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Electromagnetic Theory Electronics & Communication Engineering (EC)
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Cover Design : Tarun Tank
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First Edition : May 2019
Book Code : GATE‐EMT‐633
ISBN : 978‐93‐86699‐63‐3
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GATE examination has been emerging as one of the most prestigious competitive exam for engineers.
Earlier it was considered to be an exam just for eligibility for pursuing PG courses, but now GATE exam
has gained a lot of attention of students as this exam open an ocean of possibilities like :
1. Admission into IISc, IITs, IIITs, NITs
A good GATE score is helpful for getting admission into IISc, IITs, IIITs, NITs and many other
renowned institutions for M.Tech./M.E./M.S. An M.Tech graduate has a number of career
opportunities in research fields and education industries. Students get ` 12,400 per month as
stipend during their course.
2. Selection in various Public Sector Undertakings (PSUs)
A good GATE score is helpful for getting job in government‐owned corporations termed
as Public Sector Undertakings (PSUs) in India like IOCL, BHEL, NTPC, BARC, ONGC, PGCIL, DVC,
HPCL, GAIL, SAIL & many more.
3. Direct recruitment to Group A level posts in Central government, i.e., Senior Field Officer (Tele),
Senior Research Officer (Crypto) and Senior Research Officer (S&T) in Cabinet Secretariat,
Government of India, is now being carried out on the basis of GATE score.
4. Foreign universities through GATE
GATE has crossed the boundaries to become an international level test for entry into
postgraduate engineering programmes in abroad. Some institutes in two countries Singapore
and Germany are known to accept GATE score for admission to their PG engineering
programmes.
5. National Institute of Industrial Engg. (NITIE)
NITIE offers PGDIE / PGDMM / PGDPM on the basis of GATE scores. The shortlisted
candidates are then called for group Discussion and Personal Interview rounds.
NITIE offers a Doctoral Level Fellowship Programme recognized by Ministry of HRD
(MHRD) as equivalent to PhD of any Indian University.
Regular full time candidates those who will qualify for the financial assistance will receive
` 25,000 during 1st and 2nd year of the Fellowship programme and ` 28,000 during 3rd, 4th and 5th year of the Fellowship programme as per MHRD guidelines.
6. Ph.D. in IISc/ IITs
IISc and IITs take admissions for Ph.D. on the basis of GATE score.
Earn a Ph.D. degree directly after Bachelor’s degree through integrated programme.
A fulltime residential researcher (RR) programme.
7. Fellowship Program in management (FPM)
Enrolment through GATE score card
Stipend of ` 22,000 – 30,000 per month + HRA
It is a fellowship program
Application form is generally available in month of sept. and oct.
Note : In near future, hopefully GATE exam will become a mandatory exit test for all engineering
students, so take this exam seriously. Best of LUCK !
IMPORTANCE of GATE
Section Question No. No. of Questions Marks Per Question Total Marks
General Aptitude
1 to 5 5 1 5
6 to 10 5 2 10
Technical
+
Engineering Mathematics
1 to 25 25 1 25
26 to 55 30 2 60
Total Duration : 3 hours Total Questions : 65 Total Marks : 100
Note : 40 to 45 marks will be allotted to Numerical Answer Type Questions
Pattern of Questions :
(i) Multiple Choice Questions (MCQ) carrying 1 or 2 marks each in all the papers and
sections. These questions are objective in nature, and each will have a choice of four
answers, out of which the candidate has to select (mark) the correct answer.
Negative Marking for Wrong Answers : For a wrong answer chosen in a MCQ, there will
be negative marking. For 1‐mark MCQ, 1/3 mark will be deducted for a wrong answer.
Likewise for, 2‐marks MCQ, 2/3 mark will be deducted for a wrong answer.
(ii) Numerical Answer Type (NAT) Questions carrying 1 or 2 marks each in all the papers
and sections. For these questions, the answer is a signed real number, which needs to
be entered by the candidate using the virtual numeric keypad on the monitor (Keyboard
of the computer will be disabled). No choices will be shown for these type of questions.
