Wiley acing the gate electrical engineering part 1

Wiley’s Acing the GATE Examination in Electrical Engineering is intended to be the complete book for those aspiring to compete in the Graduate Aptitude Test in Engineering GATE in Electr

WILEY ACING THE GATE

ELECTRICAL ENGINEERING

Dr Debashis Chatterjee

Professor Department of Electrical Engineering

Jadavpur University Kolkata

Dr J S Lather Professor Department of Electrical Engineering National Institute of Technology

Kurukshetra

Dr Lalita Gupta Assistant Professor Maulana Azad National Institute of Technology

Bhopal

WILEY ACING THE GATE

ELECTRICAL ENGINEERING

Copyright © 2016 by Wiley India Pvt Ltd., 4435-36/7, Ansari Road, Daryaganj, New Delhi-110002.

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Wiley’s Acing the GATE Examination in Electrical Engineering is intended to be the complete book for those aspiring to compete in the Graduate Aptitude Test in Engineering (GATE) in Electrical Engineering discipline

It comprehensively covers all the topics as prescribed in the GATE 2016 syllabus in terms of study material, quick reference material and an extensive question bank, complete with solutions The book offers a number of useful features and the approach is logical concept building rather than only formula based, as offered by the other books generally published in this domain

The objective has been to structure this book as a complete reference covering fundamental aspects of theory before

proceeding to relevant questions A three tier approach has been adopted to create this manuscript, that is, coverage of the basic building blocks for each of the subjects followed by the exhaustive solved examples, and then practice exercises with one and two marks questions Lastly the set of questions from GATE previous years’ papers starting 2003 to cur-

rent have been provided along with their solutions and explanations as a supplement to the formal subject coverage

The book presumes basic understanding of the fundamentals of Electrical Engineering and related basic electronics

The book is divided into 10 chapters based on the units of the syllabus, wherein each chapter constitutes a subject The chapters are divided into various sections which are self-sufficient and easy to read and understand The sequence of the chapters has been arranged in such a way that almost no cross-referencing to subsequent chapter is needed The system-

atic coverage of the book by the aspirant enhances their knowledge with valuable insights into problem solving approach The authors are of the view that the systematic coverage of this text will not only enhance the GATE cracking skills

but also impart a level of knowledge to the students which might have been missed by them during formal study of the subjects in the classroom In addition, the subject centric approach of the books trains the reader to crack other parallel examinations like UPSC Exams etc

Though adequate precautions have been taken to ensure correctness of theoretical concepts, equations and related

questions, we appreciate communication regarding any inadvertent mistakes that you might come across during the course of your study

Last but not the least, the authors wish to convey thanks to the entire editorial and production team for being

patient to wait for the completion of manuscript by the authors and understanding the technicalities involved in the writing such a comprehensive book and finally for all their efforts to make this dream a reality The continuous inputs from the editorial team of Wiley India and systematic coverage and compilation by the authors, have resulted into this cohesive and complete reference text for GATE examination in Electrical Engineering

Dr Debashis Chatterjee is Professor at the Department of Electrical Engineering, Jadavpur University, Kolkata

He received his B.E in Electrical Engineering from Jadavpur University, Kolkata, M Tech in Machine Drives and Power Electronics from IIT-Kharagpur and Ph.D from Jadavpur University, Kolkata Dr Chatterjee is the author and co-author of more than 60 journal articles and conference papers in reputed publications His areas of interest are Parameter Estimation and Speed Control of Induction Machines, Control of Induction Generators, Development of Improved Harmonic Elimination Techniques of Inverters, Control of Permanent Magnet Machines

Dr J S Lather is Professor at the Department of Electrical Engineering, National Institute of Technology, Kurukshetra

He received his B.E in Electrical Engineering from SVNIT, Surat, M Tech in Control Systems from NIT, Kurukshetra and Ph.D in the area of Robust Control from NIT, Kurukshetra Dr Lather has more than 21 years of teaching, research and industrial experience to his credit He has published more than 40 research papers in national and international journals and in conferences proceedings His areas of specialisation are Control Systems, Power Systems and Semantic Computing

Dr Lalita Gupta is Assistant Professor at the Department of Electronics and Communication Engineering, Maulana Azad National Institute of Technology, Bhopal She received her B.E in Electronics and Telecommunication from

