AET MG university

Applied Electromgnetic Theory(AET)

S5 ECE

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Objectives• To analyze fields potentials due to static changes

• To evaluate static magnetic fields

•To understand how materials affect electric and magnetic fields

• To understand the relation between the fields under time varying situations

• To understand principles of propagation of

Uniform plane waves.

Review of vector analysis• Vector analysis is a mathematical tool with which

electromagnetic(EM) concepts are most conveniently expressed.

• A quantity is called a scalar if it has only magnitude (e.g.,mass,temperature, electric potential, population).

• A quantity is called a vector if it has both magnitude and direction (e.g., velocity, force, electric field intensity).

• The magnitude of a vector is a scalar written as A or …….

Vector Representation

• 2D Vector Representation

Unit Vector, Magnitude• 3D Vector Representation• Classification of Vectors

Zero, unit, like or Unlike, Equal, Co initial Vectors

• Vector Analysis (Addition, Subtraction, Multiplication)

Vector Mathematics

• Vector addition

• Vector subtraction • Vector multiplication

Scalar (dot ) product (A•B)

Vector (cross) product (A X B)

Dot Product

21. =| |

2.

3. ( )

4. ( ) ( ) ( )

5. 0 0

c c c

a a a

a b b a

a b c a b a c

a b a b a b

a

Cross Product

Cross Product

• If a, b, and c are vectors and c is a scalar, then

1. a x b = –b x a

2. (ca) x b = c(a x b) = a x (cb)

3. a x (b + c) = a x b + a x c

4.(a + b) x c = a x c + b x c

5. a · (b x c) = (a x b) · c

6. a x (b x c) = (a · c)b – (a · b)c

Detailed vector Analysis

• Resultant vector

calculation and unit vector• Vector components

representation of x & y axis• Dot and Cross product of 3D vectors

Unit vectors & Other vectors

Co ordinate Planes• The three coordinate axes determine the three

coordinate planes.• The xy-plane contains the x- and y-axes.• The yz-plane contains the y- and z-axes.• The xz-plane contains the x- and z-axes

3-D COORDINATE SYSTEMS

Orthogonal coordinate systems

• The Cartesian (rectangular) coordinate system

• The cylindrical coordinate system• The spherical coordinate system

Cartesian coordinate system

Explanations

x=length, y=breath& z be the height of the element

-∞ < x < ∞

- ∞ < y < ∞

- ∞ < z < ∞

The point A in the coordinate system can be expressed by vector equation

<∞

Cartesian coordinate system

Chapter 1 18

Differential displacement

Differential normal area

Differential volume

zyx dzdydxld aaa

z

y

x

dxdySd

dxdzSd

dydzSd

a

a

a

dxdydzdv

Cartesian Coordinates

1-19

Cylindrical Coordinate Systems

1-20

Circular Cylindrical Coordinates

Point P has coordinatesSpecified by P(z)

Explanations

ρ=radius pssing through P , φ= azimuthal angle measured from X axis & z be the height of the element

0 ≤ ρ< ∞

0≤ φ ≤ 2Π

- ∞ < z < ∞

The point A in the coordinate system can be expressed by vector equation

1-22

Summary

Chapter 1 BEE 3113ELECTROMAGNETIC FIELDS THEORY

23



Differential volume

zdzaadadld

zaddSd

dzadSd

dzadSd

dzdddv

Cylindrical CoordinatesDifferential elements

Constant Coordinate Surfaces in Spherical Coordinates

0 ≤ r< ∞

0≤ ϴ ≤ Π

0 ≤ Φ ≤ 2 Π

1-24

1-25

Spherical Coordinates

Differential Volume in Spherical Coordinates

Chapter 1 27



Differential volume

adrarddrald r sin

ardrdSd

adrdrSd

addrSd r

sin

sin2

ddrdrdv sin2


1-28

Relation between Cartesian & Cylindrical coordinate system

1-29


Point P has coordinatesSpecified by P(r)

