1 eee 498/598 overview of electrical engineering lecture 11: electromagnetic power flow; reflection...
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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
Derivation of Poynting’s Derivation of Poynting’s Theorem in the Time Theorem in the Time
Domain (Cont’d)Domain (Cont’d) Recall a vector identityRecall a vector identity
Furthermore,Furthermore,
HEEHHE
t
BHKHKHEH
t
DEJEJEHE
ci
ci
Lecture 117
Derivation of Poynting’s Derivation of Poynting’s Theorem in the Time Theorem in the Time
Domain (Cont’d)Domain (Cont’d)
t
DEJEJE
t
BHKHKH
HEEHHE
ci
ci
Lecture 118
Derivation of Poynting’s Derivation of Poynting’s Theorem in the Time Theorem in the Time
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
Derivation of Poynting’s Derivation of Poynting’s Theorem in the Time Theorem in the Time
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
Derivation of Poynting’s Derivation of Poynting’s Theorem in the Time Theorem in the Time
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
Derivation of Poynting’s Derivation of Poynting’s Theorem in the Time Theorem in the Time
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
Physical Interpretation of Physical Interpretation of the Terms in Poynting’s the Terms in Poynting’s
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
Physical Interpretation of Physical Interpretation of the Terms in Poynting’s the Terms in Poynting’s
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
Physical Interpretation of Physical Interpretation of the Terms in Poynting’s the Terms in Poynting’s
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
Physical Interpretation of Physical Interpretation of the Terms in Poynting’s the Terms in Poynting’s
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
Physical Interpretation of Physical Interpretation of the Terms in Poynting’s the Terms in Poynting’s
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
Normal Incidence on a Normal Incidence on a Lossless Dielectric Lossless Dielectric
(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
Normal Incidence on a Normal Incidence on a Lossless Dielectric Lossless Dielectric
(Cont’d)(Cont’d) Reflected waveReflected wave
zjryrzr
zjrxr
eE
aEaH
eEaE
1
1
1
0
1
0
ˆˆ1
ˆ
unknown
Lecture 1132
Normal Incidence on a Normal Incidence on a Lossless Dielectric Lossless Dielectric
(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
Normal Incidence on a Normal Incidence on a Lossless Dielectric Lossless Dielectric
(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
Normal Incidence on a Normal Incidence on a Lossless Dielectric Lossless Dielectric
(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
Normal Incidence on a Normal Incidence on a Lossless Dielectric Lossless Dielectric
(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
Normal Incidence on a Normal Incidence on a Lossless Dielectric Lossless Dielectric
(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