Bài giảng Engineering electromagnetic - Chapter XII: The Uniform Plane Wave - Nguyễn Công Phương
Wave Polarization (1)
• In the previous sections, E & H are supposed to lie in
fix directions
• However, the directions of E & H within the plane
perpendicular to az may change as functions of time and
position
• λ, v
p, S,
• The instantaneous orientation of field vectors
• Wave polarization: its electric field vector orientation as
a function of time, at a fixed point in space
• H can be found from E
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Nguyễn Công Phương
Engineering Electromagnetics
The Uniform Plane Wave
Contents
I. Introduction
II. Vector Analysis
III. Coulomb’s Law & Electric Field Intensity
IV. Electric Flux Density, Gauss’ Law & Divergence
V. Energy & Potential
VI. Current & Conductors
VII. Dielectrics & Capacitance
VIII.Poisson’s & Laplace’s Equations
IX. The Steady Magnetic Field
X. Magnetic Forces & Inductance
XI. Time – Varying Fields & Maxwell’s Equations
XII. The Uniform Plane Wave
XIII.Plane Wave Reflection & Dispersion
XIV.Guided Waves & Radiation
Uniform plane wave - sites.google.com/site/ncpdhbkhn 2
The Uniform Plane Wave
1. Wave Propagation in Free Space
2. Wave Propagation in Dielectrics
3. The Poynting Vector
4. Skin Effect
5. Wave Polarization
Uniform plane wave - sites.google.com/site/ncpdhbkhn 3
Wave Propagation in Free Space (1)
E
H
0 t
H
E
0 t
.E 0
.H 0
Uniform plane wave - sites.google.com/site/ncpdhbkhn 4
Wave Propagation in Free Space (2)
E Ex a x
Ex E( x , y , z )cos( t )
ej t cos t j sin t
j() t j j t
Ex Re Exyze (,,) Re Exyzee (,,)
j
Exs E(,,) x y z e
Es E xs a x
E Re E e j t
x xs
Uniform plane wave - sites.google.com/site/ncpdhbkhn 5
Ex. Wave Propagation in Free Space (3)
Find the time – varying function of the vector field:
o o o
Es 100 30ax 20 50a y 40 210az V/ m
If f = 1 MHz
j30o j 50 o j 210 o
Es100e a x 20 e a y 40 e a z V/ m
j30o j 50 o j 210 o j 2 10 6 t
Es(t ) 100 e a x 20 e a y 40 e a z e
j(2 106 t 30 o ) j (2 10 6 t 50 o ) j (2 10 6 t 210 o )
100eax 20 e a y 40 e a z
6 o
E(t ) 100cos(2 10 t 30 ) ax
6 o 6 o
20cos(2 10t 50 )ay 40cos(2 10 t 210 ) a z
Uniform plane wave - sites.google.com/site/ncpdhbkhn 6
Wave Propagation in Free Space (3)
Ex E( x , y , z )cos( t )
E
x E( x , y , z )cos( t ) E ( x , y , z )sin( t )
t t
j t j t j
Rej Exs e Re j E ( x , y , z ) e e
j() t
Rej E ( x , y , z ) e
ReE ( x , y , z ) j cos( t ) j sin( t )
ReE ( x , y , z ) j cos( t ) sin( t )
E( x , y , z )sin( t )
E
x Re j E e j t
t xs
Uniform plane wave - sites.google.com/site/ncpdhbkhn 7
Wave Propagation in Free Space (4)
Ex E( x , y , z )cos( t )
E
x Re j E e j t j E
t xs xs
E
H HE j
0 t s0 s
H EH j
E s0 s
0 t
.E 0
.E 0 s
.H 0 .Hs 0
