## Preface

The field of electrical and electronic engineering is vast and diverse. However, two topics hold the key to the entire field. They are ‘Circuit Theory’ and ‘Signals and Systems’. Both these topics provide a solid foundation for later learning, as well as for future professional activities.

This undergraduate textbook, the first of a two-book series, deals with one of these two pivotal subjects in detail. In addition, it connects ‘Circuit Theory’ and ‘Signals and Systems’, thereby preparing the student-reader for a detailed study of this important subject either concurrently or subsequently.

The theory of electric circuits and networks, a subject derived from the more basic subject of electromagnetic fields, is the cornerstone of electrical and electronics engineering. Students need to master this subject and assimilate its basic concepts in order to become competent engineers.

My book *Electric Circuits and Networks* (ISBN 978-81-317-1390-7), published by Pearson
Education in 2009, was an attempt to provide a solid foundation on electric circuits and networks
to the undergraduate students. However, this book was perceived as being too voluminous and too
comprehensive for a first-level course on Circuits. Hence, for better acceptability and better utilization
of the content, I decided to rewrite the material and present it in two books. Of these, the first one
on Electric Circuit Analysis has been designed to serve as a textbook for first/second level course
on circuits and the second one on Electric Network Analysis and Synthesis has been envisaged for
an advanced level course on network analysis and synthesis. The latter text is being augmented with
additional content that caters to the needs of the advanced user. This text, *Electric Circuit Analysis,* is
the first of the series, while work on the second text with new material included on network functions,
electric filters and passive network design, is currently in progress.

#### Objectives of the Book Series

*Primary Objective*: To serve as textbooks that will meet students’ and instructors’ need for a one/
two-semester course on electrical circuits and networks for undergraduate students of electrical and
electronics engineering (EE), electronics and communications engineering (EC) and allied streams.
This textbook series introduces, explains and reinforces the basic concepts of analysis of dynamic
circuits in time-domain and frequency-domain.

*Secondary Objective*: To use circuit theory as a carrier of the fundamentals of linear system and
continuous signal analysis so that the students of EE and EC streams are well-prepared to take up a
detailed study of higher level subjects such as analog and digital electronics, pulse electronics, analog
and digital communication systems, digital signal processing, control systems, and power electronics
at a later stage.

#### Electric Circuits in EE and EC Curricula

The subject of Electric Circuits and Networks is currently covered in two courses in Indian technical
universities. The introductory portion is covered as a part of a course offered in the first year of
undergraduate program. It is usually called ‘Basic Electrical Engineering’. About half of the course
time is devoted to introductory circuit theory covering the basic principles, DC circuit analysis, circuit
theorems and single frequency sinusoidal steady-state analysis using phasor theory. This course is
usually a core course for *all disciplines.* Therefore, it is limited very much in its content and depth as
far as topics in circuit theory are concerned. The course is aimed at giving an overview of electrical
engineering to undergraduate students of all engineering disciplines.

Students of disciplines other than EE and EC need to be given a brief exposure to electrical machines, industrial electronics, power systems etc., in the third semester. Many universities include this content in the form of a course called ‘Electrical Technology’ in the third semester for students of other engineering disciplines. This approach makes it necessary to teach them AC steady-state analysis of RLC circuits even before they can be told about transient response in such circuits. EE students, however, need AC phasor analysis only from the fourth or fifth semester when they start on Electric Machines and Power Systems. But the first year course on basic electrical engineering has to be a common course and hence even EE and EC students learn AC steady-state analysis before transient response.

The second course on circuits is usually taught in the third or fourth semester and is termed ‘Electric Circuit Theory’ for EE students and ‘Circuits and Networks’ or ‘Network Analysis’ for EC students. Few comments on these different course titles and course content are in order.

Traditionally, undergraduate circuit theory courses for EE stream slant towards a ‘steady-state’ approach to teaching circuit theory. The syllabi of many universities in India contain extensive coverage on single-phase and three-phase circuits with the transients in RC and RL circuits postponed to the last module in the syllabus. The course instructor usually finds himself with insufficient contact hours towards the end of the semester to do full justice to this topic. EE stream often orients Circuits courses to serve as prerequisites for courses on electrical machines and power systems.

