PUBLICATIONS A1 11 00989856

Montgomery, James Pa/Electromagnetlc bou QC100 .U556 V164;1979 C.1 NBS-PUB-C 1979

Q NBS MONOGRAPH 164

U.S. DEPARTMENT OF COMMERCE / National Bureau of Standard^

Electromagnetic Boundary-Value Problems

Based Upon a Modification of

Residue Calculus and

Function Theoretic Techniques

QC.

100
NATIONAL BUREAU OF STANDARDS

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The National Bureau of Standards was reorganized, effective April 9, 1978.
Electromagnetic Boundary-Value Problems '^^ ' Based Upon a Modification of Residue Calculus ACc O-^fC

and Function Theoretic Techniques ©QJ CO

James Patrick Montgomery ^ '^^'^ David C. Chang C

^

Center for Electronics

and Electrical Engineering

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80302

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

|j|C^ NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued June 1 979
Library of Congress Catalog Card Number: 79-600072

National Bureau of Standards Monograph 164

Nat. Bur. Stand. (U.S.), Mono. 164, 183 pages(June 1979)

CODEN: NBSMA6

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1979

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

Stock No. OO3-O03-O2O75-3 Price $5.00
I PREFACE

The work reported in this monograph is the result of an intensive collahorative effort

between the authors and KBS scientists. The contract to the University of Colorado was

initiated because of the need for an accurate, well-defined in-situ method of measiiring the

complex dielectric properties of the earth in the frequency range of 0.1 to 12. k GHz. The

work is part of a larger National Bureau of Standards project that is systematically determining the measurement capability (accuracy and precision) of selected measxirement techniques

presently used in high frequency and microwave remote sensing, as well as developing well

defined data bases and new measurement techniques for partic\ilar application of high national

need.

The particular approach presented in this monograph was chosen primarily because of the

availability of TEM type antennas that have a very large bandwidth, and because of the

desirability of making the measurements with the antennas very close (in the near field) to

the material being measured.

The needed open region mathematical techniques are systematically developed through

the solution of selected closed region problems before extension to the open region problems.

The work presents powerful new tools that can be applied to either open 'or closed

region problems and, hopefully, lays the basic theoretical ground work for the development

of the needed well-defined, accurate techniques of in-situ measurement of earth type

materials

.

The application of the presented theoretical ideas to experimental measurement systems

is currently being actively pursued.

W.E. Little

National Engineering Laboratory

National Biureau of Standards

Boulder, Colorado 80302

iii

CONTENTS

PART I

SOLUTION OF CLOSED REGION PROBLEMS

PAGE

1. INTRODUCTION (PART I) 1

2. FOUNDATION OF THE MODIFIED RESIDUE CALCULUS TECHNIQUE 4

1. Introduction 4

2. The Canonical Problem 4

3. Formulation and Solution of Composite Problems 11

3. THE TRIFURCATED WAVEGUIDE 15

1. Introduction 15

2. Formulation of the Equations 15

3. Asymptotics 18

4. Truncation of the Equations 21

5. Dielectric Loading 22

6. The Scattered Fields 25

7. Numerical Results 26

7.1 Introduction 26

7.2 The Trifurcated Waveguide 26

7.3 The Dielectrically Loaded Trifurcated Waveguide 30

4. THE N-FURCATED WAVEGUIDE 34

1. Introduction 34

2. Formulation of the Equations 34

3. Asymptotics 37

4. Truncation of the Equations 42

5. Dielectric Loading 45

6. The Scattered Fields 48

7. Numerical Results 48

7.1 Introduction 48

7.2 The N-Furcated Waveguide ' 49

7. 3 The Dielectrically Loaded N-Furcated Waveguide 51

5. OTHER CLOSED REGION PROBLEMS 53

1. Introduction 53

2. TE Eigenvalues of Ridged Waveguide 53

3. Scattering by a Dielectric Step 58

4. Dielectric Loaded N-Furcated Waveguide 58

6. CONCLUSIONS (PART I) 60

PART II

SOLUTION OF OPEN REGION PROBLEMS

CHAPTER PAGE

7. INTRODUCTION (PART II) 64

8. FOUNDATION OF THE MODIFIED FUNCTION THEORETIC TECHNIQUE 65

1. Introduction 65

2. The Canonical Problem 65

v
CHAPTER PAGE

2.1 Introduction 65

2.2 The Electric Wall Case 67

2.3 The Magnetic Wall Case . 80

3. Formulation and Solution of Composite Problems 83

9. A PARALLEL PLATE WAVEGUIDE RADIATING INTO A HOMOGENEOUS HALF-SPACE 88

1. Introduction 88

2. Formulation of the Equations 88

3. Numerical Solution 91

4. Numerical Results 92

10. A FINITE PHASED ARRAY .102

1. Introduction 102

2. Formulation of the Equations 102

2.1 Introduction 102

2.2 The Electric Symmetry Wall 102

2.3 The Magnetic Symmetry Wall 111

3. Numerical Results 115

3.1 The Electric Wall Case 115

3.2 The Magnetic Wall Case 121

3.3 Superposition of the Results for an Electric and a Magnetic Wall— 126

11. REMOTE SENSING OF THE EARTH USING PARALLEL PLATE WAVEGUIDES 132

1. Introduction 132

2. Formulation of the Equations 133

2.1 Introduction 133

2.2 The Electric Symmetry Case 133

2.3 The Magnetic Symmetry Case 136

3. Numerical Results 138

3.1 Introduction 138

3.2 The Electric Wall Case 138

3.3 The Magnetic Wall Case 143

3.4 Superposition of the Results 143

12. OTHER OPEN REGION PROBLEMS 148

1. Introduction 148

2. Flanged Waveguide Radiating into a Half Space 149

3. Scattering by a Thick Semi-Infinite Plane 153

4. Radiation from a Slot in a Waveguide Wall 157

13. CONCLUSIONS (PART II) 157

14. COMMENTS AND FINAL SUMMARY 159

BIBLIOGRAPHY 161

APPENDIX A: The Edge Condition and the Asymptotic Behavior of T(oj) 163

APPENDIX B: Asymptotic Behavior of the Perturbation Sum for the E-Plane Step— 165

APPENDIX C: Asymptotic Behavior of the Perturbation S\am for the Trifurcated

Waveguide 169

APPENDIX D: Evaluation of the Infinite Product for the Electric Wall Case 171

APPENDIX E: The Canonical Problem with a Magnetic Symmetry Boundary 174

APPENDIX F: The Trifurcated Waveguide with a Magnetic Wall 176

vi
ELECTROMAGNETIC BOUNDARY-VALUE PROBLEMS BASED UPON

A MODIFICATION OF RESIDUE CALCULUS AND FUNCTION THEORETIC TECHNIQUES

James Patrick Montgomery*

David C. Chang**

ABSTRACT

The solution of a number of electromagnetic problems, in both closed and

open systems, using the modified residue calculus and functional theoretic

techniques is presented.

The solutions start with known closed region problems and then are extended

to new closed region problems and finally to several open region problems.

Specific problems considered for the closed region are: l) the trifurcated

waveguide; 2) the dielectrically loaded trifurcated waveguide; 3) the N-furcated

waveguide; k) the dielectrically loaded N-furcated waveguide; 5) determination

of the Eigenvalues of ridged waveguide; and 6) scattering by a dielectric stop.

Open region problems considered are: l) a parallel plate radiating into a homogeneous half-space; 2) a finite phased array; 3) remote sensing of the earth

using parallel plate waveguides; h) a flanged waveguide radiating into a halfspace; 5) scattering by a thick, semi-infinite plane; and 6) radiation from a slot in a waveguide wall.

Some suggested extension of the techniques to other types of problems is

also included.

Key words: Closed systems; electromagnetic problems; fionctional theoretic

techniques; modified residue calculus; open systems; remote sensing.

PART I

SOLUTION OF CLOSED REGION PROBLEMS

CHAPTER 1. INTRODUCTION

The first part of this monograph is concerned with the analysis of closed region waveguide junction problems. This allows simplification of the techniques to be used since no branch cuts are involved in the spectral representation of the fields. However, the analysis

can be extended to open region problems in a logical manner. This is the subject of the

second part of this monograph.

The problems of interest will be confined to two-dimensional geometries for which a strictly TE or TM solution is possible.

Generally speaking, direct mode matching could be employed as a method of solution of

the problems to be presented. However, direct mode matching has several disadvantages

which often outweigh the simplicity of the method. One of the disadvantages is that many

problems have been shown to exhibit a relative convergence phenomena with regard to the

truncation of the modal representations of the various regions. For the bifiircated waveguide the solution is known (Mittra and Lee, 1971) to converge to the correct result only

when the ratio of the number of modes is chosen equal to the ratio of the heights of the

waveguides. This choice ensures the satisfaction of the edge condition of the problem and

Montgomery is with Texas Instrument Corporation, Dallas, Texas.

Chang is Professor of Electrical Engineering, University of Colorado, Boulder, Colorado.

I
hence the uniqueness of the solution. However, for more complicated structures the solution

is not generally known. Accurate solutions are still possible; however, this is often at

the expense of including an excessively large number of modes in the larger waveguide

region. For the most efficient solution one must have some guidelines in the choice of

the nmber of modes of the various regions. For many problems this is not possible without

an extensive numerical convergence study. Even with this disadvantage, mode matching is

often used because of its simplicity or generality. However, direct mode matching has

another disadvantage that is often overlooked. Direct mode matching does not use a priori

information regarding the geometry. For example, the bifurcated waveguide solution is

known exactly; yet, a direct mode matching solution is of the same order of difficulty as a nonsoluble geometry.

. For many problems there are wo analysis techniques which appear to be superior to

direct mode matching: the generalized scattering matrix technique (GSMT) and the modified

residue calculus technique (MRCT). Both of these techniques recognize that many problems

are composed of a combination of soluble problems. The GSMT and the MRCT are used to solve

problems by efficiently combining these known solutions.

The basic soluble problem used in the solution of two dimensional waveguide discontinuity problems is the bifurcated waveguide. It is well known that the solution of the bi-

.

furcated waveguide can be obtained either by the Wiener-Hopf method or the residue calculus

technique (Mittra and Lee, 1971)- Pace and Mittra (196I+) originally used these known solutions in conjunction with the generalized scattering matrix technique (GSMT) to arrive at

the solution of composite problems. These composite problems were obtained by identifying

an auxiliary problem such that it was clear that the solution of the problem was a modification of the bifurcated junction. As distances and material parameters approached limiting

values, the solution to the original problem was obtained.

The GSMT, although a numerically efficient scheme, has one particular weakness. Since

the GSMT uses truncated matrix representations of junctions, it is difficult to show that

the edge conditions of composite problems are either changed or added to the edge conditions

of the soluble problems. Since the edge conditions of the composite problem may not be

satisfied, one may not be assured of the uniqueness of the solution. However, it is frequently the case that these effects are small when calciilating quantities such as dominant

mode reflection coefficients. Van Blaricum and Mittra (1969) remedied this for a certain

class of problems. They made use of the same auxiliary problem used in the GSMT; however,

they formulated the problem in a manner where a modified residue calculus technique (MRCT)

was used. The MRCT solution was obtained by recognizing that the solution to the problem

is obtained by shifting zeroes of the original residue calculus solution of the bifurcated

waveguide. These shifted zeroes could be found asymptotically by using the edge condition

of the problem. Iterative and matrix techniques were used (Mittra and Lee, 1971) to find

a finite set of shifted zeroes or the equivalent Lagrangian interpolating polynomial representation. Only a small number of these zeroes were needed to accurately find such quantities as the reflection coefficients of the dominant modes. Additionally, because the solution explicitly satisfied the edge ^ndition, the convergence of the results was better than

the GSMT solution.

2
Recently, Royer and Mittra (1972) examined a dielectric step in a parallel plate waveguide. Since there was interest in high dielectric constants, the solution was formulated

using an extension of the MRCT. However, the asymptotic shift of zeroes could not be found.

Thus an infinite form of a Lagrangian interpolating polynomial was used instead of a shifted

zero representation. Consequently, the asymptotic form of the coefficients of the expansion

were found from an application of the edge condition. This enabled the infinite equations

to be truncated and solved in an efficient manner.

This part of the monograph studies a canonical problem of a bifurcated waveguide with

infinitely many known modes incident from all guides. The solution of the problem can be

expressed advantageously using an infinite form of the Lagrangian interpolating polynomial.

This solution can then be used to solve various composite problems. In particular, the Eplane step is solved using this representation as opposed to a shifted zero representation.

The canonical solution is then applied to junctions which have not been solved using

the MRCT previously. Solutions are given for the trifurcated waveguide as well as the

modification of the junction due to dielectric loading. The trifurcated waveguide for

arbitrary spacing of the plates has been solved by Pace and Mittra (1966) using the GSMT.

The solution of the trifurcated waveguide is then generalized to the N-fixrcated waveguide with arbitrary spacing of the conducting plates. The general solution of an arbitrary

number of plates has been given formally by Heins (19^8). Heins' solution, however, has

little practical value. More recently, Igarashi (196^) used a diagonalization procedure in

conjunction with the simultaneous Wiener-Hopf equations and obtained explicit expressions

for the fields in the various waveguides. A necessary condition for the diagonalization of

the equations was equal spacing of the conducting plates. No such restriction is necessary

for the MRCT solution of the N-furcated waveguide. The solution is also given for a particular dielectric loading of the N-furcated waveguide.

One interesting point of the MRCT solution of these problems is that multiple edge

conditions are explicitly satisfied, thus enhancing the convergence of the solution over those which have been (or might be) obtained using the GSMT.

Chapter 5 serves as a forum for discussing solutions of other closed region problems.

Among these are: the eigenvalue solution of ridged waveguide and the dielectric step in a waveguide. Additionally, further numerical results are presented for the N-furcated waveguide.

It should be noted that these concepts can be used to solve many problems which are modifications of soluble Wiener-Hopf problems other than just the bifurcated waveguide.

3
CHAPTER 2. FOUNDATION OF THE MODIFIED RESIDUE CALCULUS TECHNIQUE

1. Introduction

It is the purpose of this chapter to show that the modified residue calculus technique

can be approached in a direct manner by considering the canonical problem of a bifurcated

waveguide with infinitely many modes incident from all waveguides. The general solution

can be conveniently written in the form of a perturbation expansion. It is then shown that

this solution can be applied to composite problems such as the E-plane waveguide step.

2. The Canonical Problem

Geometry of the canonical problem is shown in figure 2.2,1. It consists of three airfilled parallel-plate waveguide sections with the width of the larger section equal to the

sum of widths of the two smaller sections; i.e., a = b+c. The solution to this problem has

been known for a number of years for a single mode incident from one of the three waveguides.

These solutions have been found either the Wiener-Hopf technique or the residue calculus

technique (Mittra and Lee, 1971). We shall discuss in this section the generalization of

this solution to the case involving infinitely many modes incident from all directions.

Let us consider the TM solution of the problem. The TE solution follows in the same manner and will not be given.

The TM fields are derivable from a scaler function ((> = H^ and the fields in each region

are given by [Mittra and Lee, 1971]

I n=0

/ N Y z -Y z .(o) na . . na A e + A e n n

mr , , cos — (x-x„)

a U

I n=0

B<°) e n

~Y uZ Y ,

z

'nb ^ ^ 'nb + B e n

n-rr , . cos — (x-x„)

= I n=0

/ N -Y z C^°) e + C e n nY z nc mr . cos — (x-x,

)

c 1

where the superscript (o) indicates an incident field and

Y V(mr/h)^ - , n7r/h > k

'nh -

nTT mr/h < k

and a time convention, e-''^^, is assumed and suppressed. We further define k = a)(y e )*5 as

o o o the wave number in air, and as the intrinsic impedance of a plane wave in air.

Matching the tangential magnetic field H (=({>) and the tangential electric field E , -1 y

2 i.e., -ik n (3/3Y)<fi, at z = z we can arrive at the equations

o o o

4
T

i

x = x,

Fig. 2.2.1: The Canonical Problem: The Bifurcated

Wavegui de

.

5
I n=0

, Co) na o , . na o A e + A e n n

I n=0

oo

I n=0

, , -Y z Y z ^Co) ^ nc o ^ ^ ^ nc o n n

cos — (x-x„) =

a 0

nil

„(o)' \b^o , „ e + B e

y vZ nb o

cos — (x-x^); x^ j< X ^ x^ + c.

cos Cx-x„) ; x < x < x„ + b

D u o — — U

and

n=0 na

, , Y z -Y z

. (o) na o . na o A e - A e n n

HTT , , COS — (x-Xq) =

-I

n=0

n=0

nc

'nb

/ N -Y z Y z c(°) e

'^^ ° - C e

° n n

^(o) ^-\b^o _ 3 ^ n n

y uZ nb o

cos — (x-x, ) ; X, < X < x, + c,

c i i — — i

COS —g(x-Xq) ; Xq £ X £ Xq + b

We may use the orthogonality of the elgenfunction expansions and eliminate coefficients to

obtain the following four equations.

/V jk z Y ^B be e

""^ " = 2jk b A^"-* e ° ° 6 mb m m o o mo

y \J- mb o

Y Z

, (o) niT . mrb na o

°° A — sm e + (.1)^+1

I ^

§ S

n=l Y ~ Y mb na

mr CO A — sm

—Y Z mrb na o m = 0,1,2,

n=l Y u + Y mb na

(2.1)

Y ^ B^°) b e e mb m m

-y i,z mb o -jk z = 2jk b A e ° ° & o o mo

Y z

. (o) niT . mrb na o

00 A — sm e

n=l Y u + Y mb na

. mr oo A — sm

_l_

, . ,m+l y n a a

Y ~ Y n=l mb na

—Y Z mrb na o m = 0,1,2, (2.2)
Y z , , jk z Y C c e e = 2ik c A e o mc m m o o mo

Y Z

. (o) nir . mrb na o A — sm e n a a

Y ~ Y n=l mc na

I. -Y z

nir . riTTD na o

00 A — sin e + I " % +^ ; m = 0.1,2,..., (2.3)

n=l mc na

/ s -Y z -jk i. „(o) mc o . o o jt Y C c e e = 2ik c A e o mc mm o o mo

Y Z

. (o) niT . nirb na o A — sin e

1 Y + Y n=l mc na

~Y Z

nir J nirb na o

00 A — sin n a a + I ^ ^ ; m = 0,1,2,.... (2.4)

Y ~ Y n=l mc na

where

.2, m = 0

m 1, m >_ 1

and 6 is the Kronecker delta, mn

Equations (2.2) and (2.4) relate the unknown modal amplitudes A and the incident

fields A , B , and C , while (2.1) and (2.3) relate the unknoim modal amplitudes A

n n n n

to B and C . n n

For m=0, (2.2) and (2.4) become identical in form since y^^^ = Y^^ -^^o' allowing one to

eliminate the summations and find

A =^c^°^ +^b(°^ (2.5)

o a o a o

K^'-iJ

which is a unique feature of the TM solution.

The key to the residue calculus technique is to construct a complete solution to the

problem according to the matching equation (2.2) to (2.4), rather than solving them

directly. Thus, let us now consider the following integral,

j> , (2:6)

2Tri w-Y i_ 2iTj oj-Y 'mb mc

7
where m = 0,1,2, ••• and T(a)) is to be constructed uniquely with the residue series of these

integrals identical with equations (2.2) and (2.^+). The contour of integration is the infinite circle in the complex o) plane. We assume that T(a)) behaves appropriately at infinity

so that the integrals exist and are zero.

Let us further assume that T(co) has simple poles at y > -Y » n = 1,2,''*. Then

na na

, ^, V, 0° RES[T,Y ] - RES[T,-Y ] ^^TUOdco ^ na^_ ^ n^^T(Y^^) = 0. (2.7)

27rj (o-Y n=l y - Y n=l y + Y mc na mc na mc

and

, , vm+l , , 00 RES[T,Y ] (-1) ^

T(a))da) ^ V

na''

2TTj ^ w-Y , ~ Y - Y -u mb n=l na mb

oo RES[T,-Y ] (_l) y na

Y + Y n=l na mb

+ (-1)^"'^T(y^^) = 0. (2.8)

where m = 0,l,2'«-. Comparing (2.7) and (2.8) with (2.1+) and (2.2) we find these two sets

of equations are identical provided that

f.y TiTiorm 1 ,(o) mir . mirb ^ma^o ^ „ (i) RES[T,-Y J = -A — sm e m = 1,2, ••• ma ma a

u -Y z (ii) RES[T,Y^^] = -A^ — sm — e m = l,2,...

(iii) T(y ) = -C^°\ c e"'^"^''^° m = 1,2,..- mc m mc

(iv) T(Y^^) = (-l)"" bI°\^^ b e"'''^^'^ m = 1,2 ...

(v) T(jk^) = 2jk^c e''^^°^° (A^ - C^°^)

We can also consider the integrals

i-]J^ ^ TUOdo) 1 T(a))da) ^2 9)

277 j ^ (jj+Y ^

2T:j ^ co+Y mb mc

where m = 0,1,2,.... Using the above properties we find the following by comparing with

(2.1) and (2.3)

Y z (vi) T(-Y )=CYceTM'^° m = 1,2,...

mc m mc

(vii) T(-Y , ) = (-l)'""'^ Y^^ b e ° m = 1,2,..- mb mb m
jk z (viii) T(-jk^) = 2jk^c[C^ - A^°^] e ° °

(ix) T(-Jk^) = -2Jk^b[B^ - A^°^] e''^°^°

Alternatively, (ix) can be replaced by A^°^ oaoao = — C + — B which is then analogous to the

relationship given in (2.5). From Appendix A it is shown that the edge condition implies

T(a)) = 0((o ) as |(o| « and hence the assumption regarding the convergence of the

integrals is justified. Because of its importance let us also consider this in our list

of properties of T(a)) (for the bifurcated case only)

(x) T((o) = 0(0)-^/^), |a)|

Thus, the question now becomes one of constructing the function T(u)) uniquely from

some of the properties listed above. Upon finding T(tD) the complete solution to the

problem is given by (ii), (vi), (vii), (viii) and (ix).

The clue to the construction is found by considering the solution with only a single

mode incident on the jimction. For simplicity say the incident mode is B^°^ = 1. Then (i)

implies that T((d) has no poles at -Yj^g^i (iii) implies that T(a)) has simple zeroes at y^^i

and (iv) implies that T(co) has simple zeroes at y -u- Hence mb

n(a),Y ) n((i),Y ) T(a)) = K H(u)) ^7^-71 (2.10)

a where

coh/nir

n(w,Yj^) = n n=l

1 - ^nh

e

and H(a)) is an entire function with no zeroes. The exponential factor has been introduced

into the infinite product in order to insure uniform convergence. However, when the

products are grouped as in (2.10) the exponential is not needed. From Mittra and Lee (page

13, 1971) we find the asymptotic expansion of the following infinite produce

, a)/ir[b In

J- + c In r]

T(a)) = K H(a)) w""^/'^ e ^ ^ , 03 |a)|

In order for condition (x) to be met we must have

H(a)) = e"'^'^^'-^ ln(b/a) + c ln(c/a)]

(2.11)

where K is a constant. However, K can be determined from (v) and (2.5),

ii(jk ,v ) n(jk Y

J

o a

-jk Z = 2jk^ c e °

°

9
Hence, the solution is

T(a)) = 2ik

^

e"^''^'"

H(a)) F(a.)

(2.12)

where F(w) = n(a),Y^)n(a),Y^)

n(a),Y^)

ci

The reflection coefficient is given by (ix)

-j2k z H(-jk ) F(-jk ) -c 0 0 o o

o " a

^ H(jk^) F(jk^) (2.13)

When the waveguides are single moded structures, we obtain |b^| = c/a, a well-known result

(Marcuvitz, 1964).

Let us now consider how (2.12) can be modified to reflect the general solution. From

(i) we see that if A^°^ ^ 0 we must introduce simple poles at -y . Similarly, from (iii) m na

and (iv) we see that we must introduce simple poles at y , and y in order to remove the mb mc zeroes of F(a)). This leads us to consider a function of the form

r

T(a)) = H(a)) F(a3) K

n - (a)-jko

) H'•• —111—

V

(b)

n=l nb

(c) (a)

' w-Y ^1 w+Y n=l nc n=l na

(2.14)

where

H(a)) = e

-co/TT±b In b/a + c In c/al

K^, g^^\ ^n^^^^'^ ^n'^

obtained by using (i), (iii), (iv), and (v)

.

-jk z , , K H(jk )F(jk ) = 2jk c e

° °(A -c'-°'^) o o o o o o (2.15)

H(-Y )F(-Y )(y +jk ) ^ na na na o n

(a)

Y Z

. (o) niT . nirb na o = -A — sin e n a a n = 1,2, (2.16)

„/ N,.(n), ^ ^^ric (c)

«(\c)^ ^\c^ y ^" nc

^ \ -y z = -C^^^ Y c e

° n nc n - 1,2, (2.17)

^^(^>r ^ '-^nb j^o^ (b)

"^^nb^^ ^^nb^ 77 ^n

nb

= (-1)'^ B^°^ Y K

b e n nb

'nb o n = 1,2, (2.18)

10
where F^'^^y ) and F^'^^ (y , ) are to be Interpreted as omitting the nth zero at either

nc nb

Y or Y 1 . and A is related to the incident field B and C according to (25).

'nc nb' o o o This represents the complete general solution to the bifurcated waveguide problem.

• 3. Formulation and Solution of Composite Problems

The key to the MRCT is the identification of an auxiliary problem. The auxiliary problem is such that the solution may be identified in terms of soluble problems. For example,

the auxiliary problem may clearly indicate that the desired solution is a perturbation of a bifurcated waveguide or a parallel plate in a homogeneous space. Mittra and Lee (1970) have

indicated a number of such problems.

Before proceeding let us illustrate the above process with a problem which has been

solved using the MRCT using the concept of shifted zeroes (Mittra, Lee, Van Blaricum; 1968).

We will solve the E-plane step using the canonical solution of section 2.

Figure 2.3.1 illustrates the E-plane step and the auxiliary geometry. Notice that the

auxiliary geometry has a recessed dielectric of finite permittivity. When 6 = and e = <»,

the auxiliary problem coincides with the original E-plane step. Notice that this recession

has identified two soluble problems: (1) a bifurcated waveguide junction at z=0, and

(2) a dielectric junction within a parallel plate waveguide at z=-6. This auxiliary

problem allows us to perturb the bifurcated solution advantageously.

For simplicity let us consider the case of TEM incidence from the smaller guide. Extension to higher order incidence or a TE solution is straightforward. The dielectric

creates reflections of any scattered modes from the junction at z=0. This may be thought

of as an incident field upon the junction. From the canonical solution we recognize that if we knew these modal amplitudes, T(a)) would be given by (2.14) with g^^^ = g^^^ ^ 0 and say

(c) ^

g = g . It is still convenient to use this form even with g unknown. In this case g °n '^n '^n ^n

represents a perturbation of the bifurcated solution due to the dielectric loading. From

(2.14) we have

f

T(a)) = H(a))F(a))

oo g

K - (w-jk ) I n=l nc(3.1)

where the notation is similar to section 2. This will be referred to as a perturbation type

expansion. Using (2.17) we have that

Sn = \ 'n^ (3-2)

where K is found from (2.17) as K = - (y c)(l-jk /y ) '''[H(y )F^'^^(y )] and involves

n n nc o nc nc nc only simple calculable functions. If we represent the component of the field in the

dielectric as

" r z

(\, = J

T) e cos — (x-b) (3.3)

y n=0 ^

11
T

b

i ±

z

(a) E-Plane Step

(b) Auxiliary Geometry

Fig. 2.3.1: The E-Plane Step and the Auxiliary

Prob lem

.

12
where

nc

mr

O

we can easily find that

C^°^ = R C = n n n

ey - r nc nc

ey + r nc nc

-2y & e Cn

(3.4)

and

r 6 + Y 5 2eY . . D = e nc nc^ ^(o) n ey - r n

(3.5) nc nc

1/2, From (210) on p. 14, we have = 0(n ) as n <», and thus we see from (3.5) and (3.2) that

for 6 = 0

1/2

g = 0(n ' D ), n -> oo ''n n ,

For the 6=0 case it is also easy to show (Mittra and Lee, page 170, 1971) that

D = 0(n"^^^"^), n-^

n

(3.6)

where

hence

1 , -1 (e-1)

A = — sin „ ; , .,

;

IT [2(e+l)

g^ = 0(n ), n^

(3.7)

(3.8)

This allows us to write (3.1) in an approximate form suitable for numerical analysis

^ oo rr CO T(w) ^ H((i))F(a)) K -((o-jk )i I +i I ^

V J

(3.9)

-1/2,

In this particular problem the edge condition has changed from T(w) = 0(a) ), |oj| to

I I -A,

T(oj) = 0(a)

'^^^ ^) , |a)| -> «>. An examination of (3.9) reveals that the multiplying term of

H(a))F(a)) must be 0(a) ), |a)| «=. This implies that any constant terms contributed by the

perturbation sums must cancel with in order for the higher term of the second sum to

dominate. Hence, we must have

K - I g„ - g I n ^ = 0. (3.10)

n=l n=N+l

This argument has assumed that

13
n=N+l "^nc (3.11)

as |aj| -> This is shown in Appendix B.