Chapter 1 30

Relation to Cartesian coordinates system

x

yz

yx

zyxr

1

221

222

tan

)(tan

cos

sinsin

cossin

rz

ry

rx

Transformation

1-32

Dot Products of Unit Vectors in Cylindrical and Rectangular Coordinate Systems

ax ay az

aρ cosφ sinφ 0

aφ (-sinφ) cosφ 0

az 0 0 1

1-33

Dot Products of Unit Vectors in the Spherical and Rectangular Coordinate Systems

Dot Products of Unit Vectors in the Spherical and cylindrical Coordinate Systems

aρ aΦ az

ar sinϴ 0 cosΦ

aϴ cosϴ 0 -sinϴ

aφ 0 1 0

Vector integration• Linear integrals• Vector area and surface integrals• Volume integrals

Line Integral• The line integral is the

integral of the tangential component of A along Curve L

• Closed contour integral (abca)

Circulation of A around L

A is a vector field

L

ldA

Surface Integral (flux)• Vector field A containing

the smooth surface S• Also called; Flux of A

through S

• Closed Surface IntegralNet outward flux of A from S

A is a vector field

S

SdA

Volume Integral• Integral of scalar over the volume VV

Vector Differential Operator • The vector differential operator (gradient

operator), is not a vector in itself, but when it operates on a scalar function, for example, a vector ensues.

zyx dz

d

dy

d

dx

daaa

zyx dz

d

d

d

d

daaa

aaa

dr

d

rd

d

dr

dr sin

Del related topics

Gradient

Computation formula

properties

Ain Aout

0 A

The flux leaving the one end must exceed the flux entering at the other end.The tubular element is “divergent” in the direction of flow.

Therefore, the operator is frequently called the “divergence” :

AA divDivergence of a vector

Divergence

Outward flux per unit volume is known is as divergence.:

Divergence

h1 h2 h3 u v w

rect 1 1 1 x y z

cylin 1 ρ 1 ρ φ z

spher 1 r rsinϴ r ϴ φ

Divergence=

Divergence• Cylindrical Coordinate System

• Spherical Coordinate System

Divergence

(a) Positive divergence, (b) negative divergence, (c) zero divergence.

Divergence• To evaluate the divergence

of a vector field A at point P(x0,y0,x0), we let the point surrounded by a differential volume

Divergence• properties of the divergence of a vector

field– It produces a scalar field

Divergence Theorem

Curl

Curl is a net orientation or circulation per unit area

Definition. The curl of a is an axial (or rotational) vector whose magnitude is the maximum circulation of A per unit area as the area lends to zero and whose direction is the normal direction of the area when the area is oriented so as to make the circulation maximum.

General

Curl

• Cartesian Coordinates

Curl• Cylindrical Coordinates

Curl• Spherical Coordinates

Properties

Physical Significance

Stokes’ Theorem

Solenoidal & Irrotational (Conservative) vector fields

Laplacian1 – Scalar Laplacian. The Laplacian of a scalar field V, written as . is the divergence of the gradient of V.

The Laplacian of a scalar field is scalar

V2

VVLaplacianV 2

V Gradient of a scalar is vectorDivergence of a vector is scalar

Laplacian• In cartesian coordinates

• In Cylindrical coordinates

• In Spherical Coordinates

zyx

zyxzyx

az

Va

y

Va

x

VV

az

Va

y

Va

x

Va

za

ya

xV

2

2

2

2

2

22

2

Laplacian

• A scalar field V is said to be harmonic in a given region if its Laplacian vanishes in that region. 02 V

Columbs Law

Columbs law in vector form

• Superposition principle in Columbs law

Electricfield intensity

Charge Distributions

Electricfield intensity at different distributions

Electricflux

Relation between E and D D=

Gauss’s Law

First Maxwell's to be derived

• Applications of Gauss’s Law (assignment)

Potential due to a point charge

• Potential due to a system of charges

Potential due to a Charge Distribution

Potential Gradient

• Maximum rate of change of potential with respect to length is known as potential gradient

