Project Background My project goal was to accurately model a dipole in the presence of the lossy Earth. I used exact image theory developed previously to numerically determine the electric fields of a infinitesimal dipole, used superposition to extend the calculations to a finite dipole, and then used the induced EMF method to determine input impedance. Model Overview Our scientific model assumed :  Flat Earth with two layers  Upper half space – “air”  Lower half space – lossy earth  Infinitesimal Vertical Dipole In this type of problem, two fields are involved :  Direct Electric Fields - fields due to antenna radiating  Diffracted Electric Fields - fields from antenna that are reflecting off the lower surfaces The solution of the diffracted fields has been a subject of research since Sommerfeld Solution Sommerfeld developed a solution involving Sommerfeld integrals in These integrals are difficult to evaluate numerically because they are :  Non-analytic  Oscillatory  Require asymptotic techniques  Valid for certain regions  Exhibit convergence problems Exact Image Theory This theory was developed at the University of Michigan in 2002 by Sarabani, Casciato, and Koh. First, the electric field components are broken down into diffraced and direct components. The reflection coefficients present in the original Sommerfeld formulation are then modified using Laplace transforms. This transforms the expression into a summation of a single and double integral which is further simplified using Bessel function identities A single non-analytic integrand remains. Note that this theory is named exact image theory because after the field derivation, the diffracted fields can be interpreted as the result from a dipole image and an infinite array of dipole images at complex distances. EIT Integrand  Rapidly Decays  Non-Oscillatory  Easy numerical evaluation Finite Length Dipoles Finite dipoles can be approximated by a sum of infinitesimal dipoles using the superposition principle. For our purposes infinitesimal dipoles were defined to be approximately 0.01 wavelength. Input Impedance Induced EMF Method : Current distributed is assumed sinusoidal  Transmission line approximation  Holds fully for an infinitesimally thin dipole  Inaccurate when dipole approaches half space (~0.03 wavelength) Modeling a Dipole Above Earth Saikat Bhadra – SURE 2005 Dr. Xiao-Bang Xu Numerical Techniques Gaussian Integration Integral Truncation Vectorized Matlab Code EIT Problems As the lower half-space approaches the material properties of free space, diffracted fields should approach zero. This effectively removes the lossy earth from the model and thus, there is no surface for the radiated fields to bounce off of. However, numerical results derived from the University of Michigan EIT model do not show that diffracted fields approach zero. Conclusion  Computational time varies with antenna location  Frequency independence  Impedance asymptotically approaches original antenna impedance  EIT model could be promising but problems need to be solved Research Applications  Antenna Design  Integral Equations & Numerical Methods References K. Sarabandi, M. D. Casciato, and I. S. Koh, “Efficient calculation of the fields of a dipole radiating above an impedance surface,” IEEE Trans. Antennas Propagat., vol 50, pp , Sept Future Work  Solve the EIT model problems  Extend the problem to dipoles of arbitrary orientation  Develop MOM techniques and combine with a source model for a more accurate current distribution Acknowledgements  Dr. Xu  Dr. Noneaker Observation Point Dipole Impedance Half Space Free Space Direct Diffracted