In order to derive the necessary equations for we consider that

T(-Y ) = Y c C mc mc m

Using (3.2) and (3.4) we have

mc mc m m m

which according to (3.1) yields the following set of linear equation for g

, N g _ "

H(-Y )F(-Y )[K-(Y +jk ) { I + g I mc mc ' o mc o

•

-, Y +Y t n=l nc mc n=N

n-l-A ] , ,,-1^.-1

; N = Y cR K g

Y +Y ; mc m m n

N+1 nc mc

(3.12)

for m = 1,2, .. .N,

From (v) and (viii) of section 2 we also have an additional equation

K F(jk )H(jk ) = 2jk — o o o o a

,(o) T(-jk ) 2jk c o (3.13)

Equations (3.10), (3.12), and (3.13) are the necessary linear equations to solve for K^, g^,

for n = 1,2,...N, and g. The scattered modes in the two regions can then be found by using

properties (v) and (iv) of section 2,

It should be noted that the concept of shifted zeroes could have been used to solve

this problem. The interested reader is referred to Mittra and Lee (1971). The interesting

point of this solution is that the perturbation expansion approach can be used to solve problems that can be solved using the shifted zero technique, but the reverse is not always true

(see Royer and Mittra, 1972).

14
CHAPTER 3. THE TRIFURCATED WAVEGUIDE

1. Introduction

This chapter is concerned with the application of the generalized MRCT to the trifurcated waveguide and the dielectrically loaded trifurcated waveguide. This type of problem

has not been solved by the MRCT previously. However, this problem has been previously

solved using the GSMT by Pace and Mittra (1966). It is shown that the satisfaction of the

edge condition at both edges by the MRCT solution improves the convergence over that

obtained using the GSMT. The solution of the trifurcated waveguide then allows one to

proceed to the more complicated case of the N-f\arcated waveguide with minimal difficulty.

2. Formulation of the Equations

Figure 3.2.1 illustrates the trifiircated waveguide geometry and the associated aioxiliary

problem.

With reference to the auxiliary problem we see that we are perturbing two bifurcated

waveguide junctions. This leads us to construct two meromorphic functions as follows:

T^(a)) = H^(a))F^(a)) o o (jj+Y n=l no

(2) o o ^- co-Y n=l nc

(2.1)

(2.2)

where

H^(a)) = e

-a)/Tr[b^ln(b^/c) + b2ln(b2/c)]

E^i^) = e

-aj/Tr[b ln(b /a) + c InCc/a)]

o o

n(a),Y, ) n(a),Y, ) b b

"i'"'

= nt... )

n(co,Y^ ) n(to,Y^)

where TEM mode incidence in any one of the four waveguide sections has been assumed. Extension to higher order TM mode incidence or TE incidence is straightforward. It may be

recognized that (2.1) and (2.2) are just special cases of the canonical solution. T^(co)

is identified with the junction at z = 0. The scattered modes from z = A produce an incident modal spectrum on the jionction at z = 0 from the coupling region c. From the

canonical solution we recognize that g^^^ = g^^^ = 0 and g^^^ = g^"'"^. Similarly for T„(a)) n n n n j 2' '

15
r

T

(a) Trifurcated Waveguide

T

t:

X

z=0 z=A

t

1

i

[ 1

1

T

("b) Auxiliary Geometry

Fig. 3.2.1: The Trifurcated Waveguide and the

Auxiliary Problem.

16
we recognize that the solution is obtained from the canon leal solution with g^^^ = g^^^ = 0

/ \ / o \ n n

and = g^ . Here, the superscripts a, b, c refer to the three different regions in

canonical problem (see Fig. 2.2.1).

For this particular problem K^^^ and K^^^ are known and are given by property (v) of

section 2, Chapter 2 as

.(1)

K

o

(2)

2jk bJC+ - B^°J

o 2 1 o 0,2

2jk c o A - C+

o o

/(H^ak^)F^ak^)

H2(jk^)F2(jk^)

(2.3)

(2.^)

where

C+ = b,/c B^°J + b,/c B^°;

o 2 o,2 1 o,l

A = c/a C"^ + b /a B^°^

o o o o,o

and B^°^ (n = 0,1,2) is the amplitude of the TEM mode incident from the b region.

o,n ^ ' » ' n

°

± The perturbation coefficients may be related to the modal coefficients in the

coupling region c of the auxiliary problem using properties (i) and (ill) of section 2,

Chapter 2.

g^l)

= K^l) C-, n > 1 n n n —

g(2) = k(2) c^ n > 1

°n n n —

(2.5)

(2.6)

where

^(1) -rnr . 1, K = sm / n c c F-(-Y )H, (-Y )(y +jk ) 1 nc 1 nc no o

, n > 1 (2.t;

k(2) = -y' c/ n ncn > 1 (2.8)

where Fi'^^(Y ) indicates that the nth zero term at y is to be omitted.

2 nc nc The equations for g^'''^ and g^^^ ma-y be derived by requiring that T^(a)) and T^{(}i) give

consistent results for the modal coefficients in the coupling region. Using properties (ii]

and (vi) of section 2, Chapter 2 and (2.5) and (2.6) we have for A = 0:

RES[T, ,Y ] = — sin

1 nc c

nirb.

K

(2)

-1

^2) n > 1 (2.9)

T„(-Y ) = Y c

2 nc nc K

1)

-1

n

n > 1 (2.10)

(2.9) and (2.10), together with the definition of T^(w) and in (2.1) and (2.2).
Then represent two infinite sets of equations for the perturhation coefficients g^'''^ and

(2) "

3. Asymptotics

In order to efficiently triincate equations (2.9) and (2.10), we shall use the asymptotic

behavior of the perturbation coefficients.

The asymptotic behavior of g^"*^^ a-^d g limiting procedure. We first consider the asymptotic form for A 7^ 0. In (ii) and (vi) of

section 2, chapter 2, we set y '^^ - mc c (page 1^), this respectively yields

The asymptotic behavior of g^"*^^ a-^d g^^^ foi" A = 0 is found by considering a double

Lde: section 2, chapter 2, we set y — + e as m °°: then let e -> 0 and A 0. Using

mc c °

C" = 0(m"^/^) (3.1) m

C = 0{m~^^^ sin -} (3.2)

m c

where the notation for the mode coefficients in the coupling region is obvious. (Note also

that c"*"mmand C correspond respectively to A in (A. 7) and 0 in (A. 5) of Appendix A.) The

_^ n m

oscillatory portion of C is necessary in order that the field be properly sing\ilar at

m

X = b + b-, . o 1

This is an important point. In general if we have N edges of various types in a large

guide of dimension a, then the asymptotic behavior as n » is

N

A = y 0' n m=l

-d+P ) m . nir n sm — X

a m

(3.3)

where x is the location of the mth region and p is the power index associated with the m m

edge condition at x^. This can be more clearly understood if we examine the field in region a

H = cj) = I A e"^^^^ cos ^

X iS.k)

y A n

Examine E which behaves as z as z 0 and x = x . Then from Mittra and Lee (page 11,

1971) we have

n A sin —X = oCn \ (3-5)

n am

If we multiply (3.3) by sin mrx^/a all terms will be oscillatory except the term m = k, and

we will pick the appropriate edge condition out of the sum. (Note that sin — = 0{l).)

With a single edge thisoscillatory term is generally implicitly stated. However, when

multiple edges exist it is important to give these terms explicitly.

From (page ih) we can find that

fl) 1/2 """"^ ^^'^^ K^^^ n

= 0{n^^ sin

c ^

18
n

C3.T)

Hence from (3.6), (3-7), (2,5) and (2.6) we have

,(1) „(,-! !!^)

n c

n c

(3.8)

(3.9)

Using (3.8) and (3.9) we can write (2.1) and (2.2) in a form more tractable for numerical

computations

.

r T^(a)) = H^(w)F^(a))

rl n

n=l nc

, ^ ^ °° n sin mrb^ /

c

+ I ^

n=i+N^ '^^^c

(3.10)

T2(co) = H2(w)F2(a))

.(2)

^2 ^n

n=l nc

« n sin mrb^ /c

n=l+N, w-Y,nc

(3.ii;

where clearly

g^"'"^ =

g^"""^

n""*" sin nfrh^/c, n >_ 1+N^

(2) -(2) -1

g^ = g n sin nirb^/c, n >_ l+N^

Before using (3.10) and (3.11) in (2.9) and (2.10) let us consider the mode coefficients in

the regions a and h, in order to insure that the field is properly singular using (3.3)

and (3.9) for the asymptotic behavior of g^"''^ and S^^^- Using properties (ii) and (vii) of

section 2, Chapter 2 we respectively have that

T (-Y , ) = (-l)"""^^ Y ^ B

• 1 mb^ 'mb^ 1 m,l (3.12)

and

niTb

RES[T^,Y ] = -A sin ^

2 ' na n a a

(3.13)

where B is the mth reflected modal coefficient in the waveguide with dimension b,

.

m,l ^ 1

19
This leads to an examination of the sxims

<" n sin nirb^/c = F —.

1 n=N

-1- +1 n - ^

m ^ «>

and

<» n sin niTb^/c

S = I

, m -> » n=N2+l n - cm

Thus let us examine the \iniversal sum

oo _i

V n sin ne

S = }. , 0) -> "

n=N

n - (0 (3. Ik)

In Appendix C it is shown that

' Sm WTT

hence from (3.12)

and thus (3.12) yields

^1^-Vb ^ "

0(^1"^''^) + 0(m-^/2(-l)"^)

B , = 0(m-3/2) + 0(m-^/2(-l)"^)

The first term therefore can be used to satisfy the edge condition at the lower edge x = b^

while the second term for the upper edge at x = (b^+b^). This result is then in agreement

with the concept of (3.3). Similarly,

_ , mrb

RES[T^,Y ] = n ' sin

2 na a

r nffb 0 sin

+ 0

r mr(b +b- ) o 1 sm

and thus from (3.13)

A = 0 f-3/2 . n sm 1 + 0 n a I sm

nirCb +bT ) o 1

Again, the first term then explicitly satisfies the edge condition at x = b^, while the

second term satisfies the edge condition at x = (b^+b^). Hence the asymptotic results

(3.8), (3.9) allow both edge conditions to be satisfied.

20
k. Truncation of the Equations

We are nov in a position to use the knowledge of the asymptotic behavior of the perturbation coefficients in truncating equations (2,9) and (2.10). This section examines two ways

of truncating the equations using the asymptotic behavior of the perturbation coefficients.

The first kind of triincation is what has commonly been used (Royer and Mittra, 1972).

This consists of merely choosing extra equations by letting the free index of (2.9) and

(2.10) take on one additional value. This yields the following simultaneous linear equations

?1

4^^

, -d) ^ m-^ sin m^b^/c

^2)

,(1)

= 2 n = 1,2,«",1+N„ (^.1)

^nc - 2

(2) (1)^ ^2 , -(2) ^

sin m^b.^/c

^n L y +y ^ ^ Y +Y m=l nc mc m=N„+l nc mc

A.

2

,(2)

o

Y + jk

nc o

n = 1,2,---,1+N^ (1+.2)

where

mrb^

nir . 1 sxn

= TpT (^-3)

H^(y^^)RES[F^,y^^](Y^^-Jk^)K^

x(2) = '-^ ^

(k.k)

H^(-Y )F^(-Y )(y +jk ) K^^^

2 nc 2 nc' nc ^ o n

Note that since we are not changing an edge condition but adding an edge condition, an

equation comparable to (3.10) in Chapter 2 is not needed.

It should be noted that with any truncation there are an infinite number of equations

which remain. As in other MRCT solutions these remaining equations can be used as a

check on accuracy of the solution.

Of course, the above choice is not the only manner of truncation. For example one can

equate the leading asymptotic terms of equations (2.9) and (2.10). This results in two equations independent of a free index. Using the asymptotic expressions for the infinite product

in Mittra and Lee (l9Tl) we find from (2.9)

I'g^'^-i^'^ I n-^sinn.b /c.I^P i(2) ^^(1) ^^ ^^ n=l " n=i+N^ 1 ^1

g(2) + -(2)

J

^-1 _ ^ p -d) = K^2)

n=l ^ n=l+N2 ± d o

21
where

and

The remaining equations are chosen as in the previous truncation.

There is yet another possible choice. This is the direct truncation of the original

equations. However, hecause of the arguments given in section 3, this solution will not

explicitly satisfy hoth of the edge conditions of the problem.

5. Dielectric Loading

Figure 3.5.1 illustrates a trifurcated waveguide with a dielectric loading in the

largest waveguide. The auxiliary problem is similar to the normal trifurcated waveguide.

With reference to the auxiliary problem we see that we can identify a function T^(a))

with the junction at z = -A.

T^(a)) = H^(a))F^(a))

O W+Y

n=l nc

(5.1)

This equation is identical to (2.1). However, the fmction (2.2) for the junction at z = 0

is modified to be

r _ „ J2) „ J3)^

o o cj-Y w+Y I m=l nc n=l na''

(5.2)

where g^'^^ corresponds to g^^^ of the canonical solution, due to the presence of the dielectric . Equations (2.3) - (2.10) apply to the dielectrically loaded case if the expression for

Tgdij) given in (5.2) is used.

Similarly we still have

g,(l),g_(,2) = 0(n-lsin

n n

niTb,

since for d ^ 0 we are not changing or adding an edge condition.

In order to accoiant for the dielectric consider the following,

the trifurcated junction and the dielectric, the field is given by

y n=0 ^ ^

/ \ Y z -Y z (°)e + A e n

mr

cos — z a

In the region between

(5.3)

22
(a) Di ele ct ri cally Loaded Trifurcated Waveguide

z=-A z=0 z=d

f

(b) Auxiliary Geometry

Fig. 3. 5.1: Dielectri cally Loaded Trifurcated

Waveguide and the Auxiliary Problem

23
and it is easily shown that

A^°^ = R A

n n n (5.M

where

e Y - r -2y d

„ na na na

R = — e n e Y + r na na

(5.5)

and where

na

rnr - ek

We will consider the case of conduction losses in the dielectric by using the complex permittivity

120 TTO (5.6)

From property (i) of section 2, Chapter 2 we have that

^(3) ,(3) ^(o) n n n

(5.7)

where

- ^

^/t^2(-^na)^2(-^na)(^na^^^o)]

(3) The asymptotic behavior of can be found using (5.T) and (5.^) to be

(5.8)

,(3) = o(n-l e-2-^/-)

n

(5.9)

(3) Because of the exponential behavior of g^ the series appearing in (5.2) can be truncated

at a finite value, say n = N . (3) An equation for g^ can now be derived using property (ii) of section 2, Chapter 2 and

(5.U), (5.7) and (5.9)

rnrb RES[T^,Y ] = — sin A

2 na a a n

mrb T -mr . o „-l sm R

a an n

-1^ (3) g^-^^ n = 1,2,---,1+N2 (5.10)

Equations (5.10) together with the appropriately modified forms of (2.9) and (2.10) represent the necessary simultaneous equations for

^n^^ ^n"^^'

Note that we are considering only the conventional method of truncation.

24
6. The Scattered Fields

The previous sections have dealt with the formulation and solution of the MRCT equations

for the perturbation coefficients. Upon finding the perturbation coefficients, we are able

to evaluate the constructed meromorphic functions at the appropriate points in the complex

plane and determine the scattered fields. This is done with the aid of the avixiliary

geometry. Using the properties (viii) and (ix) of section 2, Chapter 2, we find the following TEM coefficients of the scattered fields for the unloaded trifiircated waveguide

^ = -T_(-Jk )/(2jk b ) (6.1)

o,o 2 o o o

B^^^ = -T^(-jk^)/(2Jk^b^) + T2(-jk^)/(2jk^c) (6.2)

^o,2

" T^(-Jk^)/(2jk^b2) + T2(-jk^)/(2jk^c) (6.3)

where B^ ^

(n = 0,1,2) is the amplitude of the reflected TEM mode in the waveguide of

dimension b . n When only a single waveguide is excited with a TEM mode with an amplitude of unity,

(6.l)-(6.3) represent either (current) reflection coefficients or (current) coupling coefficients. The reader is reminded that for TEM incidence from the largest guide the TEM solution is immediate, the solution being given by properties (viii) and (ix) of section 2,

Chapter 2. Also, the TEM transmission coefficient to the larger waveguide is found immediately

from repeated use of equation (2.5).

The complete scattered fields can be fovind with the aid of the auxiliary problem and the

properties given in section 2, Chapter 2. For this monograph, only the TEM modal amplitudes are of immediate interest.

For the case of the dielectric loaded trifurcated waveguide, the results are essentially

the same as those already given except that we must add in the reflected TEM field from the

dielectric. This yields

^o,o = -^2(-J^o)/(2jk^tJ + R,A^ i6-h)

= -Ti(-jk^)/(2jVl) * T2(-jkJ/(2Jk^c) + R^A^ (6.5)

^o,2 = T^(-jk„)/(2JV2) ^ T2C-jkJ/(2jk^c) + R^A^ (6.6)

where is given by (5-5) and

and

o a o a o,o

o c o,2 c o,l

25
The remainder of this chapter concerns only TEM incidence. Other incident modes may

he included in a direct manner using the properties discussed in section 2, Chapter 2.

7. Numerical Results

7.1 Introduction

This section presents the numerical solution of the trifurcated waveguide as well as the dielectric loaded trifurcated waveguide.

One interesting aspect of, the numerical solution of the problems is the method used to

evaluate the infinite product form H((jj)F(w) appearing in (2.1^*) of Chapter 2. The method

is capable of giving results accurate to an arbitrary accuracy using only a small number of

terms in the product plus some correction terms. For the data computed in this report, 50

terms were used in the evaluation of the infinite product for 5 place accuracy. The technique used is given in Appendix D. The infinite oscillatory summations used in the construction of the meromorphic functions (see (3.10) and (3.11), for example) were summed numerically using a moving average.

Less than 50 terms were generally necessary to yield 5 place accuracy.

7.2 The Trifurcated Waveguide

The practical solution of the trifurcated waveguide using the truncated equations re- quired two numerical considerations. We must decide how to choose the ratio of N^/N^, and we must decide how large and must be for acceptable accuracy of the results.

A numerical study of the ratio, N^/N^, revealed that the final result was independent

of the ratio (as opposed to direct mode matching where the solution does depend on such

ratios). It was thus convenient to choose = N^.

The choice of how large = = must be for a given accuracy, depends on the geometry of the problem. Even for a given trifurcated waveguide we must decide how b^ and b^

are chos n, since switching b^ and b^ merely turns the waveguide upside down. Table 3.7-2.1

illustrates the TEM current reflection coefficient of an edge waveguide as a function of

N = = N_. The column B is the reflection coefficient of an incident current mode of

p 1 2 o,o

unity in waveguide section "0", while B o next column is the reflection coefficient

o ,^

when the current mode is incident from section "2". Since the geometry of the two cases

is identical, one expects the two coefficients to converge to the same value as increases.

This geometry was also considered by Pace and Mittra (1966) and their data is shown with

N corresponding to the size of the scattering matrix used.

26
Tatle 3.7.2.1 Convergence Results for the Reflection Coefficient

in the Outer Section of a Trifurcated Waveguide.

N

« B

o

i- B

0 ,2

B

0,0

(Pace)

1 0.32i+2i| 131. 91° 0.32ij-2T 131.91° 0.326 133.1°

2 O.32U25 131. 91° 0.32l+2i+ 131.93° 0.327 132.0°

3 0. 321+25 131. 91° O.32I+2I+ 131.92° 0.32i+ 132.5°

1+ O.32U25 131. 91° O.32I+25 131.91° O.32I* 132. 5°

6 0.32 1+25 131. 91° O.32I+25 131.91° O.32U 132.6°

8 0.32i+25 131. 91° 0.32i+25 131.91°

*k b = I.270I+6, k b = O.I+1I+17, k b^ = 0.20033 00 o 1 o 2

tk b_ = I.270I+6, k b, T O.I+II4I7, k b = 0,20033

o 2 o 1 00

The above MRCT data was computed using the conventional method of truncation. Note that 5 place accuracy is achieved almost immediately. This is a definite improvement over the GSMT,

although for many engineering applications the GSMT results are still quite acceptable. It

is interesting to note that the reflection coefficient without the adjacent conducting plate

is found from (2,13) in Chapter 2 to be 0.326 exp(l32.6°). Note that ^ converges faster

than B . This later case corresponds to a larger coupling region dimension, c. O , c: The reflection coefficient of the central waveguide of the above case is given in

Table 3.7.2,2.

Table 3.7.2.2 Convergence Results for the Reflection Coefficient

in the Center Section of a Trifurcated Waveguide.

*k

* N B , _£ Oil

1 0.lk2hk 152.38°

2 0.7^+21+1+ 152.37°

3 0.7^2U1+ 152.37°

k 0.7^2U1+ 152.37°

c b = I.270U6, k b = O.klkn, k b^ = 0,20033 00 o 1 o 2

Again the convergence is extremely fast.

The above data was computed using the conventional method of truncation of the equations.

The same data was computed using the asymptotic choice of the extra equations. This is shown

in Table 3.7.2.3.

27
Table 3.7.2.3 Effect of the Choice of Truncation on the Convergence

of the Reflection Coefficients B > ^ o an<i "the Coupling Coefficient B . °'° °' o J J.

N

* B

0

b"^0 ,2

* B

0

1 O.32I1I8 131..93° 0.32308 132. 35° 0.71*296 152. 5!+° 2 O.32I+26 131. 91° O.32U06 131. 95° O.7I+2I+I 152.36°

3 0.321+26 131. 91° O.32U36 131. 90° O.7I+2I+2 152.36°

k 0.32i+25 131. 91° 0.32 1+38 131. 90° O.7I+2U6 152.38°

6 0.321+25 131. 91° O.32I126 131. 91° O.7I+2I+I+ 152.37°

8 O.32I+25 131. 91° O.32I+2U 131.91° O.7I+2UI1 152.37°

*k "b = I.270I+6, k Id = O.I+1I+17, k = 0.20033 00 ol o2

tk = I.270I+6, k b = O.I+1U17, k b = 0.20033

o 2 o 1 00

Five place accuracy is again achieved; however, this choice of -the truncated equations does

not appear to be quite as good as the conventional truncation choice. This is logical since we are truncating the equations at such small indices that the asymptotic value of the equations has not been reached.

As a further comparison with the above data. Table 3.7.2.1+ gives the results of using

direct truncation (i.e., no asymptotics )

.

Table 3.7.2.1+ Convergence of Direct Truncation.

H

* B

0

B^

0 ,2

B

0 ,1

1 0.3233 131. h° 0.3257 135. 30 0.739^ 153.8°

2 O.32I+I+ 132. 0° O.32I+5 132. 30 O.7I+29 152.1°

3 O.32I+I+ 132. 0° 0.3237 131. 5° 0. 71+28 152.2°

1+ 0.321+1 131. 8° 0.3237 131. 1+° O.7I+2O 152.6°

6 O.32I+3 131. 9° O.32I+3 131. 9° 0. 71+25 152.3°

8 0.321+3 131. 9° O.32I+I+ 132. 1° 0. 71+25 152.3°

*k b = I.270I+6, k b = 0.1+1 1+17, k b^ = 0.20033 00 o 1 o 2

tk b^ = 1.2701+6, k b = O.I+II+I7, k b - 0.20033

o 2 o 1 00

This solution does not explicitly satisfy both of the edge conditions; however, it does con- verge to the correct result just as the computations made by Pace and Mittra (1966) using

the GSMT did. However, the convergence of both of these methods is much slower than either

of the methods described which satisfy both edge conditions explicitly. It is also interesting to note that the direct trvmcation appears to converge faster than the GSMT solution.

This is important to note since the MRCT solution without asymptotics can be applied as

easily as the GSMT.

The data presented thus far has been for a waveguide junction which propagates only the

TEM mode in any waveguide region. It is instructive to solve multimoded waveguide problems

using the MRCT. Table 3.7.2.5 presents data for a case where two of the waveguides support

the TM^ mode in addition to the TEM mode.

28
Tatle 3.7.2,5 Effect of Multimoding on the Convergence Results of the

Reflection Coefficients B B in the Outer Section

0,2 0,0

of the Trifurcated Waveguide and the Coupling Coefficient

B , in the Center Section.

0,1

* •!- * u B b' B

_£ r\

KJ

r\\J

1 0.38296 7i+.68° 0.31798 75.03° 0.85180 156.01°

2 0.38287 7l+.75° 0.37277 Ik. 79° 0.85159 155.99°

3 0.38287 7^1.70° 0.38013 7U.72° 0.85160 156.01°

h 0.38289 7l+.67° 0.38226 lk.6Q° 0.85l6i+ 156.02°

6 0.38290 7^.66° 0.38321 Ik. 63° 0.85166 156.02°

8 0.38289 Ik. 66° 0.38320 7^.65° 0.85165 156.02°

10 0.38289 Ik. 66° 0.38306 7^.65° 0.85165 156.02°

12 0.38289 'Jk.66° 0.38296 Ik. 66° 0.85165 156.02°

*k h„ = I.270U6, k h = O.klkn, k b = k.kl20'^ o 2 o 1 00

tk b = I.270U6, k b = O.klkn, k b^ = k.kl20'^ 00 o 1 o 2

The convergence is again quite fast and five place accuracy can be achieved. The convergence

of the recession with the larger coupling dimension, c, is somewhat slower than in the

single moded case because of the multimoding in the coupling region. Also the overmoded

data represents more of a perturbation to the bifurcated solution since the magnitude of

the reflection coefficient without the adjacent conducting plate is 0.^*^52 (from (2.13),

Chapter 2)

.

Table 3.7.2.6 illustrates this same data using the asymptotic choice of the last

equations.

Table 3.7.2.6 Effect of Multimoding on the Convergence Resiilts of

the Trifiircated Waveguide (Asymptotic Truncation)

* + * N B „ b' B , p 0,2 0,2 0,1

1 0.38695 75.92° O.i+3375 7^.52° 0.837^^ 156.62°

2 0.38310 7^.83° 0.1+0086 72.97° 0.8it925 156.15°

3 0.38327 Ik. 68° 0.3919^ 71.82° O.8530I+ 155.77°

k 0.38329 Ik. 61° 0.38780 73.65° 0.85317 155.75°

6 0.38288 7^.66° O.38U33 lk.3k° 0.85172 156.02°

8 0.38282 Ik. 62° 0.38317 7^.77° 0.851^1+ 156.08°

10 0.38290 lk.6l° O.3828I4 7i+.96° 0.85165 156.02°

12 0.38296 Ik. 66° 0.38283 Ik. 96° 0.85173 155.99°

*k b_ = I.270I+6, k b = O.U1U17, k b = k.kl20'^ o 2 ' o 1 00

tk b = I.270I16, k b = O.klklT, k b^ = U. 1+1205 00 o 1 ' o 2

From the data it is clear that for larger waveguides the differences of the conventional

and asymptotic truncation methods eire amplified with the conventional truncation method

apparently being better.

For a complete comparison of methods it is instructive to compute the multimoded data

with direct truncation. This is shown in Table 3.7.2.7.

29
Talale 3. 1.2. J Effect of Multimoding on the Convergence Results of the

Trifurcated Waveguide (Direct Truncation)

N

T) ^o,2 B

0,0

B

Q .1

1 0.3888 82.3° O.kkkj 79.1° 0.81+82 150.8°

2 0.3835 75.5° 0. ii330 78.8° 0. 851I1 ISS 1+° 3 0.3807 73.6° 0.1+167 78.3° 0.8518 157.0°

k 0.1+0il2 77.^° n ftsi fi 6 0.3829 7^.7° 0.3889 75.6° 0.8517 156.0°

8 0.3836 75.0° 0.3821 7l+.3° 0.8516 155.7°

10 0.3829 7^.7° 0.3797 73.8° 0.8517 156.0°

12 0.3825 7^.5° 0.3796 73.7° 0.8517 156.2°

*^^2 = 1.270U6, k^b^ = 0.iiliH7, k b =

0 0

It. 1+1205

tk h

o o

= 1.270it6, k b =

0 1

O.l+li+17, k^b^ = l+. 1+1205

Again the convergence of the direct truncation method is slower than either the conventional

or asymptotic truncation methods.

The choice of recession giving the larger coupling region serves as a convenient compari

son of all the truncation methods since the effects are magnified. Figiire 3.7.2.1 illustrates

graphically the convergence of the magnitude of the reflection coefficient B^ ^ given in

Tables 3.7.2.5 - 3.7.2.7. It is clear that both of the methods which satisfy both of the

edge conditions explicitly are superior to the direct truncation method, particularly for

extremely accurate results. For some engineering applications, however, the direct trmcation method is acceptable and will yield resiilts more efficient than many more conventional

methods. Figure 3.7.2.1 also illustrates that the conventional choice of the truncation

method converges faster than the asymptotic choice of the truncation.

The discussion thus far has been limited to the convergence of various reflection

coefficients. It is also interesting to examine some typical pertiarbation coefficients.

Figure 3.7.2.2 illustrates the behavior of g^"""^ for the data of Table 3.7.2.1 for = 8

and the calculation of B^ ^. It is quite clear that the asymptotic beha"vior given in (3.8)

and (3.9) is quickly achieved.

7.3 The Dielectrically Loaded Trifurcated Waveguide

The dielectric loading of the trifurcated waveguide adds an additional numerical parameter, N^. It was generally convenient to choose = = = N^. However, since the

pert\irbation coefficients due to the dielectric decay exponentially, can generally be

chosen" smaller than and N^.

