Transcript Document

EEE 498/598
Overview of Electrical
Engineering
Lecture 11:
Electromagnetic Power Flow;
Reflection And Transmission Of
Normally and Obliquely Incident Plane
Waves; Useful Theorems
1
Lecture 11 Objectives

To study electromagnetic power flow;
reflection and transmission of normally
and obliquely incident plane waves; and
some useful theorems.
2
Lecture 11
Flow of Electromagnetic Power



Electromagnetic waves transport throughout space
the energy and momentum arising from a set of
charges and currents (the sources).
If the electromagnetic waves interact with another
set of charges and currents in a receiver,
information (energy) can be delivered from the
sources to another location in space.
The energy and momentum exchange between
waves and charges and currents is described by the
Lorentz force equation.
3
Lecture 11
Derivation of Poynting’s
Theorem
Poynting’s theorem concerns the
conservation of energy for a given volume
in space.
 Poynting’s theorem is a consequence of
Maxwell’s equations.

4
Lecture 11
Derivation of Poynting’s Theorem in
the Time Domain (Cont’d)

Time-Domain Maxwell’s curl equations
in differential form
B
  E  K i  K c 
t
D
 H  J i  J c 
t
5
Lecture 11
Derivation of Poynting’s Theorem
in the Time Domain (Cont’d)

Recall a vector identity
  E  H   H    E  E    H

Furthermore,
D
 E    H  E  J i  E  J c  E 
t
B
H    E  H  K i  H  K c  H 
t
6
Lecture 11
Derivation of Poynting’s Theorem in
the Time Domain (Cont’d)
  E  H   H    E  E    H
B
 H  K i  H  K c  H 
t
D
EJi  EJc  E
t
7
Lecture 11
Derivation of Poynting’s Theorem
in the Time Domain (Cont’d)

Integrating over a volume V bounded by a closed
surface S, we have
B 
 D
V E  J i  H  K i dv  V  E  t  H  t  dv  V E  J c dv
  H  M c dv     E  H  dv
V
V
8
Lecture 11
Derivation of Poynting’s Theorem
in the Time Domain (Cont’d)

Using the divergence theorem, we obtain the general
form of Poynting’s theorem
B 
 D
V E  J i  H  K i dv  V  E  t  H  t  dv  V E  J c dv
  H  M c dv   E  H   d s
V
S
9
Lecture 11
Derivation of Poynting’s Theorem
in the Time Domain (Cont’d)

For simple, lossless media, we have
E
H 

V E  J i  H  K i dv  V   E  t   H  t  dv
  E  H   d s
S

Note that
A
A 1  2
A
A

A
t
t 2 t
 
10
Lecture 11
Derivation of Poynting’s Theorem
in the Time Domain (Cont’d)

Hence, we have the form of Poynting’s theorem
valid in simple, lossless media:
 1 2 1
2
V E  J i  H  K i dv   t V  2 E  2 H dv
  E  H   d s
S
11
Lecture 11
Derivation of Poynting’s Theorem in
the Frequency Domain (Cont’d)

Time-Harmonic Maxwell’s curl equations in
differential form for a simple medium
  E   j H  K i
  H  j E  J i

     j   j

m
     j   j

12
Lecture 11
Derivation of Poynting’s Theorem in
the Frequency Domain (Cont’d)

Poynting’s theorem for a simple medium
1
1
2
2




E

J

H

K
dv


j


E


H
 dv
i
i
V
V  2
2



     E 2    H 2 dv
V
  E 2 dv    m H 2 dv   E  H   d s
V
V
13
S
Lecture 11
Physical Interpretation of the
Terms in Poynting’s Theorem

The terms

E
dv


H
dv
m


2
V
2
V
represent the instantaneous power
dissipated in the electric and magnetic
conductivity losses, respectively, in volume
V.
14
Lecture 11
Physical Interpretation of the Terms
in Poynting’s Theorem (Cont’d)

The terms






E
dv



H
dv


2
V
2
V
represent the instantaneous power
dissipated in the polarization and
magnetization losses, respectively, in
volume V.
15
Lecture 11
Physical Interpretation of the Terms
in Poynting’s Theorem (Cont’d)

Recall that the electric energy density is given by
1
2
we   E
2

Recall that the magnetic energy density is given
by
1
2
wm   H
2
16
Lecture 11
Physical Interpretation of the Terms
in Poynting’s Theorem (Cont’d)

Hence, the terms
1  2 1  2
V  2  E  2  H  dv
represent the total electromagnetic
energy stored in the volume V.
17
Lecture 11
Physical Interpretation of the Terms
in Poynting’s Theorem (Cont’d)

The term
 E  H  d s
S
represents the flow of instantaneous
power out of the volume V through the
surface S.
18
Lecture 11
Physical Interpretation of the Terms
in Poynting’s Theorem (Cont’d)

The term
 E  J
i
 H  K i dv
V
represents the total electromagnetic
energy generated by the sources in the
volume V.
19
Lecture 11
Physical Interpretation of the Terms in
Poynting’s Theorem (Cont’d)
In words the Poynting vector can be stated as
“The sum of the power generated by the sources,
the imaginary power (representing the time-rate
of increase) of the stored electric and magnetic
energies, the power leaving, and the power
dissipated in the enclosed volume is equal to
zero.”

