1

My Summer Vacation

Integral Equations and Method of Moment

Solutions to Waveguide Aperture Problems

Praveen A. BommannavarAdvisor: Dr. Chalmers M. Butler

2SURE 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

x

y

z

Assume vector potential in z direction:

Apply Maxwell’s Equations:

Wave Equation for vector potential:

Enforce Boundary Conditions:

Separation of Variables:

( )ˆ( , ) a ,x z x z=A z

( ) ( )0, , 0z zE z E h z= =

( )2 2 a=0kÑ +

x 0

xh,me

x

z

( ) ( ) ( )a ,x z X x Z z=

4

( ) ( )0 01

, cosq q qjkz jkz z zhx q q

q

E x z a e a e a e a e xpg g¥

+ - - + - -

=

= + + +å

( ) ( )0 0

1

1, cosq q qjkz jkz z z

hy q qq q

a a kH x z e e j a e a e xpg g

h h h g

+ - ¥- + - -

=

= - + -å

qq zjkg = ( ) ( )

( ) ( )

2 22 2

2 22 2

,

,q

q qh h

zq qh h

k kk

j k k

p p

p p

ìï - >ïïï= íïï - - <ïïî

Field Components in Parallel Plate Guide

x 0

xh,me

x

z

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5

ixE

z 0

,me

z x 0

xhregion a region b

,meab

Aperture Method – Integral Equation Formulation

SURE Program 2005

Approach:

Determine general field expressions in both regions

Use Fourier Techniques to find coefficientsCoefficients will be in terms of

Apply Continuity of H to arrive at an Integral Equation

( )AE x

6

Field Components in two Regions of Guide

0i jkzxE E e-=

( )0 0

1

, cos , 0q qjkz jkz zhx q

q

E x z E e ae ae x zpg¥

-

=

= + + £å

( ) 0 0

1

1, cos , 0q qjkz jkz z

hy qq q

E a kH x z e e j a e x zpg

h h h g

¥-

=

= - + £å

Excitation

Region a

( )0

1

, sin ( ) sinh ( )cos , 0q qjkhx q q

q

E x z be k z be z x zpg g¥

- -

=

= - + - £ £ål ll l l

Region b

( )0

1

1, cos ( ) cosh ( )cos , 0q qjk

hy q qq q

j kH x z be k z j b e z x zpg g

h h g

¥- -

=

= - - - £ £ål ll l l

ixE

z 0

,me

z x 0

xhregion a region b

,meab

SURE Program 2005

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Definition of Fourier Coefficients

( ) ( )

0 , 0

,0 ,

0 ,

Ax

x a

E x E x a x b

b x h

ìï < <ïïïï= < <íïïï < <ïïî

0 01

0 , 0

cos

0 ,

qh

q

x a

E a x

b x h

pp¥

=

ìï < <ïïïï+ + = íïïï < <ïïî

å

( )0 0

1 Ab

aa E E x dx

h= - + ò

( )2

cosAb

qhq

aa E x xdx

hp= ò

( )0

1

0 , 0

sin sinh cos ,

0 ,

q Aqjkhq q

q

x a

be k be x E x a x b

b x h

pg g¥

- -

=

ìï < <ïïïï- - = < <íïïï < <ïïî

ål ll l

Region a

Region b

( )0 sin

Ajk b

a

eb E xdx

h k= - ò

l

l( )

2cos

sinh

qA

bqhq

aq

eb E x xdx

h

gp

g= ò

l

l

ixE

z 0

,me

z x 0

xhregion a region b

,meab

SURE Program 2005

8

Magnetic field in Regions-

Region a

Region b

( ) ( )

( )( )

00

1

1 1,

2 1cos cos , 0q

A

A

bjkz jkz

ya

bq qzh h

aq q

EH x z e E E x dx e

hk

j E x xdx e x zh

p pg

h h

h g

-

¥

=

æ ö÷ç= + - ÷ç ÷çè ø

+ £

ò

å ò

( ) ( )

( )( )1

cos ( ),

sin

cosh ( )2cos cos ,

sinh

0

A

A

b

ya

bqq q

h ha

q q q

j k zH x z E xdx

h k

zkj E x xdx xh

z

p p

h

g

h g g

¥

=

-= -

--

£ £

ò

å ò

ll

l

l

l

SURE Program 2005

9

Integral Equation for Aperture Electric Field

( ) 0

1

2cos cos 2 ,

sin sin

zq

q q

Ajkjkb

q qh h

aq z z

Ej e k eE x j x x dx a x b

h k h k kp p

h h h

¥

=

ì üï ïï ïï ï¢ ¢ ¢- = - < <í ýï ïï ïï ïî þåò

ll

l l

SURE Program 2005

Method of Moment Solution:

Expand into N pulses

Enforce the equation at N points (Point Matching) ORIntegrate the new expression over 1 pulse (Pulse Testing)

Set up a Matrix Equation Matrix will be square

Solve for unknown column matrix

( )AE x

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Pulse Expansion

( )

1

( )AN

n nn

E x E xp=

= å

SURE Program 2005

Make the following replacement:

Definitions:

2 21 ,( )

0 ,n n

n

x x xx

otherwisep

D Dì - < < +ïïï= íïïïî

b aN-

D =a b

( 1/ 2)nx a n= +D -

x1

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Pulse Expansion (cont.)