Pt Ravi Shankar Shukla University Raipur, M Tech in Digital Communication from Maulana Azad National Institute

of Technology, Bhopal and Ph.D from the same institute Since July 2004 she has been associated as a faculty member with the Maulana Azad National Institute of Technology, Bhopal She is a member of IEEE, IETE, ICEIT, IE

Dr Gupta has 35 research publications in national and international journals of repute Her area of specialisation is Signal Processing

ABOUT GATE EXAMINATION

The Graduate Admission Test in Engineering (GATE) is an All-India level competitive examination for engineering graduates interested in pursuing Masters or Ph.D programmes in India The examination tests the examinees in General Aptitude, Engineering Mathematics and the discipline (subject) of study in the undergraduate course The objective of GATE is to identify meritorious and motivated candidates for higher studies in Engineering and Sciences The examination serves as a benchmark for normalisation of the undergraduate engineering education in the country

The level of competitiveness can be gauged from the fact that close to ten lakh students appear in this competitive

examination every year

Admission to Higher Learning Courses

A valid GATE score is essential to become eligible for admission to the post-graduate courses in Engineering, that is, M.Tech, M.E or direct doctoral programme in the Indian higher education institutes Although qualifying the GATE examination entitles you to apply for the higher degrees; achieving qualifying score is definitely not enough if one is aspiring for admission to top institutes like the IITs, the NITs, the Indian Institute of Science (IISc) and some of the high ranked universities For this, a high GATE score is important Needless to say, a percentile of greater than 95 is perhaps the least one needs to secure admission to a top institute A total of 804463 candidates appeared for GATE

2015 out of which about 125851 students belonged to Electrical branch of engineering It is important to mention here that only 15.05% of those who appeared could qualify according to the qualifying marks set by the GATE examination committee

Financial Assistance

Selected GATE qualified candidates admitted to M.Tech programmes in Colleges/Universities all over India are eligible for obtaining financial assistance The financial assistance is awarded to Indian nationals doing the M.Tech programmes, subject to institute rules It is also available in the form of Half-Time Teaching Assistantship (HTTA)

and is tenable for a maximum period of 24 months HTTA students are required to assist the department for 8 hours

of work per week related to academic activities of the department such as laboratory demonstration, tutorials, evaluation of assignments, test papers, seminars, research projects, etc

Jobs in Public Sector Undertakings

While admission to a top institute for the Masters programme continues to be the most important reason for working hard to secure a good score in the GATE examination; another reason to appear and qualify GATE examination with good score is that many Public Sector Undertakings (PSUs) are and probably in future almost all, will be recruiting through GATE examination And it is quite likely that even big private sector companies may start considering GATE seriously for their recruitment as GATE score can give a bigger clue about competencies of candidates they are recruiting

A large number of PSUs have already started recruiting on the basis of GATE score These include companies like, Power Grid, Delhi Development Authority (DDA), Indian Oil, Bharat Electronics (BE), Bharat Heavy Electricals Limited (BHEL), National Thermal Power Corporation (NTPC), HPCL, DVC, NALCO, NLC, Central Electronics Limited (CEL), BSPHCL, Vizag Steel and Gas Authority of India Limited (GAIL) In 2015, 15 PSUs signed MoUs with IIT Kanpur to receive the GATE 2015 results across various papers

APPLYING FOR THE EXAMINATION

Eligibility

The candidates applying for GATE examination must meet the under mentioned requirements

1 A candidate is allowed to appear only in one paper The first step therefore is to select the paper you wish to appear for

2 The next step is to choose the city for appearing in the examination There are three choices to be given in the order of preference The candidate can choose a particular city as the first choice for appearing in GATE exami-

nation Having done that, he/she would know the zone the chosen first preference city belongs to The candidate

can then choose his/ her second choice only from the cities available in that zone As an additional option, a third

choice was also introduced from GATE 2014 The list of third choice cities will be as specified by each zone Note

that this third choice city may either be from the zone to which the first and second choice cities belong or from

some other zone The third choice will be considered only when the candidate can not be accommodated either

in first or second choice cities The tentative zone wise list of cities for GATE every year is given in Examination

Cities However, the GATE Committee reserves the right to add a new city or remove an existing city and allot

a city that may not be any of the choices of a candidate

completed) and Professional Society Examinations equivalent to B.E./B.Tech (completed section-A or equivalent

of such courses)