This led to the EC stream preparing a different syllabus for their second-level circuit theory course—one that was expected to orient the student towards the dynamic behaviour of circuits in time-domain and analysis of dynamic behaviour in the frequency domain. But, in practice, the syllabus for this subject is an attempt to crowd too many topics from Network Analysis and Synthesis into what should have been a basic course on Circuits.

Such a difference in orientation between the EE-stream syllabus for circuit theory and EC-stream syllabus for circuit theory is neither needed nor desirable. The demarcation line between EE and EC has blurred considerably over the last few years. In fact, students of both disciplines need good coverage of Linear Systems Analysis or Signals and Systems in the third or fourth semester. Unfortunately, Linear Systems Analysis has gone out of the curriculum even in those universities which were wise enough to introduce it earlier, and Signals and Systems has started making its appearance in EC curriculum in many universities. But the EE stream is yet to lose its penchant for AC steady-state in many Indian technical universities.

The subject of electrical circuit theory is as* electronic* as it is *electric.* Inductors and capacitors
do not get scared and behave differently when they see a transistor. Neither do they reach sinusoidal
steady-state without going through a transient state just because they happen to be part of a power
system or electrical machine.

Against this background, I state the pedagogical viewpoint I have adopted in writing this textbook.

#### Pedagogical Viewpoint Adopted in this Book

- With a few minor changes in emphasis here and there, both EE and EC students need the same Circuit Theory course.
- Introducing time-domain response of circuits before AC steady-state response is pedagogically superior. However, curricular constraints make it necessary to introduce AC steady-state analysis first and it is done that way in this book.
- ‘Lumped Linear Electrical Circuits’ is an ideally suited subject to introduce and reinforce ‘Linear System’ concepts and ‘Signals and Systems’ concepts in the EE and EC undergraduate courses. This is especially important in view of shortage of course time, which makes it difficult to introduce full-fledged courses in these two subjects. This textbook is organized along the flow of Linear Systems Analysis concepts.
- Circuit Theory is a very important
*foundation course*for EE, EC and allied disciplines. The quality of teaching and intellectual capability of students (especially the quality of teaching) varies widely in different sectors of technical educational institutions in India. Therefore, a textbook on circuit theory has to be written explaining the basic concepts thoroughly and repeatedly, with the average student in mind—not the brilliant ones who manage to get into ivy-league institutions. Such a textbook will supplement good teaching in the case of students of premier institutions and, more importantly, save the average students from life-long confusion. - The pages of a textbook on Circuit Theory are precious due to the reasons described above. Therefore, all extraneous matter should be dispensed with. The first in this category is the so-called historical vignettes aimed at motivating the students. I have avoided them and instead, used the precious pages to explain basic concepts from different points of view.
- The pre-engineering school curriculum in India prepares the students well in mathematics and physics. Engineering students have not yet become impatient enough to demand examples of practical applications of each and every basic concept introduced in subjects like Circuit Theory or Newtonian Mechanics. There is no need to keep motivating the student by citing synthetic-looking examples of complex electrical and electronic systems when one is writing on basic topics in Circuit Theory. The pages can be used for providing detailed explanation on basic concepts. The first year or second year undergraduate student is far away from a practical engineering application! I believe that a typical Indian engineering student is willing to cover the distance patiently.
- Circuit Theory is a foundation course. It is difficult to quote a practical application for each and every concept without spending considerable number of pages to describe the application and set the background. The pedagogical impact of this wasteful exercise is doubtful. However, those applications that are within the general information level of an undergraduate student should be included. Thus, applications that require long explanations to fit them into the context must be avoided in the interest of saving pages for explanations on Circuit Theory concepts.
- Circuit Theory is a basic subject. All other topics that the students are going to learn in future
semesters will be anchored on it. Hence, it should be possible to set pointers to applications in higher
topics in a textbook on Circuit Theory. Such pointers can come in the form of worked examples
or end-of-chapter problems that take up an idealized version of some practical application. An
example would be to use an idealized form of fly-back switched mode converter and to show how
the essential working of this converter can be understood from the inductance
*v*–*i*relationship. In fact, all well-known switched mode power converter circuits can be employed in the chapter which deals with the*v*–*i*relation of an inductor. Similarly, switched capacitor circuits can be introduced in the section dealing with the*v*–*i*relation of a capacitor. - Circuit Theory can be learnt well without using simulation software. Circuit simulation packages are only tools. I am of the opinion that using simulation software becomes a source of distraction in a foundation course. A foundation course is aimed at flexing the student’s intellect in order to encourage the growth of analytical capability in him.
- An argument usually put forth in support of simulation software as an educational aid is that it helps one to study the response of circuits for various parameter sets and visualize the effect of such variations. That is precisely why I oppose it in a foundation course. Ability to visualize such things using his/her head and his/her ability for mental imagery is very much essential in an engineer. Let the student develop that first. He/she can seek the help of simulation software later when he/she is dealing with a complex circuit that goes beyond the limits of mental imagery.