The answer can be a number such as 10 or – 10 (an integer only). The answer may be in
decimals as well, for example, 10.1 (one decimal) or 10.01 (two decimal) or –10.001
(three decimal). These questions will be mentioned with, up to which decimal places,
the candidates need to make an answer. Also, an appropriate range will be considered
while evaluating the numerical answer type questions so that the candidate is not
penalized due to the usual round‐off errors. Wherever required and possible, it is better
to give NAT answer up to a maximum of three decimal places.
Note : There is NO negative marking for a wrong answer in NAT questions.
GATE Exam Pattern
It is our pleasure, that we insist on presenting “Electromagnetic Theory (EMT)”
authored for Electronics & Communication Engineering to all of the aspirants
and career seekers. The prime objective of this book is to respond to
tremendous amount of ever growing demand for error free, flawless and
succinct but conceptually empowered solutions to the subject Communication
System.
This book serves to the best supplement the texts for Electronics & Communication Engineering
but shall be useful to a larger extent for Electrical Engineering and Instrumentation Engineering
as well. Simultaneously having its salient feature the book comprises :
Step by step solution to all questions.
Detailed explanation of all the questions.
Solutions are presented in simple and easily understandable language.
Video solutions for good questions.
The authors do not sense any deficit in believing that this title will in many aspects, be different
from the similar titles within the search of student.
We would like to express our sincere appreciation to Mrs. Sakshi Dhande Ma’am
(Co‐founder, GATE ACADEMY Group) for her constant support and constructive suggestions and
comments in reviewing the script.
The final manuscript has been prepared with utmost care. However, going a line that, there
is always room for improvement in anything done, we would welcome and greatly appreciate the
suggestions and corrections for further improvement.
Umesh Dhande
(Founder ‐ GATE ACADEMY Group)
PREFACE
We are glad of this opportunity to acknowledge the views and to express with all the
weaknesses of mere words the gratitude that we must always feel for the generosity of them.
We now express our gracious gratitude to the persons who have contributed a lot in
order to put forth this into device. They are to be mentioned here and they are Koushalya
Chandrawanshi, Reena Sahu, Vikas Athe and Shashikant Pandey.
Special thanks to Mr. Saket Verma Sir, who has been involved in this project from the
beginning and has given his best effort on his part. This book was not possible without his
unconditional effort.
Lastly, we take this opportunity to acknowledge the service of the total team of
publication and everyone who collaborated in producing this work.
GATE ACADEMY
ACKNOWLEDGEMENT
Chapter 1
Coordinate System & Del Operators ……………..…….……….. 1.1 – 1.32
Chapter 2
Electrostatic ……………………………….…………..……………………. 2.1 – 2.80
Chapter 3
Magnetostatic ……………………..…………………..………………….. 3.1 – 3.44
Chapter 4
Plane Wave Propagation & Polarization ……..……………….. 4.1 – 4.92
Chapter 5
Transmission Line & S‐Parameter …………………………………. 5.1 – 5.94
Chapter 6
Waveguide…………………………………….…………….……………….. 6.1 – 6.46
Chapter 7
Antenna …………………………..………………………….……………….. 7.1 – 7.72
Chapter 8
Basics of Radar & Optical Fiber ……………………….……………. 8.1 – 8.40
CONTENTS
Electrostatics; Maxwell’s equations: differential and integral forms and their interpretation,
boundary conditions, wave equation, Poynting vector; Plane waves and properties:
reflection and refraction, polarization, phase and group velocity, propagation through
various media, skin depth; Transmission lines: equations, characteristic impedance,
impedance matching, impedance transformation, S‐parameters, Smith chart; Waveguides:
modes, boundary conditions, cut‐off frequencies, dispersion relations; Antennas: antenna
types, radiation pattern, gain and directivity, return loss, antenna arrays; Basics of radar;
Light propagation in optical fibers.
