1 EEE 498/598 Overview of Electrical Engineering Lecture 11: Electromagnetic Power Flow; Reflection And Transmission Of Normally and Obliquely Incident

1

EEE 498/598EEE 498/598Overview of Electrical Overview of Electrical

EngineeringEngineering

Lecture 11:Lecture 11:Electromagnetic Power Flow; Electromagnetic Power Flow; Reflection And Transmission Reflection And Transmission Of Normally and Obliquely Of Normally and Obliquely

Incident Plane Waves; Useful Incident Plane Waves; Useful TheoremsTheorems

Lecture 112

Lecture 11 ObjectivesLecture 11 Objectives

To study electromagnetic power To study electromagnetic power flow; reflection and transmission flow; reflection and transmission of normally and obliquely of normally and obliquely incident plane waves; and some incident plane waves; and some useful theorems.useful theorems.

Lecture 113

Flow of Flow of Electromagnetic Power Electromagnetic Power

Electromagnetic waves transport throughout Electromagnetic waves transport throughout space the energy and momentum arising from space the energy and momentum arising from a set of charges and currents (the sources).a set of charges and currents (the sources).

If the electromagnetic waves interact with If the electromagnetic waves interact with another set of charges and currents in a another set of charges and currents in a receiver, information (energy) can be receiver, information (energy) can be delivered from the sources to another location delivered from the sources to another location in space.in space.

The energy and momentum exchange between The energy and momentum exchange between waves and charges and currents is described waves and charges and currents is described by the Lorentz force equation.by the Lorentz force equation.

Lecture 114

Derivation of Poynting’s Derivation of Poynting’s TheoremTheorem

Poynting’s theorem concerns the Poynting’s theorem concerns the conservation of energy for a conservation of energy for a given volume in space.given volume in space.

Poynting’s theorem is a Poynting’s theorem is a consequence of Maxwell’s consequence of Maxwell’s equations.equations.

Lecture 115

Derivation of Poynting’s Derivation of Poynting’s Theorem in the Time Theorem in the Time

Domain (Cont’d)Domain (Cont’d) Time-Domain Maxwell’s curl Time-Domain Maxwell’s curl

equations in differential formequations in differential form

t

DJJH

t

BKKE

ci

ci

Lecture 116


Domain (Cont’d)Domain (Cont’d) Recall a vector identityRecall a vector identity

Furthermore,Furthermore,

HEEHHE

t

BHKHKHEH

t

DEJEJEHE

ci

ci

Lecture 117


Domain (Cont’d)Domain (Cont’d)

t

DEJEJE

t

BHKHKH

HEEHHE

ci

ci

Lecture 118


Domain (Cont’d)Domain (Cont’d) Integrating over a volume Integrating over a volume VV bounded by bounded by

a closed surface a closed surface SS, we have, we have

VV

c

V

c

VV

ii

dvHEdvMH

dvJEdvt

BH

t

DEdvKHJE

Lecture 119


Domain (Cont’d)Domain (Cont’d) Using the divergence theorem, we Using the divergence theorem, we

obtain the general form of Poynting’s obtain the general form of Poynting’s theoremtheorem

SV

c

V

c

VV

ii

sdHEdvMH

dvJEdvt

BH

t

DEdvKHJE

Lecture 1110


Domain (Cont’d)Domain (Cont’d) For simple, lossless media, we haveFor simple, lossless media, we have

Note thatNote that

2

2

1A

tt

AA

t

AA

S

VV

ii

sdHE

dvt

HH

t

EEdvKHJE

Lecture 1111


Domain (Cont’d)Domain (Cont’d) Hence, we have the form of Hence, we have the form of

Poynting’s theorem valid in simple, Poynting’s theorem valid in simple, lossless media:lossless media:

S

VV

ii

sdHE

dvHEt

dvKHJE 22

2

1

2

1

Lecture 1112

Derivation of Poynting’s Derivation of Poynting’s Theorem in the Frequency Theorem in the Frequency

Domain (Cont’d)Domain (Cont’d) Time-Harmonic Maxwell’s curl equations Time-Harmonic Maxwell’s curl equations

in differential form for a simple mediumin differential form for a simple medium

i

i

JEjH

KHjE

mjj

jj

Lecture 1113

Derivation of Poynting’s Derivation of Poynting’s Theorem in the Frequency Theorem in the Frequency

Domain (Cont’d)Domain (Cont’d) Poynting’s theorem for a simple Poynting’s theorem for a simple

mediummedium

SV

m

V

V

VV

ii

sdHEdvHdvE

dvHE

dvHEjdvKHJE

22

22

22

2

1

2

1

Lecture 1114

Physical Interpretation Physical Interpretation of the Terms in of the Terms in

Poynting’s TheoremPoynting’s Theorem The termsThe terms

represent the represent the instantaneous power instantaneous power dissipateddissipated in the electric and in the electric and magnetic conductivity losses, magnetic conductivity losses, respectively, in volume respectively, in volume VV..