4 Candidates who are likely to complete the qualifying examination during the year of the GATE examination or later have to submit a certificate from their college Principal They have to obtain a signature from their principal along with the seal on the “Certificate from the Principal” format that will be printed on the application PDF file

which is generated after completion of the online application submission

5 Candidates who have appeared in the final semester/year exam in the year immediately preceding the year of GATE

examination (for GATE-2016, it will be 2015), but with a backlog (arrears/failed subjects) in any of the papers in their

qualifying degree should (a) submit a copy of any one of the marks sheets of the final year, or (b) have to obtain a

signature from their Principal along with the seal on the “Certificate from the Principal” format that will be printed

on the application PDF file which will be generated after completion of the online application submission

Official Website

All announcements regarding GATE can be seen on the official website of the current organizing institute There are

a large number of other websites that contain GATE relevant information

STRUCTURE OF THE EXAMINATION

The GATE examination is conducted for 22 disciplines (papers) that are listed in GATE brochure and also available

on official GATE website The syllabus for each of these is also given separately in detail The candidate is expected to select and appear in the appropriate paper as per the discipline of his qualifying degree However, he is free to choose any paper, depending on the plan for admission into higher degree and the eligibility requirements of the same

Examination Pattern

GATE examination consists of a single paper of 3 hour-duration There are a total of 65 questions for 100 marks belonging

to the following sections:

•  General aptitude: Comprises of 10 questions, out of which five questions are of 1 mark each and five for 2 marks

each These are designed to check the language and analytical skills of the aspirants

•  Subject paper: Comprises of 25 questions of 1 mark each and 30 questions of 2 marks each with Engineering

Mathematics constituting 13—15% of the total marks These are designed to check the subject knowledge of the

aspirants

The questions are a mix of multiple choice and numerical type:

•  Multiple choice questions (MCQs): These questions will have single option correct

•  Numerical answer questions (NAQs): These come with no choices and candidates are expected to answer using

a virtual keypad The numerical answer will be a real number, signed or unsigned, with due consideration being

given for a range during the answer evaluation

Marking Scheme

For 1 mark questions, 1/3 Mark is deducted for a wrong answer For a 2 mark question, 2/3 mark is deducted for a wrong answer There is no negative marking for numerical answer type questions

Mode of Examination

The GATE examinations for the papers of all streams are held on-line These include papers with codes AE, AG, AR,

BT, CE, CH, CY, GG, MA, MN, MT, PH, TF, XE, XL, CS, EC, EE, IN, ME and PI

UNDERSTANDING GATE RELATED INFORMATION

Pre-Examination

Pre-examination related information is covered in detail under different headings in the previous pages Before starting the application process, you must:

1 Ensure you are eligible for the relevant GATE examination

2 Determine the GATE paper you wish you appear for (You can appear in only one paper)

3 Choose at least two cities that are convenient for you to write the exam

4 Application for appearing in GATE has to be made online only

5 All Supporting documents should be sent online only No hard copy will be accepted

6 Payments have to be made through debit/ATM cards, credit cards or internet banking and e-challan only

7 Your choice of exam paper will determine date, and choice of available cities

Post-Examination (Normalisation of GATE Score)

Examination for CE, CS, EC, EE and ME papers is generally held in multi-sessions Hence, for these papers,

a suitable normalisation is applied to take into account any variation in the difficulty levels of the question papers across different sessions The normalisation is done based on the fundamental assumption that “in all

multi-session GATE papers, the distribution of abilities of candidates is the same across all the sessions” This assumption is justified since the number of candidates appearing in multi-session papers in GATE 2015 is large and the procedure of allocation of session to candidates is random Further it is also ensured that for the same multi-session paper, the number of candidates allotted in each session is of the same order of magnitude For the above mentioned papers; GATE score will be computed based on the normalized marks and not the actual mark obtained in the examination For all other papers, actual marks obtained in the examination will be used for computation of GTE score.

