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

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

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

4 Field Components in Parallel Plate Guide SURE Program 2005

5 Aperture Method – Integral Equation Formulation SURE Program 2005 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

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 SURE Program 2005 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

10 Pulse Expansion SURE Program 2005 Make the following replacement: Definitions: a  b x 1

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

12 Point Matching/ Pulse Testing SURE Program 2005  Point Matching - enforce this equation at N points  These N points happen to be the points already defined  x in previous equation just becomes x m  Pulse Testing – integrate the equation from x m –  to  x m +   These N points happen to be those points already defined We have 2 options:

13 Complications in point matching SURE Program 2005 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

14 Bromwich’s Formula SURE Program 2005 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, 2  ). 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.

15 SURE Program 2005  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

16 Matrix Equation SURE Program 2005  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

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

18 SURE Program 2005 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

19 SURE Program 2005 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.

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

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

22 SURE Program 2005 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

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

24 SURE Program 2005 Questions About My Summer?