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Praveen A. Bommannavar Advisor: Dr. Chalmers M. Butler

My Summer Vacation Integral Equations and Method of Moment Solutions to Waveguide Aperture Problems. Praveen A. Bommannavar Advisor: Dr. Chalmers M. Butler. Outline. Background: Waveguide derivations Integral equations – formulations Solution Methods and Results Applications and Future Work.

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Praveen A. Bommannavar Advisor: Dr. Chalmers M. Butler

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  1. My Summer VacationIntegral Equations and Method of Moment Solutions to Waveguide Aperture Problems Praveen A. Bommannavar Advisor: Dr. Chalmers M. Butler

  2. Outline • Background: Waveguide derivations • Integral equations – formulations • Solution Methods and Results • Applications and Future Work SURE Program 2005

  3. Parallel Plate Guide Derivations Assume vector potential in z direction: Apply Maxwell’s Equations: Wave Equation for vector potential: Enforce Boundary Conditions: Separation of Variables: SURE Program 2005

  4. Field Components in Parallel Plate Guide SURE Program 2005

  5. Aperture Method – Integral Equation Formulation • Approach: • Determine general field expressions in both regions • Use Fourier Techniques to find coefficients • Coefficients will be in terms of • Apply Continuity of H to arrive at an Integral Equation SURE Program 2005

  6. Field Components in two Regions of Guide Excitation Region a Region b SURE Program 2005

  7. Definition of Fourier Coefficients Region a Region b SURE Program 2005

  8. Magnetic field in Regions- Region a Region b SURE Program 2005

  9. Integral Equation for Aperture Electric Field • Method of Moment Solution: • Expand into N pulses • Enforce the equation at N points (Point Matching) OR • Integrate the new expression over 1 pulse (Pulse Testing) • Set up a Matrix Equation • Matrix will be square • Solve for unknown column matrix SURE Program 2005

  10. Pulse Expansion Make the following replacement: Definitions: x1 a D b SURE Program 2005

  11. Pulse Expansion (cont.) becomes Treat this as an equation of N unknowns. This is one good equation. How do we get (N-1) more? SURE Program 2005

  12. Point Matching/ Pulse Testing We have 2 options: • Point Matching - enforce this equation at N points • These N points happen to be the points already defined • x in previous equation just becomes xm • Pulse Testing – integrate the equation from xm – D/2 toxm + D/2 • These N points happen to be those points already defined SURE Program 2005

  13. Complications in point matching We must pay attention to the convergence of the infinite sum • In the limit that q goes to infinity, this has the form: • This converges very slowly – computationally “annoying” • Kummer’s method • Gist: subtract another series with known analytic solution from our series. • Accelerates the convergence SURE Program 2005

  14. Bromwich’s Formula It turns out that Bromwich’s Formula will fix our problem: • Subtract, then add back on… • Another complication: This identity has a VERY narrow region of convergence (0, 2p). So we have to go back to our formula and fix it up and add conditions so that our equation takes this into account. This is mostly a coding complication. SURE Program 2005

  15. Pulse testing doesn’t have this problem of convergence. The reason for this is that we integrated one more time and so in the limit that q goes to infinity, our terms have the form: • The extra q in the denominator saves the day! This series converges rapidly. • Moral: Do pulse testing whenever possible SURE Program 2005

  16. Matrix Equation • We now have N equations and N unknowns. So we solve this in a matrix equation. • Used MATLAB to calculate unknown matrix and to plot • We expect the field near the fins to spike up – property of edges in electromagnetics; also expect symmetry SURE Program 2005

  17. Plot • Dotted line: Pulse Testing • Solid line: Point Matching SURE Program 2005

  18. Other Waveguide Configurations • Easier than with short: fields have same form • Matrix is coupled • 3 regions; must enforce H twice • Matrix is coupled • 2 regions, but still must enforce H twice SURE Program 2005

  19. Coupling • Coupling occurs when we have 2 or more apertures, each having an effect on themselves as well as the other aperture(s) • This is reflected in the matrix by different regions (sub-matrices) • Matrices along the diagonal are the same as if there were only that aperture. The others are due to coupling. SURE Program 2005

  20. More Plots • Dotted line: Pulse Testing • Solid line: Point Matching SURE Program 2005

  21. More Plots • Take data and determine current on strip. • Dotted line: My data • Solid line: Adam’s data SURE Program 2005

  22. Applications / Future Work • Waveguides can model hallways in a building or cavities for other applications • Future Work • More complex geometries • Coaxial, rectangular, etc. • Slotted plates on guide • Radiation Patterns SURE Program 2005

  23. Acknowledgements • Dr. Butler • Adam Schreiber • Javier Schloemann SURE Program 2005

  24. Questions About My Summer? SURE Program 2005

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