ATTRIBUTES FOR SUCCESS IN THE EXAMINATION

As PSUs began to accept GATE score as shortlisting criteria, the number of candidates appearing for the Electrical Engineering branch has greatly increased In 2010, 52,246 students appeared for GATE in Electrical Engineering; while

in 2015, the number rose to 125,851 Even as the number of seats for M.Tech programs in IITs and NITs increase, it

is marginal as compared to the increase in the number of candidates This has increased level of competition and has brought focus on the methodical planning that students must embrace to succeed in such tough environment There are many ingredients to being successful in this examination With an aim to offer you the leading edge, we suggest the following attributes so as to enhance your chance of success

Preparing for the Examination

Knowing your Subject

The syllabus for GATE covers all the subjects essential to Electrical Engineering It takes both time and strategic planning to go through the entire syllabus which was covered in 6—7 semesters during B.Tech/B.E courses

1 The focus of GATE is on core concepts Most university syllabi include advanced topics, which are not included

in GATE Have a clear demarcation of the topics that need to be mastered

2 GATE is based mostly on numerical problem solution There is no rote learning involved Practice numericals to understand concepts, especially from the subjects that have a higher weightage

3 Be aware of the standard notations and assumptions, for those are typically not provided in the statement of the questions

4 Build on your strengths Work on the subjects you like first As you start solving GATE level problems from those

subjects, it will boost your confidence and then you can pick up subjects that you may not be very comfortable with

Preparing a Study Plan

Analyse the past few years’ papers This will give you a clear idea of which subjects and topics carry the

maxi-mum weightage A well-chartered study plan takes into account the time required for mastering both the concepts as well the application of these high weightage topics The most relevant topics for each unit and unit-wise weightage is provided in the part opener preceding each chapter opener These can be helpful in deriving focus areas

1 The study plan may include a schedule of chapter-wise time allotted to conceptual understanding and problem solving The subjects with high weightage like Network Analysis, Electrical Machines, Control Systems and

Power Systems can be allotted extra time, whereas subjects like Digital Electronics and Measurements (which

typically have less weightage) can be done in a shorter time

2 Extra time may be allotted to the topics you feel are challenging For example, many students struggle with Signals and Systems However, GATE typically deals with the easier concepts from this course, like time shift-

ing and scaling and Fourier transforms, which are simply mathematical algorithms Start from the very basics of

these topics and increase the level of difficulty slowly

3 Once you have the confidence to solve all types of problems, attempt the MCQs Time yourself and know the average time taken to solve an MCQ MCQs from some subjects, like Power Systems and Electrical Machines

may take somewhat longer to solve than others As long as the average time taken is within limits, do not worry

about these

Refining Problem-Solving Skills

The examination tests the ability to analyse information provided, make judgements, and apply the understanding of topic to arrive at the best-suited answer

1 Revision all the important formulae and unit summary would be useful before attempting practice tests or mock tests Some subjects like Network Analysis and Control Systems are required for understanding other subjects

Clear these basics first before going towards advanced topics

2 GATE questions very often have a lot of information, most of which is not actually needed to solve the problem With practice, you’ll learn to filter information down to the data required to solve the problem

3 Re-work the type of problems that were not solvable in the first attempt Use these to identify the common

mis-takes you make

4 Self-evaluate using the answer keys

•   For questions that you got correct, evaluate your approach against the approach applied Did you use the short

cuts? Can you get the same solution faster?

•  For questions that you got wrong, identify the mistake—mathematical or conceptual Rework through the questions

•  For questions that you could not answer, study the concept again

5 Attempt as many previous years’ GATE questions based on the topic and follow the above approach for self-evaluation

Ensuring Effective Preparation and Time Management

Since the syllabus to be covered is very vast, time management plays a key role in effective preparation Following steps might assist you in preparing daily schedule of activities during the course of preparation

1 Have a realistic idea of the actual time that you have University examinations or deadlines at work will reduce the time that you appear to have

2 Set small weekly targets as you progress through your study plan and focus on achieving them to avoid getting overwhelmed by the entire process

3 Avoid skipping a day entirely If another commitment appears, reduce the number of study hours but don’t break the continuity

4 Do not dwell on a problem for too long Ask for help and sort it out, or keep it aside for later

TAKING THE EXAMINATION

Maximise your Exam-Taking Efficiency

Towards the end of study plan, it is important to focus on exam preparedness and time management skills

1 If possible, attempt online Mock GATEs (some will be available on the GAOPS website) and see the time taken

to attempt the full paper Also, check the accuracy of your answers

2 Determine if either your speed or accuracy needs more work Focus on specific topics whose questions too

longer to solve Practice more of these questions If questions from a certain topic are incorrect, re-read the

concepts

3 Do NOT over attempt! It is very tempting to attempt MCQs But remember that there is negative marking involved Mark an MCQ only if you are certain of the answer

On the Examination Day!