Second Maxwell's equation to be derived

Relationship between E & V

Electric Dipole

• Dipole moment & Equipotential surface

• Calculation of electric field due to potential from the coordinate systems

(Energy stored due to point charge)

Energy stored due to a distribution of charge

Calculation of energy using distribution in terms of D in volume charge distribution

Boundary condition

• The condition that the field must satisfy at the interface separating the media are called boundary conditions

Refraction of E & D at the boundary

Module -II

Current density

• Conduction current • Displacement current

Maxwell's first equation

Wave Equations in Electricfield

Wave Equations in Magnetic Field

139

Polarization of em waves

Vertical Polarization-When E field vector of EM wave is perpendicular to

the earth, the EM wave said to be Vertically Polarized..

12/8/2015

140

• Horizontal polarization– When E field vector of EM wave is parallel to the earth,

the EM wave said to be Horizontally Polarized.

12/8/2015

141

Circular Polarization– When E and H field of the EM wave are of same

amplitude and having a phase difference of 90o, wave is said to be circularly polarized..

12/8/2015

Fig: Circular Polarization.

Lumped circuits: resistors, capacitors, inductors

neglect time delays (phase)

account for propagation and time delays (phase change)

Transmission-Line Theory

Distributed circuit elements: transmission lines

We need transmission-line theory whenever the length of a line is significant compared with a wavelength.

155

Transmission Line

2 conductors

4 per-unit-length parameters:

C = capacitance/length [F/m]

L = inductance/length [H/m]

R = resistance/length [/m]

G = conductance/length [ /m or S/m]

Dz

156

Transmission Line (cont.)

z

,i z t

+ + + + + + +- - - - - - - - - - ,v z tx x xB

157

RDz LDz

GDz CDz

z

v(z+z,t)

+

-

v(z,t)

+

-

i(z,t) i(z+z,t)

( , )( , ) ( , ) ( , )

( , )( , ) ( , ) ( , )

i z tv z t v z z t i z t R z L z

tv z z t

i z t i z z t v z z t G z C zt

Transmission Line (cont.)

158

RDz LDz

GDz CDz

z

v(z+z,t)

+

-

v(z,t)

+

-

i(z,t) i(z+z,t)

Hence

( , ) ( , ) ( , )( , )

( , ) ( , ) ( , )( , )

v z z t v z t i z tRi z t L

z ti z z t i z t v z z t

Gv z z t Cz t

Now let Dz 0:

v iRi L

z ti v

Gv Cz t

“Telegrapher’sEquations”

TEM Transmission Line (cont.)

159

v iRi L

z ti v

Gv Cz t

To combine these, take the derivative of the first one with

respect to z:

2

2

2

2

v i iR L

z z z t

i iR L

z t z

vR Gv C

t

v vL G C

t t

Switch the order of the derivatives.


160

2 2

2 2( ) 0

v v vRG v RC LG LC

z t t

The same equation also holds for i.

Hence, we have:

2 2

2 2

v v v vR Gv C L G C

z t t t


161

2

2

2( ) ( ) 0

d VRG V RC LG j V LC V

dz

2 2

2 2( ) 0

v v vRG v RC LG LC

z t t


Time-Harmonic Waves:

162

Note that

= series impedance/length

2

2

2( )

d VRG V j RC LG V LC V

dz

2( ) ( )( )RG j RC LG LC R j L G j C

Z R j L

Y G j C

= parallel admittance/length

Then we can write:2

2( )

d VZY V

dz


163

Let

Convention:

Solution:

2 ZY

( ) z zV z Ae Be

1/2

( )( )R j L G j C

principal square root

2

2

2( )

d VV

dzThen


is called the "propagation constant."

/2jz z e

j

0, 0

attenuationcontant

phaseconstant

164

Analysis of Wave• Forward and Backward wave Analysis

Properties of Transmission Line

• Characteristics impedance of Transmission line

v iRi L

z ti v

Gv Cz t

Load Impedance & Reflection Coefficient

• Reflection at any point on the transmission line

• Impedance at any point on the transmission line

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