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ENE 429Antenna and Transmission lines Theory
Lecture 2 Uniform plane waves
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Review Wave equations
Time-Harmonics equations
where
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2
������������������������������������������ E EE
t t2
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������������������������������������������ H HH
t t
2 2 0 ����������������������������
s sE E
2 2 0 ����������������������������
s sH H
( ) j j
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Time-harmonic wave equationsor
where
This term is called propagation constant or we can write
= +j
where = attenuation constant (Np/m) = phase constant (rad/m)
2 2 0 ����������������������������
s sH H
( ) j j
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Transverse ElectroMagnetic wave (TEM)
http://www.edumedia.fr/a185_l2-transverse-electromagnetic-wave.html
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Solutions of Helmholtz equations The instantaneous forms of the solutions
The phasor forms of the solutions
0 0cos( ) cos( )
��������������z z
x xE E e t z a E e t z a
0 0cos( ) cos( )z z
y yH H e t z a H e t z a ��������������
0 0
z j z z j zs x xE E e e a E e e a
��������������
0 0
z j z z j zs y yH H e e a H e e a
��������������
incident wave reflected wave
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Attenuation constant
Attenuation constant determines the penetration of the wave into a medium
Attenuation constant are different for different applications
The penetration depth or skin depth, is the distance z that causes to reduce to
z = 1
z = 1/ =
E��������������
10E e
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Good conductor
1 1
f
At high operation frequency, skin depth decreases
A magnetic material is not suitable for signal carrier
A high conductivity material has low skin depth
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Currents in conductor
To understand a concept of sheet resistance
1L LR
A wt
1 LR
t w Rsheet () Lw
1sheetR
t sheet resistance
from
At high frequency, it will be adapted to skin effect resistance
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Currents in conductor
0
0
zx x
zx x
E E e
J E e
Therefore the current that flows through the slab at t is
;xI J dS ds dydz
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Currents in conductor
;xI J dS ds dydz
00 0
wz
xz y
I E e dydz
0
0
zxw E e
0 .xI w E A
From
Jx or current density decreases as the slab gets thicker
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Currents in conductor
0xV E L
0
0
1xskin
x
E LV L LR R
I w E w w
For distance L in x-direction
For finite thickness,
R is called skin resistanceRskin is called skin-effect resistance
0 00 0
(1 )t w
z tx x
z y
I E e dydz w E e
/
1
(1 )skin tRe
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Currents in conductor
Current is confined within a skin depth of the coaxial cable
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Ex1 A steel pipe is constructed of a material for which r = 180 and = 4106 S/m. The two radii are 5 and 7 mm, and the length is 75 m. If the total current I(t) carried by the pipe is 8cost A, where = 1200 rad/s, find:
a) The skin depth
b) The skin resistance
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c) The dc resistance
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The Poynting theorem and power transmission
2 21 1( )
2 2E H d S J E dV E dV H dV
t t
����������������������������������������������������������������������
Poynting theorem
Total power leavingthe surface
Joule’s lawfor instantaneouspower dissipated per volume (dissi-pated by heat)
Rate of change of energy storedIn the fields
2W/mS E H ������������������������������������������
Instantaneous poynting vector
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Example of Poynting theorem in DC case
2 21 1( )
2 2E H d S J E dV E dV H dV
t t
����������������������������������������������������������������������
Rate of change of energy storedIn the fields = 0
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Example of Poynting theorem in DC case
2 z
IJ a
a
��������������
By using Ohm’s law,
From
2 z
J IE a
a ��������������
��������������
2 2
2 20 0 0( )
a LId d dz
a
2 22
1 LI I R
a
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Example of Poynting theorem in DC case
E H d S������������������������������������������
From Ampère’s circuital law,
Verify with
H dl I����������������������������
2 aH I ��������������
2
IH a
a
��������������
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Example of Poynting theorem in DC case
2
2 32
IS d S a d dz
a
����������������������������
2
2 2 32 2z
I I IS E H a a a
aa a
������������������������������������������
2 222
2 3 20 02
LI a I Ld dz I R
a a
Total power
W
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Uniform plane wave (UPW) power transmission Time-averaged power density
1Re( )
2avgP E H
������������������������������������������
amount of power WavgP P d S����������������������������
for lossless case, 00
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��������������j z j zx
avg x yx
EP E e a e a
201
2x
avg zE
P a ��������������
W/m2
W/m2
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Uniform plane wave (UPW) power transmission
0
z j z jxxE E e e e a
��������������
intrinsic impedance for lossy medium nje
0
1 1 z j z jz xxH a E a E e e e a
����������������������������
0 njz j z jxy
Ee e e e a
for lossy medium, we can write
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Uniform plane wave (UPW) power transmission
2
201Re
2jzx
zE
e e a
from1
Re( )2
avgP E H
������������������������������������������
2
201cos
2zx
zE
e a
W/m2
Question: Have you ever wondered why aluminum foil is not allowed inthe microwave oven?
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Polarization
UPW is characterized by its propagation direction and frequency.
Its attenuation and phase are determined by medium’s parameters.
Polarization determines the orientation of the electric field in a fixed spatial plane orthogonal to the direction of the propagation.
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Linear polarization
Consider in free space,
0( , ) cos( ) xE z t E t z a
��������������E��������������
At plane z = 0, a tip of field traces straight line segment called “linearly polarized wave” E
��������������
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A pair of linearly polarized wave also produces linear polarization
Linear polarization
0 0( , ) cos( ) cos( )x yx yE z t E t z a E t z a
��������������
At z = 0 plane
At t = 0, both linearly polarized wavesHave their maximum values
0 0(0,0) x yx xE E a E a
��������������(0, ) 0
4t
E ��������������
0 0(0, ) cos( ) cos( )x yx yE t E t a E t a
��������������
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More generalized of two linearly poloraized waves,
Linear polarization occurs when two linearly polarized waves are
More generalized linear polarization
0 0( , ) cos( ) cos( )x yx x y yE z t E t z a E t z a
��������������
in phase 0y x
out of phase 180y x
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Super position of two linearly polarized waves that
If x = 0 and y = 45, we have
Elliptically polarized wave
0 180y x or
0 0(0, ) cos( ) cos( )
4x yx yE t E t a E t a
��������������
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occurs when Exo and Eyo are equal and
Right hand circularly polarized (RHCP) wave
Left hand circularly polarized (LHCP) wave
Circularly polarized wave
90y x
0 0(0, ) cos( ) cos( )
2x yx yE t E t a E t a
��������������
0 0(0, ) cos( ) cos( )
2x yx yE t E t a E t a
��������������
90y x
90y x
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Phasor forms:
for RHCP,
for LHCP,
Circularly polarized wave
0 0( 0) yx
jjx yx yE z E e a E e a
��������������from
0( 0) ( )x yxE z E a ja
��������������
0( 0) ( )x yxE z E a ja
��������������
Note: There are also RHEP and LHEP
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Ex2 Given
,determine the polarization of this wave
( , ) 3cos( 30 ) 8cos( 90 )x yE z t t z a t z a ��������������
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Ex3 The electric field of a uniform plane wave in free space is given by , determine
50100( ) j ys z xE a ja e
��������������
a) f
b) The magnetic field intensity sH��������������
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c)
d) Describe the polarization of the wave
S��������������