Broadband PLC Radiation from a Power Line with Sag
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Transcript of Broadband PLC Radiation from a Power Line with Sag
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Broadband PLC Radiation from a
Power Line with Sag
Nan Maung, SURE 2006SURE Advisor: Dr. Xiao-Bang Xu
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OBJECTIVE
To model a radiating Catenary Line Source (Eg. An Outdoor wire with sag)
Understand Physical Interpretation of Mathematical Models
To use theoretical knowledge to test whether Model and Numerical Solutions created are Physically Reasonable
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INTENDED MODEL
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INTENDED MODELCatenary Wire Modeled by Finite-Length Dipoles
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INTENDED MODEL
Number of dipoles
Dipole Midpoints
Wire Length
/10n
(2 1)20nx n
22 n
n
xz s h
L
constantny
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THEORY
Solutions are derived based on: Superposition Helmholtz Equation Fourier Transform Techniques Sommerfeld Radiation Conditions
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ANALYSIS & VERIFICATION
Solution must make Physical sense Intermediate (simpler) Models used for
verification A Straight Line Source A Hertzian Dipole Compare Solution derived for
Catenary to Line Source Hertzian Dipole is used as basis for
model of Finite-Length Dipole
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METHOD OF SOLUTION(General)
Boundary Value Problem Define Source Type Derive Helmholtz Equation for Vector
Magnetic Potential Forward Fourier Transform Find Solution in Spectral Domain (SD) TD Solution must satisfy Sommerfeld
Radiation Condition Inverse Fourier Transform IFT Integrals must be convergent
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A Straight-Line Source
Located in upper Half-Space above Media Interface at z = 0
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SOMMERFELD INTEGRALS(Coming back to Spatial Domain)
aa (2) 2 2a0
(2) 2 2 ( ') ( ')0
a
In Region z > 0 ; z' > 0
Solution by Fourier Transform Technique
(y,z) = - j [ ( ( ') ( ') ) ( ')] 4
2( ') = - ( ( ') ( ') ) ( ) a y
ea aa
e j z z jk y ya yaa e y
a H k y y z z g
g H k y y z z k e e dk
ba ( ') ( ')
a
In Region b z < 0 ; z' > 0
Solution by Fourier Transform Technique
2(y,z) = -j ( ) a yj z z jk y y
ye ya k e e dk
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Predicted behavior of SolutionsBased on Physical Interpretation
First term in is due to an infinite line source in homogeneous medium
First term in is due to image of the line source in a PEC plane at the boundary
Second term in is correction for the fact that a PEC plane does not faithfully model the media interface and Region b
The correction term should decrease if the dielectric properties of Medium b are allowed to approach those of Medium a
aaa
eaag
eaag
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NUMERICAL RESULTS & SOLUTION CHECK
Real and imaginary parts of Correction Integral vs. Relative Dielectric of Medium b
Observed at z=15
2 3 4 5 6 7 8 9 10 11
-0.65
-0.6
-0.55
-0.5
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
eb,r
Rea
l par
t of
Cor
rect
ion
Inte
gral
in a
aa
Real part of Correction Integral term in aaa vs. eb,r
2 3 4 5 6 7 8 9 10 11
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
eb,r
Rea
l par
t of
Cor
rect
ion
Inte
gral
in a
aa
Real part of Correction Integral term in aaa vs. eb,r
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NUMERICAL RESULTS & SOLUTION CHECK
Real and Imaginary parts of Correction Integral vs. Relative Dielectric of Medium b
Observed at z = 7
2 3 4 5 6 7 8 9 10 11
-0.7
-0.65
-0.6
-0.55
-0.5
-0.45
-0.4
-0.35
-0.3
-0.25
eb,r