SURE Program 2005

( ) 0

1

2cos cos 2 ,

sin sin

zq

q q

Ajkjkb

q qh h

aq z z

Ej e k eE x j x x dx a x b

h k h k kp p

h h h

¥

=

ì üï ïï ïï ï¢ ¢ ¢- = - < <í ýï ïï ïï ïî þåò

ll

l l

0

1

... sin ( ) sin ( ) cos 22 2

Nq q qh h hn n n

n

EE x x xp p p

h=

ì üé ùD Dï ïï ï+ - - = -ê úí ýï ïê úë ûï ïî þå

becomes

This is one good equation. How do we get (N-1) more?

Treat this as an equation of N unknowns.

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Point Matching/ Pulse Testing

SURE Program 2005

Point Matching - enforce this equation at N pointsThese N points happen to be the points already definedx in previous equation just becomes xm

Pulse Testing – integrate the equation from xm – toxm +

These N points happen to be those points already defined

We have 2 options:

0

1

... sin ( ) sin ( ) cos 22 2

Nq q qh h hn n n m

n

EE x x xp p p

h=

ì üé ùD Dï ïï ï+ - - = -ê úí ýï ïê úë ûï ïî þå

0

1

(...) sin ( ) sin ( ) sin ( ) sin ( ) 22 2 2 2

Nq q q qh h h hn n n m m

n

EE x x x xp p p p

h=

ì üé ùé ùD D D D Dï ïï ï+ - - + - - = -ê úê úí ýï ïê úê úë ûë ûï ïî þå

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Complications in point matching

SURE Program 2005

We must pay attention to the convergence of the infinite sum

1

1sin ( ) sin ( ) cos

sin 2 2

zq

q q

jkq q qh h hn n m

q z z

ex x x

k k qp p p

¥

=

æöé ùD D÷ç + - -÷ê úç ÷÷ç ê úè øë ûå

l

l

In the limit that q goes to infinity, this has the form:

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1sin ( )

2qh n m

q

x xq

p¥

=

æ öé ùD÷ç + +÷ê úç ÷÷ç ê úè øë ûå

This converges very slowly – computationally “annoying”Kummer’s methodGist: subtract another series with known analytic solution from our series. Accelerates the convergence

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Bromwich’s Formula

SURE Program 2005

It turns out that Bromwich’s Formula will fix our problem:

22

sin 1sin sin ln(2sin ),0 2

1 4 2n

nn

q qq q q p

¥

=

= - < <-å

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.

[ ]21

1sin

q

qq

q¥

=

æ ö÷ç ÷ç ÷÷çè øå 22

sin 1sin sin ln(2sin )

1 4 2n

nn

q qq q

¥

=

- + --å

15SURE 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

31

1sin ( )

2qh n m

q

x xq

p¥

=

æ öé ùD÷ç + +÷ê úç ÷÷ç ê úè øë ûå

16

Matrix Equation

SURE Program 2005

We now have N equations and N unknowns. So we solve this in a matrix equation.

0

11 1 1

1 0

2

2

n

m mn n

Ea a E

a a E E

h

h

æ ö- ÷ç ÷çæ ö ÷æ ö ç ÷÷ç ç÷ç ÷÷ç÷ çç ÷÷ç÷ ç ÷÷ç ÷ç ç ÷÷ç =÷ç ÷÷ çç ÷ç ÷÷ çç ÷ ÷ç ÷ çç ÷ ÷÷ç ç÷ç ÷÷ -ç ÷è ø ç ÷è ø ç ÷ç ÷÷çè ø

K

M O M M M

L

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

17SURE Program 2005

Plot

Dotted line: Pulse TestingSolid line: Point Matching

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Other Waveguide Configurations

ixE

z 0

,

z x 0

xhregion a region b

, ab

ixE

z 0

,

z x 0

xhregion a region c

, ab

cd

region b

ixE

z 0

,

z x 0

xhregion a region b

, a

b

Easier than with short: fields have same form

Matrix is coupled3 regions; must enforce H twice

Matrix is coupled2 regions, but still must enforce H twice

19SURE Program 2005

Coupling

ixE

z 0

,

z x 0

xhregion a region c

, ab

cd

region b 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.

11 1 11 1

1 1

11 1 11 1

1 1

n n

m mn m mn

n n

m mn m mn

a a c c

a a c c

b b d d

b b d d

æ ö÷çæ ö æ ö÷ç ÷÷ ÷ç çç ÷÷ ÷ç çç ÷÷ ÷ç çç ÷ ÷ç çç ÷ ÷ç çç ÷ ÷ç ç÷ ÷çç ç÷ ÷ç ÷ ÷ç ççè ø è øççççççççæ ö æ öç ÷ ÷ç çç ÷ ÷ç çç ÷ ÷ç çç ÷ ÷ç çç ÷ ÷ç ç÷ ÷çç ç÷ ÷çç ç÷ ÷ç ÷ ÷ç çç ÷ ÷ç ç÷ ÷çè ø è øè ø

K K

M O M K M O M

L L

M O M

K K

M O M L M O M

L L

÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷

20SURE Program 2005

More Plots

ixE

z 0

,

z x 0

xhregion a region b

, ab Dotted line: Pulse Testing

Solid line: Point Matching

21SURE Program 2005

More PlotsTake data and determine current on strip.Dotted line: My dataSolid line: Adam’s data

ixE

z 0

,

z x 0

xhregion a region b

, a

b

22SURE Program 2005

Applications / Future Work

Waveguides can model hallways in a building or cavities for other applications

Future Work

More complex geometriesCoaxial, rectangular, etc.

Slotted plates on guideRadiation Patterns

23SURE Program 2005

Acknowledgements

Dr. Butler

Adam Schreiber

Javier Schloemann

24SURE Program 2005

Questions About My Summer?