After all, we do not include a long chapter on waveform generators and another one on oscilloscopes in every Circuit Theory textbook. In fact, some of the modern-day waveform generators and oscilloscopes are so complex that a chapter on each of them will not really be out of place. Yet, we do not spend pages of a Circuit Theory textbook for that. The same rule governs simulation software too.

#### Pedagogy

- Every chapter begins with a statement of chapter objectives and relative emphasis of topics covered in that chapter.
- Detailed summary covering all the important points made in the chapter is provided at the end of each chapter.
- Boxed entries highlight important concepts and reinforce them. The book is replete with worked examples illustrating the concepts explained in the text are included. Simple formula-substitution kind of worked examples are avoided.
- Large number of problems is included at the end of every chapter. Section-wise organization
of these problems is
*avoided intentionally*. I expect the student to understand the entire chapter and use all the concepts covered in that chapter (and from earlier chapters) to solve a problem if necessary. After all, no one tells him which concepts are relevant in solving a particular problem in the examination hall or in practical engineering.

#### Outline and Organization

This book contains 14 chapters.

The first three chapters address the basic concepts. The first chapter goes into the physics of two-terminal circuit elements briefly and deals with element relations, circuit variables, and sign convention. It also addresses the concepts of linearity, time-invariance and bilaterality properties of two-terminal elements. This chapter assumes that the reader has been introduced to the basic physics of electromagnetic fields in pre-engineering high school physics. It also attempts to explain the important assumptions underlying circuit theory from the point of view of electromagnetic fields. The treatment is qualitative and not at all intended to be rigorous.

The second chapter covers the two basic laws – Kirchhoff ’s voltage law and Kirchoff ’s current laws – in detail. Emphasis is placed on the applicability of these two laws under various conditions.

The third chapter looks into the *v*–*i* relationship of the resistor, the inductor and the capacitor.
Series-parallel equivalents are also covered in this chapter. This chapter analyses the *v*–*i* relations of
inductor and capacitor in great detail. The concept of ‘memory’ in circuit elements is introduced in
this chapter and the electrical circuits are divided into two classes – *memoryless circuits* and *circuits with memory.* Circuits with memory are termed as *Dynamic Circuits* from that point onwards.

The next two chapters deal with analysis of *memoryless circuits.* Chapter 4 takes up the analysis
of memoryless circuits containing independent voltage and current sources, linear resistors and linear
memoryless dependent sources using node analysis and mesh analysis methods. An argument based
on nodal admittance matrix (or mesh impedance matrix) and its cofactors is used to show that a
memoryless circuit comprising memoryless linear two-terminal elements will be a *linear system* and
that it will obey the superposition principle. Chapter 5 systematically develops all important circuit
theorems from the properties of a linear system.

After the analysis of memoryless circuits, the book moves on to elucidate sinusoidal steady-state in dynamic circuits. This part of the book starts with a detailed look at power and energy in periodic waveforms in Chapter 6. The periodic sinusoid is introduced and the concepts of its amplitude, frequency and phase are made clear. The concept of cycle-average power in the context of periodic waveforms is covered in detail.

Chapter 7 begins with a qualitative description of transient response and forced response taking an RL circuit as an example, and illustrates how the sinusoidal steady-state can be solved by using the complex exponential function. It goes on to expound on phasor theory, transformation of the circuit into phasor domain, solving the circuit in phasor domain, and moving back to time-domain. It also introduces active power, reactive power and power factor and presents the basic ideas of frequency response.

Chapter 8 takes up three-phase balanced and unbalanced circuits and includes symmetrical components as well. Unbalanced three-phase circuits and symmetrical components may be optional in ‘Basic Electrical Engineering’ course.

