EEE241: Fundamentals of Electromagnetics

EEE241: Fundamentals of Electromagnetics

Introductory Concepts, Vector Fields and Coordinate Systems

Instructor: Dragica Vasileska

Outline

• Class Description

• Introductory Concepts

• Vector Fields

• Coordinate Systems

Class Description

Prerequisites by Topic:– University physics– Complex numbers– Partial differentiation– Multiple Integrals– Vector Analysis– Fourier Series

Class Description

• Prerequisites: EEE 202; MAT 267, 274 (or 275), MAT 272; PHY 131, 132

• Computer Usage: Students are assumed to be versed in the use MathCAD or MATLAB to perform scientific computing such as numerical calculations, plotting of functions and performing integrations. Students will develop and visualize solutions to moderately complicated field problems using these tools.

• Textbook: Cheng, Field and Wave Electromagnetics.

Class Description

• Grading:

Midterm #1 25%

Midterm #2 25%Final 25%

Homework 25%

Class Description

Why Study Electromagnetics?

Examples of Electromagnetic Applications

Examples of Electromagnetic Applications, Cont’d




Research Areas of Electromagnetics

• Antenas• Microwaves• Computational Electromagnetics• Electromagnetic Scattering• Electromagnetic Propagation• Radars• Optics• etc …

Why is Electromagnetics Difficult?

What is Electromagnetics?

What is a charge q?

Fundamental Laws of Electromagnetics

Steps in Studying Electromagnetics

SI (International System) of Units

Units Derived From the Fundamental Units

Fundamental Electromagnetic Field Quantities

Three Universal Constants

Fundamental Relationships

Scalar and Vector Fields

• A scalar field is a function that gives us a single value of some variable for every point in space.

• Examples: voltage, current, energy, temperature

• A vector is a quantity which has both a magnitude and a direction in space.

• Examples: velocity, momentum, acceleration and force

Example of a Scalar Field

26

Scalar Fields

e.g. Temperature: Every location has associated value (number with units)

27

Scalar Fields - Contours

• Colors represent surface temperature• Contour lines show constant

temperatures

28

Fields are 3D

•T = T(x,y,z)

•Hard to visualize Work in 2D

29

Vector FieldsVector (magnitude, direction) at every point

in space

Example: Velocity vector field - jet stream

Vector Fields Explained

Examples of Vector Fields



VECTOR REPRESENTATION

3 PRIMARY COORDINATE SYSTEMS:

• RECTANGULAR

• CYLINDRICAL

• SPHERICAL

Choice is based on symmetry of problem

Examples:

Sheets - RECTANGULAR

Wires/Cables - CYLINDRICAL

Spheres - SPHERICAL

Orthogonal Coordinate Systems: (coordinates mutually perpendicular)

Spherical Coordinates

Cylindrical Coordinates

Cartesian Coordinates

P (x,y,z)

P (r, Θ, Φ)

P (r, Θ, z)

x

y

zP(x,y,z)

θ

z

rx y

z

P(r, θ, z)

θ

Φ

r

z

yx

P(r, θ, Φ)

Page 108

Rectangular Coordinates

-Parabolic Cylindrical Coordinates (u,v,z)-Paraboloidal Coordinates (u, v, Φ)-Elliptic Cylindrical Coordinates (u, v, z)-Prolate Spheroidal Coordinates (ξ, η, φ)-Oblate Spheroidal Coordinates (ξ, η, φ)-Bipolar Coordinates (u,v,z)-Toroidal Coordinates (u, v, Φ)-Conical Coordinates (λ, μ, ν)-Confocal Ellipsoidal Coordinate (λ, μ, ν)-Confocal Paraboloidal Coordinate (λ, μ, ν)

Parabolic Cylindrical Coordinates

Paraboloidal Coordinates

Elliptic Cylindrical Coordinates

Prolate Spheroidal Coordinates

Oblate Spheroidal Coordinates

Bipolar Coordinates

Toroidal Coordinates

Conical Coordinates

Confocal Ellipsoidal Coordinate

Confocal Paraboloidal Coordinate

Cartesian CoordinatesP(x,y,z)

Spherical CoordinatesP(r, θ, Φ)

Cylindrical CoordinatesP(r, θ, z)

x

y

zP(x,y,z)

