ELEG 648Summer 2012

Lecture #1Mark Mirotznik, Ph.D.

Associate ProfessorThe University of Delaware

Tel: (302)831-4221Email: mirotzni@ece.udel.edu

Vector Analysis Review:

A

Aa

AaA A

a

= unit vector

AaA A

1. Dot Product (projection)

)cos( ABBABA

2. Cross Product

)sin( ABn BAaBA

AaA A

A

BaB B

ABna

Orthogonal Coordinate Systems:

23

22

21

213

312

321

332211

uuu

uuu

uuu

uuu

Auuuuuu

AAAA

aaa

aaa

aaa

aAAaAaAaA

332211 uuuuuu BABABABA

)(

)()(

12213

3113223321

uuuuu

uuuuuuuuuu

BABAa

BABAaBABAaBA

321

321

321

uuu

uuu

uuu

BBB

AAA

aaa

BA

Orthogonal Coordinate Systems:

dl332211 dladladlald uuu

Sd

na

dSaSd n

321 dldldldv

dl1dl2

dl3

Cartesian Coordinate Systems:

yxz

zxy

zyx

Azzyyxx

aaa

aaa

aaa

aAAaAaAaA

zzyyxx BABABABA

zyx

zyx

zyx

BBB

AAA

aaa

BA

x

y

z

Cartesian Coordinate Systems (cont):

dzdydxdv

dydxads

dzdxads

dzdyads

dzdydxdl

dzadyadxald

zz

yy

xx

zyx

222

Cylindrical Coordinate Systems:

Azzrr aAAaAaAaA

dzdrdrdv

dzrdads

dzdrads

dzrdads

dzardadrald

zz

rr

zr

x

y

z

r

z

(r,z)

Spherical Coordinate Systems:

ARR aAAaAaAaA

dddRRdv

dRdRads

ddRRads

ddRads

dRaRdadRald

RR

R

)sin(

)sin(

)sin(

)sin(

2

2

x

y

z

R

(R)

Vector Coordinate Transformation:

z

r

z

y

x

A

A

A

A

A

A

100

0)cos()sin(

0)sin()cos(

A

A

A

A

A

A R

z

y

x

0)sin()cos(

)cos()sin()cos()sin()sin(

)sin()cos()cos()cos()sin(

Gradient of a Scalar Field:

Assume f(x,y,z) is a scalar fieldThe maximum spatial rate of change of f at some locationis a vector given by the gradient of f denoted byGrad(f) or f

)sin(R

fa

r

fa

R

faf

z

fa

r

fa

r

faf

z

fa

y

fa

x

faf

R

zr

zyx

Divergence of a Vector Field:

Assume E(x,y,z) is a vector field. The divergence of E is defined as the net outward flux of E in some volume as thevolume goes to zero. It is denoted by E

E

R

ER

ERRR

E

z

EE

rrE

rrE

z

E

y

E

x

EE

R

zr

zyx

)sin(

1

))(sin()sin(

1)(

1

1)(

1

22

Curl of a Vector Field:

Assume E(x,y,z) is a vector field. The curl of E is measureof the circulation of E also called a “vortex” source. Itis denoted by E

ERREER

aRaRa

RE

ErEEzr

aara

rE

EEEzyx

aaa

E

R

R

zr

zr

zyx

zyx

)sin(

)sin(

)sin(

1

1

2

Laplacian of a Scalar Field:

)( VAssume f(x,y,z) is a scalar field. The Laplacian is defined as and denoted by V2

2

2

22

22

2

2

2

2

2

22

2

2

2

2

2

22

)(sin

1

))(sin()sin(

1)(

1

1)(

1

V

R

VR

VR

RRR

V

z

VV

rV

rr

rrV

z

V

y

V

x

VV

Examples:

1. Given the scalar functionzeyxzyxV )2/sin()2/sin(),,(

Find the magnitude and direction of the maximum rate of chance atlocation (xo,yo,zo)

2. Determine )( V

3. Determine )( V

3. The magnetic field produced by a long wire conducting a constant currentIs given by

r

IarB o

)(

Find B

Basic Theorems:

