(2):. Finally, we have In solving this example, we further note the following. The voltage across the current source is unknown. Hence, it is not useful to write KVL for. The current across the voltage source is unknown. Hence, it is not useful to write KCL at 1 or 3. In general, we avoid writing KCL at a node, to which a voltage source is connected, unless it is mandatory to find the current through the voltage source. In addition, applying KVL in a mesh containing a current source is usually not useful, unless the voltage across the current source must be found. Example 13: Consider the following circuit involving a voltage source and a current source connected to four different resistors.  Find the value of , that is, the voltage across the resistor. Solution As in the previous examples, we label the nodes, define the directions of the currents, and define the voltages in accordance with the sign convention. In order to simplify the solution, we again do not define voltage variables separately and write all equations in terms of currents. Applying KVL and KCL consecutively, we obtain KVL( ): , KCL(4): , KCL(3): , KVL( ):  A. Finally, we have A and V. Example 14: Consider the following circuit involving a total of eight components. Find. Solution We again label the nodes, from 1 to 4, and define the current directions. Node 2 is defined as the combination of three intersection points.  Applying KCL at nodes 3 and 2, we obtain KCL(3): , KCL(2):. Then, using KVL, we derive KVL( ): , KVL( ): A. Using the updated information, we can also find , as well as as Finally, we apply KVL( ) to find as Example 15: Consider the following circuit involving six components.  Find the value of. Solution As in the previous examples, we label the nodes, define the directions of the currents, and define the voltages using the sign convention. We note that the 2 resistor is short-circuited and can be omitted in the analysis. Using KVL and KCL, we obtain KVL( ): A, KCL(2): A, KVL( ): V.  As briefly discussed before, it is generally suggested to avoid using KCL at a node with a connection to a voltage source. However, in this case, the current across the voltage source must be found in order to find the voltage value. Therefore, we apply KCL at node 2. Example 16: Consider the following circuit involving six components, in addition to a given current along a path. Find the value of. Solution Once again, we process the circuit as follows.  Using KVL and KCL, we derive KVL( ): A, KCL(2): A, KVL( ): V. Interestingly, the solution above does not depend on the resistor, and is always 24 V if this resistor is not zero (not short-circuited). If this resistor was short-circuited, then the question would be inconsistent. Exercise 2: In the following circuit, the device works for currents in the range from 4 A to 6 A. Find the range of values for. Exercise 3: In the following circuit, find the value of. Exercise 4: In the following circuit, find the value of. Exercise 5: In the following circuit, find the value of.  Exercise 6: In the following circuit, find the value of. Exercise 7: In the following circuit, find the power of the device. 2.3 When Things Go Wrong with KCL and KVL While KCL and KVL, in general, are easy to understand conceptually, they can be tricky to use, especially when circuits involve many components. In the circuit above, we again would like to find. As indicated before, when analyzing this circuit, applying KCL at node 1 or 3 would not be useful. For example, consider KCL at node 3. We must define a current between nodes 1 and 3 as follows. This way, we can write KCL as KCL(3): , which does not provide new information as must be defined in order to write this equation. The scenario becomes more interesting when we consider KCL on the other side, at node 1. We have KCL(1):. Now, combining (adding) the two equations above, we further derive In this useful equation, we observe that the new variable is eliminated. In other words, while KCL at node 1 or 3 would not be useful alone, they could be used together to derive a single useful equation. In the original solution this is not considered, as the necessary equations can already be obtained by means of KVL equations. However, in nodal analysis (see Chapter 3), where only KCL applications are allowed, we develop the supernode concept that effectively combines KCL equations at nodes, as practiced above. On the other hand, in some cases, current along a voltage source may be required, needing KCL at one of its terminals. For example, consider the power of the voltage source above. Using KCL at node 3 and borrowing A from the original solution, we obtain Hence, considering the sign convention, the power of the voltage source is found to be W. Problems also arise when applying KVL in a mesh involving a current source. We again consider the circuit above, while applying KVL in loops containing the current source. Using one of the resistors, we have the following scenario. Then KVL leads to KVL( ): , where is the voltage across the 4 A source, using the sign convention. Obviously, this equation is not useful alone, since it involves a new unknown,. Constructing KVL through the resistor, we can also obtain KVL( ):. The two equations above can be combined to eliminate as Nevertheless, the same equation could be found by applying KVL through the resistor, the resistor, and the voltage source (see the original solution, where is used instead of ). Confusion often occurs when dealing with short circuits. We reconsider the following circuit, where a resistor is short-circuited. Our aim is to find the current through the short circuit,. We label one of the related intersections node 4, whereas it is labeled node 1 in the original solution. Obviously, nodes 