V

m

V

dvHdvE 22

Lecture 1115

Physical Interpretation of Physical Interpretation of the Terms in Poynting’s the Terms in Poynting’s

Theorem (Cont’d)Theorem (Cont’d) The termsThe terms

represent the represent the instantaneous power instantaneous power dissipateddissipated in the polarization and in the polarization and magnetization losses, magnetization losses, respectively, in volume respectively, in volume VV..

VV

dvHdvE 22

Lecture 1116


Theorem (Cont’d)Theorem (Cont’d) Recall that the electric energy Recall that the electric energy

density is given bydensity is given by

Recall that the magnetic energy Recall that the magnetic energy density is given by density is given by

2

2

1Ewe

2

2

1Hwm

Lecture 1117


Theorem (Cont’d)Theorem (Cont’d) Hence, the terms Hence, the terms

represent the represent the total electromagnetic total electromagnetic energy storedenergy stored in the volume in the volume VV. .

V

dvHE 22

2

1

2

1

Lecture 1118


Theorem (Cont’d)Theorem (Cont’d) The termThe term

represents represents the flow of instantaneous the flow of instantaneous powerpower out of the volume out of the volume VV through the surface through the surface SS..

S

sdHE

Lecture 1119


Theorem (Cont’d)Theorem (Cont’d) The term The term

represents the represents the total electromagnetic total electromagnetic energy generated by the sourcesenergy generated by the sources in the in the volume volume VV. .

V

ii dvKHJE

Lecture 1120


Theorem (Cont’d)Theorem (Cont’d) In words the Poynting vector can be In words the Poynting vector can be

stated as stated as “The sum of the power generated by “The sum of the power generated by the sources, the imaginary power (representing the sources, the imaginary power (representing the time-rate of increase) of the stored electric the time-rate of increase) of the stored electric and magnetic energies, the power leaving, and and magnetic energies, the power leaving, and the power dissipated in the enclosed volume is the power dissipated in the enclosed volume is equal to zero.”equal to zero.”

SV

m

V

VVV

ii

sdHEdvHdvE

dvHEdvHEjdvKHJE

0

2

1

2

1

22

2222

Lecture 1121

Poynting Vector in the Poynting Vector in the Time DomainTime Domain

We define a new vector called the We define a new vector called the (instantaneous) (instantaneous) Poynting vectorPoynting vector as as

The Poynting vector has the same direction The Poynting vector has the same direction as the direction of propagation.as the direction of propagation.

The Poynting vector at a point is equivalent The Poynting vector at a point is equivalent to the power density of the wave at that to the power density of the wave at that point.point.

HES • The Poynting vector has units of W/m2.

Lecture 1122

Time-Average Poynting Time-Average Poynting VectorVector

The time-average Poynting The time-average Poynting vector can be computed from the vector can be computed from the instantaneous Poynting vector asinstantaneous Poynting vector as

dttrST

rSpT

pav

0

,1

period of the wave

Lecture 1123

Time-Average Poynting Time-Average Poynting Vector (Cont’d)Vector (Cont’d)

The time-average Poynting The time-average Poynting vector can also be computed asvector can also be computed as

*Re2

1HErS av

phasors

Lecture 1124

Time-Average Poynting Time-Average Poynting Vector for a Uniform Vector for a Uniform

Plane WavePlane Wave Consider a uniform plane wave Consider a uniform plane wave

traveling in the +traveling in the +zz-direction in a -direction in a lossy medium:lossy medium:

zjz

cy

zjzx

eeE

zH

eeEzE

0

0

Lecture 1125

Time-Average Poynting Time-Average Poynting Vector for a Uniform Plane Vector for a Uniform Plane

Wave (Cont’d)Wave (Cont’d) The time-average Poynting The time-average Poynting

vector isvector is

cos2

ˆRe

2ˆ

1Re

2ˆRe

2

1

2

2

02

2

2

0

*2

2

0*

zz

zz

zzav

eE

aeE

a

eE

aHES

Lecture 1126

Time-Average Poynting Time-Average Poynting Vector for a Uniform Plane Vector for a Uniform Plane

Wave (Cont’d)Wave (Cont’d) For a lossless medium, we haveFor a lossless medium, we have

2ˆ

0

0

2

0EaS zav

Lecture 1127

Reflection and Reflection and Transmission of Waves at Transmission of Waves at

Planar InterfacesPlanar Interfaces

medium 2medium 1

incident wave

reflected wave

transmitted wave

Lecture 1128

Normal Incidence on a Normal Incidence on a Lossless DielectricLossless Dielectric

Consider both medium 1 and medium Consider both medium 1 and medium 2 to be lossless dielectrics.2 to be lossless dielectrics.