Elements of vector calculus, Maxwell’s equations‐basic concepts; Gauss’, Stokes’ theorems;
Wave propagation through different media; Transmission Lines‐different types, basics,
Smith’s chart, impedance matching/transformation, S‐parameters, pulse excitation, uses;
Waveguides‐basics, rectangular types, modes, cut‐off frequency, dispersion, dielectric
types; Antennas‐radiation pattern, monopoles/dipoles, gain, arrays‐active/passive, theory,
uses.
Maxwell's equations, Time varying fields, Wave equation and its solution, Rectangular
waveguide, Poynting vector, Antenna parameters, Half‐wave antenna, Transmission lines,
Characteristic of Impedance matching, Smith chart.
GATE SYLLABUS
ESE SYLLABUS
UGC‐NET SYLLABUS
© Copyrightwww.gateacademy.co.inH Oead ffice : A/114-115, Smriti Nagar, Bhilai (C.G.), Contact : 9713113156, 9589894176
B Oranch ffice : Raipur : 79743-90037, Bhopal : 83198-88401
Learning Objectives :
After reading this chapter you should be able to :
understand basics of vector algebra
explain different coordinate system
transform vectors in different coordinate system
define differential length, surface and volume in all coordinate system
describe del operators and their significance
1.1 Scalars and Vectors
1.2 Unit Vector
1.3 Vector Addition and Subtraction
1.4 Position and Distance Vectors
1.5 Vector Multiplication
1.6 Co‐ordinate Systems
1.7 Transformation of Coordinate Systems
1.8 Relationship between Cylindrical and Cartesian Unit Vector
1.9 Relationship between Spherical and Cartesian Unit Vector
1.10 Del Operator
1.11 Gradient of Scalar
1.12 Divergence of a Vector
1.13 Curl of a Vector
1.14 Laplacian of a Scalar
Coordinate System & Del Operators
1CHAPTER
Table of Contents
© Copyrightwww.gateacademy.co.inH Oead ffice : A/114-115, Smriti Nagar, Bhilai (C.G.), Contact : 9713113156, 9589894176
B Oranch ffice : Raipur : 79743-90037, Bhopal : 83198-88401
Learning Objectives : After reading this chapter you should be able to : describe Coulomb’s law and its application explain electric field intensity due to various charge distribution describe Gauss’s law and concept of Gaussian surface describe potential difference and potential gradient explain capacitive system and capacitance of different capacitors understand electric boundary conditions apply method of images
2.1 Coulomb’s Law
2.2 Electric Field Intensity and Electric Flux
Density
2.3 Continuous Charge Distribution
2.4 Electric Field Due to Various Charge
Distribution
2.5 Gauss’s Law
2.6 Applications of Gauss’s Law
2.7 Electric Field Intensity Due to
Conducting Sphere
2.8 Electric Field Intensity Due to Non‐
conducting Sphere
2.9 Potential Difference and Absolute
Potential
2.10 Potential Due to Various Charge
Distributions
2.11 Potential Difference Due to Infinite Long
Line Charge
2.12 Potential Difference and Potential Due
to Conducting Sphere of Radius ‘R’
2.13 Equipotential Surfaces
2.14 Relationship between Electric Field
Intensity E and Electric Potential V
2.15 Electric Field and Electric Potential Due
to Dipole
2.16 Electric Current Density
2.17 Energy Density in Electrostatic Fields
2.18 Dielectric Material
2.19 Capacitances for Different Types of
Capacitors
2.20 Energy Stored in a Capacitor
2.21 Conductors
2.22 Continuity Equation : Conservation of
Charge
2.23 Electric Boundary Conditions
2.24 Poisson’s and Laplace’s Equations
2.25 Method of Images
Electrostatic
2CHAPTER
Table of Contents
