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My Summer Vacation Integral Equations and Method of Moment Solutions to Waveguide Aperture Problems

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Title: My Summer Vacation Integral Equations and Method of Moment Solutions to Waveguide Aperture Problems


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
Maxwells Equations Wave Equation for vector
potential Enforce Boundary Conditions Separati
on 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 to xm 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
  • Kummers method
  • Gist subtract another series with known analytic
    solution from our series.
  • Accelerates the convergence

SURE Program 2005
14
Bromwichs Formula
It turns out that Bromwichs 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 doesnt 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 Adams 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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