Let us place the boundary between Let us place the boundary between the two media in the the two media in the z z = 0 plane, and = 0 plane, and consider an incident plane wave consider an incident plane wave which is traveling in the +which is traveling in the +zz-direction.-direction.

No loss of generality is suffered if we No loss of generality is suffered if we assume that the electric field of the assume that the electric field of the incident wave is in the incident wave is in the xx-direction.-direction.

Lecture 1129

Normal Incidence on a Normal Incidence on a Lossless Dielectric Lossless Dielectric

(Cont’d)(Cont’d)medium 2medium 1

z

x

11, HE 22 , HE

z = 0

0,, 111 0,, 222

Lecture 1130


(Cont’d)(Cont’d) Incident waveIncident wave

zjiyizi

zjixi

eE

aEaH

eEaE

1

1

1

0

1

0

ˆˆ1

ˆ

known

1

11111

Lecture 1131


(Cont’d)(Cont’d) Reflected waveReflected wave

zjryrzr

zjrxr

eE

aEaH

eEaE

1

1

1

0

1

0

ˆˆ1

ˆ

unknown

Lecture 1132


(Cont’d)(Cont’d) Transmitted waveTransmitted wave

zjtytzt

zjtxt

eE

aEaH

eEaE

2

2

2

0

2

0

ˆˆ1

ˆ

unknown

2

22222

Lecture 1133


(Cont’d)(Cont’d) The total electric and magnetic The total electric and magnetic

fields in medium 1 arefields in medium 1 are

zjrzjiyri

zjr

zjixri

eE

eE

aHHH

eEeEaEEE

11

11

1

0

1

01

001

ˆ

ˆ

Lecture 1134


(Cont’d)(Cont’d) The total electric and magnetic The total electric and magnetic

fields in medium 2 arefields in medium 2 are

zjtyt

zjtxt

eE

aHH

eEaEE

2

2

2

02

02

ˆ

ˆ

Lecture 1135


(Cont’d)(Cont’d) To determine the unknowns To determine the unknowns EEr0r0 and and

EEt0t0, we must enforce the BCs at , we must enforce the BCs at zz = 0 = 0::

00

00

21

21

zHzH

zEzE

Lecture 1136


(Cont’d)(Cont’d) From the BCs we haveFrom the BCs we have

2

0

1

0

1

0

000

tri

tri

EEE

EEE

or

012

200

12

120

2, itir EEEE

Lecture 1137

Reflection and Reflection and Transmission Transmission CoefficientsCoefficients

Define the Define the reflection coefficientreflection coefficient as as

Define the Define the transmission coefficienttransmission coefficient as as

12

12

0

0

i

r

E

E

12

2

0

0 2

i

t

E

E

Lecture 1138

Reflection and Reflection and Transmission Transmission

Coefficients (Cont’d)Coefficients (Cont’d) Note also thatNote also that The definitions of the reflection and The definitions of the reflection and

transmission coefficients do generalize transmission coefficients do generalize to the case of lossy media.to the case of lossy media.

For lossless media, For lossless media, and and are real. are real.

For lossy media, For lossy media, and and are complex. are complex.

1

20,11

2,1

Lecture 1139

Traveling Waves and Traveling Waves and Standing WavesStanding Waves

The total field in medium 1 is The total field in medium 1 is partially a partially a traveling wavetraveling wave and and partially a partially a standing wavestanding wave..

The total field in medium 2 is a The total field in medium 2 is a purepure traveling wave traveling wave..

