Title: My Summer Vacation Integral Equations and Method of Moment Solutions to Waveguide Aperture Problems
1My Summer VacationIntegral Equations and Method
of Moment Solutions to Waveguide Aperture
Problems
Praveen A. Bommannavar Advisor Dr. Chalmers M.
Butler
2Outline
- Background Waveguide derivations
- Integral equations formulations
- Solution Methods and Results
- Applications and Future Work
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3Parallel Plate Guide Derivations
Assume vector potential in z direction Apply
Maxwells Equations Wave Equation for vector
potential Enforce Boundary Conditions Separati
on of Variables
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4Field Components in Parallel Plate Guide
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5Aperture 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
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6Field Components in two Regions of Guide
Excitation
Region a
Region b
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7Definition of Fourier Coefficients
Region a
Region b
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8Magnetic field in Regions-
Region a
Region b
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9Integral 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
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10Pulse Expansion
Make the following replacement
Definitions
x1
a D b
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11Pulse Expansion (cont.)
becomes
Treat this as an equation of N unknowns.
This is one good equation. How do we get (N-1)
more?
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12Point 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
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13Complications 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
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14Bromwichs 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.
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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
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16Matrix 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
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17Plot
- Dotted line Pulse Testing
- Solid line Point Matching
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18Other 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
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19Coupling
- 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.
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20More Plots
- Dotted line Pulse Testing
- Solid line Point Matching
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21More Plots
- Take data and determine current on strip.
- Dotted line My data
- Solid line Adams data
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22Applications / 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
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23Acknowledgements
- Dr. Butler
- Adam Schreiber
- Javier Schloemann
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24Questions About My Summer?
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