Uniform plane wave - sites.google.com/site/ncpdhbkhn 8
Wave Propagation in Free Space (5)
EHs j0 s EHHs j0 s j 0 s
HEs j 0 s
2
EEs 0 0 s
2
Es () .E s E s
.Es 0 ( .Es ) 0
2 2
EEs k0 s
k0 0 0 (wavenumber)
2 2
Exs k0 E xs
2 2 2
EEExs xs xs 2 2
2 2 2 k0 Exs d Exs 2
x y z k0 Exs
dz2
Suppose Exs does not vary with x or y
Uniform plane wave - sites.google.com/site/ncpdhbkhn 9
Wave Propagation in Free Space (6)
2
d Exs 2
k0 Exs
dz2
jk0 z
Exs E x0 e Ex( z , t ) E x0 cos( t k 0 z )
Ex( z , t ) E x 0 cos( t k 0 z )
k0 0 0
1 k0
2.998 108 3 10 8 m/s c
0 0
Ex( z , t ) E x0 cos[ ( t z / c )]
Ex( z , t ) E x cos[ ( t z / c )]
Uniform plane wave - sites.google.com/site/ncpdhbkhn 10
Wave Propagation in Free Space (7)
Ex(,) z t E x0 cos[( t z /)] c
Ex(,) z t E x 0 cos[( t z /)] c
dE
EH j xs j H
s0 sdz 0 ys
jk0 z
Exs E x0 e
1
jk0 z0 jk 0 z
Hys () jk0 E x 0 e E x 0 e
j0 0
H( z , t ) E0 cos( t k z )
y x0 0 E
0 x 0
H y 0
Ex(,) z t E x0 cos[( t z /)] c
Uniform plane wave - sites.google.com/site/ncpdhbkhn 11
The Uniform Plane Wave
1. Wave Propagation in Free Space
2. Wave Propagation in Dielectrics
3. The Poynting Vector
4. Skin Effect
5. Wave Polarization
Uniform plane wave - sites.google.com/site/ncpdhbkhn 12
Wave Propagation in Dielectrics (1)
2 2
EEs k s
k k0 r r
0
, 0 377 120
0
2
d Exs 2
k Exs
dz2
jk j
jkz z j z
Exs E x0 e E x 0 e e
z
Ex E x0 ecos( t z )
Uniform plane wave - sites.google.com/site/ncpdhbkhn 13
Wave Propagation in Dielectrics (2)
j 0() r j r
k k0 r r
k ( j ) 1 j
j 0() r j r
1/ 2
2
Re[jk ] 1 1
2
1/ 2
2
Im[jk ] 1 1
2
Uniform plane wave - sites.google.com/site/ncpdhbkhn 14
Wave Propagation in Dielectrics (3)
E E e z cos( t z ) v
x x0 p
2
2
EEx x0 z j z
Hys e e
H y
1/ 2
2 0
Re[jk ] 1 1
2
1 c
1/ 2 vp
2 r r
Im[jk ] 1 1 2 1
2 0
f r r
0
Uniform plane wave - sites.google.com/site/ncpdhbkhn 15
Wave Propagation in Dielectrics (4)
0 Ex E x0 cos( t z )
E
Ex Hx0 cos( t z )
y
H y
Uniform plane wave - sites.google.com/site/ncpdhbkhn 16
Wave Propagation in Dielectrics (5)
Ex.
Find the attenuation of a 2.5 GHz wave propagating in fresh water,
given ε’r = 78, ε’’r = 7, μr = 1.
1/ 2
2
1 1
2
k0 r r
k0 / c
1/ 2
9 2
2 2.5 10 78 7 1
1 1 21 Np/m 4.8 cm
3 108 2 78
Uniform plane wave - sites.google.com/site/ncpdhbkhn 17
Wave Propagation in Dielectrics (6)
JE
HEs j s
j
HEEEs j() j s s j s
HJEs s j s
HEJJs () j s s ds
JEJE, j
s s ds s
Uniform plane wave - sites.google.com/site/ncpdhbkhn 18
Wave Propagation in Dielectrics (7)
JEds j s
JE() j
s s
JE s s
1/ 2
2
1 1
2 E tan
s
J s
JEJE s s, ds j s
Jds j j
Uniform plane wave - sites.google.com/site/ncpdhbkhn 19
Wave Propagation in Dielectrics (8)
Good dielectrics : 1
Conductor:
jk j 1 j j ' 1 j
n( n 1) n ( n 1)( n 2)
(1x )n 1 nx x2 x 3 ...
2! 3!
