1. Broadband PLC Radiation from a Power Line with Sag Nan Maung, SURE 2006 SURE Advisor: Dr. Xiao-Bang Xu

2. 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

3. INTENDED MODEL

4. INTENDED MODEL Catenary Wire Modeled by Finite-Length Dipoles

5. INTENDED MODEL Number of dipoles Dipole Midpoints

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

7. 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

8. 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

9. A Straight-Line Source Located in upper Half-Space above Media Interface at z = 0

10. SOMMERFELD INTEGRALS (Coming back to Spatial Domain)

11. Predicted behavior of Solutions Based 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

12. NUMERICAL RESULTS & SOLUTION CHECK Real and imaginary parts of Correction Integral vs. Relative Dielectric of Medium b Observed at z=15

13. NUMERICAL RESULTS & SOLUTION CHECK Real and Imaginary parts of Correction Integral vs. Relative Dielectric of Medium b Observed at z = 7

14. A Hertzian Dipole Source Definition Helmholtz Equation Boundary Conditions Dyadic Green’s Function F.T. Solution for Dyadic Elements Sommerfeld Integrals

15. Hertzian Dipole

18. INVERSE FOURIER TRANSFORM For p = a or b; in both Regions

19. Z-DIRECTED POTENTIALS IN REGIONS a AND b

20. X-DIRECTED POTENTIALS IN REGIONS a AND b

21. 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

22. NUMERICAL RESULTS & SOLUTION BEHAVIOR Media Interface Effect for various Medium b Relative Dielectric

23. NUMERICAL RESULTS & SOLUTION BEHAVIOR Magnitude of Potential for Dipole at z’=10 LEFT: below z’ RIGHT: above z’

24. A Finite-Length Dipole Source Definition Helmholtz Equation Boundary Conditions Dyadic Green’s Function F.T. Solution for Dyadic Elements Sommerfeld Integrals

25. Finite-Length Dipole Linear Approximation Assume q small (H >> L) Approximate by a Hertzian dipole at midpoint Multiplied by length L of dipole

26. Finite-Length Dipole Linear Approximation, L=Dipole Length

28. 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

29. NUMERICAL RESULT Imaginary part of x-directed potential at z=7 Potential due to line source = 1.4839e-007 +1.9810e-007i

30. NUMERICAL RESULT Imaginary part of x-directed potential at z=7 Potential due to line source = 1.4839e-007 +1.9810e-007i

31. 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

32. FUTURE WORK Linear Approximation of Finite Length Dipole (H>> l ) Made due to time constraint A better approximation or Line Integral

33. 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)

34. FUTURE WORK Straight line originally analyzed with orientation shown Potentials were z- directed Coordinate system had to be changed for comparison with Catenary line

35. 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