1. Electromagnetic Field Evaluation of Dipole Antennas in Half-Space Robert Daniels Penn State University Clemson University SURE Program Advisor: Prof Xiao Bang Xu

2. Outline Background Information The Sommerfeld problem for dipole antennas Exact Image Theory and the Sommerfeld problem Applications Results

3. Background Infinitesimal Dipole Arbitrary Length Linear Dipole Diffracted and Direct Field Portions Radiation Pattern of Dipole Antenna Numerical Integration (e.g. Gaussian Quadrature)

4. Calculating the Electromagnetic Field Strength of Infinitesimal Dipoles Radiating in Half-Space The situation Why this calculation is important The first solution: Arnold Sommerfeld

5. Numerical Integration in Sommerfeld Solution Example integrals in diffracted portion of solution Several methods have been devised to evaluate integrals of this type  Note that these integrals have no closed form solution so they must be evaluated numerically

6. Four Main Problems with these Solutions (1) Dispute between several asymptotic techniques and their advantages (2) Lack of universal solution for all values of distance between source and observation (3) Slow convergence of numeric calculation in integrals (4) In some instances, asymptotic techniques still don’t converge

7. Introducing Exact Image Theory Introduced in the early 1980s Method of representing diffracted field sources from complex images Example: (image sources for perfect conductor)

8. EIT in Sommerfeld Problem EIT improves the convergence of Sommerfeld Integrals Uses a Laplace transform on reflection coefficients Compact form of diffracted portion of field

9. Numerical Integration and EIT Ability to numerically integrate significantly improved Gaussian Quadrature necessary

10. Results for Infinitesimal Dipole Note the improved results in calculation time

11. Soil Moisture Content and Dipole Radiation The different moisture contents in soil contribute a great deal to electric field calculation

12. EIT in Linear Dipole Antennas Superposition of Infinitesimal Dipoles

13. Arbitrary Length and Orientation Dipole Antennas We are able to construct plots like this:

14. Radiation Patterns Notice the impact of different soil moisture content levels on radiation pattern for a vertical half-wave dipole

15. Conclusions EIT Applied to Sommerfeld Integrals provides various advantages Differences in the earth boundary can contribute a great deal to antenna radiation

16. Future Work Exact representation of current distribution on linear antennas instead of sinusoidal approximation Analysis of new antenna configurations that can be derived from infinitesimal dipoles

17. Acknowledgements Professor Xiao Bang Xu Professor Daniel Noneaker Applied Electromagnetics Group

18. Evaluating Highly Oscillatory Integrals if distance between source and observation is large  standard asymptotic techniques (steepest descent method) If distance between source and observation relatively small  Alternate asymptotic techniques available in literature (methods in dispute)