2
1
jk j 1 j ... j
2 8
Re[jk ] j j
2 2
2
1
Im[jk ] 1
8
Uniform plane wave - sites.google.com/site/ncpdhbkhn 20
Wave Propagation in Dielectrics (9)
j j
2 2
2
1
1
8
1
j 1 j ( / ')
2
3
1 j 1 j
8 2 2
Uniform plane wave - sites.google.com/site/ncpdhbkhn 21
The Uniform Plane Wave
1. Wave Propagation in Free Space
2. Wave Propagation in Dielectrics
3. The Poynting Vector
4. Skin Effect
5. Wave Polarization
Uniform plane wave - sites.google.com/site/ncpdhbkhn 22
The Poynting Vector (1)
D
HJ
t
D
E. H E.J E.
t
.() E H E. H H. E
D
H. E .() E H J.E E.
t
B
E
t
BD
H. .() E H E.J E.
t t
EH
.() E H J.E E. H.
t t
Uniform plane wave - sites.google.com/site/ncpdhbkhn 23
The Poynting Vector (2)
EH
.() E H J.E E. H.
t t
E D.E
E.
t t 2
H B.H
H.
t t 2
D.E B.H
.() E H J.E
t2 t 2
D.E B.H
.() E Hdv J.E dv dv dv
VVVV t2 t 2
DSD..d dv
SV
d1 d 1
()E H .d S J.E dv D.E dv B.H dv
SVVV dt 2 dt 2
Uniform plane wave - sites.google.com/site/ncpdhbkhn 24
The Poynting Vector (3)
EH2 2
()E H .d S J.E dv dv
SVVt 2 2
SEH W/m2
EHSxa x y a y z a z
Ex E x0 cos( t z ) 2
Ex0 2
E Sz cos ( t z )
Hx0 cos( t z )
y
2
1 T E2 1 E T
Sx0 cos2 ( t z ) dt x0 [1 cos(2t z )] dt
z, av T 0 2T 0
2T 2
1EEx0 1 1 x 0 2
1 sin( t 2 z ) W/ m
2T 2 0 2
Uniform plane wave - sites.google.com/site/ncpdhbkhn 25
The Poynting Vector (4)
E E e z cos( t z )
x x0 Ex0 z
Hy ecos( t z )
E2
S E H x0 e2 z cos( t z )cos( t z )
z x y
E2
x0 e2 z[cos(2 t 2 z 2 ) cos ]
2
1 E2 1
S x0 e2 z cos ReEH ˆ W/m2
z, av 2 2 s s
j z
Es E x0 e a x
EE j
Hˆ x0ej z a x 0 e e j z a
sˆ y y
Uniform plane wave - sites.google.com/site/ncpdhbkhn 26
The Uniform Plane Wave
1. Wave Propagation in Free Space
2. Wave Propagation in Dielectrics
3. The Poynting Vector
4. Skin Effect
5. Wave Polarization
Uniform plane wave - sites.google.com/site/ncpdhbkhn 27
Skin Effect (1)
jk j 1 j j j j j
j 1 90o
1 1
1 90o 1 45o j
2 2
1 1
jk j j ( j 1 1) f j
2 2
f
z z f
Ex E x0 ecos( t z ) E x 0 e cos( t z f )
Uniform plane wave - sites.google.com/site/ncpdhbkhn 28
Skin Effect (2)
E E ez f cos( t z f )
x x0 Dielectrics
z
E Ecos t
xz0 x0 0
Conductor
z f
Jx E x E x0 ecos( t z f )
1 1 1
f
0.066
Cu
f
Cu; 50Hz 9.3 mm
4
Cu; 10,000 MHz 6.61 10 mm
Uniform plane wave - sites.google.com/site/ncpdhbkhn 29
Skin Effect (3)
1
f
v
p
vp
Uniform plane wave - sites.google.com/site/ncpdhbkhn 30
Skin Effect (4)
Ex.
Consider an 1 MHz wave propagating in seawater, σ = 4 S/m, ε’r = 81.