1. Divergence Theorem or Gauss’s Law

sdEdvEsv

2. Stokes Theorem

cs

ldEsdE

)(

Examples:

1. Verify the Divergence Theorem for

zarazrA zr 2),( 2

on a cylindrical region enclosed by r=5, z=0 and z=4

r = 5

z = 0

z = 4

Odds and Ends:

1. Normal component of field

n

E

nEEn

2. Tangential component of field

tEEn

Maxwell’s Equations in Differential Form

m

ic

B

D

JJt

DH

Mt

BE

Faraday’s Law

Ampere’s Law

Gauss’s Law

Gauss’s Magnetic Law

Faraday’s Law

sdBt

ldE

t

BE

c s

S

C

t

B

E

Ampere’s Law

sc ssdJsdD

tldH

t

DJH

t

D

J

J

H

H

Gauss’s Law

v totsQdvsdD

D

totQ

D

Gauss’s Magnetic Law

0

0

ssdB

B

B

“all the flow of B entering the volume V must leave the volume”

CONSTITUTIVE RELATIONS

EJ

HB

ED

c

r o=permittivity (F/m)

o=8.854 x 10-12 (F/m)

r o=permeability (H/m)

o=4 x 10-7 (H/m)

=conductivity (S/m)

POWER and ENERGY

Ji

E, H

V

S

n

icdi

d

JJJJEt

EHeq

Mt

HEeq

)2(

)1(

)2()1( eqEeqH

)()3( icdd JJJEMHHEEHeq

take

Using the vector identity )()()( BAABBA

0)()()4( icdd JJJEMHHEeq

Integrate eq4 over the volume V in the figure

v icddv

dvJJJEMHdvHEeq )]([)()5(

Applying the divergence theorem

0)][)()6(

v is

dvJEEEt

EE

t

HHdsHEeq

POWER and ENERGY (continued)

0)][)()6(

v is

dvJEEEt

EE

t

HHdsHEeq

222,

2

1,

2

1EEEw

tE

tt

EEw

tH

tt

HH em

0][][)()7(2

vv iv

ems

dvEdvJEdvt

w

t

wdsHEeq

0][][)()8(2

vv iv mes

dvEdvJEdvwwt

dsHEeq

0,0][

]2

1[,]

2

1[

)(

2

22

vdv ii

vevm

ss

dvEPdvJEP

dvEWdvHW

dsHEP

diems PPWt

Wt

P

Stored magnetic power (W)

Stored electric power (W)

Supplied power (W)

Dissipated power (W)

What is this term?

POWER and ENERGY (continued)

0,0][

]2

1[,]

2

1[

)(

2

22

vdv ii

vevm

ss

dvEPdvJEP

dvEWdvHW

dsHEP

diems PPWt

Wt

P

Stored magnetic power (W)

Stored electric power (W)

Supplied power (W)

Dissipated power (W)

What is this term?

Ps = power exiting the volume through radiation

HES

W/m2 Poynting vector

TIME HARMONIC EM FIELDS

]),,(~

Re[),,,(

)),,(cos(),,(),,,(tj

o

ezyxEtzyxE

zyxtzyxEtzyxE

Assume all sources have a sinusoidal time dependence and all materialsproperties are linear. Since Maxwell’s equations are linear all electricand magnetic fields must also have the same sinusoidal time dependence.They can be written for the electric field as:

),,(~

zyxE is a complex function of space (phasor) called the time-harmonic electricfield. All field values and sources can be represented by their time-harmonic form.