1 It is important to take proper rest and sleep well in the night before the examination Eat well and keep yourself healthy

2 Read the instructions carefully within the allotted time, and raise any equipment/software concerns immediately

to the invigilator

3 Read the question statements carefully What is the question asking for? Work towards the solution step by step.

4 The aptitude section is usually quite easy Work through it fast and leave more time for the technical section

5 If any question appears unfamiliar or too difficult, mark it for review Come back to these questions if time permits, and try another approach to see if you can work these out Some MCQs may become solvable by eliminating options

6 Keep a track of time If possible, set your watch to alarm after every half an hour

All the very best!

Authors

Section 1: Engineering Mathematics

Linear Algebra: Matrix Algebra, Systems of linear equations, Eigenvalues, Eigenvectors

Calculus: Mean value theorems, Theorems of integral calculus, Evaluation of definite and improper integrals, Partial Derivatives, Maxima and minima, Multiple integrals, Fourier series, Vector identities, Directional derivatives, Line integral, Surface integral, Volume integral, Stokes’s theorem, Gauss’s theorem, Green’s theorem

Differential Equations: First order equations (linear and nonlinear), Higher order linear differential equations with constant coefficients, Method of variation of parameters, Cauchy’s equation, Euler’s equation, Initial and boundary value problems, Partial Differential Equations, Method of separation of variables

Complex Variables: Analytic functions, Cauchy’s integral theorem, Cauchy’s integral formula, Taylor series, Laurent series, Residue theorem, Solution integrals

Probability and Statistics: Sampling theorems, Conditional probability, Mean, Median, Mode and Standard deviation, random variables, Discrete and continuous distributions, Poisson distribution, Normal distribution, Binomial distribution, Correlation analysis, Regression analysis

Numerical Methods: Solutions of non-linear algebraic equations, Single and Multi-step methods for differential equations

Transform Theory: Fourier transform, Laplace transform, Z-transform

Section 2: Electric Circuits

Network graph, KCL, KVL, Node and Mesh analysis, Transient response of DC and AC networks, Sinusoidal

steady-state analysis, Resonance, Passive filters, Ideal current and voltage sources, Thevenin’s theorem, Norton’s theorem, Superposition theorem, Maximum power transfer theorem, Two-port networks, Three phase circuits, Power and power factor in AC circuits

Section 3: Electromagnetic Fields

Coulomb’s law, Electric field intensity, Electric flux density, Gauss’s law, Divergence, Electric field and potential due to point, line, plane and spherical charge distributions, Effect of dielectric medium, Capacitance of simple configurations,

Biot—Savart’s law, Ampere’s law, Curl, Faraday’s law, Lorentz force, Inductance, Magnetomotive force, Reluctance, Magnetic circuits, Self and mutual inductance of simple configurations.

Section 4: Signals and Systems

Representation of continuous and discrete-time signals, Shifting and scaling operations, Linear time invariant and Causal systems, Fourier series representation of continuous periodic signals, Sampling theorem, Applications of Fourier transform, Laplace transform and Z-transform

Section 5: Electrical Machines

Single phase transformer: equivalent circuit, phasor diagram, open circuit and short circuit tests, regulation and

effi-ciency; Three phase transformers: connections, parallel operation; Auto-transformer, Electromechanical energy

conver-sion principles, DC machines: separately excited, series and shunt, motoring and generating mode of operation and their characteristics, starting and speed control of DC motors; Three phase induction motors: principle of operation, types, performance, torque-speed characteristics, no-load and blocked rotor tests, equivalent circuit, starting and speed control; Operating principle of single phase induction motors; Synchronous machines: cylindrical and salient pole machines, performance, regulation and parallel operation of generators, starting of synchronous motor, characteristics; Types of losses and efficiency calculations of electric machines

