Broadband PLC Radiation from a

Power Line with Sag

Nan Maung, SURE 2006SURE Advisor: Dr. Xiao-Bang Xu

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

INTENDED MODEL

INTENDED MODELCatenary Wire Modeled by Finite-Length Dipoles

INTENDED MODEL

Number of dipoles

Dipole Midpoints

Wire Length

/10n

(2 1)20nx n

22 n

n

xz s h

L

constantny

THEORY

Solutions are derived based on: Superposition Helmholtz Equation Fourier Transform Techniques Sommerfeld Radiation Conditions

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

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

A Straight-Line Source

Located in upper Half-Space above Media Interface at z = 0

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

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

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

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

A Hertzian Dipole

Source Definition Helmholtz Equation Boundary

Conditions Dyadic Green’s

Function F.T. Solution for

Dyadic Elements Sommerfeld

Integrals

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

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

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

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

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

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

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

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

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

A Finite-Length Dipole

Source Definition Helmholtz Equation Boundary

Conditions Dyadic Green’s

Function F.T. Solution for

Dyadic Elements Sommerfeld

Integrals

Finite-Length Dipole Linear Approximation

Assume q small (H >> L) Approximate by a Hertzian dipole at midpoint Multiplied by length L of dipole

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

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

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

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

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

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

FUTURE WORK

Linear Approximation of Finite Length Dipole (H>>l )

Made due to time constraint A better approximation or Line

Integral

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)

FUTURE WORK

Straight line originally analyzed with orientation shown

Potentials were z-directed

Coordinate system had to be changed for comparison with Catenary line

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