Since there is no existing data for the dielectrically loaded trifurcated waveguide the

following steps were taken to check the programming: (l) The dielectric was combined with

the iinloaded trifurcated waveguide using only a TEM mode interaction. This was a particularly good check for large d or small e^. The data computed agreed with this data. (2) The

results of interchanging b^ and bg yielded the same results. This is an independent check

of the new programming.

30
0.5

O

0.3

T

METHOD OF TRUNCATION

DIRECT

e 0 ASYMPTOTIC ^ ^ CONVENTIONAL

ACCURATE

RESULT

kob^= 1.27046

k^b,= 0.41417

k^b =4.41205

6 7 8 10 12

Fig. 3.7.2.1 Comparison of Methods of Truncation

for the Trifurcated Waveguide.

31
Fig. 3.7.2.2: Typical Perturbation Coefficients

for the Trifurcated Waveguide.

32
Tatle 3.7.3.1 illustrates the change of the data of Table 3.7-2.1 and 3.7.2.2 with the

dielectric parameters: k d = I.256, e = 10, and a/k = 0.01. ^ o r o

Table 3.7.3.1 Convergence Results for the Reflection Coefficient

in the Outer Section of a Dielectrically

Loaded Trifurcated Waveguide.

N

-2.

B

0,0

B

0 ,1

1 0.52772 172. 61+° O.80I+22 158.83°

2 0.52773 172. 61+° O.80I+2I 158.82°

3 0.52773 172. 61+° O.80I+2O 158.82°

k 0.52773 172. 6k° O.80I+2O 158.82°

6 0.52773 172. 61+° O.80I+2O 158.82°

*k h = I.270I+6, k h = O.I+1I+I7, k = 0.20033 00 o 1 o 2

The above data was computed using the conventional truncation method. The convergence is

comparable to the trifurcated waveguide without dielectric loading. The recession corres- ponding to the larger coupling region yielded results comparable to the non-loaded waveguide

junction.

The waveguides in the above example are single moded except for the dielectric region

which supports two modes. Table 3.7.3.2 illustrates the change of the multimoded data of

Table 3.7.2.5 due to the addition of dielectric material with the parameters k^d = 1.256,

e = 10, a/k = 0.01.

r o

Table 3.7.3.2 Effect of Multimoding on the Convergence Results for the

Dielectrically Loaded Trifurcated Waveguide

( C onvent ional Trioncation).

N B

0,2

B

0 ,1

1 0.15261 168.61° 0.88923 160.27°

2 O.II+27I 159.59° 0.88667 160.10°

3 O.II+I87 159.1+9° 0.88517 159.95°

1+ O.II+I68 159.51° 0.88521 159.96°

6 O.II+I6I 159.51° 0.88523 159.96°

8t O.II+I6I+ 159.51° 0.88522 159.96°

lot O.II+I65 159.51° 0.88521 159.96°

*k b = 1+. 1+1205, k b = 0.1+1 1+17, k b^ = 1.2701+6 00 o 1 o 2

tN3= 6

Again the convergence was comparable to the trifurcated waveguide without dielectric loading

A comparison of the above data with that of section 7.2 illustrates the dramatic

effect that dielectric loading can have on the reflection coefficients.

33
CHAPTER k. THE N-FURCATED WAVEGUIDE

Introduction

This chapter presents the extension of the results of Chapter 3 to the more complicated

case of the N- furcated waveguide and its modification due to dielectric loading. The Nfurcated waveguide is a waveguide junction which has received little theoretical attention.

The N-furcated waveguide j^lnction can he used in the synthesis of desired ratios of higher

order modes in mviltimoded waveguides. Also the N-furcated waveguide can he used as a closed region approximation to the reflection and coupling coefficients of a finite array

of parallel plate waveguides.

2. Form\ilation of the Equations

Figure h.2.1 illustrates the N-furcated waveguide and its auxiliary geometry.

The solution to this problem is found by construction N-1 meromorphic functions. The

fiinction associated with the first plate and the (N-l)th plate will only have a single perturbation sum, while the remaining plates will have functions with two pertiirbation simis.

From the canonical solution these functions are readily written.

N-1,R 1 K^-^-(.-Jk ) I n=l n ,

c

(2.1)

N-2-'

where it is understood that the appropriate geometrical factors for the (N-l)th junction

are used in the canonical solution. It is convenient to introduce an additional superscript

R to g^ which refers to the location of the pertiarbation, in this case to the right of the

(N-l)th junction. Similarly,

T^(tj) = H^(u))F^(a)) K^-(a)-jk ) I n=l ^>^-],

(2.2)

where the superscript L of g^ indicates the perturbation is to the left of the 1st junction.

For the Mth junction between 1 and N-1 we have

f M,R y- 00 C

T^(u>) = H^(a>)F^(c.)

M,R M,L ^ kM_(,_j1, )| I _in , ^ Jn ^1

'^n=l n,c,, T n=l n,c / M-1 M

^

(2.3)

Note that there are 2N-U sets of unknown g 's. n The TEM normalization constants are given according to property (V) of section 2 of

Chapter 2 by

K^H^(jkjF^(jk^) = -2jk^ b^ U°-B^°]"

o o,lj

(2.1l)

(2.5)

34
N-l

N-2

N-3

2

I

b.

f

z=0

—

b

1

i

h

h

Pn-2

a

(a) N-Furcated Waveguide

z=0

ii A

N lN-I

Nf

I!

'N-3

(.b ) Auxiliary Geometry

Fig. h.2.1: The N-Furcated Waveguide and the

Auxiliary Problem.

35
oil" o M o o Ml o o,M

(2.6;

where if is the amplitude of the TEM mode incident from the left in the mth junction (refer

to Figure k.2.1 and the subscript of the c's). The U 's are given "by the equations

o

^ Jm. ^ _^ _^ ^(o)

(2.8)

=M-1 ° =M-1

and

U° = ^U^+^B^°] (2.9)

o a o a o,l

We have assumed TEM incidence from the waveguides to the left of the junction. The solution

for the TEM scattered fields for TEM incidence from the largest waveguide is direct since

no higher order modes are excited.

In order to derive equations for the perturbation coefficients, we again make use of

the aiixiliary geometry and insist that the expressions for the modal coefficients in the

various coupling regions he consistent.

For the Mth plate we have in general coupling regions to the left and right of the

plate trimcation. From property (i) of section 2, Chapter 2, we have

gM'R

= C- , , (2.10) n n n,M-l

where

I^.R =

rn]!_ /[F(-y )H^(-y )

' (y , +jk)] (2.11) Vl "m-1 ^ ^'Vl ^ ^''^M-l "'Vl

Equations (2.10) and (2.11) are valid for M = 2,'"N-1.

For the left perturbation coefficients we find from property (iii) of section 2,

Chapter 2

M,L ^ ^+ (2.12) n n n,M

where

K^'^ = -y! „ c /[f1^^y„ „ )H,/Y„ „ )(y„ „ -jkJ] (2.13)

n,c,, M M n,c., M n,c,, n,c,, o

' M M M M

Equations (2.12) and (2.13) are valid for M = l,«-'N-2.

From property (ii) of section 2, Chapter 2, we have for the Mth plate

36
RES[T ,Y ] = ^

sin

^

[K^-1'L]-1 g^-^'^ (2.1^

where we have used (2.12). This equation is valid for M = 2,«'',N-1. Similarly, we use property (vi) of section 2, Chapter 2, and find

V-Y ) = Y c FK^^l'^J-l gM-l'R

(2.15) M 'n,c^ 'n,Cj^ M n ^n

Where we have used (2.10). This equation is valid for M = l,*«',N-2.

Equations {2.lk) and (2.15') represent the desired simultaneous equations for the pertiirbation coefficients. Note that the end plates each contribute only one kind of equation

while the central plates each contribute both kinds of equations. Hence, we have 2N-U sets

of infinite equations for the 2N-U sets of unknown right and left perturbation coefficients

3. Asymptotics

In order to effectively truncate the equations for the N-furcated waveguide, we shall

find the asymptotic behavior of the perturbation coefficients.

Using (page 1^+) we can easily find the asymptotic behavior of K^'^ to be

n

1/2 . n sm

*^M-1

(3.1)

We can also find that similar to C in (3.1), Chapter 3, the expression

n

Hence

-1 . M

n sm

"^M-l-

(3.3)

This is in agreement with the results found for the trifurcated waveguide.

The results begin to deviate from the trifurcated waveguide at this point when finding

the asymptotic behavior of the left perturbation coefficients. In order to illustrate this

consider the case of M = 1. From (ii) of section 2, Chapter 2, we have

mrb

-Ana— sin i=RES[T, ,Y ] n = l,2,«'« (3.U) a I na

Because of the concept of (3-3), the summation,

1,L CO P"

1

-, ^

W-Y— n=l n,c^

of T^(a)) must contribute more than one oscillatory term asymototically in order to meet all

of the edge conditions. In particular, this sum must contribute the necessary terms to

37
satisfy the edge condition of the plates: 2,3, •••N-1. This result can he arrived at hy

considering a sequential collapse of the recessed j-unctions. Figure U.3.1 illustrates this

concept. For the mth junction, we need to satisfy the edge condition for all edges, from n

to N-1. Hence

n,M

f -3/2 . "^Vl n sm + 0

f -3/2 . ^^^Vl ^

n sm

'M J

, ^, -3/2 . ^^^Vl V2 ^ Vl^ + O n sm

'M

Using (2.12), (2.13), and the asymptotic expansion k'^'''^ = O(n^), we oht ain

(3.5)

,, ^ N-M-1 , f M,L V J -1 . n-n

p=l ^ M M=M+1

M+p

I m (3.6)

for M = l,2,"',N-2. These multi term asymptotic forms are necessary so that all the edge

conditions are satisfied explicitly.

The argument presented ahove for the multiterm asymptotic expansions of the left perturbation coefficients is not totally rigorous. This is because we are really only in a position to argue the multiterm expansion of • The remaining expansions do not

necessarily follow, although they do appeal to the intuition.

A rigorous justification of the remaining multiterm asymptotic expansions can, however,

be presented. In order to do this, let us assume the existence of the expansions and then

show that they are necessary to satisfy all of the edge conditions. The procedure is to

examine the leading asymptotic terms of (2.li+). However, we must first consider the approximate forms of (2.l)-(2.3). Using (3.3) and (3.6) we have

^-1,R „N-1,R

^ n=l (o+Y,n, cN-2

-1

H?J-1,R

I nW-^'^

sin nTrb^_^/c^^_^

n,cN-2

(3.7)

and

38
step I Step 2

Step N-2 Step N-l

Fig. U.3.1: The Sequential Collapse of the

N-Furcated VJaveguide Junction.

39
T^(a)) - H^(a))F^(cj) K^-(-Jk,){ I n=l

n

w-Y,

n,c.

, T n sin mrb^/c^

n=l+R n,c

T T

°° 1^ sin n-n-(b-+'b_ )/c^

^2 _V^.t1'L w-y n=l+N n,c

.(3.8)

03 %-2 ^ 1

n sin n-rr (132+132+ •*

n=l+N to-Y,n,c^

and for the central plates

yo)) ^ Vu))F^(co) K^-(u)-jk^){ I 'n=l

w-Y,

^'^M-1

-1 IjM,L M,L

_^^ TD °° sin mTb-Vc^ ^ w ' g

_^ -^,R y M M-1 _^ y n u-Y n=l n,cM

-M,L

g y 1 .-h^tM,

n sin b,,, ^ /c„ M+1 M

n=l+N^'^

'M

.3.9;

-1

n sin mT(bj^^^+-..+bjj_^)/c.M

uj-Yn, cM

where N indicates the number of perturbation coefficients and the superscript refers to the

appropriate coefficient. The notation for the asymptotic pertiirbation coefficients is obvious. However, note that since there is more than one left asymptotic perturbation coefficient in general there is a subscript to distinguish the various asymptotic terms of the

same perturbation coefficient. Since there is always only one right perturbation coefficient,

no subscript is necessary.

When examining the asymptotic expansions we will keep the constants associated with the

expansions for reasons which will become obvious. Using (page 1^) and the results of

Appendix C we have the asymptotic expansion

sm- x^ K - ). g^' - s

' I M,Yn,C,, ^ n^ / mrb.a,

M-1 M M M-1 m=l m^

-1 . M

, „ m sm mTT =N^'^+1 Vl

40
- I M,L

m=l m

N-M-1

I k=l

-1 . m sin irnrM 1=1

N-M-1 , ^^"?;/^M- 1 V —M.L M-1 1=1

c;- , ^S-i - \)

M-l

Also, using the asymptotic expansion (3.6) the right-hand side of (2.14) becomes

mrb.

-mr . "^^^M ^,^M-i,L,-l M-1,L

SXn X ±K 1 g

M-l M-l n 'n

2t\C.

M

N-M-1

M-2 M-l r ^1-1,L . IniT

n-x» ^ Vl Vl k=0

r —1>

M-l i,=0

We thus find that (2.14 oscillatory terms of different arguments times the large parameter

n. The first equation is of the form

M K-) g -g /wniii sin mTTb.,/c,, , o ^1 *m ^ i,>tM,R M M-l

m=l m=l+F

where

r M,L HVI.L r -1 . , , / g_^ -Si ) r, T Sin mirb.,,T/c,, *1 ,Y»-,M»L M+1 M

m=l m=l+N

%-M-l ^^Jj^,L

"^'"^ ^^"^ ^^^Vl+'^+Vl^/'^

-1 . M

_ _TT -M-1,L

'M-l

M-l

^

'^M "m '^M-2

(3.10)

The remaining equations are all similar in form and are given by

-M,L ^ -M-1,L,

M ^1 ^2 ''^M-l

H4,L ^ H4-1,L,

M ^N-M-1 %-M ''^M-l

(3.11)

Equations (3.11) prove that multiterm asymptotic expansions are necessary for every left

perturbation coefficient. If any of these coefficients are set to zero we see that (3.11)

imply that other perturbation coefficients (which we know must be non-zero because of an

41
edge condition) must be zero. Hence, the existence of the multiterm asymptotic expansions

of the left perturbation coefficients is proved by contradiction.

Using (3. 7)- (3. 9) and Appendix C we can easily show that all of the edge conditions are explicitly satisfied just as was the case for the trifurcated waveguide.

4. Truncation of the Equations

The truncation of the equations for the N-furcated waveguide is more difficult than the

trifurcated waveguide because of the asymptotic degeneracy of equation (2.14). Two basic

choices of the truncation method are considered in this section.

The basic difference between methods is the choice of the extra equations for the

asymptotic perturbation coefficients. Both of the methods use the following equations,

which are obtained by using (3.7)-(3.9) in (2.14) and (2.15). For the first plate we obtain

,,1,L 1,L -1 . , , N g -, ^

°° m sm miTb„/CT

1 Y +Y ^1 Y +Y m=l m,c^ n,c^ m=l+N m, c^ n,c^

J m sin mir (b^+b^) /c^

1, L

^2 -,ia,L Y +Y m=l+N ni,c^ n,c^

+

T ^ » m sin mir (b„+b„+' • •+b^^ t)/c-,

%-2 ,Ll,L Y +Y m=l+N m,c-|^ n,c^

(4.1)

+ (^i'^

^ Y„ , g^'^ = kV(y, , +jk ) n = 1,2,.. •,N^'^

n n, c^ 1 n o n,c^ o

And for the central plates we obtain the following two sets of equations

42
And

M,R

,,M,L^-l nTT . " M M-1,L r ^^n ) ^ - \

y

M-1 M-1 ni=l m,Cj^_^

"'''m-I

-1 • -u / imM,L M,L

H^,R y M-1 _^ y ^ ~ S Z, M R Y +Y Y -Y

+ So Z

M M-1

°° sin 11177(13 +b )/c M+1 M+2 '

' M

m=l+N ni,c., n,c,, ^ M M-1

00 + ••• + £ > %-M-l „_,Lm,

sin mu(b^^^+b^^2"'...+b^_^)/Cj^

m=l+r''^ ^n,c,, ^n,c,, , M M-1

= _Ay_ _jk) n = 1,2,...,N^-1'^

° '^'^M-l

°

^Jyi-R M,R -1 . ^ ,

y HVIjR y M M-1

n Y -Y ^2 -,^,tM,R Y -Y m=l m,c,, ^ n,c„ m=l+F m^c,, n,c,,

' M-1 M ' M-1 ' M

j,M,L M,L -1 . T_ /

y -M,L y M+1 M

L Y +Y ^1 iLtM,L Y +Y m=l m.c,, n.c,, m=l+F m.c,, n.c,,

' M ' M 'MM

T

°° in"''' sin mTr("b,,. ^b,,, ^)/c,., Hyi,L y M+1 M+2 M

- ^2 tLtM,L y +Y m=l+F 'm.c, 'n,c^,

' M M

1. T

°° sin mTT(b,,, ^ +b,,, +• • •+b,, -, )/c,,

Hy[,L '1,1, r M+1 M+2 N-1 M

y =N-M-1 -,ri<TM,L Y +Y m=l+F m,c,, n,c,,

' M M

,,M,R,-1 M+1,R ,M,, , V

' M

Similarly, for the (N-l)th plate we obtain

43
,,N-l,Lv-l rnr . '^''^^-1 N-2,L

[X^ ) Z sm — g

.J[-1,R N-1,R -1 . ^ , gm H^-1,R y m sin rai^^^_^/c^_^

m=l 'm,c.^ „ 'n,c.. . in=l+r 'm,c., _ 'n,c 'N-2 "'^N-2 - "^'"N-2 "'^N-2

1

o n = 1,2,---,N^"^'^ (1+A)

where

RES[F,,,Y ]H^,(y ) ->M,L 2 M-1 M-1 /, X = -y c., ^ —7—7 (4.5) n n,c., , M-1 _(n), \tt / \ M-1 F.\ ' Y H,, , (y M-1 n,c,, ^ Tyl-1 'n,c., ^ M-1 M-1

F,,(-Y )H„(-Y ) M TD ^ nTTb.,j^T M n,c., M n,c.,

^M ^M+l(-Yn,c,)^+l^-\,. ) M M

The asymptotic choice of the last equations can be logically extended to the N-furcated

waveguide. Proceeding in a similar manner to the trifurcated waveguide, we find the following asymptotic form for equation (i+.l)

N

' V 1»L . -1»L r -1 . ^ ,

. \ .L.l"' smmTTb /c -h + m=l m=l+N '

^n:2 ,Li,L^" sinmTT(h2+b+...+bj^_^)/c^

m=l+N

' ^ ^1 s"'"" = ^o ik.l)

where

%1 =

- \+l ^M-l ^M+1

The asymptotic form of {h.2) has already been given in equations (3.10) and (3.11)

The asymptotic form of {k.3) is

44
V M,R , r -1 . , , 1 g +g AxjnTM Sin imTb^/c„ ^ m=l m=l+N

+ 1 g + gi A T m sin imrh /r "1 tV^^,L M+1 M

m=l m=l+N

00 +

••• + %-M-l ^=Jj^,L TM^(Vl+---+Vl)/"

^ o

(4.8)

M

Similarly, the asymptotic form of (4.4) is

N

' V N-l,R ^ -N-l,R V

1 • ^ / m=l m=l+N

c., „ N-1 *1 o N-2

and Pjj_2 was given xn conjunction with (3.10).

The conventional choice of the truncation as mentioned for the trifurcated waveguide is

apparently not possible in the case of the N-furcated waveguide because of the asymptotic

degeneracy of (2.14). Any choice other than the asymptotic choice for the extra equations

associated with (2.14) will apparently lead to numerical instabilities. Hence, we will use

a hybrid choice. That is, we will use the asymptotic equations for the extra equations

associated with (4.2), that is, eq. 11, and we will use the equations obtained by using the

next index for (4.1), (4.3) and (4.4). In the case of N=3, we will have the conventional

truncation scheme. However, for N greater than 3 we will have a true hybrid choice.

Of course, one other truncation is also possible — direct truncation. As in the case

of the trifurcated waveguide, this solution will not explicitly satisfy all of the edge conditions. However, for many applications the solution may be adequate and even better than

many other more conventional methods of solution.

5. Dielectric Loading

The dielectrically loaded N-furcated waveguide is shown in Figure 4.5.1. The modification of the N-furcated solution to account for the dielectric is similar to the modification

of the trifurcated waveguide given in Chapter 3. Only the results will be givin. T^(a))

becomes

(5.1)

45
46
1 R

Again g^' is asymptotically given by

gl'I^ = 0(n-l e-^-^^^'-) (5.2)

Using the tr-uncated form of (5-1), (^.l) is modified to be

N^'^ g^'^ n ^ ~ -1 -2mM/a

Y nn _ —1,RY m e Ay-Y ~^ ^IRy -Y m=l 'm,a n,c^ m=l+N ' m,a n,c^

rtI N

»L IsIj —1.,/

g_ ^ T

°° m sm mfrD-Zc, + i.y —Y rs

+Y

+

1

y 1 L _

Y +Y

^

m=l m,c^ n,c^ m=l+N ' ni,c^ n,c^

_^ ^

°° m""^ sin mTr(b2+b +'**+b )/c +•••+ e ' y

^ (^i'")"' ^n,c^ ^1 ^n'' = - - (5.3)

1 R

Due to the introduction of g^' , the following equations must be included in the solution

N^'^ g^;'^ n „

°° -1 -2m7Td/a

y _s + -i.R y m e

L Y +Y 1 R Y +Y m=l 'm,a 'n,a m=l+N ' 'm,a 'n,a

„1,L 1,L -1 . , , T N g '

^ ^ 00 m sm mTrb_/c^ ^ a 1,R _ r fm _-l,L r 2_1_

n ^n ^- Y -Y 1 nVT^Tl>L Y "Y m=l m,c^ 'n,a m=l+N iii,c^ 'n,a

, , 00 m sin miT(b^+* • -,)/c, —1,L Y 2 M-1 1 _ o %-2 -iLl'L Y -Y

~

(Y -Jk ) m=l+N ni,c^ 'n,a 'n,a o

n = 1,2,---,N^'^ (5.i+)

— • * * .

where

R F, (-Y ) H, (-Y ) (y +jk ) _ n 1 n,a 1 'n,a 'n,a o

^ ^ ^

n RES[F, ,Y ] (y ) (y -jk ) 1 n,a 1 'n,a 'n,a ^ o

It sho^LLd be noted that the infinite sum involving the asymptotic form of g^' makes an insignificant contribution and could be omitted.

For the dielectrically loaded N-furcated waveguide we are only considering the case of

the hybrid choice of the truncation.

47
6. The Scattered Fields

The previous sections have dealt with the formulation and solution of the MRCT equations

for the perturbation coefficients. Upon finding the perturbation coefficients, we are able

to evaluate the constructed meromorphic functions at the appropriate points in the complex

plane and determine the scattered fields. This is done with the aid of the auxiliary

geometry.

Using the properties of section 2, Chapter 2, we find the following TEM modal coefficients of the scattered fields for the N-furcated waveguide.

-T (-jk ) o,l 2jk^b^

-T^(-jk^) M-1 T^(-jk^)

B u= o 1. V ^ I o-^ M=2,..-,N-1 (6.2)

and

When only a single waveguide is excited with an amplitude of unity, (6.l)-(6.3) represent

either (current) reflection coefficients or (current) coupling coefficients. The reader is

reminded that for TEM incidence from the largest waveguide the TEM solution is immediate.

For the piirpose of this monograph, only the TEM modal coefficients are desired.

Other higher order modal coefficients can be calculated using the auxiliary geometry and

the properties of section 2, Chapter 2.

For the case of the dielectrically loaded N-furcated waveguide, the res\ilts are essen- tially the same as those given above, except we must add in the TEM reflected field from

the dielectric . This yields

-T (-jk ) B n = o\ + U° R (6.^) o,l 2jk^b^ o o

-T,,(-jk ) M-1 T (-jk ) B u= ^ I l-v ° U° R ; M=2,...,N-1 (6.5) o,M 2jk^bj^ 2jk^c^ o o

T.^

^ (-jk^) N-2 T^(-jk ) B .T = ^^^ I o-v U° R (6.6)

2jk^Cj^_^ 2jk^c^ o o

where U° is given by (2.9).

7. Numerical Results

7.1 Introduction

This section presents the numerical solution of the N-furcated waveguide as well as the

dielectrically loaded N-furcated waveguide.

48
7.2 The N-Furcated Waveguide

Since there is no existing data for an N-furcated waveguide, the following steps were

indicative of the validity of the results: (l) The results agreed with the trifurcated

waveguide for N = 3. (2) Reversing the order of b^, h^, b^^, yielded the same results.

(3) The bifurcated waveguide with a magnetic wall was solved similar to the solution given

in section 2, Chapter 2. This solution is given in Appendix E. This canonical solution was

used to solve a trifurcated waveguide with a magnetic wall (ref. Appendix F). This solution was then combined with the solution given in Chapter 3 to yield results for a symmetric

N- furcated waveguide with N = 5- These results were then in agreement giving a simultaneous

check of the magnetic wall trifurcated waveguide program and the N-f\arcated waveguide program, {h) By feeding more than one waveguide simultaneously, we can simulate a trifiarcated

waveguide with a magnetic or electric wall. These results also agreed.

Table h.T.2.1 illustrates the convergence for a case with N=5. In this example a convenient choice of the number of perturbation coefficients was N^'^ = B^^^ = N^.

Table ^.7.2.1 Convergence Results for the N-fxircated Waveguide

(Hybrid Truncation)

N

_E

* B

o ,2

» B

0, 1+ t"^2-1+ -2

3 0.830 155° 0.832 155° 0.101 -5° 0.111 7°

k 0.838 155° 0.832 155° 0.099 -5° 0.105 -5°

5 O.8U2 155° 0.835 155° 0.098 -1+° 0.098 -10°

6 0.81+2 155° 0.836 155° 0.097 -1+° 0.098 -10°

7 0.8iiO 155° 0.835 155° 0.098 -h° 0.102 -5°

8 0.838 155° 0.83it 155° 0.098 -h° 0.103 -2°

9 0.836 155° 0.831+ 155° 0.099 -5° 0.100 -1+° 10 O.83I+ 155° 0.835 155° 0.099 -5° 0.098 -5°

*k b, = k b^ = I.270I+6, k b^ = k b, = 0.1+11+17, k b^ = O.I+OO66. ol o5 o2 oh o 3

tT_ , = B , with B^°l = 1, T) ^ = B ^ with B^°? = 1.

2-1+ 0,1+ 0,2 ' 1+-2 0,2 0,1+

By the symmetry of the geometry -^q 2 ~

"^o 1+ ^^'^ '^2 1+ ~ '^1+ 2" However, because of the larger

coupling region associated with the second plate, the calculations are more accurate for the

fourth plate and hence B^ ^ and T^ ^ are more accurate. This difference is more evident

from an examination 'of the coupling coefficients. For inaccuracies less than a percent,

convergence is essentially achieved for N greater than 5. In order to calculate all of

the TEM scattering parameters to the same accuracy, one can choose the number of the perturbation coefficients to have a gradient, with the larger coupling region having more

coefficients. As an example Tor N"""'^ = N^'^ = 10, N^'^ = N^'^ = 6, and N^^'^ = N^'^ = 3, we calculate the following results for the example given in Table 1+.7.2.1.

49

)\\
2 = 0-833 155°, = 0.835 155°, = 0.099 -5°

The symmetry is obvious.

The above data is for the hybrid truncation method. Table U.7.2.2 illustrates this

same data for the asymptotic choice of the truncation.

Table k.T.2.2 Convergence Results for the K-furcated Waveguide

(Asymptotic Truncation)

N

_i

* B

0 ,2

* B

0

T^2- -2

3 0.838 15i+° 0.831 155° o.ogh 30 0.113 8°

k 0.835 155° 0.830 155° 0.093 6° 0.105 0°

5 0.83i+ 156° 0.835 155° 0.100 -h° 0.098 -11°

6 O.83I+ 156° 0.835 155° 0.102 -7° 0.098 -11°

7

0.83J+ 156° 0.83i+ 155° 0.098 -3° 0.101 -5°

8 0.83k 155° 0.833 155° 0.097 -1° 0.102 -3°

9 0.835 155° 0.83^+ 155° 0.098 -1+° 0.100 -1+° 10 0.835 155° 0.835 155° 0.100 -6° 0.098 -5°

*k b, = k b^ = I.270I+6, k b^ = k b, = O.I+1I+17, k b^ = 0.i+0066. ol o5 o2 oh o 3

tT^ , = B , with B^°l = 1, T, ^ = B ^ vith B^°? = 1. 2-4 0,4 0,2 ' 4-2 0,2 0,4

A comparison of the coupling coefficients with those of Table U.7.2.1 clearly shows that the

hybrid choice is again superior.

As a final comparison, let us compute this same data using direct truncation. This is

shown in Table i+.7.2.3.

Table ^+.7-2.3 Convergence Results for the N-Furcated Waveguide

(Direct Truncation)

N

* B

0 ,2

» B

0, k 4.k <-2

3 0.836 15^° 0.832 156° 0.111 9° 0.111 90

k 0.832 155° 0.827 156° 0.102 0° 0.102 0°

5 0.832 156° 0.831 155° 0.097 -11° 0.097 -11°

6 0.832 156° 0.83i+ 155° 0.096 -11° 0.096 -11°

7 0.833 156° 0.835 155° 0.099 -5° 0.099 -5°

8 O.83U 155° 0.83^ 155° 0.101 -2° 0.101 -2°

9 0.836 155° 0.835 155° 0.100 -3° 0.100 -3°

10 0.836 155° 0.835 155° 0.099 -5° 0:099 -5°

11 0.835 155° 0.835 155° 0.100 0.100 -^°

*k b, = k b^ = I.270I+6, k b^ = k b, = O.Ull+17, k b^ = O.I+OO66. ol o5 'o2 o4 o3

tT^ , = B , with B(°i, = 1, T, ^ = B ^ with B^°,^ = 1.