Section 6: Power Systems

Power generation concepts, AC and DC transmission concepts, Models and performance of transmission lines and cables, Series and shunt compensation, Electric field distribution and insulators, Distribution systems, Per-unit quanti-

ties, Bus admittance matrix, Gauss—Seidel and Newton—Raphson load flow methods, Voltage and frequency control, Power factor correction, Symmetrical components, Symmetrical and unsymmetrical fault analysis, Principles of over-

current, Differential and distance protection; Circuit breakers, System stability concepts, Equal area criterion

Section 7: Control Systems

Mathematical modeling and representation of systems, Feedback principle, Transfer function, Block diagrams and signal flow graphs, Transient and steady-state analysis of linear time invariant systems, Routh—Hurwitz and Nyquist criteria, Bode plots, Root loci, Stability analysis, Lag, lead and lead-lag compensators; P, PI and PID controllers; State space model, State transition matrix

Section 8: Electrical and Electronic Measurements

Bridges and potentiometers, Measurement of voltage, current, power, energy and power factor; Instrument transformers, Digital voltmeters and multimeters, Phase, time and frequency measurement; Oscilloscopes, Error analysis

Section 9: Analog and Digital Electronics

Characteristics of diodes, BJT, MOSFET; Simple diode circuits: clipping, clamping, rectifiers; Amplifiers: biasing, equivalent circuit and frequency response; Oscillators and feedback amplifiers; Operational amplifiers: characteristics and applications; Simple active filters, VCOs and timers, Combinational and sequential logic circuits, Multiplexer, Demultiplexer, Schmitt trigger, Sample and hold circuits, A/D and D/A converters, 8085 Microprocessor: architecture, programming and interfacing

Section 10: Power Electronics

Characteristics of semiconductor power devices: Diode, Thyristor, Triac, GTO, MOSFET, IGBT; DC to DC conversion: Buck, Boost and Buck-Boost converters; Single and three phase configuration of uncontrolled rectifiers, Line commu-

tated thyristor based converters, Bidirectional AC to DC voltage source converters, Issues of line current harmonics, Power factor, Distortion factor of AC to DC converters, Single phase and three phase inverters, Sinusoidal pulse width modulation

Preface v

1.3.5 Incidence Matrix 23

Important Formulas 89

Important Formulas 179

4.5.9 Synchronizing Power and Synchronizing Torque 307

5.8 PU (Per Unit System) 389

6.1.5 Sensitivity of Control Systems 495

SECTION VII: ELECTRICAL AND ELECTRONIC MEASUREMENTS 613

Important Formulas 648

8.7.2 Hybrid Equivalent Model 754

9.2 Boolean Algebra 887

SECTION X: POWER ELECTRONICS 981

10.12 Inverters 1019

MARKS DISTRIBUTION FOR GATE QUESTIONS

20090

24681012

TOPIC DISTRIBUTION FOR GATE QUESTIONS

analysis, Laplace transform, Two-port network

due to point, line, plane and spherical charge distribution, Curl, Gradient, Sinusoidal steady-state analysis, KVL, KCL, Two-port network, Resonance

power transfer theorem, Thevenin/Norton theorems, Two-port network, AC fundamentals, Curl and Vector

diagrams, Capacitor, Vector

analysis, Network theorems

ELECTRIC CIRCUITS

An electric circuit is an interconnection of electrical

ele-ments In this chapter, we will study basic concpets of

electrical circuits, fundamental laws that govern

elec-tric circuit, methods of circuit analysis, network

topol-ogy, circuit theorems, first order RL and RC circuit,

Laplace transform, second order RLC circuits,

sinusoi-dal steady state analysis, magnetically coupled circuits,

frequency response and resonance, voltage and current

sources, three-phase circuits AC power analysis two port

networks

Network is a combination of elements and it may or may

not consist of a closed path Circuit is a combination of

elements and should consist of closed path

Now, under what conditions is the network or circuit

If the distributed element length is very much less than

fre-quency of excitation, then the distributed parameters are approximated into lumped parameters The inter- connection of such lumped parameters (elements) is called electrical network At higher frequencies we cannot construct the lumped electric circuit, and hence the network theory is not applicable The network theory is valid only at low frequencies, that is, upto

1 MHz frequencies only For above 1 MHz frequencies,

we use field theory The field theory approach of

solv-ing the electrical network is valid for all frequencies

starting from zero onwards

The flow chart for use of field theory and circuit

theory for solving problems related to complicated

elec-trical networks is depicted in Fig 1.2

Complicated

electrical

network

Field theory approach (Ohms’s law and Maxwell’s equations)