1
1
2
2
2
2









dv
E

J

H

K
dv

j


E


H
dv



E


H


i
i



V
V
2
2

V
  E 2 dv    m H 2 dv   E  H   d s  0
V
V
S
20
Lecture 11
Poynting Vector in the Time
Domain

We define a new vector called the (instantaneous)
Poynting vector as
• The Poynting
vector has units of
W/m2.
S  EH


The Poynting vector has the same direction as the
direction of propagation.
The Poynting vector at a point is equivalent to the
power density of the wave at that point.
21
Lecture 11
Time-Average Poynting Vector

The time-average Poynting vector can be
computed from the instantaneous Poynting
vector as
1
S av r  
Tp
Tp


S
r
,
t
dt

0
period of the wave
22
Lecture 11
Time-Average Poynting Vector
(Cont’d)

The time-average Poynting vector can also
be computed as

1
*
S av r   Re E  H
2

phasors
23
Lecture 11
Time-Average Poynting Vector
for a Uniform Plane Wave

Consider a uniform plane wave traveling in
the +z-direction in a lossy medium:
E x  z   E0 e
H y z  
E0
c
24
 z  j z
e
e
 z  j z
e
Lecture 11
Time-Average Poynting Vector for a
Uniform Plane Wave (Cont’d)

The time-average Poynting vector is
S av


2
E0  2 z  1 
1
*
 Re E  H  aˆ z
e
Re  * 
2
2
 
 aˆ z
E0
2
2
e
 2 z
Re 

2
25
 aˆ z
E0
2
2
e  2 z cos 
Lecture 11
Time-Average Poynting Vector for a
Uniform Plane Wave (Cont’d)

For a lossless medium, we have
 0
  0
S av  aˆ z
26
E0
2
2
Lecture 11
Reflection and Transmission of
Waves at Planar Interfaces
medium 1
medium 2
incident wave
transmitted wave
reflected wave
27
Lecture 11
Normal Incidence on a Lossless
Dielectric



Consider both medium 1 and medium 2 to be
lossless dielectrics.
Let us place the boundary between the two
media in the z = 0 plane, and consider an
incident plane wave which is traveling in the +zdirection.
No loss of generality is suffered if we assume
that the electric field of the incident wave is in
the x-direction.
28
Lecture 11
Normal Incidence on a
Lossless Dielectric (Cont’d)
x
medium 1
medium 2
 2 , 2 , 2  0
1, 1, 1  0
z
E1 , H 1
E2, H 2
z=0
29
Lecture 11
Normal Incidence on a Lossless
Dielectric (Cont’d)

Incident wave
known
 j1 z
ˆ
E i  a x Ei 0 e
Hi 
1
1
aˆ z  E i  aˆ y
Ei 0
1
e
 j1 z
1
1 
1
1   11
30
Lecture 11
Normal Incidence on a Lossless
Dielectric (Cont’d)

Reflected wave
unknown
 j1 z
ˆ
E r  a x Er 0 e
Hr 
1
1
 aˆ z  E r  aˆ y
31
Er 0
1
e
 j1 z
Lecture 11
Normal Incidence on a Lossless
Dielectric (Cont’d)

Transmitted wave
unknown
 j 2 z
ˆ
E t  a x Et 0 e
Ht 
1
2
aˆ z  E t  aˆ y
Et 0
2
e
 j 2 z
2
2 
2
 2    22
32
Lecture 11
Normal Incidence on a Lossless
Dielectric (Cont’d)

The total electric and magnetic fields in medium
1 are

E 1  E i  E r  aˆ x Ei 0 e
 j1 z
 Er 0 e
 j1 z

 Ei 0  j1z Er 0  j1z 
H 1  H i  H r  aˆ y 
e

e

1
 1

33
Lecture 11
Normal Incidence on a Lossless
Dielectric (Cont’d)

The total electric and magnetic fields in medium
2 are
 j 2 z
ˆ
E 2  E t  a x Et 0 e
H 2  H t  aˆ y
34
Et 0
2
e
 j 2 z
Lecture 11
Normal Incidence on a Lossless
Dielectric (Cont’d)

To determine the unknowns Er0 and Et0, we
must enforce the BCs at z = 0:
E1 z  0  E 2 z  0
H 1 z  0  H 2 z  0
35
Lecture 11

Normal Incidence on a
Lossless Dielectric (Cont’d)
From the BCs we have
Ei 0  Er 0  Et 0
Ei 0
1

Er 0
1

Et 0
2
or
2  1
Er 0 
Ei 0 ,
2  1
22
Et 0 
Ei 0
2  1
36
Lecture 11
Reflection and Transmission
Coefficients

Define the reflection coefficient as
Er 0 2  1


Ei 0 2  1

Define the transmission coefficient as
Et 0
22


Ei 0 2  1
37
Lecture 11
Reflection and Transmission
Coefficients (Cont’d)



Note also that 1    
The definitions of the reflection and
transmission coefficients do generalize to the
case of lossy media.
For lossless media,  and  are real.
0   2
 1    1,

For lossy media,  and  are complex.
 2
  1,
38
Lecture 11
Traveling Waves and Standing
Waves
The total field in medium 1 is partially a
traveling wave and partially a standing
wave.
 The total field in medium 2 is a pure
traveling wave.