1 and 4 have the same voltage. However, this does not mean that there is no current between them. In fact, applying KCL at node 1, we derive KCL(1):. We note that there is no current flow through the resistor. Using A, we obtain A. One can check this value by applying KCL at the new node 4, KCL(4): , where A, as found before. Dealing with too many current or voltage values may also lead to confusion in circuit analysis. We consider the following circuit, where the current values need to be found. Applying KVL, we obtain the two equations KVL( ): , KVL( ):. In addition, we obtain KVL( ): or This final KVL equation is correct; unfortunately, it does not provide new information compared to the previous two equations. Indeed, combination of the first two KVL equations to eliminate leads to which is exactly the same as the whole KVL through. To sum up, only two of the three KVL equations above provide useful information, while the third one is redundant. In order to solve the problem above, we note that application of KCL at nodes 2 and 4 gives KCL(2): , KCL(4):. With these two equations, the total number of equations reaches four, while there are five unknowns ( , , , , and ). The missing equation can be obtained by applying KVL through the voltage source. For example, one can derive KVL( ):. At this stage, we can list all the useful equations once more as or in matrix form, Solving the matrix equation leads to A, A, A, A, and A. Obviously, the question above seems difficult to solve when considering five unknowns. In fact, a systematic application of KCL (see Chapter 3) or KVL (see Chapter 4) would lead to only three unknowns, as well as three equations to find them. As set out above, a solution with heuristic applications of KCL and KVL may lead to linearly dependent equations that often lead to an obvious equality after substitutions, insufficient numbers of equations, if the same variable is eliminated by substitution from all equations, looping through the same equalities if a node or mesh is excessively used in deriving alternative equations. All these pitfalls can be avoided by choosing suitable strategies, such as nodal and mesh analysis, as described in the next chapters. 2.4 Series and Parallel Connections of Resistors KCL and KVL provide all the information required to solve a given basic circuit. On the other hand, their application to alternative connections of resistors leads to shortcuts that can be employed to simplify the analysis. 2.4.1 Series Connection Consider a series connection of two resistors with resistances and. Using KVL, we have where and. Then where is the equivalent resistance of the combination of two resistors. We further note that the powers of the resistors are given by Hence, the consumed power is divided into two parts that are proportional to the resistance values. Obviously, when there are resistors all connected in series, we derive the equivalent resistance as 2.4.2 Parallel Connection Consider a parallel connection of two resistors with resistances and. Using KCL, we derive where and. Hence, we obtain where represents the equivalent resistance of the combination of two resistors. One can also derive as the equivalent resistance. We note that, if and , we have and Therefore, the total resistance of a parallel connection is always smaller than the individual resistance of each resistor. It is remarkable that, when , we obtain , showing that is short-circuited. On the other hand, when , one can derive as the overall resistance. When two resistors are connected in parallel, the total current is divided into two parts as Obviously, the current tends to flow though the smaller resistance. Then the consumed power is distributed as inversely proportional to the resistance values as If there are resistors all connected in parallel, the equivalent resistance can be written as Example 17: Consider the following circuit involving seven resistors connected to a 35 V voltage source. Find. Solution By using the formulas for series and parallel connections, we systematically find equivalent resistances to reduce the circuit. We have Then the current is simply Example 18: Consider the following circuit involving a one-dimensional infinite array of resistors connected to a 2 V voltage source. Find. Solution First we assume that the equivalent resistance of the array is. Then, considering just the first loop, we have which can be solved to obtain. Therefore, Example 19: Consider the following circuit involving a 60 V voltage source connected to a network of resistors.  Find. Solution Instead of applying KVL and KCL directly, we first reduce the circuit by considering series and parallel connections of resistors. We have Then, after finding the main current to be A, we trace back the circuit as follows. Hence, we obtain Hence, we obtain Exercise 8: In the following circuit, find the value of. Exercise 9: In the following circuit, find the value of. Exercise 10: In the following circuit, find the value of. 2.5 When Things Go Wrong with Series/Parallel Resistors Major issues arise when considering resistors that appear to be connected in series or parallel, but in fact are not. We consider the following three different circuits. In the first case (the circuit on the left), and are connected in series, leading to total resistance. Similarly, and are connected in series, leading to. Then the overall resistance is given by In the second case (the circuit in the middle), and are connected in parallel, while and are similarly parallel. Therefore, the overall resistance is which is obviously different from. Only in some cases, such as when , do we have. In the third case (the circuit on the right), where or , none of the resistor pairs are connected in