1 Exact

2 Complicated

3 Too many variables (E, H, D, J)

4 Distributed circuit

Circuit theory approach (Ohm’s law, Kirchoff’s law)

1 Approximate

2 Simple

3 Less number

of variables (V, I)

4 Lumped circuit

theory on analysis of electrical network

1.1.1 Electrical Quantities and Units

1.1.1.1 Ohm’s Law

Before, the discussion on Ohm’s law, let us recapitulate

the mechanism of energy flow through the conductor

presence of free electrons and their mobility through

the conductor In the absence of electric field (E), there

is no net momentum of electrons in a conductor so

there is no net current, that is i = 0 In the presence

of axial electric field (E), force is exerted on the free

electrons, that is

F =E⋅e

C and net charge Q = ne, where n

is the number of free electrons Figure 1.3 shows the flow of

electrons under the influence of an electric field through the

conductor with length l and cross-sectional area A

In the presence of electric field, different free

elec-trons will move with different velocities But only one

velocity called drift velocity is defined It is an age velocity of all the free electrons present with in

aver-a conductor given by v

d = mE m/s where m is the

E is the applied electrical field The direction of force will be opposite to the field direction because the free electrons have a negative charge The net momentum of charge will exist and will be opposite to the direction of the field

In the presence of an electric field, all the free electrons will have the drift velocity So that the kinetic energy (K.E.) associated with each electron

is given by

K.E =

12

e)

In the absence electric field (E), drift velocity is zero and hence the kinetic energy is zero So, the total energy (W ) of electron is equal to the potential energy

T.E = P.E + K.E

W = P.E JoulesSince the conductor is an open circuit at room tem-

the thermal energy and they begin to move in different direction in a random manner Hence the net momentum

or net flow of free electrons in any direction is zero So the net flow of charge is zero and the current zero

dqdt

The time rate of flow of electric charges is defined as

an electrical current (i), that is

i

d Qdt

=( ) Ampere

Since Q is negative, so the current will flow in the site direction to that of electron motion, that is, in the direction of the applied electric field (E)

oppo-Current density (J) is defined as the current per unit cross-section area and is given by

JiA

Since A is a scalar, the direction of J is in the direction

of the current, that is, in the direction of E

According to the Ohm’s law in the field theory form

or point form, there exists a linear relation between the

current density (J) and the electric field (E), that is

Therefore, power dissipation

0

s

s = Slope

The limitations in the description of conductivity and

Ohms’ law based on network (field) theory are:

tem-pearture is kept constant otherwise temperature

increases the free electrons will acquire extra

ther-mal energy, which leads to the increase in collisions

(s) decreases.

colli-sions between the free electrons and the positive ions

(larger in size) increases, which leads to the fall in the

drift velocity and hence loss in K.E This lost energy

is dissipated in the form of heat, which results in a

voltage drop (V) across the conductor The amount

of power dissipated within the conductor is

AmVm

To overcome these, we define Ohm’s law in circuit

theory, which states that the voltage across many types

of conducting materials is directly proportional to the

current flowing through the element (material)

where the constant of proportionality R (resistance) is

constant, that is, temperature is constant

We have from Eq (1.1) that

iAVl

=s.

lAi

The resistivity of the material is defined as

r s

=1

Another form of Ohms law is

where G is the conductance, expressed in Siemens or

1R

=

Therefore,

dqdt

1.1.1.2 Power and Energy

The rate of change of energy is called power

PdWdtdWdqdqdtVi

Using Eq (1.2), we have

PVR

(1.7)

The current across the inductor is

iL

V dt

t

L =1

didt

didt

induc-−i

i0

The convention for representing current voltage and

resistor is shown in Fig 1.5(a) and the voltage-current

characteristics are depicted in Fig 1.5(b)

+V

From the current-voltage characteristics we can see that

the resistor is a passive, linear, bilateral and time

invari-ant element

Inductor (L)

When conductor is bound in the form of coil (Fig 1.6), it

will exhibit an opposition, called inductance

Li

+

−

When a time varying current flows through an inductor,

produced by it is given by

The amount of flux produced is proportional to the

current through the coil

The voltage across the inductor is

So the voltage-current relation in an inductor is linear

Hence, the relation

didt

is yet another form of Ohm’s law

Comparison of different electrical paramters in circuit and field theory is given in Table 1.1