Lecture 1140

Traveling Waves and Traveling Waves and Standing Waves Standing Waves

(Cont’d)(Cont’d) The total electric field in medium The total electric field in medium 1 is given by1 is given by

zjeEa

eeeEa

eeEaEEE

zjix

zjzjzjix

zjzjixri

10

0

01

sin21ˆ

1ˆ

ˆ

1

111

11

traveling wave

standing wave

Lecture 1141

Traveling Waves and Traveling Waves and Standing Waves: Standing Waves:

ExampleExample

medium 2medium 1

z

x

z = 0

0,, 10101 0,,4 20202

01 20

2

3

1

3

2

Lecture 1142

Traveling Waves and Traveling Waves and Standing Waves: Standing Waves: Example (Cont’d)Example (Cont’d)

-2 -1.5 -1 -0.5 0 0.5 10.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

z/0

No

rma

lize

d E

fie

ld

Lecture 1143

Standing Wave RatioStanding Wave Ratio The The standing wave ratiostanding wave ratio is defined as is defined as

In this example, we haveIn this example, we have

1

1

min1

max1

zE

zES

23

113

11

S

Lecture 1144

Time-Average Poynting Time-Average Poynting VectorsVectors

1

2

02

1

1

2

02*

1

2

0*

21ˆ

2ˆRe

2

1

2ˆRe

2

1

izraviavav

izrrrav

iziiiav

EaSSS

EaHES

EaHES

Lecture 1145

Time-Average Poynting Time-Average Poynting Vectors (Cont’d)Vectors (Cont’d)

2

2

02*2 2

ˆRe2

1

i

ztttavavE

aHESS

We note that

2

22

12

2

22

12

21

1

212

212

212

1

2

12

12

11

2

2141

11

11

Lecture 1146

Time-Average Poynting Time-Average Poynting Vectors (Cont’d)Vectors (Cont’d)

Hence, Hence,

tavraviav

avav

SSS

SS

or

21

Power is conserved at the interface.

Lecture 1147

Oblique Incidence at a Oblique Incidence at a Dielectric InterfaceDielectric Interface

11, 22 ,

0z

ri EEE 1 tEE 2

ir t

Lecture 1148

Oblique Incidence at a Oblique Incidence at a Dielectric Interface: Parallel Dielectric Interface: Parallel

Polarization (TM to z)Polarization (TM to z)

tt

rr

ii

zxjkttt

zxjkrrr

zxjkiii

ezxEE

ezxEE

ezxEE

cossin0

cossin0

cossin0

2

1

1

sinˆcosˆ

sinˆcosˆ

sinˆcosˆ

Lecture 1149

Oblique Incidence at a Oblique Incidence at a Dielectric Interface: Parallel Dielectric Interface: Parallel

Polarization (TM to z)Polarization (TM to z)

it

i

it

it

coscos

cos2

coscos

coscos

12

2

12

12

Lecture 1150

Oblique Incidence at a Oblique Incidence at a Dielectric Interface: Dielectric Interface:

Perpendicular Polarization (TE Perpendicular Polarization (TE to z)to z)

tt

rr

ii

zxjkt

zxjkr

zxjki

eyEE

eyEE

eyEE

cossin0

cossin0

cossin0

2

1

1

ˆ

ˆ

ˆ

Lecture 1151

Oblique Incidence at a Oblique Incidence at a Dielectric Interface: Dielectric Interface:

Perpenidcular Polarization Perpenidcular Polarization (TM to z)(TM to z)

ti

i

ti

ti

coscos

cos2

coscos

coscos

12

2

12

12

Lecture 1152

Brewster AngleBrewster Angle

The Brewster angle is a special The Brewster angle is a special angle of incidence for which angle of incidence for which =0.=0. For dielectric media, a Brewster For dielectric media, a Brewster

angle can occur only for parallel angle can occur only for parallel polarization.polarization.

Lecture 1153

Critical AngleCritical Angle

The critical angle is the largest The critical angle is the largest angle of incidence for which angle of incidence for which kk22 is is real (i.e., a propagating wave real (i.e., a propagating wave exists in the second medium).exists in the second medium). For dielectric media, a critical For dielectric media, a critical

angle can exist only if angle can exist only if 11>>22..

Lecture 1154

Some Useful TheoremsSome Useful Theorems

The reciprocity theoremThe reciprocity theorem Image theoryImage theory The uniqueness theoremThe uniqueness theorem

Lecture 1155

Image Theory for Current Image Theory for Current Elements above a Infinite, Flat, Elements above a Infinite, Flat,

Perfect Electric ConductorPerfect Electric Conductor

actualsources

images

electric magnetic

Lecture 1156

Image Theory for Current Image Theory for Current Elements above a Infinite, Flat, Elements above a Infinite, Flat,

Perfect Magnetic ConductorPerfect Magnetic Conductor

m

actualsources

images

electric magnetic

h

h

Documents

1 EEE 498/598 Overview of Electrical Engineering Lecture 11: Electromagnetic Power Flow; Reflection And Transmission Of Normally and Obliquely Incident