Electronics and Communication Engineering : Electromagnetiic theory, THE GATE ACADEMY

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Syllabus Electromagnetic Theory

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Syllabus for Electromagnetic Theory

Elements of vector calculus: divergence and curl; Gauss and Stoke’s theorems, Maxwell’s

equations: differential and integral forms. Wave equation, Poynting vector. Plane waves:

propagation through various media; reflection and refraction; phase and group velocity; skin

depth. Transmission lines: characteristic impedance; impedance transformation; Smith chart;

impedance matching; S parameters, pulse excitation. Waveguides: modes in rectangular

waveguides; boundary conditions; cut-off frequencies; dispersion relations. Basics of

propagation in dielectric waveguide and optical fibers. Basics of Antennas: Dipole antennas;

radiation pattern; antenna gain.

Analysis of GATE Papers

(Electromagnetic Theory)

Year Percentage of marks Overall Percentage

2013 5.00

9.12%

2012 12.00

2011 9.00

2010 7.00

2009 8.00

2008 8.00

2007 10.67

2006 12.00

2005 8.71

2004 9.34

2003 10.67

Contents Electromagnetic Theory

THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, [email protected] © Copyright reserved. Web: www.thegateacademy.com Page I

CC OO NN TT EE NN TT SS

Chapter Page No

#1. Electromagnetic Field 1 – 49 Introduction to Vector Calculus 1 – 7

Material and Physical Constants 7 – 8

Electromagnetic (EM Field) 8 – 18

Divergence of Current Density and Relaxation 18 – 22

The Magnetic Vector Potential 22 – 27

Faraday Law 27 – 29

Maxwell’s Equation’s 29 – 36

Assignment 1 37 – 39


Answer keys 43

Exlanations 43 – 49

#2. EM Wave Propagation 50 – 86 Introduction 50

General wave equations 50 – 51

Plane wave in a Dielectric medium 51 – 53

Poynting Vector 53 – 54

Phase Velocity 55 – 64

Wave Polarization 64 -71



Answer keys 79


#3. Transmission Lines 87 – 127 Introduction 87 – 98

Transmission & Reflection of Waves on a Transmission Line 98 – 100

Impedance of a Transmission Line 100 – 106

The Smith Chart 107 – 108

Scattering Parameters 108 – 109

Strip Line 109 – 113



Answer keys 120


#4. Guided E.M Waves 128 – 161 Wave Guide 128 – 130

Transverse Magnetic Mode 130 – 133

Transverse Electric Mode 133 – 142

Circuter Wave Guide 142 – 152

Contents Electromagnetic Theory

THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, [email protected] © Copyright reserved. Web: www.thegateacademy.com Page II



Answer keys 157


#5. Antennas 162 – 199 Inroduction 162

Hertzian Dipole 162 – 165

Field Regions 165

Radiation Pattern 165 – 166

Radiaton Intensity 166 – 168

Antenna Radiation Efficiency 168 – 170

Antenna Arrays 170 – 175

Solved Examples 176 – 188



Answer keys 194


Module Test 200 – 215 Test Questions 200 – 207

Answer Keys 208

Explanations 208 – 215

Reference Books 216

Chapter 1 Electromagnetic Theory

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CHAPTER 1

Electromagnetic Field

Introduction to vector calculus

Cartesian coordinates (x, y, z), x , y , z Cylindrical coordinates ( , , z), , , z Spherical coordinates (r, , ) , r , , Vector calculus formula

Table 1.1 S. No Cartesian coordinates Cylindrical

coordinates Spherical coordinates

(a) Differential displacement dl = dx + dy + dz

dl = d + d

+dz

dl = dr + rd + r sin d

(b) Differential area ds = dydz = dxdz

= dxdy

ds = d dz

= d dz = d d

ds = r sin d d = r sin dr d = r dr d

(c) Differential volume dv = dxdydz

dv = d d dz dv = r sin d d dr

Operators

1) V – gradient , of a Scalar V

2) .V – divergence , of a vector V

3) V – curl , of a vector V 4) V – laplacian , of a scalar V

DEL Operator

=

(Cartesian)

=

(Cylindrical)

=

(Spherical)

Gradient of a Scalar field V is a vector that represents both the magnitude and the direction of maximum space rate of increase of V.

V =

=

=



The following are the fundamental properties of the gradient of a scalar field V:

1. The m gnitude of V equ ls the m ximum r te of ch nge in V per unit dist nce. 2. V points in the direction of the maximum rate of change in V. 3. V t ny point is perpendicular to the constant V surface that passes through that point. 4. If A = V, V is s id to be the sc l r potenti l of A. 5. The projection of V in the direction of unit vector |a| is V. |a| and is called the directional

derivative of V along |a|. This is the rate of change of V in direction of |a|.