2

H= J m1

2

E = J m1

if it cannot generate or amplify energy, that is, if the energy which it has supplied since the begin-ning of time cannot exceed the energy which was fed into it

Current across the capacitor is

idqdt

C=

q=C⋅V

dVdt

Voltage across the capacitor

VC

i dt

t

C =1

Power in the capacitor

dVdt

dVdt

Energy of the capacitor is given by

dVdt

The energy stored in the capacitor at any instant depends

on the voltage across the capacitor at that instant The

charge-voltage characteristic curve is shown in Fig 1.9

From the characteristic curve, we can see that capacitor

is a linear, passive bilateral and time invariant element

−q

VV

C

Cq

1 + V

2

s

@l

Vi

elements

the same for current flowing in either direction

Resistor, capacitor and inductor are the examples

of bilateral elements An element is said to be

dif-ferent for two directions of current flow Diode is a unilateral element

Note: In the linear-time invariant case, resistors,

on characteristic curve is negative, then the element

is active; otherwise it is passive Every linear element

is bidirectional If the characteristic curve is similar in opposite quadrants, then the element is bidirectional;

otherwise, it is unidirectional

1.1.2.1 V-I Characterictics of Elements

are as shown in Fig 1.12(b), then the element is linear, passive and bilateral

−I

−V

II

0V

+

−(a) (b)

Figure 1.12

in Fig 1.13, then the element is non-linear, passive and unilateral For example, diode

IV

where E(t) is the difference between the energies

entering and leaving the circuit, that is, the net

energy supplied to the network If E(t) equals the

energy stored in circuit then the circuit is neither

generating, nor losing energy Such a passive circuit

is called a loss-less network Otherwise, the passive

circuit is lossy For example, resistor, capacitor,

inductor, diode, bulb, transformer

such a circuit can supply excess energy In other

words, when the elements are capable of

deliver-ing energy independently for a long time

(approxi-mately infinite time) or when element is having

property of internal amplification then element

is called as active element For example, voltage

sources, current sources, transistors, op-amps

amplitude of the response is always directly

propor-tional to the amplitude of the excitation (Fig 1.10)

−I

−V

I0

pro-portional to the amplitude of the excitation, the

circuit is called non-linear

A circuit is time-invariant if the relation between

its response and its excitation is applied; otherwise,

the network is time-varying

lumped if the physical dimensions of all its

compo-nents are negligible compared with the wavelength

of the electromagnetic signal inside the

compo-nent In such a network, since Kirchhoff’s

volt-age and current relations hold; the current within

any branch is the same at all points of the branch

between its terminating nodes [Fig 1.11(a)]

of the elements are comparable to the signal

wave-length, the spatial variations of voltages and

cur-rent along the wires and in the components must

be taken into consideration In such circuits, called

distributed network, Kirchhoff’s laws are no longer

valid, and the more general laws of Maxwell must

be applied [Fig 1.11(b)]

From the above results,

con-verse need not be true

1.1.3 Current and Voltage Sources

There are two types of energy sources:

• V (ideal, practical)

• I (ideal, practical)

• Voltage-controlled voltage source (VCVS)

• Voltage-controlled current source (VCCS)

• Current-controlled voltage source (CCVS)

• Current-controlled current source (CCCS)

1.1.3.1 Independent Sources

Ideal Voltage SourceIdeal voltage source delivers energy at specified volt-age (V

resistance of ideal voltage source is zero [Fig 1.18(a)] In

an ideal voltage source, the terminal voltage is dent of the terminal current so the source will be there like a non-linear element [Fig 1.18(b)]

S− S

If I increases, V decreases as shown in Fig 1.19(b) Note that the terminal voltage is a function of the terminal current Now, when I = 0, V = V

through a passive element can be zero so that two voltages are equal and vice versa [Fig 1.19(c)]

in Fig 1.14, then the element is non-linear, passive

Figure 1.14

in Fig 1.15, then the element is non-linear, active

Figure 1.15

as shown in Fig 1.16, then the element is non-linear,

active, unilateral For example, bipolar junction

in Fig 1.17, then the element is linear, active and

Figure 1.17