4 2
8.9 10 1
(2 106 )(81)(8.85 10 12 )
1 1
0.25 m
f ( 10)(46 10 7 )(4)
2 1.6 m
6 6
vp (2 10 )(0.25) 1.6 10 m/s
Uniform plane wave - sites.google.com/site/ncpdhbkhn 31
Skin Effect (5)
j
j
j j
'
1
j 1 45o
f
2 45o 1 1
j
z f z/
Ex E x0 ecos( t z f ) E x 0 e cos( t z / )
E
x
H y
Ex0 z/ z
Hy ecos t
2 4
Uniform plane wave - sites.google.com/site/ncpdhbkhn 32
Skin Effect (6)
x
z/
Ex E x0 ecos( t z / ) Conductor
L
Ex0 z/ z
Hy ecos t Dielectrics
2 4 Jx0
1
S Re EH ˆ z
av2 s s 0
|Jxs|
2
1 Ex0 2z /
Sav e cos b
22 4 y δ
1
E2 e 2z /
4 x0
b L 12 2z / 1 2
S S dS Ex0 e dxdy bLE x 0
L,, avS z av 0 0
4z0 4
Uniform plane wave - sites.google.com/site/ncpdhbkhn 33
Skin Effect (7)
1 x
S bLE2
L, av4 x 0 Conductor
JEx0 x 0 L
1 2 Dielectrics
S bLJ Jx0
L, av4 x 0
b z
I J dydz 0
0 0 x
|Jxs|
z /
Jx J x0 ecos t z /
z// jz b
Jxs J x0 e e y δ
(1 j ) z /
Jx0 e
b J b
I J e(1 j ) z / dydz J be(1 j ) z / x0
s0 0 x0 x0
1 j 0 1 j
Jx0 b
I cos t
2 4
Uniform plane wave - sites.google.com/site/ncpdhbkhn 34
Skin Effect (8)
x
Jx0 b
Icos t Conductor
2 4 L
IJ x0
J cos t Dielectrics J
b 2 4 x0
1 z
S () J 2 bL 0
L
|Jxs|
2
J x0 2
bLcos t
2 4 b δ
1 y
S J2 bL
L, av4 x 0
(if the current is distributed uniformly throughout 0 < z < δ)
1
S J2 bL
L, av4 x 0
(if the total current is distributed throuthout 0 < z < ∞)
Uniform plane wave - sites.google.com/site/ncpdhbkhn 35
Skin Effect (9)
LL
R
S 2 a
103
RCu, 1MHz, a 1mm, l 1km 41.5
(5.8 107 )(2 )(10 3 )(0.066 10 3 )
Uniform plane wave - sites.google.com/site/ncpdhbkhn 36
The Uniform Plane Wave
1. Wave Propagation in Free Space
2. Wave Propagation in Dielectrics
3. The Poynting Vector
4. Skin Effect
5. Wave Polarization
Uniform plane wave - sites.google.com/site/ncpdhbkhn 37
Wave Polarization (1)
• In the previous sections, E & H are supposed to lie in
fix directions
• However, the directions of E & H within the plane
perpendicular to az may change as functions of time and
position
• λ, vp, S,
• The instantaneous orientation of field vectors
• Wave polarization: its electric field vector orientation as
a function of time, at a fixed point in space
• H can be found from E
Uniform plane wave - sites.google.com/site/ncpdhbkhn 38
Wave Polarization (2) y
z j z
Es()E x0 a x E y 0 a y e e E
Ey0
z j z
Hs()H x0 a x H y 0 a y e e
H Hy0
Ey0 Ex0 z j z
ax a y e e
x
H E
1 x0 x0
S Re[EH ˆ ]
z, av2 s s
1
ReE Hˆ (a a ) E H ˆ ( a a ) e2 z
2 x0 y 0 x y y 0 x 0 y x
ˆ ˆ
1 EEx0 x 0 EEy0 y 0 2 z
Re e az
2 ˆ ˆ
1 1 2 2 2 z 2
Re Ex0 E y 0 e a z W/m
2 ˆ
Uniform plane wave - sites.google.com/site/ncpdhbkhn 39
Wave Polarization (3)
j z
Es()E x0 a x E y 0 a y e
E(z , t ) Ex0 cos( t z ) a x E y 0 cos( t z ) a y
E(z ,0) Ex0 cos( z ) a x E y 0 cos( z ) a y
E(z, 0)
Ex0 Observer
a
location
Ey0
b
z
Wave travel
Uniform plane wave - sites.google.com/site/ncpdhbkhn 40
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