Example: Find the gradient of the following scalar fields:

(a) V = e sin 2x cosh y (b) U = z cos (c) W = r sin cos

Solution

(a) V =

= e cos x cosh y e sin x sinh y e sin x cosh y

(b) U =

= z cos z sin cos

(c) W =

= sin cos sin cos sin

Divergence of vector

A at a given point P is the outward flux per unit volume as the volume shrinks about P.

Hence,

divA = . A = lim ∮ .

(1)

Where, V is the volume enclosed by the closed surf ce S in which P is loc ted. Physic lly, we

may regard the divergence of the vector field A⃗⃗ at a given point as a measure of how much the field diverges or emanates from that point.

.A =

=

( A )

=

(r A )

(A sin )

From equation (1), ∮ A ds

= ∫ . A dv



This is called divergence theorem which states that the total outward flux of the vector field A through a closed surface S is same as the volume integral of the divergence of A. Example

Determine the divergence of these vector field:

(a) P = x yz xz (b) Q = sin z z cos

(c) T =

cos r sin cos cos

Solution

(a) P =

P

P

P

=

x(x yz)

y( )

z(xz)

= xyz x

(b) Q =

( Q )

Q

Q

=

( sin )

( z)

z (z cos )

= sin cos

(c) T =

(r T )

(T sin )

(T )

=

r

r(cos )

r sin

(r sin cos )

r sin

(cos )

=

r sin r sin cos cos

= cos cos

Curl of a vector field provides the maximum value of the circulation of the field per unit area and indicates the direction along which this maximum value occurs.

That is,

curl A = A = lim (∮ .

)

------------- (2)

A = |

A A A

|

=

|

A A A

|



=

|

r r sin

A rA r sin A

|

From equation (2) we may expect that ∮ . = ∫ (

).

This is called stoke’s theorem, which states that the circulation of a vector field A around a (closed) path L is equal to the surface integral of the curl of A over the open surface S bounded by L. Example Determine the curl of each of the vector fields of previous Example. Solution

(a) = (

) (

) (

)

= ( ) ( ) ( )

= ( )

(b) = *

+ *

+

*

( )

+

= (

) ( )

( )

=

( ) ( )

(c) =

*

( )

+

*

( )+

*

( )

+

=

[

( )

( )]

[

( )

( )]

[

( )

( )

]

=

( )

( )

(

)

= (

)

(

)

(a) Laplacian of a scalar field V, is the divergence of the gradient of V and is written as .

=

=

(

)

=

(

)

(

)

If = 0, V is said to be harmonic in the region. A vector field is solenoid if .A = ; it is irrot tion l or conserv tive if A =

. ( ) = ( ) =



(b) Laplacian of vector A̅

A⃗⃗ = is lw ys vector qu ntity

A⃗⃗ = ( A ) ̂x ( A ) ̂y ( A ) ̂z

A Sc l r qu ntity A Sc l r qu ntity

A Sc l r qu ntity

V =

........Poission’s Eqn

V = ........Laplace Eqn

E = ⃗⃗

E

....... wave Eqn

Example

The potential (scalar) distribution is given as

V = y x if E0 : permittivity of free space what is the change density p at the point (2,0)?

Solution

Poission’s Eqn V =

(

) ( y x ) =

x x x x x y =

At pt( , ) x x x =

=

Example

Find the Laplacian of the following scalar fields,

(a) V = e sin 2x cosh y (b) U = z cos (c) W = r sin cos

Solution

The Laplacian in the Cartesian system can be found by taking the first derivative and later the second derivative.

(a) V =

=

x( e cos x coshy)

y(e sin x sinh y)

z( e sin x cosh y)

= e sin x cosh y e sin x coshy e sin x cosh y = e sin x cosh y

(b) U =

(

)



=

( z cos )

z cos

= z cos z cos =

(c) W =

(r

)

(sin

)

=

r

r( r sin cos )

r sin

( r sin sin cos )

r sin cos

r sin

= sin cos

r

r cos sin cos

r sin

r sin cos cos

r sin

cos

r

= cos

r ( sin cos cos )

= cos

r( cos )

Stoke’s theorem

Statement:- closed line integral of any vector A⃗⃗ integrated over any closed curve C is always

equal to the surface integral of curl of vector A⃗⃗ integr ted over the surf ce re ‘s’ which is enclosed by the closed curve ‘c’

∮ A⃗⃗ . d ⃗ = ∫ ∫( x A⃗⃗ ) dS⃗

The theorem is valid irrespective of

(i) Shape of closed curve ‘C’ (ii) Type of vector ‘A’ (iii) Type of co-ordinate system. Divergence theorem

∯ A⃗⃗

dS⃗ = ∭ V⃗⃗ . A⃗⃗ dv

S

V

S

C

Electronics and Communication Engineering : Electromagnetiic theory, THE GATE ACADEMY

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scalar v

v divergence

d d ds

v points

projection of v

scalar field v

d d dz dv

direction of unit v