θ

z

rx y

z

P(r, θ, z)

θ

Φ

r

z

yx

P(r, θ, Φ)

Coordinate Transformation

• Cartesian to Cylindrical(x, y, z) to (r,θ,Φ)

(r,θ,Φ) to (x, y, z)

• Cartesian to CylindricalVectoral Transformation



• Cartesian to Spherical(x, y, z) to (r,θ,Φ)

(r,θ,Φ) to (x, y, z)

• Cartesian to Spherical Vectoral Transformation


Page 109

x

y

z

Z plane

y planex plane

xyz

x1

y1

z1

Ax

Ay

Unit vector properties

0ˆˆˆˆˆˆ

1ˆˆˆˆˆˆ

xzzyyx

zzyyxx

yxz

xzy

zyx

ˆˆˆ

ˆˆˆ

ˆˆˆ

Vector Representation

Unit (Base) vectors

A unit vector aA along A is a vector whose magnitude is unity

A

Aa

zyx AzAyAxA ˆˆˆ

Page 109

x

y

z

Z plane

y planex plane

222zyx AAAAAA

xyz

x1

y1

z1

Ax

Ay

Az

Vector representation

Magnitude of A

Position vector A

),,( 111 zyxA

111 ˆˆˆ zzyyxx

Vector Representation

x

y

z

Ax

Ay

AzA

B

Dot product:

zzyyxx BABABABA

Cross product:

zyx

zyx

BBB

AAA

zyx

BA

ˆˆˆ

Back


Page 108

Multiplication of vectors

• Two different interactions (what’s the difference?)– Scalar or dot product :

• the calculation giving the work done by a force during a displacement

• work and hence energy are scalar quantities which arise from the multiplication of two vectors

• if A·B = 0– The vector A is zero

– The vector B is zero = 90°

ABBABA cos||||

A

B

– Vector or cross product :

• n is the unit vector along the normal to the plane containing A and B and its positive direction is determined as the right-hand screw rule

• the magnitude of the vector product of A and B is equal to the area of the parallelogram formed by A and B

• if there is a force F acting at a point P with position vector r relative to an origin O, the moment of a force F about O is defined by :

• if A x B = 0– The vector A is zero

– The vector B is zero = 0°

nsin|||| BABA

A

B

ABBA

FrL

Commutative law :

ABBA

ABBA

Distribution law :

CABACBA )(

CABACBA )(

Associative law :

))(( DCBADBCA

CBABCA )(

CBACBA )(

CBACBA )()(

Unit vector relationships

• It is frequently useful to resolve vectors into components along the axial directions in terms of the unit vectors i, j, and k.

1

0

kkjjii

ikkjji

jik

ikj

kji

kkjjii

0

zyx

zyx

zzyyxx

zyx

zyx

BBB

AAA

kji

BA

BABABABA

kBjBiBB

kAjAiAA

Scalar triple product CBA

The magnitude of is the volume of the parallelepiped with edges parallel to A, B, and C.

CBA

A

BC

AB

],,[ CBABACACBACBCBACBA

Vector triple product CBA

The vector is perpendicular to the plane of A and B. When the further vectorproduct with C is taken, the resulting vector must be perpendicular to and hence in the plane of A and B :

BA

A

BC

AB

BA

nBmACBA )( where m and n are scalar constants to be determined.

0)( BnCAmCCBACACn

BCm

BACABCCBA )()()( Since this equation is validfor any vectors A, B, and CLet A = i, B = C = j:

1

CBABCACBA

ACBBCACBA

)()()(

)()()(

x

z

y

VECTOR REPRESENTATION: UNIT VECTORS

yaxa

zaUnit Vector

Representation for Rectangular

Coordinate System

xaThe Unit Vectors imply :

ya

za

Points in the direction of increasing x

Points in the direction of increasing y

Points in the direction of increasing z

Rectangular Coordinate System

r

z

P

x

z

y


Cylindrical Coordinate System

za

a

ra

The Unit Vectors imply :

za

Points in the direction of increasing r

Points in the direction of increasing

Points in the direction of increasing z

ra

a

BaseVectors

A1

ρ radial distance in x-y plane

Φ azimuth angle measured from the positive x-axis

Z

r0

20

z


ˆˆˆ

,ˆˆˆ

,ˆˆˆ

z

z

z

zAzAAAaA ˆˆˆˆ

Pages 109-112Back

( ρ, Φ, z)