2-k 0,4 0,2 ' 4-2 0,2 0,4

It is interesting to note the symmetry of the coupling coefficients in the above table.

This is apparently due to the symmetry of the equations. A comparison of the direct

50
triincation method with the asymptotic method of truncation shows that the convergence of the

data computed from the larger coupling region function is about the same. However, a comparison of the data computed from the smaller coupling region shows that we can apparently

order the methods of truncation (with the best method first) as follows: (l) hybrid,

(2) asymptotic, and (3) direct. This is the same conclusion arrived at for the trifurcated waveguide.

It should be noted that the above example is. the same as the first trifurcated example

treated in Chapter 3, except the waveguide has been folded about a symmetry plane. This

allows us to ascertain the acciiracy of the N-furcated results by exciting the waveguides in

a symmetrical manner so as to simulate an electric symmetry wall and hence a trifurcated

I waveguide. Exciting the 2nd and kth waveguide with unit amplitude we find

, = 0.835 exp(l55°) + 0.099 exp(-5°)

o , 4

B^ , = O.lhS exp(l52°)

o , H

This is compared with the value of 0.7^2 exp(l52°) given in Chapter 3, which is very good

j indeed.

' Similarly, exciting the 1st and 5th waveguides with unit amplitude we find

I ^o,5 = 0-168 exp(92°) + 0.223 exp(l6l°)

B = 0.321+ exp(l32°)

0,5

j This is compared with the value 0.32k exp(l32°) given in Chapter 3. Hence, three place

acciiracy is clearly obtained.

7.3 The Dielectrically Loaded N-Furcated Waveguide

Since no existing data is available for the dielectrically loaded N-furcated waveguide,

checks similar to those described for the N-furcated waveguide were performed.

Table i+.7.3.1 illustrates the change of the data of Table U.7.2.1 with the inclusion of

dielectric loading with the parameters: k^d = I.256, = 10, o/k^ = 0.01.

Table U.7.3.I Convergence Results for the Dielectrically Loaded N-Furcated Waveguide

(Hybrid Truncation)

N

_E

* B

0 ,2

*o,h 2- I4 <- 2

1 0.815 153° 0.875 158° 0.098 -23° 0. 061 -122°

2 0.8219 156° 0.870 159° 0.066 -28° 0 0I+3 -19°

3 0.857 158° 0.860 158° 0.061 -27° 0 06I; -h°

h 0.866 159° 0.859 158° 0.059 -27° 0 062 -18°

5 0.870 159° 0.863 158° 0.057 -27° 0 062 -36°

6 0.870 159° 0.861+ 158° 0.057 -27° 0 062 -36°

7t 0.868 159° 0.863 158° 0.057 -27° 0 061 -26°

8t 0.866 159° 0.862 158° 0.058 -27° 0 061 -22°

9+ 0.86U 158° 0.862 158° 0.059 -27° 0 059 -26°

lOt 0.862 158° 0.863 158° 0.059 -27° 0 058 -29°

\

=

0 5 I.270U6, k^bg = k b, 0 h = O.I+1I+17, k

0 3 = 0 I40066

nI'R = 6.

= ^o,k = 1' = ^0,2 ^o!u

=

51
As with the unloaded R-furcated waveguide, the results for the smaller coupling region again

exhibit better convergence. In order to ascertain the accuracy of the results, we can again

simulate an electric symmetry wall at x = a/2 by exciting the waveguides appropriately. Exciting the 2nd and kth waveguides with \mit amplitude we have

B , = 0.863 exp(l58°) + 0.059 exp(-2T°) 0,4

= O.80U exp(l59°)

This compares with the value O.80U exp(l59°) obtained in Chapter 3.

Similarly, exciting the 1st and 5th waveguides with unit amplitude we have

B = O.1U9 exp(-158°) + O.kQh exp(l62°)

o »

5

= 0.527 exp(lT3°)

This compares to the value 0.528 exp(lT3°) obtained in Chapter 3. Thus three place accuracy

is again obtained.

52
CHAPTER 5. OTHER CLOSED REGION PROBLEMS

1 . Introduction

This chapter considers the application of the MRCT to four additional problems in order

to illustrate the ease of applying the method to various kinds of problems . For example,

the problem of finding the eigenvalues of a single ridged waveguide is outlined.

Another problem discussed is the scattering by a dielectric step. This problem has

been solved by Royer and Mittra (l9T2). However, it is believed the solution outlined would

be easier to derive than Royer 's.

Another problem considered is the variation of the reflection coefficient of a rectangular waveguide with a change in the permittivity and conductivity of the dielectric loading.

Numerical results are given.

2. TE Eigenvalues of Ridged Waveguide

Figure 5.2.1 illustrates a cross sectional view of a single ridged waveguide as well as

the associated auxiliary problem. (The dimensions have been changed from those of Chapter 2

in order to conform with standard notation used with ridged waveguide. ) The basic difference

of this problem in comparison with the problems solved previously is that there are no

sources. The problem is homogeneous.

Montgomery (l97l) illustrates how the transverse resonance argument is applied to the

ridged waveguide eigenvalue problem. In order to find the dominant TE^^ mode eigenvalue, we

must make the boundary condition at the symmetry wall of the magnetic type. In this formulation, we must then make the reflection coefficient at the first dielectric -1. This implies

oo. For the boundaries at x = a/2 and x = s/2 to be electric conductors the reflection

coefficients must be +1 in the limit for the auxiliary problem solution to approach the

desired solutions.

The solution essentially proceeds as that of the E-plane step given in Chapter 2, except

that T(cj) must have two additional terms. From Chapter 2, we have

c- b d

oo g <» g

n

T(aj) = H(aj)F(a))

r c- D a

° ^n=l '^-^c n=l '^^^nb n=l ""^d^'

(2.1)

From Chapter 2, section 3, we can easily find that for a right-angle corner,

g= = 0(n--'/6)

in the limit as °°' Since g^ aJid g^ decay exponentially we can truncate these series

without any loss of generality. Hence we can write

r T(w) H(to)F(cj)

r N g^

° V=l "-\c

-T/6 N

_^ -c V n

(2.2)

N.

/,\4-'V n=l+N 'nc n=l 'nb n=l 'nd

53
-— a

s

d

1

Symmetry boundary

(a) Ridged Waveguide Geometry

(b) Auxiliary Problem

Fig. 5.2,1: Ridged Waveguide

54
Again from Chapter 2, section 3 we find that

K - I g„ - g I n ' - g - ^ g = 0. (2.3)

n=l n=l+N n=l n=l

c

In order to arrive at the additional equations, we require that the modal coefficients be

consistent in the various region. From properties (i) and (ii) in section 2 Chapter 2, we

have the following equations,

, (o) niT . mrd ''^nb^''^ RES[T,-Y^^] = .B;-ii^sin^ e (2.Ua)

RES[T, Y^^] = -B^ ^ sin

^

e"'"^'''' (2.Ub)

where n = 1,2,3, •••N, , and where B^°^ is the coefficient of the nth mode incident on the b n junction from the region with dimension, b. is similarly the nth modal coefficient

away from the junction in the region with dimension, b. But from the boundary condition

S

at X = -"2" we must have

B^°) -(a-s)Y

n nb^ , — , =e (2.5) n

Combining (2.U) and (2.5) we have that

-(a-2s)Y

RES[T,-Y^^] = e RES[T,y^^] (2.6)

Similarly for the region with dimension, d, we have

where n = 1,2,*«*,N^. But from the magnetic symmetry condition at x = o we have that

-f-=-e (2.8) n

Hence, combining (2.7) and (2.8) we have that

-2sY -, T(Y^<i) = e T(-Y^^) (2.9)

Equations (2.6) and (2.9) express the interaction of the higher order modes with the boundaries at X = a/2 and s/2.

55
Similarly, we can easily find

, \ -Y s/2

T(y_) = C e (2.10a)

nc n nc

Y s/2

T(-Y ) = C Y C e

'^'^ (2.10b)

nc n nc

but from the boundary condition at the conductor we have

C = C^°^ (2.11)

n n

Hence,

One should note that this equation has the most pronounced effect on the solution since this

equation accounts for the change in the edge condition. Equations (2.6) and (2.9) account

for higher order mode interaction with the edge conductor and the ridge symmetry plane. But

since these modes decay exponentially, the effect is not generally great. This can also be

seen from (2.6) and (2.9). As n ^ °°, we have that g^, g^ ^ 0, exponentially. However these

coefficients do become important in the calculation of higher order mode eigenvalues.

For the first few eigenvalues, generally all the higher order modes are cut off except

for the TEM to x mode. Hence, let us concentrate on the TEM equations.

From Chapter 2, we have

B =^C^°^ +^D^°^ (2.13a)

o b o bo

T(jk^) = 2jk^c (B^-C^°^) (2.13b)

T(-jk ) = 2jk c (C -B^°^ (2.13c)

o o o o

T(-jk ) = -2jk d (D -B^°^) (2.13d)

o o o o

From these equations, we can arrive at the following equations

B = 0 +$d(°^ +^C^°^ (2.lUa)

o bo bo

C =^Mo^o ^(o)^£^o ^Co) (2.lUb)

o o b T(jk^) o b T(jk^) o

, ^ , T(-jk ) / N -, T(-jk ) , V C =B^°U^-r-4- D^°^ -- ^<' ? C^°^ (2.ll*c) o o b T(jk^) o b T(jk^) o

56
These equations can also "be written in matrix form

B

= [Q]

o

O

However, from the boundary conditions we have

o

Co) o

(o)

o

(o) o

(o) o

(o)

Thus using (2.l6) in (2.15)

( -jk^(a-s)

e

0

0

= [W]

[I] - [Q][W]

B

0 0

-e 0

( 1 B

o

D

o

C

I o J

= [0]

(2.15)

(2.16;

(2.17)

where [l] is the unit diagonal matrix. For a solution to exist, we must have

det = {[I] - [Q][W]} = 0. (2.18)

In essence, this is the equation for k^. The complete solution to the problem is then found

from equations (2.l8), (2.12), (2.9) and (2.6). Note that all of these equations are homogenous. Hence, a solution exists only if the determinant of the associated matrix of coefficients is zero. The eigenvalues are found "by finding the values of k^ for which this is

true. The associated eigenvectors (arbitrarily normalized) are found by recourse to (2.U),

(2.7), (2.10) and (2.li+).

This section has thus outlined the solution of an eigenvalue problem using the MRCT.

This solution should prove to be beneficial when accurate higher order eigenvalues are desired. Additionally, the dominant eigenvalues can be found from the zeroes of the determinant of a small matrix because of the efficient truncation due to the use of the asymptotic

s

of the unknowns.

57
3. Scattering by a Dielectric Step

Royer and Mittra (l9T2) have considered the solution of the scattering by a dielectric

step using the MRCT. A MRCT solution was used because of interest in high permittivity

dielectric materials. However, the method of obtaining the equations does not appear to be

straight forward. It is the purpose of this section to outline how the extension of the MRCT

can be applied to obtain the equation in a straight forward manner. Figure 5.3.1 illustrates the dielectric step and the associated auxiliary problem. Note

that the auxiliary problem is different from that of Royer and Mittra (1972).

With reference to the auxiliary geometry, we see that we have three distinct junctions.

The first at z = is just a bifurcated waveguide. The second is the junction at z = 0,

which is just a junction between air and a dielectric filling the waveguide. The third is

the junction at z = A^. This jiinction is just a bifurcated waveguide with slab loading.

This junction can be solved in a manner similar to the way the normal bifurcated waveguide was solved in Chapter 2. The main difference is that the modal propagation constants in the

partial slab loaded guide must be found numerically, wheieas they are known in closed form

with no dielectric loading. Hence the solution is obtained by construction two holomorphic

functions. Each function will be of a doubly modified form. The four infinite sets of

equations are found from the consistency of modal coefficients in the regions 0 <_ x <_ b and

b <_ x <_ a where -A^ 5. 5. '^^^ asymptotic behavior of the perturbation coefficients must

be such that when A^, A^ 0 that the edge condition at the dielectric is satisfied. Royer

and Mittra solved the case of TE incidence and thus the edge condition is

Ey = 0(p")

= O(p^-l)

H = O(p^-l)

z

- p ^ 0

where

2 -11

A = — cos —

TT 2

e-1

e+1

where p is the radial distance from the dielectric edge.

k. Dielectric Loaded N-Furcated Waveguide

This section serves as a forum to present some numerical data for the problem already

considered in Chapter k, section 5- In particular, we consider the case of N = the

junction being symmetrical about the center. This particular solution can be considered as

an approximation of the coupling of two waveguides above a homogeneous earth. However, as

shown in part 2 of this monograph this is not generally very accurate. If, however, a

sample can be obtained and placed in a waveguide, this analysis applies provided one replaces

the free space wavelength by the guide wavelength.

58
X

i i

2

(a) Dielect ric Step Geometry

A2 J

a = 00 1^

'/}/////, 1

(t) Auxil iary Geometry

Fig. 5.3.1: The Dielectric Step

59
From this discussion, it is logical to compute argand diagrams for the reflection and

transmission coefficients as a function of the earth's parameters. Figiare 5.U.1 illustrates

the change of the coupling coefficient, i.e., magnitude (dB) and phase versus the

parameters and a. Note that the data is normalized to the case of no dielectric (a subscript w indicates the dielectric is present, a subscript w/o indicates the dielectric is

not present). The resolution for nominal measurement accuracies is quite acceptable. The

resolution for nominal measurement accuracies is quite acceptable. The resolution is

better for the lower permittivity and conductivity cases. However, since there is a larger range in the magnitude than phase, the vertical resolution will not be as great as the horizontal resolution of the figure. Note that the constant curves tend to be

vertical lines for the lower conductivities and hence the main resolution is in £ . This

r can be somewhat remedied by considering the reflection coefficient argand diagram for

figure 5.^.2. Notice that the range of the phase is about a third of that of figure 5.^.1.

However, the range of the magnitude change is greater. These diagrams can be used simultaneously to obtain increased accuracy. One interesting combination of figures 5-^.1 and

5.i+.2 is shown in figure 5- ^-3. This argand diagram uses only the amplitude data of the

transmission and reflection coefficients. No phase data is used. For nominal acc\iracies

of the amplitudes the resolution is again acceptable and no phase data is required.

However, use of phase data will in general allow increased resolution.

This data tends to indicate that remote sensing of the earth with waveguide horns is

reasonable at frequencies where the horns are not excessively large.

The data presented has been for dimensions rather small compared to a wavelength. This

is due to the limitation of the closed region analysis. A broader range of dimensions is more feasible for the associated open region problem. This is the subject of part 2 of this

monograph.

CHAPTER 6. CONCLUSIONS (PART I)

This part of the monograph has presented the MRCT solution of a new class of closed

region problems. The approach has been to solve a canonical problem of a bifurcated waveguide with known incident fields. The solution of a composite problem is readily found from

an associated auxiliary problem.

The particular class of problems solved are problems associated- with the N-furcated

waveguide junction. The convergence of the MRCT solutions is rapid requiring only a few

perturbation terms for any particular meromorphic function constructed.

Data computed using the closed region analysis tends to indicate the usefulness of waveguide horns in remote sensing of the parameters of a homogeneous earth.

It should be noted that the approach used in this report is straight forward to apply

to most problems which can be solved using the GSMT. The advantage of the MRCT is that the

edge condition of a particular problem can be either changed or edge conditions added explicitly. This enhances the convergence of the solution over the GSMT.

60
iTJdB -iVoldB

Fig. 3,k.l: Variation of the Coupling Coefficient

of Tvo Parallel Plate Waveguides with

the Earth's Parameters (k b =k b^=1.0708

o 1 o 5

k^b2=k^b^ = 0 .1+ , k^b^ = 0.2, k^d=0 . 31^^16 ) .

61
(VSWR dB)^-(VSWR dB)^

LO

2j0-

-2.0

^ -4.0

<

{

°= -&0

o> <,

-8.0-

-10.0 -

2.0 3.0 40 52. 60

€ =2.0

:5%=0.00l

*^'^ly:0.05

o/k^OS

€^=150

Fig. 5.1^.2 Variation of the "Reflection Coefficient

of Two Parallel Plate Waveguides with

the Earth's -Parameters (k b =k b^=

o 1 o 5

1.0T08, k h^=k^-b, =0 .U , k \) = 0.2, ' o 2 d 4 o 3

k d=0 . 3lUl6 )

.

o

62
lTjdB-tT„/JdB

Fig. 5.^.3: The Combined Argand Diagram (k b^=k 'b .=

o 1 op

1.0708, k b^=k b,=O.U, k b^=0.2, k d=

o 2 o 4 o 3 o

0 .3liil6 ) .

63
PAET II

SOLUTION OF OPEN REGION PROBLEMS

CHAPTER 7. INTRODUCTION

This part of the monograph is concerned with the analysis of open region waveguide

prohlems. Up iintil now, we have confined our discussion to closed region problems. This

allowed testing of the techniques expected to "be employed on open region problems as well

as solving some interesting closed region problems. The basic analysis was simpler since

branch points were not encountered. In addition to strictly open region problems, we shall

discuss composite problems containing both open and closed region parts.

Historically, semi-infinite waveguide problems like those to be discussed have been

solved using the Wiener-Hopf technique or modifications of the Wiener-Hopf technique (Mittra

and Lee, 1971). However, we have chosen to exploit the techniques developed in the first

part and extend the MRCT to the open region case. Actually, one still has to solve the same equations; however, it is believed that the solution is more straightforward using the

modified function theoretic technique (MFTT). The first use of this method was reported by

Kostenicek and Mittra (l97l). They solved the problem of radiation of a parallel-plate

waveguide into a dielectric slab. It is interesting to note that in Kostelnicek' s original

technical report (Kostelnicek and Mittra, 1969) that the solution of this problem was obtained by limiting arguments applied to the associated closed region problem which was

solved using the MRCT.

In what follows we shall complete the derivation of the technique suggested by

Kostelnicek and Mittra (l97l) and bring it full circle by uniting it with the MRCT. In

this process, the open region analogue of the GSMT is in essence used. However, all edge

conditions are satisfied explicitly in order to enhance the convergence of the solution

over that which would normally be obtained.

The vehicle which allows one to solve a certain class of modified open region problems

is the canonical problem of a semi-infinite parallel plate waveguide. An infinite number of

known discrete modes are assumed to be incident from the interior of the waveguide, as well

as assuming fields with known arbitrary spectra incident on the waveguide junction from the

exterior. This solution (given in Chapter 8) is just the superposition of solutions which

are given in many texts (for example: Mittra and Lee, 1971). However, the combined solution

has never been given and is important in the solution of modified semi-infinite problems.

The problem of radiation from a flanged waveguide is given as a first example of the

technique in Chapter 8. This particular problem was solved using the MFTT by Itoh and

Mittra (l97l). However, the techniques of this monograph allow a simpler derivation of

the solution.

In Chapter 9, the MFTT is applied to the problem of radiation of a semi-infinite parallel

plate waveguide into a homogeneous half space. This problem is in many respects similar to

the problem solved by Kostelnicek and Mittra (1971); however, there are important differences.

First, the problem of singularities is extensively studied and efficient numerical schemes to

64
solve the integral equation are derived. Kostelnicek used the most basic form of point

matching and subsequently had to invert a much larger matrix than necessary. Secondly, the

problem of radiation into a half space is interesting in itself because of the physical

results. Thirdly, Kostelnicek used an incorrect form for his infinite products and hence

apparently did not satisfy the edge condition (Montgomery, 1973).

The problem of a finite phased array is solved in Chapter 10. The importance of this

solution is that it is not necessary to assume the array is flushed-mounted to an infinite

ground plane in order to obtain a solution. Also it is interesting to note that the

solution only involves the solution of simultaneous linear equations as opposed to an

integral equation. This particular solution is extremely importajit to the array designer

because it corresponds more closely to actual practice than the assumption of an infinite

ground plane. It is also possible tO' solve the problem of a finite array with a finite

ground plane; however, the solution will not be given. Comparison with the results obtained

by Lee (1967) for the case of a finite array with an infinite grovind plane yields some interesting results.

Chapter 11 combines the results of Chapters 9 and 10 to give the solution of a finite

array of waveguides radiating into a homogeneous half space (in this case considered to be a model of the earth's surface). Argand diagrams for the variation of the reflection coefficient and coupling coefficient of a two element array as a function of the earth's permittivity and conductivity are given.

Chapter 12 outlines the solution of several other open region problems. Among these

problems is the radiation of a flanged waveguide into a half space. It is interesting to

note that Kostelnicek and Mittra (1971 ) indicated that such a solution was not possible

using the MFTT, indicating that a complete understanding of the method did not exist at that

time

.

CHAPTER 8. FOimDATION OF THE MODIFIED FUNCTION THEORETIC TECHNIQUE

1. Introduction

It is the purpose of this chapter to show that the modified function theoretic techniques can be approached in a direct manner by considering the canonical problem of a semi-infinite parallel plate waveguide with an infinite number of waveguide modes incident

from the exterior. The general solution is obtained from a non-homogeneous Hilbert problem

and can be written in a manner similar to the pertxarbation expansion discussed in Chapter

1. In order to illustrate the method, the solution of a flanged parallel plate waveguide

radiating into free space is given.

2. The Canonical Problem

2.1 Introduction

Because of their importance, we will consider two canonical problems: (l) a semiinfinite parallel plate waveguide with an electric symmetry boundary (ref. Figure 8.2.1),

65
T

b

z=z.

electric boundary

Fig. 8.2.1; Parallel Plate Waveguide

T

b

magnetic boundary

Fig. 8.2.2: Parallel Plate Waveguide

66
and (2) a semi-infinite parallel plate waveguide with a magnetic symmetry wall (ref. Figure

8.2.2). Both of these problems have been solved for the case of a single incident waveguide

mode or plane wave incidence from the exterior. The solution to be given here represents a

general superposition of these solutions. The form of the solution is of particular advantag

when solving composite problems.

2.2 The Electric Wall Case

Let us consider the TM solution of the geometry shown in Figure 8.2.1. The TE solution

follows in a similar manner and will not be given.

The TM fields are derivable from (\> = E and the fields in each region are given by

y

^ n=0 n n

cos mrx

= / C°(X)e-Y^ + C(A)e^" cos X(x-b) dX

z<0; 0<x<b

z < z , x > b — o —

(2.2.1)

(2.2.2)

= / A°(X)e^^ A(X)e-^^

o

^ cos Xx dX z > z , X > 0 — o — (2.2.3)

where the superscript (o) indicates an incident field and

Y ^ = /(nir/b)^ - nb o

and

o

A time convention of e'^'^^ has been assumed and suppressed.

The branch of y is chosen as shown in Figure 8.2.3 so that Re(Y) > 0. The inverse

function X = /y^+k^ corresponding to the upper half of the X-plane is defined as shown in

Figure 8.2.h.

Matching the tangential fields at z = z^ we can arrive at the equations

o

/ A°(X)e ° + A(X) cos Xx dX = (2.2. U)

o

^

-YZ YZ

C°(X)e ° + C(X)e ° cos X(.x-b) dX; x > b,

n=0 n n mrx ^ ^ ^ ^ cos ^r—; 0 < X < b. b — —

67
Im X

-TT2<arg /<y

org 7 = -7r/2

org 7 = 0

org y-ir/Z

X plane

path of integration in 2.2.2,

2.2.3

org / = 7r/2

1 if >

ReX

k arg/=0

org y = -7r/2

Fig. 8.2.3: Branch Cut Selection of y=:/\''^^^^^

68
69
and

/o A°(x)e ° - A(A)e °

Y cos Ax dA =

-/

o

-yz yz

C°(A)e ° - C(A)e °

Y cos A(x-b) dA; x > b.

(2.2.5)

-I Y n=0 nb

r

, V -Y , z Y -uZ fg(o)^ nb o

_ g ^ nb o n n

rnrx _ cos -r—; 0 < X < b. b — —

We may use the orthogonality of cos nTrx/b and eliminate coefficients to find the following

two equations in B , B^°^, A(A), A°(A). m m

Y T,z mb o

Yz^

°° O / \ o

, „ fflD o / ^ vm+l r A (A) A sin Ab e dA

Y , b e B e = (-1) I mb mm ' Y •, - Y o mb

-YZ

°° / \ o ^_^jm+l

j A(AJA sin Ab e dA_

o ^mb

(2.2.6)

^(o) "Vb^

Y , b e B e mb mm

YZ °° O / \ o

o _

^ i)TM"'"-'- j

A_ L^l\ _siiL_Ab_e dA^ Vb Y

-YZ,

^ i)TM"''-'- j A(^)'^ sin Ab e dA

o ^mb ^ (2.2.7)

where

m 11, m > 1

and m=0, 1, 2, ... . It should be noted that the integrals are not Cauchy principal

values since the apparent pole is actually a removable singularity.

The equations involving C(A), C°(A), A(A), and A°(A) are more difficult to obtain

because of singularities.

Consider multiplying {2.2.h) and (2.2.5) by cos a(x-b) and integrating x from b to <».

From the orthogonality of the eigenfunctions in region c we have

/ cos a(x-b) cos A(x-b) dx = ^ 6(A-a)

b

(2.2.8)

Using Gel 'fand and Shilov (196^+), we can also find

/ cos a(x-b) cos Ax dx = ^ cos Ab 6(A-a) -

A sin Ab

a2 - a2

(2.2.9)

70
In toth (2.2.8) and (2.2.9) > <S(*) is the Dirac delta function. Using these results we may

find

rz rz TrC(a)re =Trr cos at A (a)e

" „o,,., . ^^o ,f^s, . "^^o (2.2.10)

_ -pY I A (X)X sm Xh e dX_^ j A(X)A sin Xb e dA

o Y-r o Y + r

and

-rz -rz

ir C°(oi) r e ° = IT r COS ab A(a)e °

Yz -Yz

+ I A°(X)X sin Xb e ° dX

J

A(X)X sin Xb e ° dX (2.2.11)

o Y+r o Y-r

where

r = -

o

Again, -Tr/2 <_ arg (r) <_ ir/2. Note that the Cauchy principal value is used in (2.2.10)

and (2.2.11). This interpretation of the meaning of the integrals may be found by considering

the transform pair

CO F(a) = / cos a(x-b) f(x) dx (2.2.12a)

b

CO f(x) = - / cos a(x-b) F(a) da (2.2.12b)

O

We may consider that f(x) is our original equation, either (2.2.^+) or (2.2.5). Then

the integrations involved in (2.2.10) and (2.2.11) must be interpreted in a manner such that

(2.2.12b) will yield the original result. In doing this we use the following integral

(Erdelyi; 195^)

00 mr r COS X(x-b) dX TT . / x, N PV j

^ = — sm a(x-bj

2 -v 2 2a

o a'^ - X^

Equations (2.2.7) and (2.2.11) relate the incident fields and the unknown spectrum A(X)

Equations (2.2.6) and (2.2.10) relate and C(X) in terms of A(X).

The solution of these equations is found in a manner similar to Itoh and Mittra (l9Tl).

Consider a function T(l) with branch cuts and as shown in Figure 8.2.5- Then

consider the integrals

(-1)"^'*'^ J

T(a)) do) 1

I

T(a)) dco

2Trj Z (jj - Y T. 2TTj Z 0) - r

^ mb

71
where m = 0,1,2, ... ; and S is the contour shown in Figure 8.2. 5. Then

(-1)"^''^ I

T(a)) d(o ^ (-iT'"^

J

T"(u)) - T^Co)) XdX

^

(-1)°^''^

J

T''(-a)) - T~(-u)) AdX

2Tri o 0) + Y n

(JJ mb

- ^(\h) = ° (2.2.13)

where we have used T (y , ) = t''"(y , ) = T(y -u ) in order to insure that we don't have principal mb mh mb

value integrals. Also we have transformed variables from w to X via XdX = wdw, i.e.,

X = /to'^+k^. We note that on and the value of X is chosen as a positive and real

number according to the path of integration given in (2.2.13).

Similarly

1

I

T(u)) do) _

I

T~(a)) - T"^(a)) XdX

2TTj E 03 - r 2irj o 0) - r to

_^ _1_ °j T"^(-a)) - T"(-a)) XdX

2TTj O 00 + r 0)

- I

[T*(r) + T"(r)] = 0 (2.2.1U)

Comparing (2.2.13) and (2.2,li+) with (2.2.7) and (2.2.11) we find

(i) (-1)""'^T(y . ) = Y , t c B(°)e''"^'°, m = 0,1,2,... mb mb m m

UJZj (ii) T (-o3)-T~(-oj) = 2Trj o) sin Xb A°(X)e °, oieL^

-OJZ ((iii) T~(a))-T (o)) = -2TTj o) sin Xb A(X)e °, 0)eL^

-o)z

(iv) T (u)+T~(o)) = -2Tro) cos Xb A(X)e °

-0)Z

+2Tro) C°(X)e °, ojeL^

We can also consider the integrals

(-1)°^"^^ J

T(o)) do) 1

I

T(o)) do)

2T:j E 0) + Yjj^^ 2Trj E 03 + T

Using the above properties and comparing the results obtained from these integrals we find

upon comparing with (2.2.6) and (2.2.10) the following properties

72
Imo)

Fig. 8.2.5: The Integration Contour, E

73
(v) [-ir^^Ti-y ^) = -Y ^ b e B e^^^'°, m = 0,1,2, mb mb mm 555

(vi) t'^(-u)+T (-0)) = 2770) cos Xb A°(X)e ° - 2Trw C(X)e °, toeL1

Applying the edge condition (Mittra and Lee, 1971) we can easily show that

-3/2, B (-1)"' = 0(m

m

I , m °°

Hence from (v) it follows that

(vii) T(a)) = 0((jj~"^^^), |a)| -> "

The original problem of solving the integral equation is now reduced to that of con- structing a function T(a)) satisfying the properties given above.