© Copyrightwww.gateacademy.co.inH Oead ffice : A/114-115, Smriti Nagar, Bhilai (C.G.), Contact : 9713113156, 9589894176
B Oranch ffice : Raipur : 79743-90037, Bhopal : 83198-88401
Learning Objectives : After reading this chapter you should be able to : explain Biot Savart’s law and Ampere’s law define scalar and magnetic vector potential define force on moving charges and current elements understand magnetic boundary conditions
3.1 Biot-Savart’s Law
3.2 Ampere’s Circuital Law-Maxwell’s Equation
3.3 Applications of Ampere’s Circuital Law
3.4 Magnetic Field Intensity H
due to Straight Conductor of Finite Length
3.5 Magnetic Field Intensity H
at the Centre of a Circular Conductor
3.6 Magnetic Flux Density : Maxwell‘s Equation
3.7 Scalar Magnetic Potential
3.8 Force on a Charged Particle and Lorentz Force Equation
3.9 Force Equation Between Parallel Conductors
3.10 Magnetic Torque and Moment
3.11 Magnetic Boundary Conditions
Magnetostatic
3CHAPTER
Table of Contents
© Copyrightwww.gateacademy.co.inH Oead ffice : A/114-115, Smriti Nagar, Bhilai (C.G.), Contact : 9713113156, 9589894176
B Oranch ffice : Raipur : 79743-90037, Bhopal : 83198-88401
Learning Objectives : After reading this chapter you should be able to :
define wave equation and its propagation characteristics
explain wave propagation in different mediums
understand reflection of plane wave at normal incidence and oblique incidence
describe Poynting vector and power density
describe polarization and its type
understand polarization at media interface
4.1 Introduction
4.2 Maxwell’s Equations
4.3 Representation of Wave Equation
4.4 Uniform Plane Wave Equation
4.5 Solution of Plane Wave Equation
4.6 Wave Propagation in Lossy Dielectric
4.7 Wave Propagation in Lossless Dielectric
4.8 Wave Propagation in Free‐space
4.9 Wave Propagation in Good Conductor
4.10 Reflection of Plane Wave at Normal
Incidence
4.11 Electric and Magnetic Field Patterns for
Two Different Media
4.12 Poynting Vector and Poynting Theorem
4.13 Power Density Vector in Two Different
Media
4.14 Reflection of Plane Wave due to Oblique
Incidence
4.15 Parallel Polarization
4.16 Brewster Angle for Parallel Polarization
4.17 Perpendicular Polarization
4.18 Brewster Angle for Perpendicular
Polarization
4.19 Polarization
4.20 Polarization at Media Interface
Plane Wave Propagation & Polarization
4CHAPTER
Table of Contents
© Copyrightwww.gateacademy.co.inH Oead ffice : A/114-115, Smriti Nagar, Bhilai (C.G.), Contact : 9713113156, 9589894176
B Oranch ffice : Raipur : 79743-90037, Bhopal : 83198-88401
Learning Objectives :
After reading this chapter you should be able to :
write transmission line equation
describe lossless and distortionless transmission line
understand reflection coefficient and standing wave pattern for different load conditions
explain stub matching
understand smith chart
describe scattering parameters of two port
5.1 Introduction
5.2 Equivalent Circuit
5.3 Types of Transmission Lines
5.4 Equation of Transmission Line
5.5 Transmission Line Parameters
5.6 Lossless Transmission Line
5.7 Distortionless Transmission Line
5.8 Low Loss Transmission Line
5.9 Input Impedance of Transmission Line
5.10 Reflection Coefficient
5.11 Standing Wave Ratio
5.12 Maximum & Minimum Input Impedance
5.13 Relations for Reflection and
Transmission Coefficients
5.14 Reflection and Transmission Coefficients
for Different Load
5.15 Location of Voltage Maxima and Minima