series or parallel. This is because, none of the pairs share a common current or common voltage. For general values of the resistors, this circuit must be analyzed via KVL and KCL or via nodal/mesh methods. Interestingly, when , we have , independent of the value of. 2.6 What You Need to Know before You Continue Here are a few key points before proceeding to the next chapter. KCL: The sum of currents entering ( ) and leaving ( ) a node should be zero. When writing a KCL equation, we use plus and minus signs for entering and leaving currents, respectively. KVL: The sum of voltages in a closed loop is zero. When writing a KVL equation, we always choose the clockwise direction. Then the voltages of the components are added by considering their signs according to their first terminals. Current and voltage directions: When analyzing circuits, we define the current and voltage directions arbitrarily, while enforcing the sign convention for all components. When a current/voltage value is found to be negative, we understand that the initial assumption is not correct, although this is not a problem. Useless equations: We usually avoid applying KCL at a node to which a  voltage source is connected. Similarly, a KVL in a mesh containing a current source is usually not useful. Series/parallel resistors: Combinations of resistors can often be simplified by considering series and parallel connections before the circuit is analyzed via KCL and KVL. In the next chapter, we focus on nodal analysis, which is a systematic way of applying KCL at nodes. Such methods are essential for deriving required numbers of equations, so as to avoid redundant equations and variables, when analyzing circuits. Chapter 3 Analysis of Resistive Networks: Nodal Analysis KCL and KVL are fundamental laws for analyzing circuits. However, as discussed in the previous chapter, their application may not be always clear. Specifically, for large circuits involving many components and connections, it may be difficult to derive necessary relationships between voltages and currents, while avoiding duplications and linearly dependent equations. In this chapter, we focus on nodal analysis, which is a higher-level tool based on a systematic application of KCL. Given any circuit, a proper application of nodal analysis guarantees the derivation of necessary equations for the analysis. We also discuss the generalization of nodes to further use the benefits of nodal analysis for complex circuits. 3.1 Application of Nodal Analysis We start by briefly listing the major steps of nodal analysis as follows. Ground selection: We select a reference node with zero voltage. Any node can be selected, but it usually better to choose one with more connections than the others. The node selected is called the ground of the circuit. All voltages at other nodes are defined with respect to the ground. Constructing equations: We use only KCL at nodes, except the ground, to derive all equations. KVL is not preferred in nodal analysis unless necessary. We write all equations in terms of node voltages. Solution: Next, we solve the equations to find the node voltages. Analysis: Finally, by using the node voltages, we find the desired voltage, current, and/or power values. Once again, we emphasize that a formal voltage definition requires two points. On the other hand, if there is a node where the voltage is defined as zero, it becomes practical to define node voltages as if they are independent values. Example 20 Consider the following circuit involving a current-dependent current source, a voltage source, and two resistors. Find , which flows through the voltage source. Solution This circuit can be analyzed using KCL and KVL, as usual. After labeling nodes and defining directions, we have the following circuit. Then, applying KCL and KVL, we derive KCL(1): , KVL( ):. Finally, using Ohm's law, As an alternative solution, we now apply nodal analysis, selecting node 2 as the reference node.  Using Ohm's law, we have. Then, using KCL, KCL(1): , leading to V. Therefore, A. We note that only one KCL equation has been sufficient to analyze the circuit. Specifically, two of the nodes in the earlier analysis are not used in nodal analysis. Some important points are as follows. Node 2 is made the reference node (ground) with zero voltage. In general, KCL need not be applied at a ground. In fact, needing to apply KCL at a ground is usually an indicator that another node has been skipped by mistake in the analysis. Node 3 is also not used directly in nodal analysis, because its voltage is already known due to the voltage source. In general, if the voltage at a node is easily defined, one does not need to write a KCL equation at that node. In fact, applying KCL at a node with a directly connected voltage source should be avoided, unless it is mandatory (e.g., if one must find the current through the voltage source). Application of KVL should be avoided in nodal analysis since a proper set of KCL equations should be sufficient to solve the circuit. KVL can be used to find other quantities after all node voltages are obtained. Two facts provide a deeper understanding of nodal analysis. Setting zero voltage at the ground is merely a choice. Indeed, one could assign any voltage (e.g., 10 V), which would shift voltage values at all other nodes by 10. On the other hand, real quantities, such as component voltages, currents, and powers, do not depend on  this selection. Selecting a node as ground is also completely arbitrary. One can choose any node as a reference, provided that the voltages are defined accordingly. As mentioned above, certain selections (e.g., choosing nodes with more connections) can simplify nodal analysis. Example 21 Consider the following circuit. Find. Solution Applying nodal analysis, we select a ground and define other node voltages accordingly. We note that only one node needs to be defined and used in the analysis, since two other nodes already have well-defined voltage values (i.e.,  and 70 V). We also have using Ohm's law. Applying KCL, we have KCL(1): , leading to or Then, using the relation between and , we obtain Finally, we have Example 22 Consider the following circuit. Find. Solution As in the previous examples, we select a ground and define other node voltages accordingly.  