222zAAAAAA

Magnitude of A

Position vector A

Base vector properties

11 ˆˆ zz

Dot product:

zzrr BABABABA

Cross product:

zr

zr

BBB

AAA

zr

BA

ˆˆˆ

B A

Back


Pages 109-111


Spherical Coordinate System

r

P

x

z

y

a

a

ra

The Unit Vectors imply :

Points in the direction of increasing r



ra

aa

ˆˆˆ,ˆˆˆ,ˆˆˆ RRR


Pages 113-115Back

(R, θ, Φ)

AAARA Rˆˆˆ


222 AAAAAA R

Magnitude of A

Position vector A

1ˆRR

Base vector properties

Dot product:

BABABABA RR

Cross product:

BBB

AAA

R

BA

R

R

ˆˆˆ

Back

B A


Pages 113-114

zr aaa ˆˆˆ aaar ˆˆˆ zyx aaa ˆˆˆ

RECTANGULAR Coordinate Systems

CYLINDRICAL Coordinate Systems

SPHERICAL Coordinate Systems

NOTE THE ORDER!

r,, z r,,

Note: We do not emphasize transformations between coordinate systems


Summary

METRIC COEFFICIENTS

1. Rectangular Coordinates:

When you move a small amount in x-direction, the distance is dx

In a similar fashion, you generate dy and dz

Unit is in “meters”


Differential quantities:

Differential distance:

Differential surface:

Differential Volume:

dzzdyydxxld ˆˆˆ

dxdyzsd

dxdzysd

dydzxsd

z

y

x

ˆ

ˆ

ˆ

dxdydzdv

Page 109

Cylindrical Coordinates:

Distance = r d

x

y

d

r

Differential Distances:

( dr, rd, dz )

Cylindrical Coordinates:

Differential Distances: ( dρ, rd, dz )

zadzadadld ˆˆˆ

zz addsd

adzdsd

adzdsd

ˆ

ˆ

ˆ

Differential Surfaces:

Differential Volume:

Spherical Coordinates:

Distance = r sin d

x

y

d

r sin

Differential Distances:

( dr, rd, r sind )

r

P

x

z

y


Differential quantities:

Length:

Area:

Volume:

dRRddRR

dldldlRld R

sinˆˆˆ

ˆˆˆ

RdRddldlsd

dRdRdldlsd

ddRRdldlRsd

R

R

R

ˆˆ

sinˆˆ

sinˆˆ 2

ddRdRdv sin2

dRdl

Rddl

dRdlR

sin

Pages 113-115Back

Representation of differential length dl in coordinate systems:

zyx adzadyadxld ˆˆˆ

zr adzadradrld ˆˆˆ

adrardadrld r ˆsinˆˆ

rectangular

cylindrical

spherical

METRIC COEFFICIENTS

Example

• For the object on the right calculate:

• (a) The distance BC• (b) The distance CD• (c) The surface area ABCD• (d) The surface area ABO• (e) The surface area A OFD• (f) The volume ABDCFO

AREA INTEGRALS

• integration over 2 “delta” distances

dx

dy

Example:

x

y

2

6

3 7

AREA = 7

3

6

2

dxdy = 16

Note that: z = constant

In this course, area & surface integrals will be on similar types of surfaces e.g. r =constant or = constant or = constant et c….

Representation of differential surface element:

zadydxsd ˆ

Vector is NORMAL to surface

SURFACE NORMAL

DIFFERENTIALS FOR INTEGRALS

Example of Line differentials

or or

Example of Surface differentials

zadydxsd ˆradzrdsd ˆ

or

Example of Volume differentials dzdydxdv

xadxld ˆ

radrld ˆ

ardld ˆ

zz

yx

yxr

ˆˆ

cosˆsinˆˆ

sinˆcosˆˆ

zz

yx

yxr

AA

AAA

AAA

cossin

sincos

Back

Cartesian to Cylindrical Transformation

zz

xy

yxr

)/(tan 1

22

Page 115

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EEE241: Fundamentals of Electromagnetics