Using (iv) and (iii) we can easily find that

—CO z T"(to) = T''(a)) e^^^^ - 2wj o) sin Ab e^^^ e ° C°(X), weL^ (2.2.15)

This relates the discontinuity across to just the known incident field. We may then com- bine (2.2.15) and (ii) to give

T (o)) = T"^(a)) G(w) + g(a)) (2.2.16)

where

G(a)) =

J2Xb

e*^ , (jjeL^

1,

g(w) =

-wz

o • • Tu J Ab o _o / , \ _ -2ttj 03 sm Xb e*^ e C (X), ojeL^

+277 j 0) sin Xb e ° A°(X),

where we recall X = (oj^+k^) ^ has to take a positive value on and Lg according to the

integrals defined in (2.2.13) and (2.2.1^+). Equation (2.2.16) is a non-homogeneous Hilbert

problem whose solution may be found using the theory of singular integral equations

(Muskhelishveli , 1953). The solution is facilitated by first considering the associated

homogeneous problem

X (o)) = X'^(o)) e^^"*"^, weL^; X (o)) = x'^(o)), ojeL^ (2.2.17)

in the absence of any incident field, i.e., A^°^(X) = C^°^(X) = B^°^ = 0. The solution of

(2.2.17) is foTind from a direct application of the Plemelj formulas (Muskhelishveli, 1953).

74
In X(aj) ^ b r dt

0)2 + k2 ^ l: /t^+ k^(t-w)

o 1 o

(2.2.18)

Here, the sign denotes the integration path from jk^ to °° along the (-) side of L^.

details of this integration may he found in Appendix G with the result being

The

In X(a)) _ b 1_

0)2 + k2

^ Ln

- /o)^+k^

hence

X(o)) = H(o))exp

o

w-Zoj'^+k'^

Ln

vhere H(o)) is an unknown entire function found from (i) with B

(o) _ m

(2.2.19)

0. Note that X(o)) has

only a branch cut singularity. Hence

H(o)) = H^(a!) n(o),Y^)(a)-jk^)

where H^(o)) is an entire function which can be found from condition (vii). Before proceeding with the solution it is worth discussing the meaning of the multivalued function

'o) - ^"2+k^

/o)^+k^ Ln

o

appearing in (2.2.19) (Abramowitz and Stegun, p. 6T, I965). This function must be interpreted so that it only has a branch cut L^ with a discontinuity as given by (2.2.17) as well

as being continuous across L^.

Consider the phase of the argument of the log as we traverse L^ and L^ as shown in

Figure 8.2.6. Figure 8.2.7 illustrates the associated variation of the phase of argument

of the logarithm.

Recalling the definition of the branch cut of (o)2 + k2) ^ given in Figure 8.2.^+, we can write the following explicit forms for X(o)) on the top and bottom of L^ and L^. For o)eL^,

X (o)) = H(o))exp

x'*"C(o) = H(o))exp

^ |4^2| IT ' O

'

IT ' O

'

In

0)-i4)^+k'^o

In

o)+/o)'^+k'^o -jk

(2.2.20)

where

75

77
(() = arg

a

o along and

along L

(2.2.21)

Hence

X"(u))

exp

it' o' c a (2.2.22)

Similarly, for oieL^

X"((d) = H(w) exp I /uj^+k^l In

w+/a)^+k^

o

f

X'^(co) = H(a)) exp ^1

TT ' O

' In

w+ZcD'^+k^

(2.2.23)

where <()' and 4' are defined similar to A and 4 . Thus

a c ^a c

X~(a)) _

X"'(a))

= exp (4 '+4 ')] , /w^+k^ ((|) '+<!)' ) , ojeLIT ' o ' a c 2

(2.2.2it)

Clearly then we must have

and

|) + (j) = 2ir c a

a c

for (2.2.17) to hold. Hence we must have

*a - 2'

3l

2

and

^a' = 2' *c'

Thus in order for X(a)) to be the solution to (2.2.17) we must choose a branch cut along the

negative imaginary axis with the argument of the logarithm taking on either 3tt/2 or -tt/2

along the cut depending on the direction of approach.

We can now rewrite (2.2.16) using (2.2.17) as follows

78
X"((jo) X'^(a)) X"(aj)

(2.2.25)

This can be solved using the Plemelj formulas

T(.) = x(.)(p(.) . ! K^iiLtMt _ / £!!l(tldt

X~(t)(t-a)) X (t)(t-a))

where and are defined in Figure 8.2.5, and

.(1)

-coz (1) / \ . J^b o _,o, , V g (u) = -co sin Xb e e C (A)

(2)

"'^^ g (o)) = u si'n XTd e A (X)

(2.2.26)

(2.2.27)

where P(a)) is to be fomd. Using condition (i)

m/ > / 1 \m-+l -u -0(0) ^mb^o

T(y^>,) = (-1) Y^-K ^ e B e mb mb m m

(2.2.28)

Clearly then P(aj) is just a perturbation sum of the form

P(a.) =

(ji) - jk n=l 0) - Y , o nb

where K and g can be related to B^°^ using (2.2.28). Hence

o n m

/DVU) b/cD^+k^ oj-ZcD'^+k^

T(to) = H^(aj) n(a),Y^)exp|—jj:

° Ul a)-Y . l: X-- •

1)/+N „(2)

(.t; dt _ j £

(t)(t-a)) l;!; x(t) (t-w)

(2.2.29)

where we have used the fact that on we have x''"((jj) = X (oi) = X(a)). We note that the term

(-j) in (2.2.19) for the expression of X(a)) has been removed from the logarithmic sign in

(12.2.29). Consequently, the log function now takes a principle value between -p and p.

It is seen that the singularities of the integrands are of two types: a simple pole at a)=t

and a branch cut in X(t) along L-|^, This will be important in later chapters when choosing

an efficient numerical scheme.

From condition (vii) we can easily find

ja)b

2

where = 0.577216... is Euler's constant.

79
Equation (2.2.29) is very reminiscent to the perturbation expansion used in connection

with modification of the bifurcated waveguide. There are two primary differences: (l) the

homogeneous solution has changed form to reflect the removal of a conductor to infinity,

and (2) the summations associated with the regions which become infinite become integrals.

Equation (2.2.29) represents the complete general solution of a semi-infinite parallel plate

waveguide

.

2.3 The Magnetic Wall Case

Let us now consider the TM solution of the geometry shown in Figure 8.2.2. Since many

of the details are similar to that of the dielectric wall case, only the distinctive results

will be presented.

The TM fields are derivable from ij) = and the fields in each region are given by

* =

I fB(°) e"'2n-1.2b^ ^ ^ ^ B n n n=l ^

^2n-l,2b^

o

^ C°(A)e"^^ + C(X)e'^^lcos A(x-b) dAJ

sin k,x z<z,0<x<b nb — o — —

z < z , X > b — o —

sin Ax dA, z > z , X > 0 — o —

(2.3.1)

(2.3.2)

(2.3.3)

where

nb

(2n-l)Tr

2b

In a manner similar to the electric wall case we can find the following integral

equations

yz

'^2m-l,2b^o , , ^m+1 7 A°(A)A cos Ab e ° dA ^ ^2m-l,2b ^m ^ = ^'^^ i ^2m-l,2b - ^

^ i)TM'*'-'- I A(A)A cos Ab e dA

^2m-l,2b ^ ^

2m-l , 2b m

' °(A)A cos Ab e ° dA

'2m-l,2b + Y

(2.3.M

-yz

m+1 r A(j^)A cos Ab e ° dA (-1)""^

/

o Y2m-l,2b - Y

(2.3.5)

where m = 1,2,... . Note that as in the electric wall case the Cauchy principal value is not

required for (2.3.^) and (2.3.5). Also

80
rz

IT CCa)r e = IT r sin ah A°(a) e °

YZ -YZ

_^ J

A (XJX cos Xh e dA j A(X)X cos Ab e dA_ ^2 36)

o Y-r o Y+r

-rz -rz irC(a)re =TTr sin ab A(a) e

YZ -YZ

I

A (A)X cos Al3 e dA _^ py j A(A)A cos A"b e dA_ ^2 3 o Y+r o Y-r

Note that the principal value is again required for the equations associated with both of

the open regions.

The solution to equations (2.3.^) - (2.3-7) is foimd by considering the following

integrals

(-l)"^"^-*- IT(a)) dcj 1

I

T(co) do)

2^^^ ^ '^±^2m-l,2b' ^ " *

^

where T(a)) has branch cuts and as shown in Fig\ire 8.2.55 and m = 1,2,3,... . is the

same contour as the electric wall case. Then comparing the results of the above integrals

with (2.3.U) - (2.3.T) we can find

r-\ ! ^ x, -0(0) ~^2m-l,2b^o ^ _ _ (1) (-1) T(Y2^_i^2b^ = ^2m-l,2b ^ \ ^ ' "1=1.2,3,...

uz

(ii) T (-a))-T~(-(jj) = -2Trj w cos Ab A°(A)e °, coeL^

-uz

(iii) T~(u)-T (w) = -2Trj o) cos Ab A(A)e °,

(iv) T'''(a))+T~(co) = 2tt w sin Ab A(A)e

-(OZ o

-(OZ

-2ir 0) C (A)e , ueL

1

I \ / -I \'^'^^m/ \ -u T> 2m-l,2b o , „

(v) (-1) T(-Y2^_i^2b) = -Y2m-l,2b ^ V ' ^=l.-2,,

OJZ (vi) T (-a))+T"(-(o) = -2tt u sin Ab A°(A)e °

uz

^Ptt 0) C(A)e °, ojeL^

Also the edge condition requires

(vii) T(a)) = 0(0)"-'"^^), |(jd| ^ co

Using (iii) and (iv) we can easily find

81
T (w) = -e^^'^^T'^(a)) - 2TM C°(A)e ° e^^^ cos Xb, toeL, (2.3.8)

This relates the discontinuity across L^^ to the knovm incident field. We may then combine

(2.3.8) with (ii) to give

T (o)) = 1^(0)) G(u) + g(a)) (2.3.9)

where

G(a3) = '

j2Xb

-e , weL,

1, weL,

'Xb o ~'^^o

-27703 cos Xb e'' C°(X)e °, toeL^

-wz

+2Trj(ji) cos Xb A (X)e , (jJEL,

Equation (2.3.9) is a non-homogeneous Hilbert problem similar to the electric wall case. The

primary difference with the electric wall case is the presence of a minus sign in the

homogeneous problem.

X (u) = -X'^(w) e^^''^, weL,

(2.3.10)

X (o)) = X (w),

The solution to (2.3.10) clearly involves the factorization of exp(j2Xb) as in the electric

wall case. However, we must also have a minus sign discontinuity. Such a function is by

inspection /w-jk . Hence

X(a)) = H (o)) n (w,Y„, ) /co-jk

odd (2.3.11)

exp

b/w^+k^

o

'a3-/a)2+k2

In

The general solution is now easily shown to be

T(w) = H2(a)) n (ui,y^^)/ui-2k^ expj —

odd

'co-/a)2+k2

In

y

, r g^'^(t) dt . g^^^t) dt

n=l ^2n-l,2b X (t) (t-o)) X(t) (t-u).

(2.3.12)

^2^^^ is found from (vii) to be

E^ii^) = exp \^ 1-C -In

e

r2k b- o jcob

2

82
This choice of li2M is the correct one in order for condition (vii) to hold. Note that the

asymptotic behavior of the odd infinite product is quite different from the complete infinite

product. This combined with the behavior of the term /oj-jk and the perturbation sum ensures

-1/2 ° that T(u) = 0(0) ' ), |a)|-x».

Using (i) we can relate g to B^°^ " n n

(-l)"''^(^2m-l,2b) = (-1>" «2(^2m-l,2b) ^^"^ (>2m-l,2b'^2b)

odd

*'^2m-l,2b-j^ W2m-l

In

r„ , -(2m-l)TT/2b-) . > 2m-l,2b , ITT +2

'2m-l,2h

, „(o) '^2m-l,2b^o

^m = ^2m-l,2b ^ \ ^ (2.3.13)

Similarly using (2.3.9) we can find

(1)/- \ • -vi. j^b „o^,. ^^o ^ g (w) = JO) cos Ab e C (A)e , coeL^

("2") "'^^ g (w) = +(1) cos Xb A (X)e ,

3. Formulation and Solution of Composite Problems

The key to the modified function theoretic technique is the identification of an auxiliary problem. The auxiliary problem is such that the solution may be identified in terms of

soluble problems.

Before proceeding to other problems let us illustrate this process with the open region

analogue of an E-plane step — a flanged parallel plate waveguide. This problem has been

solved by Itoh and Mittra (1971) using this same technique, but it is believed that a derivation based on the concepts of this work are perhaps clearer. Also Kostelnicek and

Mittra (1969) indicated that a solution was possible as well as sketching the equations.

Figure 8.3.1 illustrates the flanged waveguide and the associated geometry. Notice

that the associated geometry has a recessed dielectric of finite permittivity. As 6 ^ 0

and the auxiliary problem coincides with the original flanged problem. This auxiliary problem allows us to perturb the parallel plate solution effectively.

Consider the case of a TEM waveguide mode incident on the junction, then from (2.2.29) we can write

T(a)) = X(a))

K

(0- jk

(1)

(t) dt

X (t)(t-a)) (3.1)

where

X(u)) = H^(aj)(a)-jk )n(a),Y,)exp

1 o b

b/w^+k^ foj-Zw^+k^

In

83
electric wall

(a) Auxiliary Problem

k X

b

^—^ 1

electric wall

(b) The Flanged Waveguide Geometry

Fig. 8.3.1: The Flanged Parallel Plate Waveguide

84
where

H^(u)) = expj— 1-C -In

e 2ttJ 2 J

Clearly the other terms are not necessary since B^°^ = 0, m > 0 and A°(X) = 0 imply that

(2) TM

g = 0, m > 0 and (t) = 0. m

K is known from (i) of section 2.2 to be given by

o

T(jk ) = -2jk b = K

o o o [u)-jk

0) = jk

where B^°^ = 1. o From (vi) of section 2.2 we have

T(-u)) = -WTT C(X), weL^ (3.2)

_+, where T (-w) = T (-w) = T(-a)) for ueL^, since (3.1) only has a branch cut L^. Also from

(2.2.27)

g^-'-^(a)) = -a> sin Ab e^^^ C°(X) (3.3)

where X = tjHhk^ with ImX>_0. However, the junction at z = -5 can be solved to give an

additional relation between C°(y) and C(y), namely

C"(X) = C(X) R(X) (3.4)

where

where

r = /X^ - Ek^

o

.(1) Hence we may combine (3.2) - (3.4) to give the following integral equation for g (to)

(1)/ \ -sin Xb jXb„/,x^/ . g^ '(u) = e-^ R(X)X(-a)) r g^^^t) dt

to+jkik J- -

o X (t)(t+a))

weL, (3.5)

Consider the change of variable

,(1)(^) = -sin Xb eJ'^^R(X)X(-u))G(aj)

Tr(aj+jk^)

(3.6)

Then (3.5) becomes, for weL^,

85
G(.) = K + (.-.jk ) / (3.7)

t + 0)

where

jXb,

Q(j^) = +sln Xb e-^ R(X)X(-aj)

(3.8)

Tr(a)+ jk^) X (u)

Equation (3.7) is identical (with a slight change of notation) to the equation derived by

Kostelnicek and Mittra (1969) and later solved by Itoh and Mittra (1971).

In order to solve (3.7) effectively we should use the asymptotic behavior of G((o) . To

this end consider the field in the dielectric.

=

I D(A) e^^ cos X(x-b) dX

o

Using the other field relations we can easily find

D(X) = e^^-"*^)^

^

C°(X) (3.9)

And from Mittra and Lee (1971) we can easily show for 6=0

D(X) = 0(X"-^^^"'^), Ixl - (3.10)

where

A = — sin (e-1)

2(e+l)

For the case of e -> «=, A = 1/6. Then from (3.3), (3.6), (3.9) and (3.10) we have

G(t) = 0(t"^) (3.11a)

and from (3.8)

Q(t) = 0(t ^) (3.11b)

This is in agreement with Itoh (1972). Using Stieltges transforms we can show

/

Q^*-^ = o(.-i) + o(.-i-^ (3.12)

t + 0)

as °°. Hence we see that the constant terms in (3.7) must cancel as |a)| «=, or

K + /_ Q(t)G(t) dt = 0

or if we write Q(t)G(t) = G t ^ for t > t eL^ we have

o 1

86
K + / Q(t)G(t) dt+c/ tdt=0 (3.13)

° L, t 1 o

This equation is similar to (3.10) of chapter 2 derived for the E-plane step. Equation (3.13)

in conjunction with (3.7) is the solution to the problem since all the modal coefficients

and plane wave spectra are readily found from T(w) with G(ui) determined. The method of solution of the integral equation will be discussed in the next chapter in conjunction with

another problem.

87
CHAPTER 9- A PARALLEL PLATE WAVEGUIDE RADIATING INTO A HOMOGENEOUS HALF-SPACE

1. Introduction

This chapter is directed to the solution of a parallel plate waveguide radiating into a homogeneous half space. This problem has been solved for the case of a dielectric slab by

Kostelnicek and Mittra (1969), (l9Tl). The solution as given here has three significant

areas of research which warrant the inclusion of the problem: (l) it is believed that the

method of formulation and solution is more systematic and simpler to understand than

Kostelnicek; (2) the details of solving the integral equation efficiently are looked into

carefiilly; and (3) the results are physically interesting and have not been obtained before

this work.

2. Formulation of the Equations

Consider the TM solution of the geometry shown in Figure 9 •2.1. For simplicity we will

assume TEM incidence, the general TM solution follows directly.

From Chapter 8 we see that T(w) is given by

T(w) = X(w)

K

r g^^^t) dt

-jk^ " 1+ X(t)(t-u))

O Li-

(2.1)

where

b/w^+k^

X(to) = H^(u))(a)-jk^)n(w,Y^)exp| In

and

H ru)) = exp(— |l-C -m

1 [TT e

k b

o

271 2

where B^°^ = 1.

m

From (iii) of section 2.2 of Chapter 8 we have

T~Ul) - T'^(a)) = -277j to sin Ab A(A), weL^

but for weL^ and (2.2.l6) of Chapter 8 we have

thus

T (u)) = -TTLQe"^'^^ A(X), ojeL (2.2)

Also from (2.2.27) of Chapter 8 we have

( 2 ) o g (to) = u sin Xh A (X), toeL^ (2.3)

88
electric boundary

Fig. 9.2.1: Parallel Plate Waveguide Radiating into

a Homogeneous Half Space

89
But the spectral densities A (A) and A(A) are related "by the reflection coefficient

a"(A) = A(A) R(A) (2.1+)

where

r = /A^ - £k^

o

where the "branch of r is chosen such that Rer>0. Conduction losses in the dielectric are considered hy using the complex permittivity

e = e - j 120TTa/k i2.k)

It should be noted that any layered media can he taken care of by replacing R(A) by its

appropriate value. However, care should be taken that any new singularities introduced

(i.e., poles of R(A)) are properly taken into account. For example, when Kostelnicek and

Mittra (1969) solved the case of a slab they found it necessary to shift the path of integration from to a horizontal path from jk^ to °° + jk^- A detailed study of the variation

of the half space solution in the first quadrant revealed that the original path, L^, was

the best choice, since it apparently gave the smoothest solution.

Before proceeding it will prove to be convenient to change tu to -w in the integral of

(2.1) giving

T(a)) = X(a))

K

u)-jk /

R^^h-t) dt

- X(-t)(t+Oj)

Now we may combine (2.2), (2.3) and {2.k) to give the following integral equation for

(2)

g (-0)), caeL^.

(2) / N _ sin Ab ^ ^ -j Ab ^ g (-w) = — R(A) e " X (o))

K (2)/ , s o

I g (-t) dt

u-jk I- X(-t)(t+a))

° h J

(2.5)

Considering the change of variable

K sin Ab R(A)

[2)/ V _ _o

g (-w) =

TT((jJ-jk^)

e"-^^^ X"(a)) G(a))

we transform (2.5) to

Q(t)G(t) dt QM = 1 + (to-jk ) /

o' J- t + w ^1

where

X _ 1 sin A b R(A)e X (co) ^^"^ = — T^^) xCT

(2.6)

90
This is the equation derived by Kostelnicek and Mittra (1971) except for a slight change in

notation.

The asymptotic hehavior of G(t) is found hy examining (2.3) and {2.h). It is easily

shown that

Qdo) = 0 0) «> (2.7)

and

G((d) = 0(1) , |u)| «> (2.8)

Because of the exponential behavior of (2.7) it is not necessary to include the asymptotic

behavior of G(a)) in the solution and the integration limit on can be truncated at a finite

value

.

The fields are readily derived from T((i)) upon having found G(a)); in particular the TEM

reflection coefficient is given by

-T(-jk^)

^o " 2jk b (2.9)

3. Numerical Solution

The solution of (2.6) requires a careful examination of the integrals which must be

approximated numerically. An examination of the kernal of (2.6) reveals the following:

(i) Q(u) has zeroes of second order at w =

Yj^i^» n = 1,2... .

(ii) Q(a)) has a zero in the complex plane whenever R(A) = 0. For a half space this

occurs at the pseudo-Brewster angle given by:

e 'k r o ^ . 1 , 2(l+e 1+e

° r r

where e = e - Je' and we have assumed e'/z << 1. Note that since Re(Y) must be

r r r r greater than zero on the top sheet, for ^ 0, this root is on the improper

Riemann sheet (though it is quite close to the branch cut).

(iii) Due to the term sin A b, Q(a)) goes to zero as /oi-jk^ as w -> Jk^-

Hence equation (2.6) is a "smooth" equation. However, upon finding G(a)) we desire to calculate the TEM reflection coefficient which in turn involves an evaluation of the integral

/ (3.1)

From (iii) we see that as t jk^ the integrand will behave as 1/ A-jk^. Hence, careful

attention should be given to the branch point t = jk^.

91
The integration was broken into a sequence of finite intervals with the end points being

the waveguide propagation constants, y^-^- The origin was also included as an end point.

Since the first segment included the branch point the following Gaussian quadrature

(Abramowitz and Stegnn, 19^5) was used:

)fhUz= ^ I (3.2) a /b-y i=l

where

y. = a + (b-a)x.

1 1

where x.111 = 1 - and is the ith positive zero of P_ (x) and w. = 2nf^'^^ where w!^'^^ are

2n 1 1 1 the Gaussian weights of order 2n. Equation (3.2) allows the square root sing\ilarity to be

taken care of quite satisfactorily hence allowing a good approximation of (3.1). Although

this effectively increases the order of the integrand of (2.6) slightly, no degradation of

convergence was observed.

Between the remaining end points regular Gaussian quadrature was employed.

It should be noted that Itoh and Mittra (1971) used a pulse function basis with the

exception of the vicinity of the branch point.

k. Nimerical Res\ilts

Table 9.^.1 illustrates the convergence of the TEM reflection coefficient as a function

of the nimber of internals, N, and the number of points, M^, within the nth interval for the

case k b = 1.2566, k d = 3. 1^159, e = 10, a/k = 0.001.

o o r o

Table 9.^.1 Convergence of Reflection Coefficient for a Parallel Plate

Waveguide Radiating into a Half Space.

N M

_2

M3 B

0

3 2 2 2 0.5397 63.22°

3 k 1+ 0.ii366 65.51°

3 8 h 1+ O.U385 65.i+6°

3 8 8 8 0.1+379 66.11°

3 l6 8 8 0.1+385 66. ll+° 3 i6 16 16 0.1+386 66.16°

with no dielectric 0.281+6 88.1+2°

Note that the reflection coefficient converges quite fast and four place acciiracy is

achieved with as few as 32 matching points. However, quite acceptable accuracy is achieved

with as few as 16 points. This appears to be a considerable savings over Kostelnicek and

Mittra (1969), although they used an alternate path of integration on which the solution

varied greater than on the path , though they did avoid the poles of R(A).

92
Figiore 9.h.l illustrates the behavior of G(a)) for the example whose results are given

in Tahle 9.k.l. Note that the asymptotic behavior given by (2.8) is quickly achieved beyond

0) = 0.6. The most radical behavior occurs near the origin; however, even this change is

less than 10%. Notice that G(w) is quite well behaved at w 0.3, which is the location of

the pseudo-Brewster angle. Note also that G(a)) is well behaved near o) = y^^. The phase of

G{iii) was a maximum of 8° near the origin, with a nominal value of less than a degree.

Since the distance between the half space and the waveguide was a half wavelength,- the

exponential decay along was sufficient to restrict matching intervals to no more than

three.

Another case is given in Table 9- ^-2 where the waveguide width has been increased to

k b = k.'J12h, and the distance from the waveguide to half space has been decreased to

0

k^d = 0.5 (d - O.OBa). The parameters of the dielectric are still = 10 and a/k^ = 0.001.

Table 9- ^-2 Convergence of Reflection Coefficient for a Parallel Plate

Waveguide Radiating into a Half Space.

N M, M, M„ B __l_2_3_ii_5_D_7 0 Skkkhkh— 6222222 — 0.1+333 -!+9.68°

O.kklQ -50.16° 6888Ul+li 6888888 — 0.Uli08 -50.13° — O.Ul+08 -50.13°

6 16 8 8 It U 1+ — 0.I+I408 -50.13° T8888888 O.kkoQ -50.12°

vith no dielectric O.O616 82.62°

More matching intervals are necessary because the waveguide width is greater as well as

the distance, d, being smaller. Four place accuracy is again achieved with excellent results being achieved with as few as 2k matching points.

The solution of a related problem of the radiation of a parallel plate waveguide into a perfectly conducting sheet has been solved using wedge diffraction techniques (Ruddack, Tsai,

and Burnside, 1969). Figure g.k.2 illustrates good agreement between this theory (with

R(A) = e ) and the wedge diffraction resiilts. However, it should be expected that our

result should be better when d/X is less than 0.15 A. (it shoiild be noted that Ruddack'

s

data is subject to the error of reading graphical data.)

Figure 9- ^-3 shows the variation of the reflection coefficient of a O.H X waveguide as a function of distance to the half space for three sets of permittivity and conductivity. Note

that quite significant deviations from the free space case are observed, with the deviations

becoming larger as the distance, d, is decreased.

Figure 9-^.^ illustrates the variation of the reflection coefficient of a O.k X waveguide as a function of the half space parameters at a constant distance, d, of 0.1 X. Note

that quite significant changes in the reflection coefficient as the permittivity and con- ductivity vary. In order to more fiilly understand Figure g.k.k. Figure 9.U.5 illustrates

the variation of the Fresnel reflection coefficient at the half space interface for normal

incidence. Figure 9.^-6 shows the data of Fig-ure 9-^-^5 after the substraction of a parallel

93
1.2

i.l

1.0

0.99

0.98

1 0.97

o

0.96

0.95

0.94

0.93

0.92

jl.O

^ c o o o

kob = 1.2566

kod - 3.14159

-- 10

/y a/ko = 0.001

a; plane

® L| ©

0 1 2

1.0

(jj along L|

2.0

Fig. 9.^.1: Variation of the Perturbation Spectrum

along Li

94
T—T—I—I—I—

r

T—I—I—

r

lO

T —

J L

in

d

J I L

d

Qoo

d

0) p

c(5 H

O

P

o

o

<U H +> O

H -P H O «H 0) 05 0 O ^

(h «H -P

03 ^ C

0)

o

c

(U •HO

•H

o o

•H -P

-Po <u O p! +5 Pti -P

03 (1) ft

bD to <J C c3 •H 0) 4J P ^ 03 (U 4-3 <U -H (l> O t:) ^ S O o3 CO K

O

o

-P •H -P O

O C

0) M ^3 03 H (U t3 -P

<H > C m

0) 03 O -H W IS O O

CVJ

ON

W) •H

'9

95
0 i 1 I I I 1 1 1—

0 0.2 0.4 0.6 0.8 1.0 d/X

Fig. 9.^. 3a: Variation of the Reflection Coefficient

of a Parallel Plate Waveguide Radiating

into a Dielectric Half Space as a Function

of Distance and D'ielectric Parameters

96
-40

-60

Fig. 9.i+.3b

0.2 0.4 0.6

d/X

0.8 1.0

Variation of the Reflection Coefficient

of a Parallel Plate Waveguide Radiating

into a Dielectric flalf Space as a Function

of Distance and Dielectric Parameters

97
o

00

0) -p

H

a ct) H W

(u

oo

H •H •H

0) 5h H +> +J H O C

cd 0) 0)

f^

H rH

cd 0) 0

fl^ •HO

•H

05

cd (U fHO to

-p

-P •H

O

0) •H bO

C

o

C

O

•H •H •H

«H +i -P «H 03 CJ 0) •H G o ?3 o cS K

0}

o <D •H CO <U P •H o3 -P O ^3 d)

<U t>0 0) H 01 o o3 <H l> 03 ^1 <P 03 Ph 03 IS W Ph

•H

98
q:

o o o o

cvj to ^ <P (S9aj69p) y 6jv

99
o

CD

o

o 1 1—1

•HO

H •H

Coc , M

O

MW <U el le W •HO

ie

ft . « o cd

-p

O

-P -p

(U •H •H tin O O

•H bO SH

ft . •H

o

-p •H

o cd -p o TP nc

ce

o fin •H

(V 03

o

0) •H

W

H 03

0) <D CO > CJ Jh

03 0) rs ft -p

0) 0) <u S

CJ -p «H oJ H H o3 o3 Oh

0\

•H

100
FREE SPACE

WAVEGUIDE

ADMITTANCE

PARALLEL PLATE

TRANSMISSION

LINE

|—o— —o—

I

FREE SPACE

TRANSMISSION

LINE

(Plane wove

approximation '.