Using Ohm's law, we have. Then, applying KCL at node 1, we derive KCL(1): V A. In the above, and 4 A currents are leaving node 1 so that they are written as negative on the left-hand side of the equation. For the current flowing in the first branch, we have two options, leading to the same result. Defining the current as entering node 1, its value should be. This value should be treated as positive in the KCL equation. Defining the current as leaving node 1, its value should be. This value should be treated as negative in the KCL equation, making it overall. Hence, the choice of current directions will change neither the KCL equation nor the result. In order to complete the solution and find , we appply KCL at node 2, yielding KCL(2): V, leading to Example 23 Consider the following circuit. Find. Solution Using nodal analysis, we again select a ground and define other node voltages accordingly. Using KCL at nodes 1 and 2, we obtain two equations with two unknowns as KCL(1): , KCL(2):. Finally, solving the equations, we get V, V, and  A. Example 24 Consider the following circuit. Find. Solution We again start by selecting a ground and defining the other node voltages accordingly.  Following the selections above, we note that. Applying KCL at nodes 1 and 2, we derive KCL(1): , KCL(2): V. Then we have V and A. Example 25 Consider the following circuit with three sources and four resistors. Find the power of the resistor. Solution Using nodal analysis, we define only two nodes to construct equations.  In the above,. Applying KCL, we have KCL(1): , KCL(2):. Then we obtain V, V, and W. Example 26 Consider the following circuit. Find. Solution  Using nodal analysis, we have just two nodes where we apply KCL. Applying KCL at node 1, we have KCL(1): , where is used. Similarly, at node 2, we have KCL(2):. Solving the equations, we obtain V and Example 27 Consider the following circuit. Find the power of the current-dependent current source.  Find the power of the current-dependent current source. Solution Using nodal analysis, we again have two nodes where we apply KCL. First, we note that , according to the selection of the reference node. Using KCL at nodes 1 and 2, we derive KCL(1): V and A, KCL(2): V. The power of the current-dependent current source can be found by multiplying its voltage and current (keeping consistent with the sign convention), yielding Example 28 Consider the following circuit involving four different kinds of sources.  Find. Solution First, we note that A. Using nodal analysis, we can define the ground and label the nodes as follows. We further note that KCL is not required (and not useful) at node 1, since V. In addition, one should avoid applying KCL at node 3 since a voltage source (dependent source in this case) is connected to this node. Instead, using the definition of the voltage source, we have where. Therefore,  or as the first equation. For the solution, we apply KCL at the only remaining node (node 2), yielding KCL(2): V. Finally, we obtain V and A. Example 29 Consider the following circuit. Find. Solution The circuit looks complex at first glance, but can easily be solved using nodal analysis.  As shown above, using a suitable ground location, we have only one node where we need to apply KCL, and voltages at all other nodes are already known. We have KCL(1): V, leading to Exercise 11 In the following circuit, find the value of. Exercise 12 In the following circuit, find the power of the resistor. Exercise 13 In the following circuit, find the power of the resistor. Exercise 14 In the following circuit, find given that A. Exercise 15 In the following circuit, find the value of. Exercise 16 In the following circuit, find the power of the 8 A source. Exercise 17  In the following circuit, find the value of. Exercise 18 In the following circuit, find the value of. Exercise 19 In the following circuit, find the value of. Exercise 20 In the following circuit, find the value of. 3.2 Concept of Supernode KCL can be generalized to any arbitrary surface: the sum of currents entering and leaving a closed surface must be zero. An application of this generalized form is to define supernodes that contain several nodes, as well as components, in nodal analysis. A supernode enclosing a voltage source is particularly useful, especially when there are nodes (attached to this voltage source) at which voltages cannot be defined easily. Example 30 Consider the following circuit. Find. Solution Solution Using nodal analysis, we define the ground and label the nodes as follows. In the above, it is straightforward to write a KCL equation at node 1. On the other hand, a KCL equation at node 2 requires the current along the 10 V source, which cannot be defined in terms of node voltages. Interestingly, for this circuit, there is a shortcut without resorting to a supernode. Considering node 3, one can claim that the current along the voltage source is actually , since the source is serially connected to a resistor. However, this is a very special case, and we often find nodes where KCL is not trivial to write. In order to facilitate nodal analysis for this circuit, we combine nodes 2 and 3, leading to a supernode that contains the volta