39
Lecture 11
Traveling Waves and Standing
Waves (Cont’d)

The total electric field in medium 1 is given
by

E 1  E i  E r  aˆ x Ei 0 e

1  e
 j1 z
 e
 j1 z

 aˆ x Ei 0 1   e
 j1 z
 e
 aˆ x Ei 0
 j1 z
 j 2 sin 1 z 
 j1 z
 e
 j1 z

standing
wave
traveling
wave
40
Lecture 11

Traveling Waves and Standing
Waves: Example
x
medium 1
medium 2
1   0 , 1  0 , 1  0  2  4 0 , 2  0 , 2  0

2  0 2
1  0
z
1
2


3
3
z=0
41
Lecture 11
Traveling Waves and Standing
Waves: Example (Cont’d)
1.4
Normalized E field
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
-2
-1.5
-1
-0.5
z/l 0
42
0
0.5
1
Lecture 11
Standing Wave Ratio

The standing wave ratio is defined as
S

E1 z 
E1 z 

max
1 
1 
min
In this example, we have
1 1
3 2
S
1 1
3
43
Lecture 11
Time-Average Poynting Vectors
S 
av i
S 
av r




2
Ei 0
1
*
 Re E i  H i  aˆ z
2
21
2
1
2 Ei 0
*
 Re E r  H r  aˆ z 
2
21
S   S   S   aˆ 1    2
2
av 1
av i
av r
Ei 0
2
z
1
44
Lecture 11
Time-Average Poynting Vectors
(Cont’d)
S   S 
av 2
av t


1
*
 Re E t  H t  aˆ z 
2
2
Ei 0
2
2 2
We note that
2

1 
 2  1   1  2  1 2  2  1 2 
1
   
 1  

2
1
1   2  1   1 
2  1 



2
2

1  412  1  22 
 
 
 
2
1  2  1   2  2  1  2
45
2
Lecture 11
Time-Average Poynting Vectors
(Cont’d)

Hence,
S   S 
av 1
av 2
or
S   S   S 
av i
av r
av t
Power is conserved at the interface.
46
Lecture 11
Oblique Incidence at a Dielectric
Interface
z 0
1 , 1
 2 , 2
t
r
i
E1  Ei  Er
E2  Et
47
Lecture 11
Oblique Incidence at a Dielectric
Interface: Parallel Polarization (TM to z)
Ei  E0 xˆ cos  i  zˆ sin  i  e
 jk1  x sin i  z cos i 
Er  E0xˆ cos  r  zˆ sin  r  e  jk1  x sin r  z cos r 
Et  E0 xˆ cos  t  zˆ sin  t  e  jk2  x sint  z cost 
48
Lecture 11
Oblique Incidence at a Dielectric
Interface: Parallel Polarization (TM to z)
 2 cos  t  1 cos  i

 2 cos  t  1 cos  i
2 2 cos  i

 2 cos  t  1 cos  i
49
Lecture 11
Oblique Incidence at a Dielectric
Interface: Perpendicular Polarization
(TE to z)
Ei  E0 yˆ e
 jk1  x sin i  z cos i 
Er  E0 yˆ e
Et  E0 yˆ e
 jk1  x sin r  z cos r 
 jk2  x sin t  z cos t 
50
Lecture 11
Oblique Incidence at a Dielectric
Interface: Perpenidcular Polarization
(TM to z)
 2 cos  i  1 cos  t

 2 cos  i  1 cos  t
2 2 cos  i

 2 cos  i  1 cos  t
51
Lecture 11
Brewster Angle

The Brewster angle is a special angle of
incidence for which =0.
 For
dielectric media, a Brewster angle can
occur only for parallel polarization.
52
Lecture 11
Critical Angle

The critical angle is the largest angle of
incidence for which k2 is real (i.e., a
propagating wave exists in the second
medium).
 For
dielectric media, a critical angle can exist
only if 1>2.
53
Lecture 11
Some Useful Theorems
The reciprocity theorem
 Image theory
 The uniqueness theorem

54
Lecture 11
Image Theory for Current Elements above a
Infinite, Flat, Perfect Electric Conductor
electric
magnetic
actual
sources
 
images
55
Lecture 11
Image Theory for Current Elements above a
Infinite, Flat, Perfect Magnetic Conductor
electric
magnetic
h
actual
sources
m  
h
images
56
Lecture 11