Rea
l Com
pone
nt o
f C
orre
ctio
n In
tegr
al in
aaa
Real Component of Correction Integral term in aaa vs. eb,r
2 3 4 5 6 7 8 9 10 110.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
eb,r
Imag
inar
y pa
rt o
f C
orre
ctio
n In
tegr
al in
a^a
^a }
Imaginary part of Correction Integral term in aaa vs. eb,r
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A Hertzian Dipole
Source Definition Helmholtz Equation Boundary
Conditions Dyadic Green’s
Function F.T. Solution for
Dyadic Elements Sommerfeld
Integrals
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Hertzian Dipole
Unit Vector Source
J ( ) ( ')x y z r r BBBBBBBBBBBBB B
'
Creates a Magnetic Vector Potential
A ( )4 '
jk r re
r x y zr r
BBBBBBBBBBBBB B
BBBBBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBB B
'
For a General Currrent Distribution J
A J ' '4 '
jk r r
v
er r dv
r r
BBBBBBBBBBBBB B
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB B
'
Introduce the Dyadic Green's Function
( , ')4 '
jk r re
G Ir r
r r
BBBBBBBBBBBBB B
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB B
A J ' ( , ') 'v
r r G dv r rBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
2 2
A for a dipole directed in p direction located at r'
can be obtained by solving the Differential Equation
( )A = ( ') ( ') ( ')
p
pk Il x x y y z z
BBBBBBBBBBBBBBBBBBBBBBBBBBBB
BBBBBBBBBBBBBB
Since the Magnetic Vector Potential due to
Current Distribution in a volume ' can be found
by the integral of the scalar product
A J ' ( , ') 'v
v
r r G dv r rBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
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Hertzian Dipole
The Dyadic Green's function in regions a and b
can be written as ; p = a or b
0 0
( , ') 0 0
pxx
p pyy
p p pzx zy zz
G
G G
G G G
r rBBBBBBBBBBBBBB
First subscript is direction of Vector Potential
Second subscript is direction of Source
Indicate that a horizontal (x or y directed)
dipole gives rise to z directed potential.
2 2
Helmholtz equations for regions a and b:
In Region a where source is located
( ) ( , ') ( ') (1a)a
ak G I r r r rBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
2 2
In Region b
( ) ( , ') 0 (2a)b
bk G r rBBBBBBBBBBBBBB
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Hertzian Dipole
For an x-directed dipole, taking F.T
of (1a) and (2a) wrt x and y
2 2 2
Define:
a a x yk k k
2 2 2
Define:
b b x yk k k
22
2
22
2
In Region a ; z > 0 ; z' > 0
( ')
0
axxa a
azxa
G z zz
Gz
22
2
22
2
In Region b ; z < 0 ; z' > 0
0
0
bxxb
bzxb
Gz
Gz
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INVERSE FOURIER TRANSFORMFor p = a or b; in both Regions
00
1G ( , ') G ( , ') ( )
2
ppxxxx z z J d
r r
210
1 1G ( , ') cos G ( , ') ( )
2 -j
ppxxzx
x
z z J dk
r r
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Z-DIRECTED POTENTIALS IN REGIONS a AND b
') 2a10
z-directed potential in Region a
G ( , ') cos ( )2
a j z zzx Se J d
r r
' 210
z-directed potential in Region b
G ( , ') cos ( )2
bb j z j zbzx Se e J d
r r
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X-DIRECTED POTENTIALS IN REGIONS a AND b
| '| ')a a
00
x-directed potential in Region a
G ( , ') -j ( )4 | ' | 4
ajk j z zaxx
a
e eR J d
r r
r rr r
'
00
x-directed potential in Region b
G ( , ') (1 ) ( )4
b
j zb j zbxx
b
ej R e J d
r r
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PHYSICAL INTERPRETATION OF
First term is potential due to dipole in Infinite Homogeneous Medium
Second Term represents Reflection (Medium Interface Effect)
Second Term should decrease if dielectric properties of Medium b to approach those of Medium a
Potential should decay away from the wire
Gaxx
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NUMERICAL RESULTS & SOLUTION BEHAVIOR
Media Interface Effect for various Medium b Relative Dielectric
3 4 5 6 7 8 9 10 11 121