Chapter 9 addresses the issue of expanding a periodic waveform along the imaginary axis in signal space at discrete points. Fourier series in trigonometric and exponential forms are covered in detail in this chapter. This chapter (i) explains how a periodic waveform can be expanded in terms of sinusoids and why such an expansion is necessary (ii) shows how such an expansion may be obtained for a given periodic waveform, and (iii) shows how the expansion can be used to solve for the forced response of a circuit.

Expansion of input functions along imaginary axis in signal space for aperiodic waveforms through Fourier transforms is not included in this text. It will be included in the second text of the series.

The next three chapters deal with the time-domain analysis of dynamic circuits using the differential equation approach. Chapter 10 is one of the key chapters in the book. It takes up a simple RL circuit and uses it as an example system to develop many important linear systems concepts. The complete response of an RL circuit to various kinds of inputs such as unit impulse, unit step, unit complex exponential, and unit sinusoid is fully delineated from various points of view in this chapter. The chapter further expounds on the need and sufficiency of initial current specification, the concepts of time constant, rise and fall times, and bandwidth.

The response of a circuit is viewed as the sum of *transient response and forced response* on the
one hand and as the sum of zero-input response and zero-state response on the other. The role of
various response components is clearly spelt out. The application of superposition principle to zero-state
component and zero-input component is examined in detail.

Impulse response is shown to be an all-important response of a circuit. The equivalence between impulse excitation and non-zero initial conditions is established in this chapter. The chapter also shows how to derive the zero-state response to other inputs like unit step and unit ramp from impulse response in detail. The tendency of inductance to keep a circuit current smooth is pointed out and illustrated.

The notions of DC steady-state, AC steady-state and periodic steady-state are made clear and demonstrated through several worked examples. The chapter ends with a general method of solution to single time-constant RL circuits in ‘transient response + forced response’ format as well as in ‘zero-input response + zero-state response’ format. This chapter places emphasis on impulse response as the key circuit response, keeping in perspective the discussion on convolution integral in the second text planned under this series.

Chapters 11 and 12 take up a similar analysis of RC and RLC circuits respectively. Further, these chapters gradually introduce the concept of sinusoidal steady-state frequency response curves through RC and RLC circuits and set the background for Fourier series in a later chapter. Specific examples where the excitation is in the form of a sum of harmonically related sinusoids containing three to five terms are used to illustrate the use of frequency response curves and their linear distortion. The conditions for distortion-free transmission of signals are briefly hinted at in Chapter 11. A detailed coverage on distortion-free transmission of signals will appear in a chapter on Fourier transforms in the second text planned under this series.

Inconvenient circuit problems like shorting a charged capacitor, opening a current carrying inductor, connecting two charged capacitors together, and connecting an uncharged capacitor across a DC supply require inclusion of parasitic elements for correct explanation. Parasitic elements are emphasized at various places in chapters dealing with time-domain analysis.

Chapter 13 expands an arbitrary input signal along a line parallel to the vertical axis in a signal plane i.e., in terms of damped sinusoids of different frequencies rather than in terms of undamped sinusoids of different frequencies. This expansion is illustrated graphically in the case of a simple waveshape to convince the reader that an aperiodic signal can indeed be obtained by a large number of exponentially growing sinusoids and that there is nothing special about expansion of a waveshape in terms of undamped sinusoids. This expansion of signals leads to Laplace Transform of the signal. Properties of Laplace Transform, use of Laplace Transform in solving differential equations and circuits, transfer functions, impedance functions, poles, and zeros follow. This chapter also includes a graphical interpretation of frequency response function in the s-plane. Stability criterion is re-visited and circuit theorems are generalized.

Chapter 14 is on magnetically coupled circuits. It introduces the mutual inductance element, building on the properties of perfectly coupled linear transformer and ideal transformer. Transient response of coupled coil circuits and sinusoidal steady-state in such circuits are also covered in this chapter. Applications of two-winding transformer in impedance matching, design of tuned amplifiers etc., are explained.

#### Prerequisites

The student-reader is expected to have gone through basic level courses in electromagnetism, complex algebra, differential calculus and integral calculus. These are covered in the pre-engineering school curricula of all boards of senior/higher secondary school education in India.