lecture 12 plane waves in conductor, poynting theorem, and power transmission
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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 PresentationTRANSCRIPT
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Lecture 12Lecture 12 Plane Waves in Plane Waves in Conductor, Poynting Theorem, and Conductor, Poynting Theorem, and Power TransmissionPower Transmission
ENE 325ENE 325ElectromagnetElectromagnetic Fields and ic Fields and WavesWaves
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Review (1)Review (1) Wave equationsWave equations
Time-Harmonics equationsTime-Harmonics equations
wherewhere
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������������������������������������������ 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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Review (2)Review (2)
wherewhere
This This term is called term is called propagation constantpropagation constant or we or we can write can write
= = +j+j
where where = attenuation constant (Np/m) = attenuation constant (Np/m) = = phase constant (rad/m)phase constant (rad/m)
( ).j j
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Review (3)Review (3)
The instantaneous forms of the The instantaneous forms of the solutionssolutions
The phasor forms of the solutionsThe 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
Attenuation constant determines the penetration Attenuation constant determines the penetration of the wave into a mediumof the wave into a medium
Attenuation constant are different for different Attenuation constant are different for different applicationsapplications
The penetration depth The penetration depth or or skin depthskin depth, , is the distance z that causes to reduce to is the distance z that causes to reduce to
z = -1z = -1
z = -1/ z = -1/ = - = -..
E��������������
10E e
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Good conductorGood 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 conductorCurrents in conductor
To understand a concept of sheet To understand a concept of sheet resistanceresistance
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 conductorCurrents 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 conductorCurrents 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 conductorCurrents 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 conductorCurrents in conductor
Current is confined within a skin depth of the coaxial cable.
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Ex1Ex1 A steel pipe is constructed of a A steel pipe is constructed of a material for which material for which rr = = 180 and 180 and = = 44101066 S/m. The two radii are 5 and 7 S/m. The two radii are 5 and 7 mm, and the length is 75 m. If the total mm, and the length is 75 m. If the total current current I(t)I(t) carried by the pipe is carried by the pipe is 8cos8costt A, where A, where = = 12001200 rad/s, find: rad/s, find: a)a) skin depthskin depth
b)b) skin resistanceskin resistance
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c) c) dc resistancedc resistance
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The Poynting theorem and The Poynting theorem and power transmissionpower 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 Example of Poynting theorem in DC caseDC 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 Example of Poynting theorem in DC caseDC 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 Example of Poynting theorem in DC caseDC 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 Example of Poynting theorem in DC caseDC 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) Uniform plane wave (UPW) power transmissionpower transmission Time-averaged power densityTime-averaged power density
1Re( )2
avgP E H
������������������������������������������
amount of power avgP P d S����������������������������
for lossless case, 00
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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) Uniform plane wave (UPW) power transmissionpower 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) Uniform plane wave (UPW) power transmissionpower transmission
2
201Re2
jzxz
Ee e a
from
1Re( )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?