Uniform plane wave
description
Transcript of Uniform plane wave
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Uniform plane wave
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2Radiation from a current filament
SR
ej
jkR
4)ˆ(ˆ)( JRRJrE
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3Spherical waves from a source of finite dimension
R
eE
jkR
4
Transmitted waves from a finite sized source behave like spherical waves.
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4Concept of plane waves
jkReE R
eCRE
jkR
4)(
)(
)()(
4)(4)(
RRjk
RRjkRRjk
eC
R
eC
RR
eCRRE
RR
R
)()( RRjkeCRRE
jkReCRE )(
EM waves transmitted from a finite sized source spread spherically in the space. At great distances from the source, EM waves behave as a plane wave locally.
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5E & H in source free region
0
B
D
DJH
BE
t
t1. In source free region, E and H can be obtained easily.2. With J and ρ equal to zero, a source free wave equation is ob-
tained.
0
0
H
E
EH
HE
j
j02
22
EE
EEHE
k
kj
0
0)(22
222
EE
EEEEE
k
kk
22 k
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022 EE k
• If the source current away from the field point has the direction of x-axis, the electric field can have only x component. If the extent of the source is infinite along x and y directions, variations of the field along those directions become zero.
022
2
xx Ek
dz
Ed
jkzx
jkzxx eEeEE 00
The first term represent the wave propagating toward +z direction, while the second stands for the wave propagating to the opposite direction. Considering only the wave propagating in the positive direction,
jkzx
x
jkzx
eEz
E
jj
eE
0
0
1ˆ
1ˆ
ˆ
yy
EH
xE
3770 wave impedance of free
space
• If the propagating direction is in +z-axis, the Helmholtz equation be-comes a simple expression.
02
2
2
2
y
E
x
E xx
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7Plane wave propagating general direction
0
0
22
2
22
xx Ek
dz
Ed
k EE 1. If the propagating direction is other than x, y or z direction, the phase term in the exponential function can be obtained by the inner product of the k-vector and the position vector.
2. the k-vector can be compared to the normal vector of equi-phase plane.jkz
x eE 0xE
rkrk ExE ˆ
0
ˆ
0ˆ jkjkx eeE
)ˆˆˆˆ( zyxkk zyx kkkk
rkxE jx eE 0ˆ
)point nobservatio: , npropagatio waveof direction:ˆ( rk
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)ˆˆˆˆ( zyxkk zyx kkkk rkxE jx eE 0ˆ
EkEkE
H
ˆˆ
j
jk
j
The magnetic field can be obtained.
EkEk
EEErk
rkrkrk
jej
eeej
jjj
0
000 )()(
Magnetic field of a plane wave
AAA uuu )(
Vector identity
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9Wave propagation in dielectrics
1. If the wave is propagating into a medium other than free space, the molecules in that medium vibrate under the action of electric field. Meanwhile, the energy of the wave is dissi-pated and the medium become heated.
2. The dissipation can be modeled by a complex permittivity.
EE
EE
JJEEEJH
jjjj
jjjj
jj displcond
jj
3. The ratio of the real part of εr to the imaginary part is called the loss tangent of that media.
tan
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4. From the equations on the left, it can be seen that the phase of displacement current leads that of conduction current by 90 degree.
5. That is, electric field propagates first, then charges move un-der the action of that electric field.
6. Helmholtz equation in a lossy medium becomes,
displcond JJEEEJH jj
EJ
EJ
jdispl
cond
0)(
0)(
22
2
22
xx Ej
dz
Ed
j
EE
j
jeEE zxx
)(, 220
Propagation constant in a lossy dielectric
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12
222222
2222
22
)2
)(2)1
)(,
jj
jj
4
)/(12
22
2222
동순복호 1)/(12
1)/(12
2
)()/(1
04
)/(1
0)(
2
22
2/1224222422
2
2
2222
22222
tt
tt
zjzxx eEE 0
To obtain the approximate expression of α and β, we consider the two extreme cases of
① Good dielectric : ② Good conductor :
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동순복호 1)/(12
2/12
Case 1)
22/1
2
2/12
22
)/(8
112
/2
1
2)/(4
112
/2
1
2
1)/(2
11
2
)/(2
11)/(1
동순복호
22
8
11)/(
8
11
22
Plane wave in good dielectrics
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-5 -4 -3 -2 -1 0 1 2 3 4 5-1
-0.5
0
0.5
1
Plane wave in good conductors
동순 복호 1)/(12
2/12
depth)skin ;(1
2
동순 복호 /2
)/()/(1
2/1
2
zj
zzjz ee
00 EEE
δ is called as the skin depth of the medium at the given frequency. If the electric field pene-trate into a lossy medium as much as one skin depth, its strength decreases by 1/e. That is its strength becomes 36.7% of the original value.
conductor
depthskin :
Case 2)
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17Example
Find the skin depth of sea water at the frequency of 1MHz. In sea water
jjj
1 1890
)1085.8)(81)(102(
4126
good conductor
]m[25.02
]m[6.12
depth) skin;(1
2
.81,]/[4 rmS
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18Example
Calculate the resistance of a round copper of 1mm radius and 1km length at DC and 1MHz.
48.5)108.5(10
1076
3
2 r
lRDC
5.41)108.5)(10066.0(102
10
2 733
3
MHz1 a
lR
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19Wave polarization
022 EE k 1. The Helmholtz equation has three components (x, y, z).2. But the divergenceless condition imposes that ∇∙E = 0,
which is a constraint among the vector components of E-field. That is, all the component of Ex, Ey, Ez are not independent.
00
z
E
z
E
x
E zyxE
0
zzyyxxzyx EkEjkEjkz
E
z
E
x
EE
From the above condition, only two components among the Ex, Ey, Ez are independent. That is, two kinds of independent po-larizations comprise an arbitrary E-field.
a changing direction of electric field observed at a position.
Among the components of an electric field vector, only two of them is independent.
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20PolarizationExample of a electric polarization of a wave propagating in +z direction.
x
y Linear polarization
Circular polarization
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21Polarization diversity antenna