]),,(~Re[),,,(

]),,(~

Re[),,,(

]),,(~

Re[),,,(

]),,(~

Re[),,,(

]),,(~

Re[),,,(

]),,(~

Re[),,,(

tj

tj

tj

tj

tj

tj

ezyxtzyx

ezyxJtzyxJ

ezyxBtzyxB

ezyxHtzyxH

ezyxDtzyxD

ezyxEtzyxE

)sin()cos( tjte tj Euler’s Formula

PROPERTIES OF TIME HARMONIC FIELDS

]),,(~

[Re[]]),,(~

[Re[ tjtj ezyxEjezyxEt

]),,(~

[Re[1

]),,(~

[Re[ tjtj ezyxEj

dtezyxE

Time derivative:

Time integration:

TIME HARMONIC MAXWELL’S EQUATIONS

tj

mtj

tjtj

tjtjtj

tjtjtj

eeB

eeD

eJeDt

eH

eMeBt

eE

~Re

~Re

~Re~

Re

~Re

~Re

~Re

~Re

~Re

~Re

mB

D

Jt

DH

Mt

BE

mB

D

JDjH

MBjE

~~

~~

~~~

~~~

Employing the derivative property results in the following set of equations:

TIME HARMONIC EM FIELDSBOUNDARY CONDITIONS AND CONSTITUTIVE PROPERTIES

The constitutive properties and boundary conditions are very similarfor the time harmonic form:

0)~~

(ˆ

~)~~

(ˆ

~)

~~(ˆ

0)~~

(ˆ

12

12

12

12

BBn

DDn

JHHn

EEn

s

s

EJ

HB

ED

c~~

~~

~~

Constitutive Properties

General Boundary Conditions

0~

ˆ

~~ˆ

~~ˆ

0~

ˆ

2

2

2

2

Bn

Dn

JHn

En

s

s

PEC Boundary Conditions

TIME HARMONIC EM FIELDSIMPEDANCE BOUNDARY CONDITIONS

If one of the material at an interface is a good conductor but of finiteconductivity it is useful to define an impedance boundary condition:

HnjHnZJZE

jjXRZ

ssst

sss

~ˆ

2)1(

~ˆ

~~

2)1(

1,

2,

1>> 2

POWER and ENERGY: TIME HARMONIC

0~

2

1,0]

~~2

1[

]~

4

1[,]

~4

1[

)~~

(

2*

22

*

vdv ii

vevm

ss

dvEPdvJEP

dvEWdvHW

dsHEP

diems PPWWjP )(2

Time average magneticenergy (J)

Time average electric energy (J)

Supplied complex power (W)

Dissipated real power (W)Time average exiting power

CONTINUITY OF CURRENT LAW

JDt

Jt

DH

][][)(

0

B

D

Jt

DH

t

BE

0)( A

vector identity

JDt

][0

Jt

][0

tJ

jJ

time harmonic

SUMMARY

mBD

Jt

DHM

t

BE

mBD

JDjHMBjE

~~~~

~~~~~~

0)~~

(ˆ~)~~

(ˆ

~)

~~(ˆ0)

~~(ˆ

1212

1212

BBnDDn

JHHnEEn

s

s

0)(ˆ)(ˆ

)(ˆ0)(ˆ

1212

1212

BBnDDn

JHHnEEn

s

s

EJ

HB

ED

c~~

~~

~~

EJ

HB

ED

c

2

)1( jjXRZ sss

0,0][

]2

1[,]

2

1[

)(

2

22

vdv ii

vevm

ss

dvEPdvJEP

dvEWdvHW

dsHEP

0~

2

1,0]

~~2

1[

]~

4

1[,]

~4

1[

)~~

(

2*

22

*

vdv ii

vevm

ss

dvEPdvJEP

dvEWdvHW

dsHEP

Frequency DomainTime Domain

Electromagnetic Properties of Materials

Primary Material Properties

r o=permeability (H/m)

o=4 x 10-7 (H/m)

r o=permittivity (F/m)

o=8.854 x 10-12 (F/m)

=conductivity (S/m)

Electrical Properties

Magnetic Properties

Secondary Material Properties

Electrical Properties

rn Index of refraction

e Electric susceptibility

Magnetic Properties

m Magnetic susceptibility

Electric Properties of Materials

+

Eext

Eext

-

+

li

Qi

No external field

Applied external field

iii lQp

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

Bulk material (N molecules)

N

iii

N

ii lQpP

11

Electric dipole momentof individual atom ormolecule:

Net dipole moment or polarization vector:

Eext

Eext

Electric Properties of Materials (continued)

N

iii

N

ii lQpP

11

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

Bulk material (N molecules)

Eext

Eext

E

E

EE

PED

ro

eo

eoo

o

)1(EP eo

?