FRESNEL

ADMITTANCE

Fig. 9.h.l[: Approximate Equivalent Circuit

101
1

I

impedance of a single isolated waveguide radiating into free space. The similarity of the

data of Figure 9* ^-6 with the Fresnel reflection coefficient is now ohvious. Figure 9- ^-7

suggests the possihility of establishing an equivalent circuit for a waveguide radiating into

a half space. The transfonner allows for the scale change and is dependent on the waveguide

width 2"b and the distance d primarily. The line length, 1^, is just the physical electrical

distance 2k^d. The second line length, Si^, is a rather complicated function of e, a, and d.

For a given height it is possible to arrive at empirical formulas for the circuit parameters.

This would suggest that the more complicated structures such as rectangular and circular

waveguides radiating into a half space can be modeled with approximate equivalent circuits,

with the parameters of the circuits being determined experimentally.

CHAPTER 10. A FINITE PHASED ARRAY

1. Introduction

Waveguide phased arrays have received much attention in the last few years because of

properties such as fast scan capabilities, multimode operation, and reliability. Perhaps

the easiest analysis of planar phased arrays has been the application of Floquet's theorem

to an infinitely periodic array (Amitay, Galindo, and Wu; 1972). However, many arrays are

small enough that such an analysis is not valid. For finite arrays, one common method of

analysis has been the moment method. One common approximation in these studies has been

the assumption of an infinitely large perfectly conducting ground plane (or some approximation to it). It is the purpose of this chapter to sue the modified function theoretic technique to study a finite phased array with no ground plane. It should be noted that this

analysis could also be easily applied to a finite array with a ground plane of finite or

infinite extent.

2. Formulation of the Equations

2.1 Introduction

For simplicity in the solution we will assume that the array has a symmetry plane parallel

to the waveguide walls. This is not a limitation of theory but is only a convenience. The

solution of a completely aperiodic array can be found in a straightforward manner.

Thus we will consider the solution of the two problems illustrated in Figures 10.2.1

and 10.2.2, the only difference being the symmetry wall boundary condition.

2.2 The Electric Symmetry Wall

Fig\ire 10.2.1b also illustrates the auxiliary problem. Note that the problem can be

further separated into two kinds of problems: (l) the interior problem, and (2) the exterior problem. The interior problem is the one associated with the first N-1 plates,

and is solved using the theory of Chapter 2 for modifications of the bifiircated waveguide.

102
N-lN-2-

.N

N-l

z-0

electric boundary

(a) Final GeometryFig. 10.2.1: The Finite Array with an Electric Symmetry

Boundary

103
N-H

N-2—

t

N

Nl-i^— c

T

C

N-l

c

N-2

~~| Z=0

electric boundary

(h) Auxil iary GeometryFig. 10.-2.1: The Finite Array with an Electric Symmetry

Boundary

104
N

N-l

N-2

,N

z=0

magnetic boundary

(a) Final GeometryFig. 10.2.2: The Finite Array with a Magnetic Symmetry

Boundary

105
N-2—

t

N

NT

N-I

I

z=0

magnetic boundary

(b) Auxiliary Geometry

Fig. 10.2.2: The Finite Array -with a Magnetic Symmetry

Boundary

106
The exterior protlem is the one associated with the Nth plate. This is seen to be just semiinfinite waveguide with an internal modification. With these thoughts in mind we can write

the following N holomorphic functions:

T^(a)) = F^(a)) K

(1)

1,R

-(a)-jk ) I ^n

n=l (ji+y

(2.2.1)

^n=1 n,c

M,R

=n

M

n=l n ,

c

M-1

(2.2.2)

where M = 2,3,...,N-1, and

Tj^(w) = X(a))

oo N,L

o u-Y

^=1 "'^N-lJ

(2.2.3)

In equations (2.2,l) and (2.2.2), F(a)) is given "by

F^(co) = H^(to)

n(a3,Y, )n(a),Y, ) 1 _2_

n(a),Y ) 1

where

H^(cd) = expj-^ [b^lnb^/c^ + b^lnbg/c^]

and

Fj,(a3) = Hj^(a3)

n(tj,Y )n(a),Y, ) M-1 M+1

n(w,Y^ ) M

where

H,,(a)) = exp-i— [c,, , Inc., ,/c,, + b,,, Inb,,. ^ /c,,] M IT M-1 M-1 M M+1 M+1 M

In equation (2.2.3), X(a)) is the homogeneous solution given in Chapter 8 for a semi-infinite

waveguide with a half height of c^ ^, divided by (cj-jk^)

From Chapter 2, we see that we can write the following equations:

(-1)'^*-'"T (-Y ) = Y c

^ ' M^ '^n.c,, / ^n,c,, , M-1^ n M-1 M-1

(2.2.U)

107
where M = 2,3,...,N-1, and

niT c K^'^ = =^ sin ^/[FJ-Y )(y +jk )]

Also for M = 1,2,3, ... ,N-2, we have

mr c RES[T^,Y. „

] = ^sin

—-^ tK^^-l.L^l M.1,L

(2.2.5) M n,c,, c., c., M M M

where

M-1 M-1 M-1

(n) where F (y ) implies that the zero at y is omitted from the infinite product.

'^''^M-l '^''^M-1 For the case M = N-1, (2.2.5) becomes

HES[T,, ,,Y_ ] = ^

sin [k!'^^^^^ g?'^ (2.2.T)

"''^N-l "N-1 =N-1

where

= (-l^'^nc W[^^'^(^n c )

• (^nc "^^^^ 'N-1 ' N-l ' W-1

where X^'^^(y ) implies that the zero at y is omitted from the infinite product.

'^'^N-l '^'''n-1 Equation (2.2.7) is found hy using property (i) of Chapter 8 in conjunction with the results

of Chapter 2. From property (v) of Chapter 8 and the results of Chapter 2, we also have that

Equations ( 2 . 2 . it )- ( 2 . 2 . 8) represent an infinite set of simiiltaneous equations for the perturbation coefficients, g^'''^ (M = 2,3,...,N) and g^'^ (M = 1 , 2 , . . . ,N-1 ) . Note that the recession

of plates in the solution of the closed region case is opposite to the recession chosen in

Chapters 3 and h. However, this is only a minor change as will be discussed later in this

chapter.

For this particular problem, K^'^^ (M = 1,2,...N) are known and can be related to the

o incident TEM modal coefficients as follows (ref. Chapter 2):

.(1) _ o.,, V Ttt Jo)

where

1 c^ o,2 c^ o,l

108
and where B^°^ is the incident TEM modal coefficient from the nth waveguide above the symo,n

metry boundary. Also

= 2Jk^b3[U2-B(°)]/F,(jkJ

where

2 c^ 0,3 1

and in general for M up to N-1, we have

where

Ti = Vi (o) .flizin

- c^ o,M+l c^ M-1

And for the case M=N, we have

.(N) _ K;

= -2jk^ c^_^ Vi/^(j^o) (2.2.11;

where

U

- A_b(o) ^. !n^„

" =N-1 "n-1

In order to solve ( 2. 2. 1+ )-(2. 2. 8) efficiently, we are motivated to investigate the

asymptotic behavior of the various perturbation coefficients. This procedure is essentially

identical to that discussed in Chapters 3 and h and thus only the results will be presented.

As in Chapters 3 and k, we can find that only a single asymptotic perturbation term is

necessary for the right perturbation terms, that is, we will replace S^'^ by the following

for n > N^'^

M,R ^^,R, , sn -1 . M-1 „

I = g

' (-1) n sm — (2.2.12)

M

where M = 1,2,..., N-1. Notice that since c,, ^ = c,,-b,,,-,, (2.2.12) can also be written

M-1 M M+1

M,R Hyi,R -1 . ''^M+l = _g n sm —M

which is in agreement with equation (3.3) of Chapter k.

109
We again find that multi-term asymptotic expansions for the left pertiirbation terms are necessary to satisfy all of the edge conditions explicitly, namely for n > N^'''^ we will

M L replace g^' by the following:

M,L -M,L/ T vn -1 . 1

g^ = g^' (-1) n sm——

M-1

+ i^'^(-l)V^ sin nTT -^^i^^ + + (2.2.13)

^M-1

_^ -^,L, -, sn -1 . ^W'-Vi^

M-1

where M = 2,3, ... ,N.

Upon the substitution of (2.2.12) and (2.2.13) into equations (2 . 2.1 )-(2. 2. 3) and the

subsequent substitution into ( 2. 2. U)-( 2. 2. 8) we arrive at an efficiently truncated linear

system of equations for the perturbation coefficients.

However, we must still decide how to choose the additional equations for the asymptotic

perturbation coefficients. We may, however, use the results of the truncation study of

Chapters 3 and h, and use what we call the hybrid truncation method. Essentially this

choice of truncation chooses the "n+1 equation" of equations (2.2.U), (2.2.6), and (2.2.7).

However, (2.2.5) is asymptotically degenerate for reasons outlined in Chapter k. Hence, the

true asymptotic form of (2.2.5) is used, yielding M+1 equations.

For brevity, we will not give the explicit form of the equations. The interested reader

is instead referred to Chapter h.

Upon finding the perturbation coefficients, the waveguide fields as well as the fields

in free space are readily found using the properties of the functions as given in Chapters 2

and 8. In particular, the reflected TEM modal coefficients are given by:

-T^(-jk) T^(-Jk) N-1 T (-jk ) ^oM = -^^ —- I — (2.2.11.)

2jk^ b^ 2jk^ c^_^ n=l 2Jk^ c^_^

for M = 2,3,...,N. The summation is only used for M <_ N-1. However, for the case M=l we have

2jk^ b^ 2Jk^ c^_^ n=2 2Jk^ c^_^

In both (2.2.11+) and (2.2.15), B is the reflected TEM modal coefficient in the nth

o,n

waveguide above the symmetry plane.

Other waveguide modal quantities can be easily found by recourse to the properties of

the canonical function given in Chapters 2 and 8 and the use of the auxiliary geometry.

The far-field radiation pattern is also of interest for a finite array. From equation

(2.2.16) of Chapter 8 and property (iii) of the same chapter, we can easily find that the

spectral density for z > 0 is given by

110
A(A) = , caeL,1

(2.2.16)

Hence,

^ «> T„(a)) -J Ac,, ^

, / V -If N N-1 -uz , d).(x,z) = — e e cos Ax dA

o

(2. 2. IT)

This integral can easily be evaluated asymptotically in the far field using the method of

steepest descents (Mittra and Lee, 1971 ) to give

A k r

-J(k r-Tr/l+)

2t\

Tj^(jk^ cos 6)e (2.2.18)

where 6 is the polar angle measured from the z axis. It is interesting to note that the

factor cos 0, which is present for an infinite array problem, is not present in (2.2.18).

2 . 3 The Magnetic Symmetry Wall

Figure 10.2.2 illustrates the geometry of interest in this section as well as the auxiliary problem. The two basic components of the solution are: (l) the bifurcated waveguide

with a magnetic symmetry boundary, and (2) a semi-infinite parallel plate waveguide with a

magnetic symmetry wall. The first problem's solution has been given in Appendix E, while

the second problem's solution has been given in Chapter 8.

From Appendix E, we can easily find that for the first plate we have

T^(a)) = F^(cj) K

n=l '^"'^2n-l,2cJ

l-"

(2.3.1)

where

F^(a)) = H^(a)) n n=l

(l-'^/^2n-l,2b^)(l-'^/^nb2^

(I-w/y2n-l,2c

1

where

H^(a3) = expj-^ b^ m - . b^ m -

And in general we have for the Mth plate (M = 2,3,...,N-l)

^n=l

M,R

g.M,L

'^"^^2n-l,2c,, n=l '^~^2n-l ,2c,,

' , M M-1

(2.2.2)

where

111
F^M = Hj^(a)) H n=l

M

where

H^(<.) = expf^ M-1 c,, M+1 c. 'M 'M

For the Nth semi-infinite plane, we use the results of Chapter 8 and easily find that

Tj^(a)) = X(a)) I n=l ^ ^2n-l,2c

(2.3.3)

N-1

where X(u) is the homogeneous solution of the semi-infinite parallel plate waveguide with a magnetic symmetry wall.

We may arrive at simultaneous equations for the pertur.bation coefficients by a similar manner used in the previous section and in Chapter 2. That is, we require that the expressions for the same modal coefficient in a given region of the auxiliary problem be consistent,

whichever holomorphic function is used.

Hence we may find for M = l,2,...,N-2

M' '2n-l,2c ] M

, f. . r M+1,Lt-1 M+1,L

k cos (k c,, ^ ) [KT J g n,c,, n,c,, M-1 n ^n

M M

:2.3.h)

where

_ (2n-l)Tr

n,cM2c

M

vn 2

(-1) Y C

^ ' ^2n-l,2c,, -, M-1 M-1

F^'^^Y )(y -1

M ^'^2n-l,2c,, /^'^2n-l,2c,, -, ^ M-1 M-1

and for M = 2, 3, • .

. ,N

^ ' '^2n-l,2c„ / ^2n-l,2c,, , M-1 n ^ ^n

'M-1 M-1

(2.3.5)

where

k cos (k c,, ^ ) n,c,, n,c,, M-1

'M 'M

^M^-Y2n-l,2c„)(^2n-l,2c/j^'

M M

(2.3.6)

112
And from the open region part of the auxiliary geometry we find

RES[T,, ,,Yo n o ] = ^ cos k c,, „ • [j^.L(o)]-1 N,L

(2.3-7) N-l"2n-l,2c^_^-' ^"^ " n

where

^,L(o) ^2n-l,2c^_.^

2n-l,2cjj_^

Note that for this particular problem that K^^^ is known and given by

where M = 1,2,,..,N-1 and where B^°2.-, is "the incident TEM modal coefficient from the M+lth

o,M+l

waveguide. For M = 1, we have

o 2^V2 ^o,2/^l<J^o) ^2.3.9)

Note that the equations for these quantities are slightly different from equation (2.2.9)

and (2.2.10) for the electric symmetry boundary case. This is due to the fact that the

coupling regions of the aiuciliary problem cannot support a TEM mode.

Equations ( 2. 3- ^)-(2. 3. 7) constitute an infinite set of linear equations for the perturbation coefficients. In order to solve these equations efficiently, we investigate the

asymptotic behavior of the pertvirbation coefficients. Again the procedure is identical to

that discussed in Chapters 3 and k.

For n > N^'^ we will use

M,R -1, - vn , _ g^ = g n (-1) cos k Cj^_3_ (2.3.10)

M

for M = 1,2,...,N-1. Similarly, for n > N^'^ we will use

M,L H^,L -If , vn , I = g.,' n (-1) cos k b^ , ^n "^M-l n,c,, ^ 1

M-1

+ i^'J n-^(-l)^ cos k^ „ (b +b,) (2.3.11) =M-2 n,c^_^ ^"1 -2

+ ••• + g^'^ n ^(-1)'' cos k^ ^ (K+b^+..-+b^ J 1 n,c,, , Id M-1 M-1

113
With these asymptotic expressions, the proof that all of the edge conditions are satisfied

explicitly follows closely to that of the electric wall case. Upon the substitution of (2.3.10) and (2.3.11) into equation (2 . 3. 1 )-( 2. 3 . 3) and with

the subsequent substitution into (2. 3- ^ )-( 2. 3 . T ) we arrive at an efficiently truncated

linear system of equations for the perturbation coefficients. The extra equations for the

Mth parallel plate region follows by use of *(page ik) and the results of appendix C. We

obtain from the left side of (2.3.^+)

Res(T,„Y

M' '2n-l,2c

/ 2 , >n -1/2 r, l^ ^'^M-l -l . , 1.

^Zy, T" " cosL{n-:r-) —-—] sin(n-;r-)

1Tb.M+1

M

W

K + I ° M=l

00 n'^'''^ M-1 <»

M=n'^'«

^ " ^ M=l " j=l J M,L ^'=M-1 ^ J +1 +1

81

j=2

-M,L

h-1

'M Jl=l

M

(2.3.12)

from the right side of (2.3.'+) we obtain

J- „

V ^^cCi, ^ UvM+I'LtI M+1,L „ f 2Tr , ,n 2 , 1, '^^M-1 r^^+l^L . MM 1 M+2 M M j=l M 1=1

(2.3.13)

Equating (2.3.12) and (2.3.13) produces the extra equations in the same mannaer as in

chapter h. Again, the extra equations arise from the oscillatory terms of different

arguments times the large parameter n. Note that equation (2.3.11) can be expressed as

a sine sieras, placing the asymptotic result more in conformity with appendix A. The

choice of the extra equations for the asymptotic perturbation coefficients is the hybrid

truncation method discussed in the previous section.

For brevity, we will not give the explicit form of the extra equations. The interested

reader is urged to compare the preceding with equations (3.10) and (3.II) in Chapter k. Upon finding the perturbation coefficients, the waveguide fields as well as the fields

in free space are readily found using the properties of the functions as given in Chapters

2 and 8. In particular, the reflected TEM modal coefficients are given by:

2J^Vi

for M = 1,2,...,N-1, and where B ^ is the reflected TEM modal coefficient in the M+lth

o,M+l

waveguide

.

Other waveguide modal quantities can easily be found be recourse to the properties of

the canonical function given in Chapters 2 and 8 and the use of the auxiliary geometry.

114
The far-field radiation pattern is also of interest for a finite array. From equation

(2.3.!+) of Chapter 8 and property (iii) of the same chapter, we can easily find that the

spectral density for z > 0 is given by

-j^<=N-l

-T^(a.) e ^

1

^^'^ = VT^ (2.3.15)

Hence,

00 —J (j)Q (fr^(x,z) = ^ / ^

Tj;(a))e

N-l^-Yz ^.^ (2.3.16)

o

This integral can easily "be evaluated asymptotically in the far field using the method steepest descents (Mittra and Lee, 1971) to give

f-r- -j(k r-Tr/U) -jk c sin 6

'''a^'''^^ WFT ^ ^N^J^o * ^ (2.3.17)

o

3. Numerical Results

3.1 The Electric Wall Case

This section presents the results of two studies. The first study is an examination of

how the closed region results of Chapters 3 and h converge to the open region results. The

second study considers the convergence of the open region results as a function of the numher

of perturbation coefficients.

Figures 10. 3.1 - 10.3.1.3 illustrate the variation of the dominant mode parameters for

a trifurcated waveguide with k b^ = 1.2T0U6, k b-, = O.U1U17, and with k b variable from

o 2 o 1 00

0.2 to 20. (Note that the indices of the trifurcated waveguide dimensions must have 1 added

to them to correspond to the current notation. ) The data calculated using the open region

analysis is shown for comparison. Note that of all the parameters that the reflection co- efficient of the waveguide with dimension k^b.^ = O.U1I+17 converges the fastest. (The phase

is not shown but it converged even faster with a maximum deviation of only k° from the open

region solution.) However, the reflection coefficient of the waveguide with k^b^ - 1.270^+6

converges much slower. The same is true for the coupling coefficient. All of the data is

observed to oscillate about the values computed using the open region analysis.

This data reaffirms the conclusions reached by Mittra and Richardson (l970) that the

closed region problem generally converges slowly to the open region problem.

Table 10.3.1.1 illustrates the convergence of some dominant mode parameters as a function

of the number of perturbation coefficients (N^ h N^'^ e n'^'''^) using the open region analysis

for the case of k b^ = 1. 270^16, k b„ = O.UlUlT.

115
0.7

0.6

0.5

0.4

o'

CD 0.3

0.2

0.1

0.1

I i n ; T 1 1

\—I I I I

I

1 1 1 1 1 I I I

kgOs * 1.27046

kQ b2 ' 0.41417

kQ b| I VARIABLE

^0,3 - '

^0,2 ' ^0,1 - °

\

OPEN REGION

RESULT

1 t

I I I I I I I I J L I I I I

i.O 10.0 100.0

kob|

Fig. 10.3.2.1a: Reflection Coefficient of Trifurcated

Waveguide as a Function of bj.

116
160

140

120

tn

a>

S* 100

o*

m

o> <

80

60

40

20

0.1

I I I I I

I T 1 1 I I I I

I T 1 1 I I I I

kgbj s 1.27046

kg l>2 > 0.41417

kQ b, 3 VARIABLE

^0,3 - '

"0,2 • »0, 1 - °

OPEN REGION

RESULT

_l I I ' ' ' ' J I I I I I I J I I I I 1 I

1.0 10.0 100.0

kQb,

Fig. 10. 3. 2. lb: Reflection Coefficient of Trifurcated

Waveguide as a Function of

117
118
-20

M -30

91

w

-40

0.1

T 1 1

1 I I I T

1 1 1—I I I I

OPEN REGION

RESULT

,-3

liQb3 = 1.27046

kjj bj = 0.41417

kg b| = VARIABLE

T = Bq 2 WITH

0,3 ^0, 1 - ^0,2 -°

J I I I I I -I I

I I I - 1I J J I I ' ' '

1.0 10.0 100.0

kQb,

Fig. 10.3.2.2b Coupling Coefficient of Trifurcated

Waveguide as a Function of

119
0.90

d

to

0.80

0.70

0.1

T 1 1 I I I I I

I I -I I I r I I

1.0

-I 1—I—I I

I I I

I

I I I I I

OPEN REGION

/ RESULT

02

.1 =^0^3=0

J I I ,1 till J I I I

10.0 100.0

Fig. 10.3.2.3: Reflection Coefficient of Trifurcated

Waveguide as a Function of "bj

120
Ta"ble 10.3.1.1 Convergence of Open Region Solution

* t + N B , b' T

p 0,1 0,2

5 0.25909 90.222° 0.8i+558 155.26° O.53I+97 -28.770°

7 0.25909 90.226° 0.81+557 155.26° 0.53503 -28.771°

"^T = B ^ with B^°] = 1, * with B^°,^ = 1, t with B^°^ = 1.

0,2 0,1 0,1 0,2

Clearly five place accuracy is achieved with only a few perturhation coefficients for

this case. However, the second waveguide has a width of only O.O66X. Table 10.3.1.2 illustrates the convergence of the dominant mode parameters for the case - 1.1+137, ^q^2

~

2.827IH, k^b^ = 2.827I+2. This particular array was examined by Lee (1967). His data is also

shown in the table. Note that the convergence is slower than the previous case where the

waveguides were smaller. However, excellent results are still obtained. Also note that the

data is in closer agreement to Lee for the central waveguides, which is to be expected since

Lee used an approximation to a flush-mounted infinite groijind plane while in our analysis no

gro\and plane is assumed.

Table 10.3.1.2 Convergence of Open Region Solution

N B^

0,1^ B+

0, 2 3 T

,t1 ^2

5 0.2193 73.6° 0.31 1+8 73.6° 0.2766 80. 6° 0.1663 -107. 8° 0.0918 71+.^5°

7 0.2181 73.1+° 0.3085 7l+.l° 0.2656 82. 90 O.I6I+9 -107. 5° 0.0891 76..5°

9 0.2195 73.7° 0.3126 73.8° 0.2720 81. 6° 0.1665 -107. 8° 0.0923 71+. 8°

11 0.2198 73.7° 0.3122 73.8° 0.2702 81. 90 0.1668 -107. 8° 0.0932 71+.,6°

13 0.2197 73.7° 0.3120 73.8° 0.2700 82. 0° 0.1667 -107.,8° 0.0929 71+.^7°

15 0.2195 73.7° 0.3122 73.8° 0.2711 81. 8° 0.1665 -107.,8° 0. 0921+ 7I+.,8°

Lee'

s

Data 0.2160 7l+.3° 0.3032 75.1° 0.2503 86. 2° 0.1625 -106. 8° O.O86I+ 77.,7°

tT^ = B „ with B^°,^ 0,2 0,1

= B ^ with B^°,^

0,3 0,1

= 1

2 = 1

xCurrent reflection coefficients.

Figure 10.3.1.1+ illustrates the far field radiation patterns of this same array. Note

that the patterns have nulls near the angles expected from separable array theory. However,

note that the null at 58° has noticeably filled due to the differences in aperture illumination because of mutual coupling.

3.2 The Magnetic Wall Case

This section presents results similar to section 3.2 for the magnetic wall case. Figure 10.3.2.1 illustrates the variation of the TEM reflection coefficient of a trifurcated waveguide with a magnetic symmetry wall (ref. Appendix F) with k^b^ = I.270I+6,

k^b^ = 0.1+11+17, and k^b^ variable. The data using the open region analysis is shown for

121
inV

o

•HP

eS P

•HOX>O

COa

-Pp

h

P-,-

w

P - p (U

a) H Ch W O H

ono

•H

(0)

122
Fig. 10.3.2.1a: Reflection Coefficient of Trifurcated

Waveguide with Magnetic Wall as a

Function of "b-^

123
1 70 j 1 1 1—I

I I I I

I 1 1—I—I—I I I 1

1

1 1 I—I—I I I I

160 -

IQO I I I I ' I t I I I I I I I

' I ' ......

0.1 1.0 10.0 100.0

Fig. 10.3.2.1b: Reflection Coefficient of Trifurcated

V7aveguide with Magnetic Wall as a

Function of b-^

124
125
comparison. Since the center region does not support a TEM mode only the reflection coefficient of the guide with k^^j^ ~ Q.^l^lT is shown. Note that the convergence is slower than

the electric wall case for the same geometry (ref. Figure 10.3.1.3). The data also oscillates

atout the value predicted using open region analysis.

Tahle 10.3.2.1 illustrates the convergence of this same data as a function of the numher

of perturbation coefficients (N ') using the open region analysis.

Table 10.3.2.1 Convergence of Open Region Results

N B - Oj2

T 0.81226 lUl+.TO°

9 0.81227 ll+U.70°

The data illustrates that five place accuracy is achieved with only a few perturbation coefficients . Table 10.3.2.2 illustrates the convergence of the dominant mode parameters for the case k^b^ = l.i+137, k^bg = 2.827^+1, k^b^ = 2.827^2. The convergence is quite good. Also, the

comparison with Lee's (1967) data is again quite good considering the difference in the

presence of a ground plane.

Table 10.3.2.2 Convergence of Open Region Results

N

J2.

B

0, 2* B

0 ,3*

t"^

5 O.I30U 75 7° 0.1981 72 1° O.I1I+7 -111.5°

7 O.I30I+ 75 7° 0.1970 73 k° O.lll+l -110.2°

9 0.130n 75 7° 0.1972 73 0° O.III+7 -111.2°

11 O.I30U 75 90 0.1972 72 90 O.III+6 -111.1°

13 0.130i+ 75 90 0.1971 73 1° O.llhk -110.6°

Lee ' s Data 0.1299 76 8° 0.1877 77 5° O.IIU6 -107.6°

tT = B ^ with = 1

0,3 0,2

*C\urrent reflection coefficients

Figure 10.3.2.2 shows the far-field radiation pattern of this same array.

3 . 3 Superposition of the Results for an Electric and a Magnetic Wall

Little data exists for the coupling of parallel plates without a ground plane. However,

Dybdal, Rudduck, and Tsai (1966) solved the problem of coupling between two parallel plates

using wedge diffraction techniques. A comparison of their data with that calculated using

this theory is shown in Fig\xre 10.3.3.1. Note that the modified function theoretic technique predicts resonant effects whenever the separation is a multiple of 0.5 A. At these

separations, the wedge diffraction techniques used by Dybdal, et al. is inadequate because

126
-40

0.4 0.8 1.6 2.0 2.4

I d/X )

Fig. 10.3.3.1: Mutual Coupling Between Two Parallel

Plate Waveguides

127
160 T 1 1 1 T

-200

— Free Space Phase = knd

o This Theory

Fig. 10.3.3.2: Phase of Mutual Coupling Betveen Tvo

Parallel Plate Waveguides

128
o

o

o

o

129
o

o

o

o

130
of the modes at cutoff in the inner region with dimension d. Figure 10.3.3.2 shows the phase

of the coupling coefficient for this same case. Note that the phase behaves according to

the geometrical distance except near the resonances, where the phase progression is slower

than free space.

