Integro-Differential Equation Solution Method for Current on a Thin Wire Yuriy Goykhman Adam Schreiber Advisor: Dr Butler

Outline I. Derivation of the Equation II. Properties of the Kernel III. Solution Method IV. Results V. Conclusions and Future Work

Relating A and Φ to E

Definition of a Kernel

Deriving the Integral Equation I Total axial current is independent:

Deriving the Integral Equation II Plug in equations for A and into to obtain:

Properties of the Kernel Even function Similar to the delta function

Graph of the Kernel

Integration of the Kernel Singular integral is difficult to integrate numerically Singularity extraction 1) find a function that we can integrate that has the same limit as K(z-z’) at singularity 2) Subtract and then add back the integrated form 3) The result is very smooth and can be easily integrated numerically

Resultant Function

Solution Method Break down the current into triangles

Breaking I into Triangles

I as a Sum of Triangles

Getting N-equations N unknowns Need N equations to solve for I Use pulses to get N equations otherwise

Getting the Matrix Equation Right Hand Side of the Equation: Ohm’s Law:

Computing Matrix Elements

Properties of the Matrix N by N square matrix Symmetric Only need to know 1 row or column

Solving for I Examples 1) Uniform Excitation at h =.25λ 2) E = z excitation at h =.25λ

E = 1 excitation

E = z excitation

Conclusions and Future Work Reliable method for accurately calculating a current distribution on a wire subject to various excitations Extend to analysis of tuned antennas to achieve properties peculiar to a specific application, e.g., broadband communications antennas Extend to curved wires Antenna Arrays

Acknowledgements Dr Butler Mike Lockard Dr Noneaker Dr Xu Clemson University NSF

Questions???