What are the assumptionshere? er 1 Static permittivity

or relative permittivity

Electric Properties of Materials (continued)Conductivity

x

y

z E

E

J

J=current density=qvvz whereqv=volume charge density andvz= charge drift velocity

When subjected to an external electric field E the charge velocity is increased and is given by

e v eq v

Where e is called the electron mobility. The current density is thus given by

E

E q Je v

Where is called the conductivity. Its units are S/m

m

1

Where is called the resistivity.Material Conductivity (S/m)Silver 6.1 x 107

Glass 1.0 x 10-12

Sea Water 4

Electric Properties of Materials (continued)

1. Orientational Polarization: molecules have a slight polarization even in theabsence of an applied field. However each polarization vector is orientated randomlyso the net P vector is zero. Such materials are known as polar; water is a good example.

2. Ionic Polarization: Evident in materials ionic materials such as NaCl. Positive andnegative ions tend to align with the applied field.

3. Electronic Polarization: Evident in most materials and exists when an appliedfield displaces the electron cloud of an atom relative to the positive nucleus.

812 OHr

Magnetic Properties of Materials

No magnetic field: randomoriented magnetic dipoles

N

iiii

N

ii ndsIMM

11

ˆ

Net magnetic dipolemoment or magnetization vector:

Ii Mi

I iM

i

I iM

i

IiM

i

Applied external magnetic field

BextBext

I i

M i

I i

M i

I i

M i

I i

M i

N

iiii ndsIM

1

0ˆ

Magnetic dipoles randomly orientedresulting in zero net magnetization vector:

Magnetic dipoles tend to align withexternal magnetic field resulting in non-zeronet magnetization vector:

N

iiii ndsIM

1

0ˆ

Magnetic Properties of Materials (continued)

H

H

HH

MHB

ro

mo

moo

o

)1(

HM mo

?

What are the assumptionshere? mr 1 Static permeability

or relative permeability

Applied external magnetic field

BextBext

I i

M i

I i

M i

I i

M i

I i

M i

N

iiii

N

ii ndsIMM

11

ˆ

Magnetic Properties of Materials (continued)

1. Diamagnetic: Net magnetization vector tends to appose the direction of the applied field resulting in a relative permeability slightly less than 1.0Examples: silver (r=0.9998)

2. Paramagnetic: Net magnetization vector tends to align in the direction of the applied field resulting in a relative permeability slightly greater than 1.0Examples: Aluminum (r=1.00002)

3 Ferromagnetic: Net magnetization vector tends to align strongly in the direction of the applied field resulting in a relative permeability much greater than 1.0Examples: Iron (r=5000)

.

Classification of Materials

1. Homogenous or Inhomogenous: If the material properties are independent of spatiallocation then the material is homogenous, otherwise it is called inhomogenous

2. Isotropic or Anisotropic: If the material properties are independent of the polarizationof the applied field then the material is isotropic, otherwise it is called anisotropic.

3. Linear or non-Linear: If the material properties are independent on the magnitudeand phase of the electric and magnetic fields, otherwise it is called non-linear

),,( zyx Inhomogenous

ED

E

E

E

D

D

D

z

y

x

zzzyzx

yzyyyx

xzxyxx

z

y

x

anisotropic

...33

21 EEEED oo

Classification of Materials

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-

+

-+

-+

-+

-+

-+

-+

-+

-+

-+

-+

-+

-+

-+

-+

-+

-+

-+

-+

-+

-+

-+

-+

-+

-+

-+

-+

-+

-+

-+-

+-+-+-+-

+-

+-

+

-+

-+

-+

-+

-+

-+

-+

-+

-+-

+-

+-

+

-

+

-

+

-

+

-

+

-

+-

+-

+-

+

-+

-

+

-

+

-

+

t=t1t=t2

t=t3

t=t4 t=t5

t=t6A material’s atoms or molecules attempt to keep up with a changing electric field. This results in two things: (1) friction causes energy loss via heat and (2) the dynamic response of themolecules will be a function of the frequency of the applied field (i.e. frequency dependantmaterial properties)