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9x 10
-8
Medium B Relative Dielectric ebr
el
Mag
nitu
de o
f In
tegr
al T
erm
in G
xxa
Media Interface Effect for various ebrel
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NUMERICAL RESULTS & SOLUTION BEHAVIOR
Magnitude of Potential for Dipole at z’=10 LEFT: below z’ RIGHT: above z’
2 3 4 5 6 7 8 91.2
1.4
1.6
1.8
2
2.2
2.4
2.6x 10
-8
z component of field point
Mag
nitu
de o
f G
xxa
Graph of Gxxa for a source located at z=10
11 12 13 14 15 16 17 181.9
2
2.1
2.2
2.3
2.4
2.5
2.6x 10
-8
z component of field point
Mag
nitu
de o
f G
xxa
Graph of Gxxa for a source located at z=10
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A Finite-Length Dipole
Source Definition Helmholtz Equation Boundary
Conditions Dyadic Green’s
Function F.T. Solution for
Dyadic Elements Sommerfeld
Integrals
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Finite-Length Dipole Linear Approximation
Assume q small (H >> L) Approximate by a Hertzian dipole at midpoint Multiplied by length L of dipole
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Finite-Length Dipole Linear Approximation, L=Dipole Length
| '| ')a a
00
x-directed potential in Region a
G ( , ') -j ( )4 | ' | 4
ajk j z zaxx
a
e eL R J d
r r
r rr r
') 2a10
z-directed potential in Region a
G ( , ') cos ( )2
a j z zzx L Se J d
r r
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Finite-Length DipoleLinear Approximation, L=Dipole Length
'
00
x-directed potential in Region b
G ( , ') (1 ) ( )4
b
j zb j zbxx
b
ej L R e J d
r r
' 210
z-directed potential in Region b
G ( , ') cos ( )2
bb j z j zbzx L Se e J d
r r
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BEHAVIOR OF SOLUTIONS
How does deviation from a straight line (amount of sag) affect potentials above and below the Catenary line
Compare to potentials created by straight line source
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NUMERICAL RESULT
Imaginary part of x-directed potential at z=7 Potential due to line source
= 1.4839e-007 +1.9810e-007iaaa
0 0.1 0.2 0.3 0.4 0.5 0.6 0.71.595
1.6
1.605
1.61
1.615
1.62
1.625
1.63
1.635
1.64x 10
-7
Sag of Catenary Wire
Rea
l Par
t of
Gxxa
Real Part of Gxxa vs Sag of Catenary; at z=7
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NUMERICAL RESULT
Imaginary part of x-directed potential at z=7 Potential due to line source
= 1.4839e-007 +1.9810e-007iaaa
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-4.795
-4.79
-4.785
-4.78
-4.775
-4.77
-4.765
-4.76
x 10-7
Sag of Catenary Wire
Imag
inar
y P
art
of G
xxa
Imaginary Part of Gxxa vs Sag of Catenary; at z=7
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COMPARISON OF CATENARY MODELLED BY DIPOLES TO STRAIGHT
LINE
Real and Imaginary parts of two potentials are observed separately
As amount of Sag is decreased: Re( ) Re( ) Im( ) Im ( )* At field points below the two
sources
axxG
aaaaxxG
aaa
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FUTURE WORK
Linear Approximation of Finite Length Dipole (H>>l )
Made due to time constraint A better approximation or Line
Integral
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FUTURE WORK
Earth is assumed Lossless Dielectric Could also be studied as Lossy
Dielectric Better understanding of how to
compare a problem with 2-D Geometry (Infinite Straight Line) to 3-D Geometry (Dipole)
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FUTURE WORK
Straight line originally analyzed with orientation shown
Potentials were z-directed
Coordinate system had to be changed for comparison with Catenary line
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ACKNOWLEDGEMENTS
Dr. Xiao-Bang Xu, SURE Advisor Dr. Daniel L. Noneaker, SURE
Program Director National Science Foundation 2006 SURE Students and
Graduate Assistant Karsten Lowe