In section 3-1 and 3.2, we presented data for an array examined by Lee (196t). We shall now consider the superposition of that data to obtain the characteristics of the complete

five element array. Table 10.3.3.1 shows the complete comparison of all the scattering

coefficients with Lee's data. The waveguide's are numbered 1-5 from the edge. Note that

good correlation is obtained with Lee's data even though his data is including the effect of

a simulated ground plane. However, there is a trend for the coupling coefficients to decay

slower without the presence of the ground plane.

Figure 10.3-3.3 illustrates the variation of the active reflection coefficient of each

element as the array is scanned. Note that the reflection coefficient varies from element

to element, with the edge element reflection coefficients being asymmetrical.

Figiire 10.3.3.^ illustrates the individual patterns of each element of this array as a result of mutual coupling coupling among all elements as well as the isolated element pattern.

Note that the edge element patterns are asymmetrical. Note that the element patterns have a

Table 10.3.3.1 Scattering coefficients of Lee's array.

Reflected

Mode

Waveguide in Guide

Excited No. This Theory Lee (1967)

3 1 O.O92U 7^.81° O.O86I+ 77-7°

3 2 0.1665 -107.8° 0.1625 -106.8°

3 3 0.2195 73.67° 0.2160 lh.3°

3 h 0.1665 -107.8° 0.1625 -106.8°

3 5 O.092U 7^.81° O.086U 77.7°

2 1 0.1712

2 2 0.2212

2 3 0.1713

2 k 0.0909

2 5 0.0583

1 1 0.2333

1 2 0.1712

1 3 O.O92U

1 h 0.0583

1 5 o.oim

-105.2° 0.1630 -102. 3°

7^.37° 0.2165 75. 6°

-105.3° 0.1625 -106. 8°

72.1+° 0.0867 73. 8°

-93.93° 0.0501 -90=>

78.1° O.218U 82. 5°

-105.2° 0.1630 -102. 3°

lh.9° O.086U 77. 7°

-93.9° 0.0502 -90=i

103. 5° 0.035i+ 10k. 8°

131
finite value at 6 = ± 90°, in contrast to Lee (196T) who predicts zero values. This method

of analysis would he partictilarly powerful in predicting wide angle scan performance of an array in the H plane, where the assumption of an infinite ground plane would produce nulls

at e = ± 90°. Note that the element grains are greater for angles away from the center

elements. Also note that for these values the behavior resembles closely that of an isolated

element with no ground plane. Also observe that for angles toward the center of the array

that the element patterns tend to be more iiniform particularly for large angles of observation.

CHAPTER 11. REMOTE SENSING OF THE EARTH USING PARALLEL PLATE WAVEGUIDES

1 . Introduction

The remote sensing of the earth's subsurface properties is commonly done at low frequencies in order to get the desired penetration. At these frequencies loops and dipoles

are commonly used. Ward (196T) discusses the application of elementary source theory to

this problem.

For the remote sensing of the earth's properties nearer the surface, moderate frequencies are used. Loops and dipoles are still commonly used; however, the antenna dimensions

are no longer small compared to a wavelength. Chang (l97l) has analyzed many of these problems using the numerical solution of integral equations.

At higher frequencies, waveguides can often be used instead of the more conventional

loops and dipoles. Waveguides have been commonly employed at higher frequencies; however,

they have been confined to measuring the Fresnel reflection coefficients. Greater sensitivity should be possible if the near fields of the antenna are allowed to interact with the

earth. This will cause the characteristics of the antennas themselves to change as a fiinction of the environment. From a theoretical viewpoint, such waveguides are more prone to

an exacting analysis than dipoles and loops because waveguides do not have gap corrections

and other feed modeling problems. Additionally, one might obtain increased sensitivity by

using two waveguides instead of one. The coupling between the waveguides would provide this

increased sensitivity.

The analysis presented in the first part of this monograph was confined to closed

region problems. If a sample of the earth can be obtained conveniently, then the analysis

of part 1 applies directly. In this case if one replaces the free space wavelength by the

guide wavel ength J tJie analysis applies to rectangular waveguide. However, generally the

determination of the earth's properties must be done remotely.

This chapter considers the problem of a finite array illiaminating a homogeneous half

space. In essence, this chapter combines the solutions of a waveguide radiating into a

half space (Chapter 9) and a finite phased array (Chapter lO). The solution is new and

gives physically interesting results for such problems as the coupling of two waveguides

above a half space.

132
2. Formulation of the Equations

2.1 Introduction

As in Chapter 10, we will assume a symmetry boundary parallel to the waveguides. This

is not necessary, but is convenient. Hence, we will consider the superposition of the results from two problems: (l) the electric symmetry case and (2) the magnetic symmetry case.

2.2 The Electric Symmetry Case

Figure 11.2.2.1 illustrates the auxiliary problem. As in the case of finite phased

array the problem can be further separated into two kinds of problems: (l) the interior

problem, and (2) the exterior problem. The interior problem is identical to that of

Chapter 10 while the exterior problem is a modification of the results of a single waveguide radiating into a homogeneous half space given in Chapter 9-

Clearly then, the N holomorphic functions are identical to (2.2.l)-(2.2.3) of Chapter

10, with the exception of the function associated with the Nth plate. on the open region.

This function is appropriately modified to account for the higher order modes incident

internally on the junction as well as the scattered field from the half space. From

Chapter 8, we can find:

Tj^(tj) = X(a))

f /o

co-j

^ N,L

Ul '"^n,c^_^ X(t) (t-a))^'

(2.2.1)

where X(aj) is the homogeneous solution as given in Chapter 8, section 3- Changing t to -t

in the integral we have

N,L

Tj^(a)) = X(a)) o

I _

I S^^^-t) dt

'^-'^^o n=l ""^n,c^_^ L" X(-t) (t+o))

(2.2.2)

.(2), From Chapter 9i we can easily derive the integral equation for g (t) by considering

the exterior problem. From Chapter 9»

T (u) = -TToje

jXc N-1

A(X), ojeL" (2.2.3)

and

g^^^((jD) = oj sin ^c^_^ A°(A), ueL^ [2.2.k)

But from the boundary condition at the dielectric we can relate A(X) and A (X).

A°(A) = R(X) A(X) (2.2.5;

where

133
I o • • •

CM

oIIM

OcoOO

oo

cdPi

as <

0) -p

Ho

•H -P

^^H

O (1) ^ 'HO

OJ c3 HP O O -P

bO

CM

CVJ

•H

134
where

r = /A^ - ek^

o

with the branch of V chosen such that Re(r) ^0.

Using (2.2.2)-(2.2.5), we may arrive at the following

.(2)

sin Ac

(-0)) = N-1

R(A) e

-jAcN-1 N o =n

to-jk a)-Y o n=l n,c

- /

R'^\-t) dt

- X(-t) (oi+t)

N-1

;2.2.6)

Considering the change of variable

(2), . "^'^ ^Vl R(A) -^^Vl , ^, .

we transform (2.2.6) to

G(a)) = + (oi-jk ) ^n Q(t) G(t) dt

t + (jj

, (jjeL (2.2.7)

where

Q(a)) = - sin Acj^_^ R(A) e X (u)

^(w-jk^) X(-a))

Equation (2.2.7) is the desired integral equation for G(a)).

We must also modify (2.2.8) of Chapter 10 to reflect the addition of the integral to

Tj^(co). Hence,

T (-Y ) = (-1) Y M-1 n °n

= -X(-Y, ^

) -—+ y +jk

'n Y +JK Y +Y

^'Vl ° "1=1 n,Cj^_^ m,Cj^_^

Q(t) G(t) dt

^1 ^'^-1

(2.2.8)

where n = 1,2,3, ... . Equation (2.2.8) is also an integral equation for G(a)),

135
All other equations associated with the interior protlem remain as given in Chapter 10

For this particular problem, the TEM coefficients are known (ref. to ( 2. 2. 9)-( 2. 2. 11

)

of Chapter 10). These equations all remain valid even in the presence of the half space.

In order to solve the integral and algehraic equations efficiently, we must consider

the asymptotic behavior of the various perturbation coefficients. However, because the

presence of the half space does not change or introduce any edge condition, all the asymptotic forms given in Chapter 10 ( (2.2. 12)-(2. 2.13) ) are still valid.

All remaining details of the analytical solution are the same as in Chapter 10 when

using the new expression for Tj^(a)).

2.3 The Magnetic Symmetry Case

Figiire 11.2.3-1 illustrates the aioxiliary problem of magnetic symmetry wall case. As

in the case of the electric wall, only the exterior problem results need to be changed from

the results of Chapter 10. Hence, we need to change only the Nth holomorphic function (i.e

(2.3.3) of Chapter 10 ). From Chapter 8, (2.3.12) we have that

T^(to) = X(a))

^n .(2),

r g' 'It) dt

i+ X(t) (t-oj n=l '^"^2n-l,2Cj^_^

(2.3.1)

where X(a)) is the homogeneous solution for the magnetic wall case. Changing t to -t in

the integral we have

T^(a)) = X(a)) I =n

tn=l " ^2n-l,2cj^_^

_ r K^^h-t) dt

•'- X(-t) (t+o)) (2.3.2)

(2) We can arrive at an integral equation for g (t) by recalling from Chapter 8 that

T~(u)) - T^(a)) = -2-nj to cos XCj^_^ A(A), weL^ (2.3.3)

and

j2Ac

_^ Tjj(w) = -e (2.3.i+:

Combining (2.3-3) and (2.3-^) yields.

Tjj(a)) = -jTTu e

j Ac N-1 A(A), weL

1 (2.3.5)

also from (2.3-13) of Chapter 8

( 2 ) o g (o)) = +0) cos A (a), (jjeLg (2.3.6)

But from the boiindary condition at the half space we can relate A(A) and A (A;

A°(A) = R(A) A(A) (2.3.7)

136
oIIM

CVJ

o

-o

c:oO

"53 c

CPoE

0)o

cdft

t>>co

cd^ <M ^ H < 03

(U P

o

•H 'H C !h •H -PO

CD ^ H O dJ

<)-l -H e

0) 03 H

rQ

O

O +J

PL) -H

•H

rH

t)0

s:! •H

-P

cd •H

00

CVJ

•H

CMI CO

—

137
where

and where

r = A2 -

o

with e being the complex permittivity of the half space. Using (2.3.5)-(2.3.7)5 we may

arrive at the following

.(2)

(-0)) =

-j' cos Ac

N-1

R(X) e

-iXcN-1 ^

N,L

- /

n=l '2n-l,2c„ , N-1 1

- X(-t) (t+o))

(2.3.8)

Again a change of variable is made.

M>{-^) = ^

R(X) e X (w) G(a))

Thus we transform (2.3.8) to

G(a,) =

I ^n _ r Q(t) G(t) dt

^ (jd-y^.. , ^ i- t + 0) n=l '2n-l,2Cjj_^

(2.3.9)

where

-j Ac

N-1 X (co) Q(.) = J cos Xc^_^R(A) e

Equation (2.3.9) is the desired integral equation for G(a)). All other equations given

in section 2.3 of Chapter 10 are valid.

3. Numerical Results

3.1 Introduction

The numerical solution of the integral equations derived in sections 2.2 and 2.3 was accomplished using~Ga?ilssian quadrature similar to that described in Chapter 9-

3.2 The Electric Wall Case

This section presents the results of two studies. The first study is an examination of

how the closed region results of Chapters 3 and h converge to the open region results. The

second study considers the convergence of the open region res\ilts as a function of the number

of perturbation coefficients and truncation of the integral equation.

138
The evolution of closed region problems into open region problems is interesting due to

two reasons: (l) It is interesting to examine a problem which can be solved both in the open

and closed region cases to see which problem is "easier" to solve, and (2) It provides a

check on the open region solution against previous closed region results.

Figures 11.3.2.1 (a)-(c) illustrate the variation of the dominant mode parameters for

a dielectrically loaded trifurcated waveguide with k^^^ ~ 1.270^6, k^^^ = O.klklJ, and with

k b variable. Also, e = 10, a/k = 0.01, and k d = 1.256. The open region data is shown 00 r o o for comparison. As in Chapter 10, the reflection coefficient for the smallest waveguide

(k^b^ = O.Ul^lT) converges fastest to the open region result. However, the reflection

coefficient of the waveguide with ^^^2 ~ 1- 270^^6 and the coupling coefficient between the

two guides both converge much slower to the open region solution. However, all of the data

computed is observed to oscillate about the values computed using the open region analysis.

It should be noted that the convergence to the open region solution is about the same as

that shown in Chapter 10 where the dielectric half space is not present. The dielectric

is quite lossy and 0'2X^ away from the horn apertiire. Hence, in order for the edge wall

distance to become secondary, the distance to the half space, must be an even smaller '-fraction

of a wavelength than 0.2A^.

Table 11.3.2.1 illustrates the convergence of some dominant mode parameters as a function

of the number of intervals, N, along and the number of Gaussian quadrature points, M^,

within the nth interval for the case e = 10, a/k = 0.01, k d = I.256, k b = I.270U6,

^

r ''o 'o 'ol

k^bg = 0.itlUl7, and with N ' = N ' = 5-

Table 11.3.2.1 Convergence of Open Region Solution

N \

M2 Mj^ B

5 0 ,1^ B

0 ,2' T

t

2 16 8 — 0.19399 177. 68° 0.87189 159.M6° 0.50981 -57. kh°

3 16 8 8 — 0.19385 177. 66° 0.87150 159. 39° 0.50993 -57. 51°

i+ 16 8 8 8 0.19385 177. 66° 0.871^+9 159. 39° O.i+0993 -57. 51°

5 16 8 8 8 8 0.19385 177. 66° 0.871^9 159. 39° 0.50993 -57. 51°

3 16 16 8 — O.19U12 177. 66° 0.871^+7 159. 39° 0.51002 -57. 52°

3 16 16 16 — O.19U13 177. 67° 0.871^+8 159. 39° 0. 51002 -57. 52°

tB , with B^°^ = 1, B^°l = 0.

0,1 0,1 0,2

tB _ with B^°l = 1, B^°,^ = 0.

0,2 0,2 0,1

tT = B ^ with B^°] = 1, B^°^

0,2 0,1 0,2

= 0.

Essentially five place accuracy is achieved with as few as three matching intervals and

32 match points.

Table 11.3.2.2 illustrates the convergence of the same geometry considered in Table

1 R 2,L 11.2.3.1 except as a function of the number of perturbation coefficients, = N

' N

and with N 2, = 16, = 8.

139
190

180

170

V

S" 160

o"

CD

T 1 1 1—I I I I

I 1 1 1 1—I I I I I T 1 1 1—I I I I

OPEN REGION RESULT

150

<

140

130

120

«>3

«r if

kg b3 = 1.27046

kn t, I 0.41417 0 "2

0 «>,

€, = 10 , a/ko = 0.01

,

kg b, = VARIABLE

(o) (o) le)

B0,2 Bo.l =Bo,3 = 0

J I I I I I I J I I

' ' I J I I I I I I

0.1 1.0 knbi 10.0 100.0

(a) Phase of Bq ,

2

"T 1 1 1 !—I I I [ "1 1 1 1 1 I I I 1 1 1 1 I I I

1.0

cj 0.9

o

ID

0.8

OPEN REGION RESULT -

Kq b, = VARIABLE

Cr = 10 , CT/ko = 0.01 , kgd = 1. 256

(0) "0,2 - ^0,1 - ^

J I U fill ' I I I I

0,3 ' 0

' '

0.1

Fig. 11.3.2.1

1.0 10.0 100.0 kob|

("b) Magnitude of Bo,2

Scattering Paraneters of Dielectri cally

Loaded Waveguide as a Function of

140
(c) Magnitude of B0j3

Fig. 11.3.2.1: Scattering Parameters of Dielectri cally

Loaded Waveguide as a Function of

141
0.3 -

0.2

0.1

-i I I—I—I—I I I

-30

-40 -

-50 -

0)

S" -60

D

-70

w<

-80

-90

-100

0.1

T 1—r—I I I I I

1 1—I—I I I I I

ko ^3 1.27046 kg bj • 0.41417 kg b| ' VARIABLE

Cr 10 , a/k, I 0.01

T.Bo,3 WITH Bo, 2

-

B0.i"« Bo.r> 0

OPEN REGION RESULT

I I I 1 I I I I -I I I I I I J L. I'll

1.0 10.0 100.0

-1 1 r-

(e) Magnitude of T

I I I

I

1 T 1

I I I I I 1 1 1 1—I I I I

OPEN REGION RESULT

kg b3 = 1.27046

€, ,<r' t"© " I * VARIABLE / y / = 10 , <7/ko = 0.01, ^

k„d = 1.256

(0) T • Bo,3 WITH Bo,2 '

"0,1 ' °0,3 - °

I 'I I I ' ' I J I I I I I I

I I I I 1 I I

1.0 ko b,

lO.O 100.0

(f) Ph a s e o I T

J

ig. 11.3.2.1: Scattering Parameters of Dielec tri cally

Loaded Waveguide as a Function of \>-^

142
Tatle 11.3.2.2 Convergence of Open Region Solution

p 0,1* 0,2*

5 0.19399 177.68° 0.87189 159.^6° 0.50981 -'^7.kk°

7 0.19^+00 177.68° 0.87188 159.i+6° 0.50896 -57.i+3°

9 0.19^02 177.68° 0.87186 159.i+6° 0.50993 -57.^+3°

*Same as TaMe 11.3.2.1

This clearly illustrates that the convergence of the solution is quite good.

3. 3 The Magnetic Wall Case

This section presents results similar to section 3.2, vith one exception. No closed

region data is presented since the problem of an N-furcated waveguide with dielectric loading and a magnetic symmetry wall was not implemented on the computer. However, an alternate

check of the solution is available. Figure 11.3.3.1 illustrates the variation of the re- flection coefficient of a truncated parallel plate waveguide parallel to a magnetic symmetry

wall as a function of the distance from the waveguide aperture to a conducting half space.

The reflection coefficient for the case of no dielectric is shown for comparison. One notes

that the reflection coefficient for the case with the dielectric oscillates about the no dielectric case, symmetrically and with progressively smaller amplitude.

Table 11.3.3.1 shows the convergence of the dominant mode reflection coefficient for the

following parameters: e = 10, a/k - 0.01, k d = 1.256, k b = I.250U6, k b_ = OAlklf

1R 2L ^ 'ol ^'o2

and N' =N' =N. Nis the number of segments along L and M is the number of matching

points within the nth

p

interval.

In

Table 11.3.3-1 Convergence of Open Region Solution

N N M3 B

0,2

7 3 k k h — 0.7^^20 11+7.98°

7 3 6 6 6 — 0.75396 li+7.91°

7 3 8 8 8 — 0.75322 11+7.80°

7 3 10 10 10 O.753OI+ 11+7.81°

7 3 12 12 12 0.7^+313 11+7.81°

7 h 12 12 12 12 0.75313 11+7.80°

7 3 16 8 16 — 0.75317 11+7.81°

5 3 16 8 16 — 0.75316 11+7.80°

9 3 16 8 16 — 0.75317 ll+7.80°

Again, excellent convergence is observed, both with respect to approximation of the

continuous as well as discrete parts of the problem.

3. 1+ Superposition of the Results

The title to this chapter suggests the use of waveguides to remotely sense the parameters of the earth. In this case, we are suggesting a locally plane approximation as well

as a homogeneous half space.

143
0.2 -

0.1 -

0 0.5 1.0 1.5 2.0

d/\

(a) Magnitude of Reflection Coefficient

0 0.5 1.0 1.5 2.0

d/X

(b) Phase of Reflection Coefficient

Fig. 11.3.3.1: Variation of the Reflection Coefficient

of the Magnetic Wall Case as a Function

of Distance

144
0

bO

H

o

-P

le

o ec

w 1—1

-p -PO •H H (U (—1 o el

01

<l-t rH •H o le O +J

-p

H cd

03

o

o

•H Oi fl o o

•H •H •H

Cm -P Ch

M

u

<D

Ca

O

e •H

O le at C •H CO O u

•H

o

w

-P > K -p O Eh <u

(U >^ 0 e H c6 O 0} «H a U

<U «H ft O <; w H«

•H

145
o

CD

"rH

"r*i 1—' e .

0) -p CJ

cd f\ \ \U H r—

1

[—1

ft . H M

le

vj

ii 1 UJ

1 % H (-1

^4 -P

1) ^

cs3 fJ . Ph

*r-t 'h -P >-i G r\

0)e erf •rn H rri r-t

>-* o pis •H O

p 03 u E-i (D m

r-( cC erf Ch <;

dJ «H (U CO K O 0) CJ 13 erf CD 'd -P •H ft -p CO

0) o S CD Cm erf

(L) > rH H erf erf Ph

OJ

• H

H

bO •H

P-4

(S99j6ap) 2*0 g

146
1—

I—I

\—I—

r

T—I—r—

r

I I I I

I I I I II I I

130

toI

-140

0)

T3

<

-150

cr/k =0.0000^^ /

a/k^=O.OOIcr/k^=0.003

o-/k^=0.005

o-/k =0.007

0

a/ k =0.01

0

o-/k,=0.5

o.sX

-160 J

I I ' J I I I I 1 I I I I I I I I I I I I I

0.1 0.15 0.2

ITI

0.25 0.3 0.35 0.4

Fig. 11.3.U.3: Coupling Between Two Parallel Plate

Waveguides Radiating into a Dielectric

Half Space as a Function of the Dielectric

Parameters

147
Figure 11.3.^.1 illustrates reflection coefficient of one of two 0-^^^ waveguides

spaced 0-5^^ apart with respect to their centers, at a distance of O.IX^ away from a half

space. The variation of the permittivity and conductivity cause the reflection coefficient

to change quite noticeably. In fact, the variation is quite similar to that of a single

O.UA^ waveguide given in Chapter 9, in figure 9- ^-3, as indeed it should be. Figure 11.3.U.2

illustrates this same data matched to the impedance of a single waveguide looking into free

space. Again, this is quite similar to the data of Chapter 9? and resembles quite closely

the nomally incident Fresnel reflection coefficient. Figure 11.3.^.3 illustrates the

the coupling coefficient between the two waveguides as a function of variation of the half

space parameters. One should observe that the phase of the coupling coefficient varies over

about 20° while the magnitude varies between about 0.15 to 0.3. This is to be compared with

about hO° of change in the phase of the matched reflection coefficient and an amplitude of

the matched reflection coefficient varying from about 0.15 to 0.6.

Hence, in this particular case it appears as if the pair of antennas is of little further

help in solving the inverse problem of determining and a, as compared with a single antenna.

However, this is not to say that the coupling coefficient may not be useful in this determination. Additionally, the coupling coefficient might prove to be more sensitive to variation

for such problems as layered earth models or buried dielectric anomalies.

CHAPTER 12. OTHER OPEN REGION PROBLEMS

1. Introduction

The purpose of this chapter is to illustrate the ease of application of the modified

fiinction theoretic technique to some additional open region problems. In particular the

following problems are solved using the modified function theoretic technique: (l) a flanged

waveguide radiating into half space, (2) scattering by a thick semi-infinite plane, and

(3) radiation from a slot in a waveguide.

A flanged waveguide radiating into a grounded dielectric sheath has been solved by

Wu (1969) using moment methods. Also Kostelnicek and Mittra (1969, 1971) discuss the solution of the problem of a flanged waveguide radiating into a dielectric slab using the modified function theoretic technique. In their 1971 paper they make the erroneous statement

that the associated homogeneous Hilbert problem cannot be solved in closed form. Indeed

the solution of this problem is quite straightforward using the techniques we have developed

in chapter 8.

The scattering by a thick semi-infinite plane has been solved by Lee and Mittra (1968)

using the generalized scattering matrix technique. The solution given in this chapter

serves to illustrate the use of the modified function theoretic technique when the incident

field is a plane wave. Another problem of interest is the radiation from a slot in a waveguide wall. This

problem is a simple extension of the problem of a waveguide radiating into a half space

given in Chapter 9-

148
2. Flanged Waveguide Radiating into a Half Space

The geometry of this prohlem and its auxiliary problem are shovm in Figure 12.2.1. From

Chapter 8 we can clearly write the holomorphic function T(a)) as

(

T(a3) = X(u)

K .(1) (2),

["-j^o X-(t)(t-a)) X(t)(t-a3)

^

J

^(t) dt _ 1^'(t) dt

(2.2)

where

where

X(oo) = H^(a))(w-jk^)n(a),Y^)exp-j

^ In

H^(a)) = expj^ i-c^-m

k "b o

2tt 2 /

From Chapter 8 we have that

(1)

(w) = -u sin Xb e*^^^ C°(X), weL-,

( 2 ) o g (o)) = 0) sin Xb A (X), weL^

(2.3a)

(2.3b)

Also we have

T~(a))-T'''(a)) = -2irj to sin Xb A(X), toeL^ (2.1|)

and

T (-u)+T (-cd) = 2Tra)[cos Xb A°(X) - C(X)], weL^ (2.5)

However, the spectra are related at z=d and at z=-6 by the following

C°(X) = R^(X) C(X) (2.6)

where

R,(X) e'o) - r' -20)6 e

e'o) + r'

and

A"(X) = R^(X) A(X) (2.7)

where

R^(X) £0) - r -2a)d e

eu) + r

Using the property of X((jj) that

X"(a)) = X"^(u))ej^^^, oieL,

149
(a) The Auxiliary Problem
we can immediately write (2.1+) as

K

X (a))2je sin At PV / dt _ p^'^t) dt

X'(t)(t-a)) X(t)(t-a)) ,

+ 2TTj g^''-^a))e"'^^^ cos Xb = -27rj u sin Xh A(X), weL^

The principal value of the integral associated with has to be taken,

and (2.3b) we arrive at the following integral equation

Then using (2.7)

X~(a))e"j^^ sin Ab

K

u-jk PV /

g^^^t) dt _

I g^^^t) dt

L~ X (t)(t-a)) X(t)(t-u)

+ ^ g^^^(a))e"j^^ cos Ab = ^ R^^(A) g^^Va)), ueL^ (2.8;

Equation (2.8) is just the extension of equation (2.5) of Chapter 9-

Similarly for weL^ we know that

X~(a)) = X (u) = X(w)

hence we can write (2.5) explicitly as

K

X(a)) J g^^^t) dt _ 1^ ^^^ht) dt

'^"^^o L" X"(t)(t-a)) X(t)(t-a))

sin Ab cos Ab g^^^o,) - e-j^^ R-^(A)g^^^-a)) (2.9)

Equation (2.9) is just the extension of equation (3.5) of Chapter 8.

Equations (2.8) and (2.9) are simultaneous integral equations for the unknown functions

g^'^\iii) and g^^^(a)). Note that in each equation we have a Cauchy principle value integral

in contrast to the previous cases where the integrals existed in the usual Riemann sense. We

will not give any numerical results here but we are in a position to discuss the asymptotic

behavior of the unknowns. However, before doing this it is convenient to make a transformation of variables similar to those used for the flanged guide and the radiation into a half

space. Thus consider

' (2)

K sin Ab R.(A) , s

( \ _ _o A -jAb xpl2j / V (,-0)} = e X (ojjG (oj;, weL^

•ir(cD-jk^)

(2.10)

and

.(1) M = -K sin Ab . , , / ^ v —2 e^^^ R^(A)X(-co)G^^^(ai), ujeL^

TT(a)+jk^)

(2.11)

151
Using (2.10) and (2.1l), (2.8) becomes

I Q^^^t)G^^^t) dt ^

I Q^^^(t)G^^^-t) dt

t - w t - 0)

(2) + TT cot Xb Q^^^u) G^^^o)) = ^—4^, u)eL,

(jj-jk 1

(2.12)

where

-sin Xb R^(X)e'^''^ X(-u)

i (w) = ^

1T((jD+jk^) X ((jj)

and

sin Xb R.(X)e X (-w)

r^o)) = ^

, oieL.

Tr(u+jk^) X(a))

Similarly, using (2.10) and (2.1l), (2.9) becomes

uj-jko L

r q^^\t)G^^\t) dt , . Q^^^t)G^^^-t) dt

t - 03 J+ t - 03

TT cot Xb Q^^Va,)G'^^^-03) + 03eL^

03- J k^ 2

(2.13)

Now from Chapters 8 and 9 we have already found that

Q^^^(-03) = o(^ ^1 LI 03eL,

and

G^2)(^) = 0(1), 03 °°, 03eL^

(2) and we know that integrals involving G (03) can be trmcated because of the exponential

(2) decay of Q (03). Also we have previously found that

.(1),

(03) = 0(0}"'"), |o3| -^oo, 03eL^

and

G^"'-^03) = 0(03 ^) , LI ^ 00, ueL.

152
where

1 • -1

A = — sin

IT

e'-l

2(e'+l)

For the edge condition to be explicitly satisfied at z = 0, x = h, we can write

(2.11+)

In order to insTire that the edge condition is explicitly satisfied we must insure that

sin Ah A(X) = 0(a"^^^"^), | X |

Then from {2.k) or equivalently (2.12) we see that the following term must vanish in order

that G^^^o)) = 0(0)"^).

1 + /_ Q

L, ^^^t)G^^^t) dt + / l^2^-t)G^^^t) dt = 0

or

(u ) 0)"^

1+ /° Q^^\t)G^^^t) dt + G^+ /_ Q^2^-t)G^2^t) dt = 0 (2.15)

Equation (2.15) is merely the extension of equation (3.13) of Chapter 8.