4. Dispersive or non-dispersive: If the material properties are independent of frequencythen the material is non-dispersive, otherwise it is called dispersive

Electric Properties of Materials Frequency Behavior (Complex Permittivity)

0~

~~

~~~~

~~

B

D

JJDjH

BjE

ci

EJ

HB

ED

sc

o~~

~~

~)(

~ *

0)~

(

~)~

(

~~~~

~~

*

*

H

E

EJEjH

HjE

si

o

)()()(* j

is called the complex permittivity

0)~

(

~)~

(

~~~)(

~

~~

*

H

E

EJEjjH

HjE

si

o

loss termDielectric constant

0)~

(

~)~

(

~~)(

~

~~

*

H

E

JEjH

HjE

ieff

o

s

eff jj )()()(

Frequency Behavior of Sea Water

Electric Properties of Materials Frequency Behavior (Complex Permittivity)

0)~

(

~)~

(

~)(

~~~

~~

*

H

E

EJEjH

HjE

si

o

0)~

(

~)~

(

~~~)(

~

~~

*

H

E

EJEjjH

HjE

si

o

0)~

(

~)~

(

~~~~

~~

*

H

E

EJEjH

HjE

effi

o

saseff )(

0)~

(

~)~

(

~~~~

~~

*

H

E

JJJH

HjE

effid

o

EJ

J

EjJ

effeff

i

d

~~

~

~~

Displacement current

Source current

Effective electric conduction current

Electric Properties of Materials Frequency Behavior (Complex Permittivity)

0)~

(

~)~

(

~~~~

~~

*

H

E

EJEjH

HjE

effi

o

saseff )(0)

~(

~)~

(

~~)1(

~

~~

*

H

E

JEjjH

HjE

ieff

o

0)~

(

~)~

(

~~))tan(1(

~

~~

*

H

E

JEjjH

HjE

ieff

o

0)~

(

~)~

(

~~~

~~

*

H

E

JJH

HjE

icd

o

EjjJ effcd

~)1(

~

effeff )tan(

Electric Properties of Materials Frequency Behavior (Complex Permittivity)

saseff )(

0)~

(

~)~

(

~~~~

~~

*

H

E

JJJH

HjE

effid

o

EJ

J

EjJ

effeff

i

d

~~

~

~~

Displacement current

Source current

Effective electric conduction current

)1(~~

eff

effd JJ

Good Dielectric

)1(~~

eff

deff JJ

Good Conductor

Wave Equation

0

B

E

JEt

EH

t

HE

Ht

E

t

HE

)(][

t

J

t

E

t

EE

JEt

E

tE

2

2

AAA

2)( Vector Identity

t

J

t

E

t

EEE

2

22)(

t

J

t

E

t

EE

2

221

Time Dependent Homogenous Wave Equation (E-Field)

1

2

22

t

J

t

E

t

EE

Wave EquationSource-Free Time Dependent Homogenous Wave Equation (E-Field)

1

2

22

t

J

t

E

t

EE

0,0 J

Source Free

02

22

t

E

t

EE

Source-Free Lossless Time Dependent Homogenous Wave Equation (E-Field)

0Lossless

02

22

t

EE

Wave EquationSource-Free Time Dependent Homogenous Wave Equation (H-Field)

1

2

22 J

t

H

t

HH

0,0 J

Source Free 02

22

t

H

t

HH

0,0,0 J

Source Free and Lossless 02

22

t

HH

Wave Equation: Time Harmonic

1

2

22

t

J

t

E

t

EE

0,0 J

Source Free

02

22

t

E

t

EE

0Lossless

02

22

t

EE

Time Domain Frequency Domain

~1~~~~ 22 JEjEE

0,0 J

Source Free

0~~~ 22 EjEE

0Lossless

0~~ 22 EE

“Helmholtz Equation”