5.16 Stub Matching
5.17 Single Stub Matching
5.18 Transient Analysis in Transmission Line
(Repeated Reflection)
5.19 The Smith Chart
5.20 Constant Resistance Circles
5.21 Constant Reactance Circles
5.22 Microstrip Transmission lines
5.23 S‐Parameter
5.24 S‐Matrix for Lossless Transmission Line
Transmission Line & S-Parameter
5CHAPTER
Table of Contents
© Copyrightwww.gateacademy.co.inH Oead ffice : A/114-115, Smriti Nagar, Bhilai (C.G.), Contact : 9713113156, 9589894176
B Oranch ffice : Raipur : 79743-90037, Bhopal : 83198-88401
Learning Objectives :
After reading this chapter you should be able to :
derive waveguide equation
understand different modes of propagation
dispersion relation of waveguides
understand field pattern in waveguides
6.1 Introduction
6.2 Comparison Between Waveguide and Twin Wire Transmission Line
6.3 Modes of Wave Propagation
6.4 Rectangular Waveguide
6.5 Rectangular Waveguide Equations
6.6 Modes of Propagation in Rectangular Waveguide
6.7 Field Components in TM Mode
6.8 Field Components in TE Mode
6.9 Rectangular Waveguide Parameters
6.10 Summary of Guided Parameters in Rectangular Waveguide
6.11 Power Transmission and Attenuation in Waveguide
6.12 Power Loss in Waveguide
6.13 Degenerate Modes
6.14 Mode Filter
6.15 Circular Waveguide
Waveguide
6CHAPTER
Table of Contents
© Copyrightwww.gateacademy.co.inH Oead ffice : A/114-115, Smriti Nagar, Bhilai (C.G.), Contact : 9713113156, 9589894176
B Oranch ffice : Raipur : 79743-90037, Bhopal : 83198-88401
Learning Objectives :
After reading this chapter you should be able to :
understand concept of radiation
derive equation of radiation
describe characteristic of antenna
describe antenna array and its parameter
define polarization loss factor
apply the Friis transmission equation to a free space
7.1 Introduction
7.2 Concept of Radiation
7.3 Isotropic Radiator
7.4 Hertzian Dipole
7.5 Power Radiated From a Hertzian Dipole
7.6 Half Wave Dipole Field Expressions
7.7 Quarter Wave Monopole Antenna
7.8 Small Loop Antenna
7.9 Antenna Parameters
7.10 Antenna Arrays
7.11 Uniform Linear Arrays
7.12 Array Factor For Different Patterns
7.13 Parabolic Reflector or Parabolic Microwave Dish
7.14 Principle of Pattern Multiplication
7.15 Polarization Loss Factor & Efficiency
7.16 Friis Equation
Antenna
7CHAPTER
Table of Contents
© Copyrightwww.gateacademy.co.inH Oead ffice : A/114-115, Smriti Nagar, Bhilai (C.G.), Contact : 9713113156, 9589894176
B Oranch ffice : Raipur : 79743-90037, Bhopal : 83198-88401
Learning Objectives : After reading this chapter you should be able to : describe basics of Radar and block diagram of Radar system tell frequency bands used in Radar explain radar range equation define optical fiber and its applications understand light propagation through optical fiber
8.1 Introduction to Radar
8.2 Principle of Radar
8.3 Block diagram of Radar
8.4 Common Parameters of Radar Pulse
8.5 Pulse Repetition Frequency
8.6 Pulse Radar
8.7 Applications of Radar
8.8 Limitations of Radar
8.9 Radar Frequency Bands
8.10 Resolution
8.11 Radar Range Equation
8.12 Optical Fiber Fundamentals
8.13 General Optical Fiber Communication System
8.14 Advantages of Optical Fiber Communications
8.15 Disadvantages of Optical Fiber Communications
8.16 Electromagnetic Spectrum
8.17 Applications of Optical Fiber Communication System
8.18 Ray Transmission Theory
Basics of Radar & Optical Fiber
8CHAPTER
Table of Contents