Before concluding this section, it is in order to "briefly discuss the method of niimerical solution that one might use in solving (2.12) and (2.13). Note that in contrast to the

equations obtained in the solution of flanged waveguide and the solution of a waveguide

radiating into a homogeneous half space that equations (2.12) and (2.13) require the evaluation of Cauchy principle value integrals. Hence any numerical approximation technique used

for the solution of these equations must account for the principle value integrals. As noted

by Kostelnicek and Mittra (1969, 1971) one possible alternative is to change the paths of

integration. However, when doing this the new path of integration is rather arbitrary and

may introduce more numerical difficulty than the original path.

3. Scattering by a Thick Semi-Infinite Plane

Fig-ure 12.3.1 illustrates the geometry of the thick semi-infinite plane as well as the

auxiliary problem. For simplicity we are only solving the electric bo\indary case. In general, incidence at an arbitrary angle requires that the magnetic symmetry problem be solved

in addition to the electric case. However, the method is clearly illustrated from just the

electric solution.

From Chapter 8 we see that the solution of the problem may be found from the holomorphic

function

T(a)) = X(co)

n=0 '^-Ynb

r R^^ht) dt

1+ X(t)(t-a))

^2

(3.1)

153
Electric Symmetry Boundary

(a) .Auxiliary ProlDlem

Electric Symmetry Boundary

(b) Half Plane Geometry

Fig. 12.3.1: The Thick Half Plane

'154
where X((i3) is the homogeneous solution. Since the incident field is a plane wave we have

A°(A) = A 6(w-jk cos 9 ), ojeL^ o o o 1

where 6(«) is the Dirac delta function and hence

J2) (-0)) = -0) sin Xh A 6(u)-jk cos 6 ), weL^

o o o 1

Thus

(2),^, ^, (2)/ -jk cos 6 A sin(k b sin 9 )

J g ( t ) dt _J g C-t) dt o o o o o_

L+ X(t)(t-u)) X(-t)(t+w) X(-jk cos 9 )(tjj+jk cos 9 ) o o o o

(3.2)

Note that in the limiting case as 9^ -> 0°, that the electric solution f\irnishes the complete

solution to the problem.

In order to find an equation for g^ we use the knowledge that z = -6 that

B^°^ = B R

m mm (3.3)

where

R =

^"^mh ^mh ^'^''^mh mh mb

where

mb

mir - ek

Then using (v) of Chapter 8, section 2, we have

(-1)"^''^T(-Y , ) = -Y , b mb mb m m

= -Y ^ b E R-^ B^°^ mo mm m

i3.h)

but from (i) of Chapter 8, section 2, we have

(-ir^T(r„,) r^b c B<°' mb mm

(-1)

Y , ^

mb

-x'°>(j.„) g„

m > 0

m = 0

(3.5)

155
where ^^"^^(y^j^^^) indicates that the mth zero at is to be omitted. Hence we can -write in

general

m m

(3.6)

where -"^jj^ is defined by (3.5). Using (3.6) in (3.^) we have

jk cos 6 A sin (k b sin 6 ) y n _|_ o o o o o' ~

n y v,+Y X(-jk cos e )(jk cos e -Y ^) n=0 mb nb ^ ^ „ ^ i^i,' o mb

= -Y ^ b e R """X mb m m m

, m = 0,1,2,,

(3.7)

Equation (3.7) is an infinite matrix equation for g^.

In order to truncate (3.7) efficiently, let us investigate the asymptotic behavior of g^.

We may follow a procedure similar to that used for the E-plane step in Chapter 2 and find that

= 0(n-l-^)

where

A 1 . -1

A = — sm IT 2(e+l)

Also if g^ = g m ^ for m > N, then we can easily show that

I n=0

g„ + g

<»

T

. jk cos 8 A sin(k b sin 9 ) VIn-1-A , o o o o o + = 0

n=N+l X(-jk cos 6 ) o o

(3.8)

From Chapter 10 we know that the far field for z > 0 is determined by

T(jk^ cos e)e

-ik b sin 6 o (3.9)

and thus upon solving for g^ and g we can easily find the far field scattering pattern.

One should note that as 6^ 0 that the term involving A^ becomes singular and it would

appear that we do not have a proper solution. However, for the case of 6^ = 0, the waveguide

walls are orthogonal to the incident electric field and the equations given in Chapter 8 are

incomplete. In this case there will be an additional term

jk z -2jk b A e ° ° 6 o o mo

on the left hand side of (2.2.6) of Chapter 8 and an additional term will be present in

(2.2.10) of Chapter 8. Thus we have that

(_1)^(_,^^) = t e''"'^'^ -2jk„ b A^ e^''°'° mb m m mo

(3.10)

156
In this case (3.7) must be modified to be

-I

n=0 ^mb''"^n'b

= Y X. t e R B^°^ - 2jk b A 6 mb m m m " o o mo

:3.ii)

h. Radiation From a Slot in a Waveguide Wall

This section serves as a forum for presenting some results using the theory of Chapter 9' In particular, if one considers the case of a waveguide radiating into a half space,

a slot in the waveguide wall can be simulated by superposition of the case where the half

space is allowed to become either a perfect electric or magnetic conductor.

For TEM excitation this type of slot is known as a series slot, because the equivalent

circuit is just a series admittance.

Figure 12.^1.1 shows the series conductance and susceptance as a function of slot width

for the case of (2b/A) = O.278. For this particular case the slot is resonant at a slot

width of about 0.2 wavelengths.

CHAPTER 13. CONCLUSIONS (PART II

)

This part of the monograph has presented the MRCT and MFTT solution of a new class of

open region problems. The approach has been to solve a canonical problem of a semi-infinite

parallel plate waveguide with known incident fields. The solution of composite problems

is readily found from the associated auxiliary problem. Many of the problems solved in this

part were combined open and closed region problems for which the results of part I of this

monograph were also applicable. An example is the finite phased array. The aixxiliary

problem may be recognized to be composed of an N-furcated waveguide and a semi-infinite

parallel plate waveguide.

This particular problem is essentially the open region analogue of the N-furcated

waveguide and its modifications as given in part I of this monograph.

The convergence of the solutions is rapid and only requires a small number of perturbation coefficients or a limited representation of a continuous perturbation spectrum.

Several physically interesting problems were solved. Among these were the problem of

radiation of a waveguide into a homogeneous half space. This solution indicates that remote

sensing of the earth is quite feasible even when the earth is in the near field of the waveguide apertiire. Even in the near field, one is able to relate the reflection coefficient to

the normally incident Fresnel reflection coefficient. The primary difference is that the

argand diagram is rotated. This same problem was also solved for the case of two waveguides.

For the case of a homogeneous earth, no particular advantage of measuring the coupling coefficient was observed. However, this may not hold true for such problems as the remote sensing

of dielectric anomalies or in layered media.

Another problem of physical interest is the problem of a finite phased array without a ground plane. The analysis to date has assumed an infinite ground plane (or some approximation

157
1 .6 |—I

1 1 1 1 r—11 1 11 1 1 1 11 1 1 r

0 0.1 0.2 0.3 0.4

SLOT WIDTH (WAVELENGTHS)

Fig. 12.U.1: Series Slot Admittance

158
to it) in order to simplify the analysis. This is a good approximation if one is only

interested in the array patterns near broadside. However, for wide angle scanning arrays

the correlation is increasingly bad because of ground plane effects. It should also be

noted that this analysis can be easily extended to the case of a finite ground plane.

Solutions are also indicated for more complicated problems such as a flanged waveguide

radiating into a layered media.

CHAPTER ih. COMMENTS MD FINAL SUMMARY

This monograph has endeavored to fill the gaps that existed in the modified residue

calculus and modified function theoretic techniques. The key to this realization has been

the identification of certain canonical problems. In part I, the canonical problem was the

bifurcated waveguide filled with homogeneous media. For part II, the canonical problem was

the semi-infinite parallel plate waveguide. These choices of canonical problems were made

because of the cartesian nature of problems were to be solved. This choice of canonical

problems is by no means an implied limitation of the MRCT and MFTT. For example, a wide

range of problems dealing with the modification of semi-infinite circular or coaxial waveguide can be solved in the same manner. For example, one can solve the problem of a non- contacting coaxial short by recognizing that it is a modification of the coaxial bifurcated

jiinction. Such a solution would involve the construction of two holomorphic functions, one with a single modification and the other with a double modification.

One can also solve a wide class of modified semi-infinite circular and coaxial open

region waveguide problems. The primary difficulty in this case compared to the closed

region is that the solution of the associated homogeneous Hilbert problem must be obtained

niomerically. However, efficient techniques for this factorization may be found in Weinstein

(1969) as well as Mittra and Lee (l97l). This canonical problem admits the possibility of

solving such problems as a flange^ circular waveguide radiating into a homogeneous half

space. Also the problem of a flanged (or un-flanged) coaxial waveguide radiating into free

space or a layered half space can be solved using the techniques.

Other canonical problems which offer interesting possibilities are the open and closed

region problems concerned with the excitation of surface wave on a grounded dielectric slab

by a semi-infinitely long parallel-plate waveguide filled with the same dielectric. This

problem cannot be solved in closed form, but Bates and Mittra (1968) have given efficient

numerical schemes for the factorization. This canonical problem allows one to solve two

interesting problems. The first is the diffraction and scattering of waves from a dielectric

step in a waveguide. This problem was solved by Royer and Mittra (19TI) and is also discussed in Chapter 5 of this monograph. The open region canonical problem allows one to

solve the open region analogue of the dielectric step, the semi-infinite dielectric waveguide. This problem has not yet received a satisfactory analytical solution. One cannot,

however, solve, the coaxial and circular analogue of these parallel plate problems because

the hybrid natirre of the mode structure does not permit a solution of this form.

One area which was not explored in this monograph was the ultimate use of asymptotics.

Most of the numerical solutions given displayed several place accuracy with only a few

perturbational terms or only a few sample points of a continuous perturbational spectrum.

159
The logical course one can follow from this is to solve problems using the MRCT and the MFTT

using only the asymptotic terms. Such solutions should easily have two place accuracy and

he quite sufficient for many engineering tasks. This technique might be comparihle to say

the geometrical theory of diffraction where nominally two place accuracy is obtained (Yee,

Felson, and Keller, I968). In fact, an investigation into the connection between these two

techniques might prove very fruitful.

Yet another area of investigation appears to be the very nature of the solutions themselves. In essence, both the MRCT and the MFTT seek solutions by expanding the spectral

representations of the fields using their singularities. In this case the singularities

are either simple poles or branch points. This is very similar to the singularity expansion

method (SEM) expounded by Bam (1973) for solving electromagnetic transient problems. This

leads one to ask the question if more complicated problems which do not have Wiener-Hopf

type canonical problems can be solved using the same basic technique. One would then

depend on a numerical technique such as the method of moments to solve the canonical

problem.

Both the MRCT and the MFTT have their foundation in the generalized scattering matrix

technique (GSMT). As an alternative to the development of the MRCT and the MFTT one might

also consider extending the GSMT to include asymptotic terms. Such terms would compensate

for the major weakness of the GSMT: the failure to change the edge condition to conform

with the known asymptotic solution. In fact, this particular technique might prove to be more powerful than either the MRCT or the MFTT since it is not limited to problems which are basically two dimensional in nature.

In is hoped that these comments will be useful to the researcher interested in the ex- tension of these techniques.

ACKNOWLEDGMENT

It is a pleasure to acknowledge the assistance of many individuals. To Professor Leonard

Lewin for his invaluable comments on analytical aspects of the project; to Mr. William E.

Little for his support and overall supervision of the project, to R. Lewis for a very careful

reading of the manuscript, and to P. Wacker, S. W. Maley, N. Nahman for many helpful suggestions. The excellent typing of J. R. Jasa and S. C, Foote as well as the illustration of

J. Spencer are gratefully acknowledged. This monograph resulted largely from the first

author's graduate work at the University of Colorado.

160
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161
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162
Appendix A: The Edge Condition and the

Asymptotic Behavior of T(a))

I 1-1/2

From Mittra and Lee (l97l) it is easy to show that E must behave as |x-x^| as -1/2

X -> x^ with z = z . Similarly, E must behave as |z-z | as z -> z and x = x^ . Thus,

1 o ' z ' o ' o 1

according to the field expression given in section 2.2, chapter 2 of part I, we have

x->-x^ n=l

00 Y Z / <lim y nB e ° cos ^

(x-x ) a (x.-x) (A.l)

n bo'

Y Z V r, nc o nir / V / \-l/2 /. lim } nC e cos — (x-x^ ) a (x-x^ ) (A. 2)

''t n c 1 1

x-»-x^ n=l

" -Y z -(z-z )mr/a . ,^

Him ) nA e sin — (x^-x )e a (z-z ) (A. 3) ''^ n a 1 o o

g->-z n=l

o

In order to find the asymptotic behavior of A , B and C , one only needs to realize that

n n n the following two summations;

00 , CO , V -1/2 . ^ V -1/2 -<Sn 1 n cos 6n and I n e n=l n=l

can be approximated for small 6 , by integrals of the form

00 oo , f -1/2 . A f -1/2 -6n ^

j n cos on dn and J n e dn

1 1

-1/2 -l/2 which are known immediately as (it6/2) and (Tr6) when 6->-0. Thus, by setting

5 = x^-x in (A.l), 6 = (x-x^) in (A. 2) and 6 = (z-z^) in (A. 3), we find

_ nb o , T vu -3/2t , , , B„e = 0[(-l) n J , (A.U)

C^e = 0[n-3/2] , (^.5)

. 'na o . nirb „r -3/2n - A^e sin —= 0[n ' ] , (A. 6)

2

Now since sin (nirb/a) = 0(l), we obtain by multiplying sin(mrb/a) on both sides of (A. 6), an alternative form

A^e ° = 0[n"3/2 sin(mrb/a)] . (A.T)

163
Using property (vii) of section 2 and {A.k) we see that

T(a)) = 0(0)"^/^) (A. 8)

for (i) = -Y , , m -> Using property (vi) of section 2, Chapter 2 and (A. 5), (A. 8) is true mo

for 0) = -Yjj^^j m Similarly using property (ii) of section 2 and (A.T)? we have

RES[T,Y ] = 0(n-^/2 ^.^2 r^b) na a

for n -y This is equivalent to saying (A. 8) holds as u y , n «> (see Royer and Mitt

1972). Hence we have that

T(u)) = 0(a) ' ) L| ^ «= (A. 9)

164
Appendix B: Asymptotic Behavior of the Perturbation

Sum for the E-Plane Step

Consider

-1-A

S =

I S (B.2)

n=N+l (1) - n

In order to examine this sum for |a)| ->• <=° we follow Evgrafov (1961) and examine the contour

integral

gl _ 1_ C

ot cot air da ^-g g)

I

where the contour is shown in Figure B-1.

Following Evgrafov (1961) we first evaluate the integral "by residues

-1-A ,. Sl =

I £ + TT co"^"^ cot cou (B.3)

n=N+l (D - n

for R ^ <=° and Re u > 9. But let us now seek an alternate representation of (B.2). Note

that

cot air -> -j sign (a. ) a. I^oo (B.3)

^ imag ' ' imag

and hence let us examine

-1-A, ^ .... , -1-A

gl _ ^ f a (cot aTr+j)da + 1. /" ci

;

+

1 r a ^( cot ctTT-j )da ^ 2^ r a ^dg

25 i- oj - a 2 0) - a

where ct is the contour (6, 0+iR, <=°+iR) and C. is (6, 0-iR, "-iR) as R ^ «> O D

cot aiT ± J = 0(e

and thus

gl _ zL t g ^cot otTr+jjda _^ 1_ r a (cot aTr-jJda 2jJ oj-a u-a

, -1-A, , -1-A,

+ — ( g + i r

ct o-g 2 J+ a)-a 2 w-a

^6

165
Pim

a -plane

0+iR

-KH—

I 2 N

0-iR

N+l

\

\

\

\

/

/

/

Fig. B-1: Contour of Integration for Asymptotic

Evaluation of Sum.

166
Now consider deforming the contours, , C to ( 9 ± ie , «> ± ie ) as e -> 0 . If Im cj > 0 then

If Im (jd < 0

CO -l-A Ira da 1 (• a da . -l-A

2 U a,-a = 2 { a)-a " ^

0 „ fc)

-l-A «> -l-A,

1 ( a da _ 1 f a da

2 •,- (jJ-a ~ 2 g

(jJ-a

-l-A CO _i_A

1_ r a da _ 1 fr a da

2 1+ (jj-a ~2 i

and if Im u = 0

_ • (jj-a ^ oj-a

-l-A °° -l-A Ira da Ira da , . -l-A

2 bi-a 2 ^ u-a

1_ j- a da ^. i [ Of 9^ _ py r da

2 ;!,+ u-a 2 cj-a w-a

For our purposes it is sufficient to consider two cases: (l) Re o) > G, and

(2) Re to < 9. For Re w > 9 we have

-l-A , , = ) + TTO) cot OJTT n=N+l

°° -l-A, , 9+j°° -l-A, , = PY C g da _ 1^ f a ( cot aTT+j Jdg

„ tj-a 2j { to-a

9-joo _i_A, - 1 r a (cot aTT-,]jda

2j

-1-A^ 9+j°° -1-A^

_ /• a da r a da

u 9 (cD-a;(l-e )

e_joo _i_A

9 (a,-a)(eJ^''^-l)

as 0) -> «> we use the residus term at a = o) and the principle value integral as the leading

terms

.

S = 0(u)"^) + 0(0)"^"^) (B.5)

167
For arg w = it, we have

-1-A

Si = y 2

n=N+l -

"

~ -1-A, e+j«> -i-A,

_ r a gg r a dot

e (co-a)(l-e-2j°'^)

e-j -1-A,

e (a)-ct)(eJ2"^-l)

as 0) we use the first integral as the leading term. By changing variables we recognize

the integral to be a hypergeometric function (Gradshteyn and Ryzhik, I965). From the

hypergeometric function, it is easy to show that again we have (B.5).

Hence (3.11) is j ustified

.

168
Appendix C: Asymptotic Behavior of the Perturhation

Sum for the Trifurcated Waveguide

Consider

s = ? ""^

(C.l)

n=N "

-

In order to examine this sum for |u)| «> we follow Evgrafox (1961) and examine the contour

integral

^1 _ 1 f sin a6 cot aTr da , s 2j ^ a(a - 0))

where E is shown in Figure B-1 (I'eplace 6 "by 9^ to eliminate confusion). Evaluating the

integral "by residues

00 sin ne . TT sin 0)6 cot uir Si =

I ^j^^ f; nU-w)

n=N

for R =» and Re oi > 6^. In a similar manner to Appendix B we can find the following

(C.3)

alternate representation of (C.2).

sin

a(a - 0))

00 _ 9+-ioo 0_-ico d -D^T r Sin aS da ^ o^ .or / r, ) \ S'- = PV J —7 N— + •••+ ••• (C.k)

o 00

for arg u = 0 and where the second and third integrals are similar to those given in

Appendix B and are not given since only the leading terms of the asymptotic expansion

of S are desired.

Let us examine the integral and the residue term as the leading terms

CO CO 00 PY r sin a6 da _ ^ r sin a6 da ^ i py f Do sin a6 da 00 y o The first term is order u The second integral can be evaluated asymptotically by

changing variables and using asymptotic expansions of the sine and cosine Integrals

given in Abramowitz and Stegun (1965). This yields

_„ r sm aQ da PV J - TT cos 0)0

•'j a - 0) o

Thus

S = 0(o) + 0

sm o)(Tr-e)

ir :—^

0) sin o)ir (C.5)

as 0) ^ =». It should be noted that the second term on the right in expression (C.5) vanishes

if arg 0) = IT in (C.l)

.

169
The result presented here is applicahle to the electric wall case. A similar res\ilt,

applicable to the magnetic wall case, is given by

1 f sin a6 tan air _ tt sin inQ tan (ott y (m-l/2) ^ sin(m+l/2)9

2,i i a(a-a)) uj ^ m - m - 1/2

I m=n

= _ p_v^

1 f

Bin(t+ )e ^ 1 r sin^

" 9-0, * e o o

2

J 2

•' 9 9 o o

Retaining the principal value integral and the residue term once again as the leading

terms, we obtain

00 —I

y (m-l/2) sin(m+l/2)e ^ oCw""'-) + 0

"(^-Q)

(C 7) ^ m-(jo-l/2 0) cos (joir m=n

Again, the second term on the right disappears for arg to = ir.

170
Appendix D: Evaluation of the Infinite Product

for the Electric Wall Case

Consider the product

CO (1-u/y , )(l-a)/Y ) p(w) = n

''''

n=l na

(D.l)

This can be written as

N-1 (I-oj/y Jil-^/y )

n-1 na

(D.2)

where is the remainder. Following a procedure similar to Kostelnicek and Mittra

(1961) we have

R./w) = exp] I 'n=N

In 1-— + In 1-—

t ^ncJ

- In 1—

na'

(D.3)

for I ''^Z Yjji^ I » I'^/'^'nc^' ^'^^''^na^ < 1 we can expand the logarithmic terms

f

CO 00 m

(jD E/.) = expj-I I ^

*-n=N m=l m m m m

nb nc na

(D.U)

Now consider the expansion

_2_

m niT

m

rnr

(m/2+l)m/2

2! niT 'nb

(m/2+2)(m/2+l)m/2

3!

b b

rnr P=l P rnr

niT

2p-2

(D.5)

where k = 1 is convenient,

o Thus

^00 00 m 00

t \ ! ^ 2p-2+m

^ ln=Nm=l"^ p=l P

^"^^

2p-2+m 2p-2+m>

(D.6)

171
Using Davis (1962) we find that we can evaluate sums of reciprocal powers of integers

using polygamma functions.

(D.T)

n=N n (m-l)!

N is chosen large enough that an asymptotic expansion of i|) is used. Only two or three

terms of this series are generally needed to find the polygamma function. Reversing

orders of summation of (D.6) we have

EM = exv\-l — I C ^ U=2 p=l P

(m) b 2p-2+m

+

c_ 2p-2+m

2p-2+m

\, 2p-2+m - \, n=N n ^ n=N

-1-

'''nb ^nc na

(D.8)

Note that the linear term of u is isolated. Let us consider this last term for a finite

upper limit and use (D.5)

M

I n=N

1 +^

Y Y Y

nb nc na

M

= I n=N

'b 'b nir

2p-2

c c 2p-2a a p=l " niT nir P=l P mr

2p-2

but b + c = a thus as M ^- ».

I n=N Y Y Y nb nc na p=2

(1)

P

.^.2p-l 2p-l

2p-l"

a_

77 1 1 1 n=N „2p-l/

(D.9)

The s\immation over p and n are fastly convergent. In order to determine how many powers

of 0) are necessary the remainder term (of u) can be approximated by using a procedure

similar to Kostelnicek (1969). Using the approximation - nir/h in the sum

" m

^ m=M n=N m m m

Y Y Y

^ nb nc na'

(D.IO)

172
we find

M

r ^ +

* »

Nttj

(D.ll)

where

m=M

This enables the remaining sum to be truncated accurately.

173
Appendix E: The Canonical ProlDlem with a Magnetic Symmetry Boundary

Consider Figoire 2,2.1, except let the "boundary at x=x^ "be a magnetic wall. We can find the TM fields from (j) = H where

y

r rA(o) 2n-l,2a . '2n-l,2a . , , s /t? -,\

) [A e ' + A e ' ]sin k (x-x ) (E.l) n n na o n=l

V rr, o ^2n-l,2h „ ^2n-l,2h . . , / x o^ = ) [B e ' + B e ' Jsin k ^(x-x ) (E.2)

n n nb o n=l

V r^lo) nc , _ nc T niT , > on

) [C e + C e J cos — (x-x^ ) (E.3)

'-^ n n c 1

n=0

where

= ( 2n-l ) TT _ ( 2n-l ) it na 2a ' nb ~ 2b

Note that regions A and B cannot support a TEM mode.

We proceed in an identical manner as section 2 and find that the solution may be

found from a meromorphic function T(u)) which has the properties:

(i) RES[T,Y^ , ^ ] = k cos k b A e 2n-l,2a o 2n-l,2a na na n

n = 1,2,---

(ii) RES[T,-Y^ , ^ ] = k cos k b A^°)e^^''"^'^^^° 2n-l,2a na na n

n = 1,2,---

(iii) T(y^J = -Y^^ c C^^^e"''^^'^ n = 1,2,...

(iv) T(jk^) = -2jk^ c C^°^e"^^°^°

^ ^' ^^^2n-l,2b^ ^2n-l,2b ^ n

®

+ Y z (vi) T(-Y )=Y cCe'^'^" n = 1,2,..'

nc nc n

jk z (vii) T(-jk ) = 2jk c C e °

°

o o o

Y 2

r ( n sn+lm/- \ -u r, 2n-l,2b o ^ ^ (viix) (-1) T(-Y2^_^^2b) = -^2n-l,2b ^ V " = l'^'-"

(ix) T(a)) = 0(aj"-'"^^) LI -> <=°

174
T(a)) can te constructed as follows:

T(aj) = H(a))F(a)) K^+((o-jk^)j I -f^—

'n=l ''^nc

where

(b)

y n=l '^-^2n-l,2b n=l '^•'^2n-l,2a

(a)

(E.M

H(a)) = exp

-0)

Id In — + c In —

a a

(E.5)

and

F(a)) = n=l (1 - '^/^2n-l,2a^

(E.6)

k 5 g^^\ g^^^j g^^^ are related to the incident fields by (iv), (iii), (v) and (ii).

o n n n

175
Appendix F: The Trifurcated Waveguide vith

a Magnetic Wall

Figure F-1 illustrates the magnetic wall trifurcated waveguide and the avixiliary

problem.

The solution is obtained by constructing two mermdrphic functions.

T^(a)) = H^(a))F^(a))

(l) m,n

K^l) . (.-jk ) I

° n=l '^^^2n-l,2c

(F.l)

+ (a)-jk ) I

(2)

h

n=l '2n-l,2c^

(F.2)

where T^(w) is identified with the junction at z = 0 and T^iui) is identified with the

junction z = A. H^(a)), E^im) , F^(a)), Fgdu), K^'^^ and K^^^ are given by (E.i|) —(E.6) with

only a change of geometrical factors necessary.

(1) (2) We can derive two infinite equations for g^ and g^ by requiring that the expressions for the modal coefficients in the coupling region be consistent.

RES [\'^2n-l,2c^ =4'^[^n'^]"' ^ cos k b

nc nc o

(_1)^-^1t ^ ^' 2^(-Y )=-Y c g^^hK^^h-^ '^2n-l,2c^ '2n-l,2c ^ ^n ^ n

^

(F.3)

(F.li)

where

n n n

g(2) = k(2)

n n n

(F.5)

(F.6)

and K^"*^^ and K^^^ are found from properties (ii) and (v) of Appendix E. In order to truncate equations (F.3) and {F .k) we find the asymptotic behavior of

g^"""^ and g^^^. From (F.5) and (F.6) we can find

(1) (2) -1 , , ^n . . . g ,g = 0(n (-1) cos k b ) n ^n n,c o (F.T)

This choice allows the efficient solution of equations.

176
a

Vmagnetic wall

(a) The Trifurcated Waveguide

z

= 0 7J= A

I

\

>

i,

I a

1

("b) Auxiliary Geometry

Fig. F-1 The Trifurcated Waveguide (with a Magnetic

Symmetry Wall) and the Auxiliary Problem.

177
;4A (REV. B-78)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS MN 164

2.Gov*t Accessiwj No. 3. R«cip>mt's Accession No.

4. TITLE AND SUBTITLE

Electromagnetic Boundary-Value Problems Based Upon

a Modification of Residue Palruluja and 'Piinr-Unn

Theoretic Techniques

5. Publication Date June 1979

PerfOfrntOROfganization Code

7. AUTHOR(S)

James Patrick Montgomery and David C. Chang

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

National Bureau of Standards

Department of Commerce

Boulder, Colorado 80302

1«. Pfoje<^ask/Wort( ibit Ho. 7203900

... .....1, . ..... '

11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS rs(ree(. City, si^ie. ziP)

Same as No* 9.

13. Type of Report & Period Covered

14. SpoiuKjrlng Agoncy Code

15. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number : 79-600072

I I Document describes a computer program; SF-185, FIPS Software Summary, is attached.

16. ABSTRACT (A 200-word or leas factual sumtnary of most significant information. If document includes a significant bibtiograpfiy or

literature survey, mention it here.) The solution of a number of electromagnetic problems, in both close I and open systems, using the modified residue calculus and functional

theoretic techniques is presented.

The solutions start with known closed region problems and then are

extended to new closed region problems and finally to several open regioi

problems

.

Specific problems considered for the closed region are: 1) the trifurcated waveguide; 2) the dielectrically loaded trifurcated waveguide;

3) the N-furcated waveguide; 4) the dielectrically loaded N-furcated

waveguide; 5) determination of the Eigenvalues of ridged; waveguide; and

6) scattering by a dielectric st6p*

Open region problems considered aret 1) a parallel plate radiating

into a homogeneous half-space; 2) a finite phased array; 3) remote

sensing of the earth using parallel plate waveguides; 4) a flanged waveguide radiating into a half-space; 5) scattering by a thick semi-inf init((

plane; and 6) radiation from a slot in a waveguide wall.

Some suggested extension of the techniques to other types of

problems is also included.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unlees a proper name;

separated by semicolons)

Closed systems; electromagnetic problems; functional theoretic

techniques; modified residue calculus; open systems; remote sensing,

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