General Solution: Point Source

02

22

t

EE

)(1

)(1

),( rtfr

crtfr

ctrE

01

2

22

t

EE

rr

rr

x

y

z

r

(r)

pointsourcerr atrEatrEatrEtrE ˆ),(ˆ),(ˆ),(),(

01

01

2

22

2

22

t

EE

rr

rr

t

EE

rr

rr

Solution:

Inward traveling spherical waveOutward traveling spherical wave

0

General Solution: Point Source

y

z

r

(r)

pointsource

)(1

)(1

),(c

rtf

rcrtf

rctrE

sec

1 mc

Wave speed

In free space:

sec103

1085.8104

11 8

127

mc

oo

Same as the speed of light!

General Solution Case: Time HarmonicRectangular Coordinates

0~~ 22 EE 0

~~ 22 EE

Wave Number

0~

~~~2

2

2

2

2

2

2

Ez

E

y

E

x

E

0

0

0

22

2

2

2

2

2

22

2

2

2

2

2

22

2

2

2

2

2

zzzz

yyyy

xxxx

Ez

E

y

E

x

E

Ez

E

y

E

x

E

Ez

E

y

E

x

E

Separation of Variable Solutions

022

2

2

2

2

2

xxxx E

z

E

y

E

x

E

Assume Solution of the form: )()()(),,( zhygxfzyxEx

0)()()()()()()()()()()()( 2 zhygxfzhygxfzhygxfzhygxf

0)]()()()()()()()()()()()([)()()(

1 2 zhygxfzhygxfzhygxfzhygxfzhygxf

0)(

)(

)(

)(

)(

)( 2

zh

zh

yg

yg

xf

xf

function of x

function of y

function of z

constant

Separation of Variable Solutions

0)(

)(

)(

)(

)(

)( 2

zh

zh

yg

yg

xf

xf

function of x

function of y

function of z

constant

2222

2

2

2

)(

)(

)(

)(

)(

)(

zyx

z

y

x

zh

zh

yg

yg

xf

xf

2222

2

2

2

0)()(

0)()(

0)()(

zyx

z

y

x

zhzh

ygyg

xfxf

Separation of Variable Solutions

2222

2

2

2

0)()(

0)()(

0)()(

zyx

z

y

x

zhzh

ygyg

xfxf

Solutions:

zzjzzj

yyjyyj

xxjxxj

eBeAzh

eBeAyg

eBeAxf

33

22

11

)(

)(

)(

purely real purely imaginary

complex

xxjxxj eBeA 11

Traveling andstanding waves Evanesent waves

xx eBeA 11

Exponentially modulatedtraveling wave

xxjxxxjx eeBeeA 11

xxjxxxjx eeBeeA 11

oror

)sin()cos( 11 xDxC xx

Separation of Variable Solutions: Examples

case a. x, y, z all real (forward traveling waves)

zzjyyjxxjxox eeeEzhygxfzyxE )()()(),,( Plane Waves

case b. xreal (forward traveling wave), x(real standing wave), z imaginary (evanesent wave)

zyy

xxjxox eyDyCeEzhygxfzyxE ))sin()cos(()()()(),,( 11

Surface Waves

z

x

y

Separation of Variable Solutions: Examples

case c. xreal (forward standing wave), x(real standing wave), z real (traveling wave)

))sin()cos())(sin()cos((),,(

))sin()cos())(sin()cos((),,(

))sin()cos())(sin()cos((),,(

1111

1111

1111

yDyCxBxAeEzyxE

yDyCxBxAeEzyxE

yDyCxBxAeEzyxE

yyxxzzj

zoz

yyxxzzj

yoy

yyxxzzj

xox

Guided Waves

Unknown constants A1, B1, C1 , D1, x, y, z

Found by applying boundary conditions anddispersion relation. Namely:

z

x

y

b

hPEC Walls

22222

0),2/,2/(

0),,2/(

0),2/,(

zyx

z

y

x

zhybxE

zybxE

zhyxE

Stay tuned we will solve the complete solution for modes in a rectangular waveguide in a later lecture.