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ENE 325 Electromagnetic Fields and Waves. Lecture 12 Plane Waves in Conductor, Poynting Theorem, and Power Transmission. Review (1). Wave equations Time-Harmonics equations where. Review (2). where This term is called propagation constant or we can write = +j - PowerPoint PPT Presentation
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27/02/51
Lecture 12 Plane Waves in Conductor, Poynting Theorem, and Power Transmission
ENE 325Electromagnetic Fields and Waves
27/02/51
Review (1) Wave equations
Time-Harmonics equations
where
22
2
������������������������������������������ E EE
t t2
22
������������������������������������������ H HH
t t
2 2 0 ����������������������������
s sE E
2 2 0 ����������������������������
s sH H
( ) j j
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Review (2)
where
This term is called propagation constant or we can write
= +j
where = attenuation constant (Np/m) = phase constant (rad/m)
( ).j j
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Review (3) 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) skin depth
b) skin resistance
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c) 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 avgP P d S����������������������������
for lossless case, 00
12
j z j zxavg x yx
EP E e a e a
��������������
201
2x
avg zE
P a ��������������
W
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?
