1. Modeling a Dipole Above Earth Saikat Bhadra Advisor : Dr. Xiao-Bang Xu Clemson SURE 2005

2. Overview Objective Problem Background & Theory Results Problems in the EIT Model Concluding Remarks

3. Objective Accurate modeling of a dipole  Linear Antenna  Lossy Earth  Material Properties Scientific Model  K. Sarabandi, M. D. Casciato, and I. Koh Efficient Calculation of the Fields of A Dipole Radiating Above an Impedance Surface

4. Solving Electromagnetic Problems The Emag Bible : Maxwell’s Equations  Available in integral and differential forms Vector Potential  Links Magnetic and Electric Fields

5. Antenna Environment Inhomogeneous Materials Time Varying Non-flat & non-Euclidean surfaces Location : Austin, TX Layered Materials

6. Simplifications Simplify math and assume :  Flat Earth Model  Two Layers Upper half space – “air” Lower half space – lossy earth  Euclidean (rectangular) geometry  Infinitesimal Vertical Dipole Superposition to extend to finite dipoles

7. Electric Field In this type of problem, two fields are involved  Direct Electric Fields Fields due to antenna radiating Solution in closed form & well documented  Diffracted Electric Fields Fields from antenna that are reflecting off the lower surfaces Subject of research since 1909 Observation Point Dipole Impedance Half Space Free Space

8. Original Solution – Diffracted Fields Arnold Sommerfeld (1909) Sommerfeld Integrals  Non-analytic  Numerical integration difficult  Requires asymptotic techniques  Valid for certain regions  Convergence difficult

9. Original Solution – Diffracted Fields cont’d

10. Exact Image Theory Solution Sarabandi, Casciato, Koh (2002) Source Equation :

11. Separate diffracted and direct components Reflection Coefficients transformed using Laplace transform Bessel function identities EIT Formulation

12. EIT Solution – Diffracted Fields Observation Point Dipole Impedance Half Space Free Space Direct Diffracted

13. EIT Solution Integral Advantages  Rapidly Decays  Non-Oscillatory  Easy numerical evaluation after exchange of integration and differentiation

14. Exact Image Theory Observation Point Dipole Impedance Surface Free Space Direct Diffracted Dipole

15. Finite Length Dipoles Sarabandi’s model uses infinitesimal dipole  Finite dipole can be approximated by a sum of infinitesimal dipoles Superposition Principle

16. Calculating Input Impedance Induced EMF Method : Current distribution assumed to sinusoidal  Transmission line approximation  Inaccurate when dipole comes close to half space

17. Numerical Techniques Gaussian Integration  Useful in many emag problems  Handles singular integrands better  More accurate than rectangular, trapezoidal, and Simpson’s rule Integral Truncation  Can’t numerically evaluate an infinite integral Vectorized Code

18. Results Computational time varies with antenna location Frequency independence Asymptotically approaches original antenna impedance

19. Results cont’d

20. Problems of the EIT Model Recall the breakdown of electric field into diffracted and direct components Diffracted fields should go to zero if the half- space is removed  There is no longer any surface for waves to bounce off of  Numerical Results disagree Currently finding theoretical errors of the model

21. Problems of the EIT Model cont’d

22. Concluding Remarks EIT model could be promising but problems need to be solved Research Applications  Antenna Design  Integral Equations & Numerical Methods

23. Future Work Solve the EIT model problems Extend the problem to dipoles of arbitrary orientation Develop more accurate model of current distribution  Investigate different source models

24. Acknowledgments Dr. Xu Dr. Noneaker

25. Questions?

26. Environmental Variables Time varying Inhomogeneous Materials (x,y)  Water  Grass  Concrete Layered Materials (z)  Trees, Grass, Soil Non-flat surfaces Amorphous (non-Euclidean) geometries Mutual Coupling