DETERMINATION of CUTOFF FREQUENCIES of SIMPLE WAVEGUIDES USING FINITE DIFFERENCE METHOD a Thesis Presented in Partial Fulfillme

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DETERMINATION OF CUTOFF FREQUENCIES OF SIMPLE WAVEGUIDES USING

FINITE DIFFERENCE METHOD

A Thesis

presented in partial fulfillment of requirements

for the degree of Master of Science with emphasis

in Electromagnetics

in the Department of Electrical Engineering

The University of Mississippi.

SRIDHAR KOLAGANI

December 2012

ABSTRACT

Waveguides are used to transfer electromagnetic energy from one location to another.

Within many electronic circles, waveguides are commonly used for microwave RF signals; the

same principle can be used for many forms of waves from sound to light. They have been used in

many technologies like acoustic waveguide speaker technology, high-performance passive

waveguide technologies for remote sensing and communication, optical computing, robotic- vision, biochemical sensing and many more.

Modern waveguide technology employs a variety of waveguides with different cross sections and perturbations, the cutoff frequencies and mode shapes of many of these waveguides are ill-suited for determination by an analytical method. In this thesis, we solve this type of

waveguides by employing the numerical procedure of finite difference method. By adopting

finite difference approach with an application of eigenvalue method, we discuss about few

different types of these waveguides in determining the cutoff frequencies of supported modes,

and extracting the possible degenerate modes and their field distributions. To validate the method

and its accuracy, it is applied to the two well known rectangular waveguides, viz. PEC

Rectangular Waveguide and Artificial Rectangular Waveguide (consists of PEC and PMC walls)

and compared with the analytical solutions.

Dedication: To my family.

iii

ACKNOWLEDGEMENTS

I would like to express my sincere gratitude to my advisors Dr. Richard K. Gordon and

Dr. W. Elliott Hutchcraft for the continuous support of my Master’s study and research, for their patience, motivation, enthusiasm, and immense knowledge. Their guidance helped me in all the time of research and writing of this thesis.

I would like to thank Dr. Ramanarayanan Viswanathan, Professor and Chair of Electrical

Engineering for being one of my committee members and for the financial support he has given me during the course of my degree at the University of Mississippi.

I would also like to thank Dr. Allen W. Glisson, Professor and former Chair of Electrical

Engineering, for the financial support he has given me during the course of my degree at the

University of Mississippi.

Last but not least I would like to thank my colleagues and friends: Dr. Chandra S. R.

Kaipa, Dr. Yashwanth Reddy Padooru, Naveen Sankranthi, Sarathi B Yedugani, Koushik Gattu,

Anvesh Kaliki, Sai Kiran, Prudhvi Duggirala, and Rambabu Sankranthi for their hope and support.

TABLE OF CONTENTS CHAPTER PAGE

Abstract … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . ii

Dedication… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . iii

Acknowledgement… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . . iv

List of Tables … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . viii

List of Figures … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . ix

1. FINITE DIFFERENCE METHOD … … … … … … … … … … … … … … … … … … … … … … .1

1.1 Introduction … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … .1

1.2 General Concepts of Finite Difference Method… … … … … … … … … … … … … … … .2

1.3 Types of Meshing … … … … … … … … … … … … … … … … … … … … … … … … … … … … 5

1.4 FD approach for solving PDE … … … … … … … … … … … … … … … … … … … … … … . .8

1.4.1 Method I … … … … … … … … … … … … … … … … … … … … … … … … … … … … … .9

1.4.2 Method II … … … … … … … … … … … … … … … … … … … … … … … … … … … … .10

2. WAVEGUIDES … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … .12

2.1 Introduction … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … 12

2.2 Waveguide theory … … … … … … … … … … … … … … … … … … … … … … … … … … … .13

2.2.1 Waveguide Advantages … … … … … … … … … … … … … … … … … … … … … … 15

2.2.2 Waveguide Disadvantages … … … … … … … … … … … … … … … … … … … … . .15

2.3 Rectangular Waveguides … … … … … … … … … … … … … … … … … … … … … … … … .16

2.3.1 TE Wave … … … … … … … … … … … … … … … … … … … … … … … … … … … … .19

2.3.2 TM Wave … … … … … … … … … … … … … … … … … … … … … … … … … … … … .21

3. DETERMINATION OF WAVENUMBERS OF THE SUPPORTED MODES BY

WAVEGUIDES USING METHOD I … … … … … … … … … … … … … … … … … … … … . .23

3.1 Introduction … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … 23

3.2 Different cases of nodes used in Matrix filling … … … … … … … … … … … … … … . .24

3.2.1 Interior nodes … … … … … … … … … … … … … … … … … … … … … … … … … … .25

3.2.2 TE case and PEC boundary … … … … … … … … … … … … … … … … … … … … .26

3.2.3 TM case and PEC boundary … … … … … … … … … … … … … … … … … … … … 27

3.3 PEC Rectangular Waveguide … … … … … … … … … … … … … … … … … … … … … … .28

3.3.1 TE Wave … … … … … … … … … … … … … … … … … … … … … … … … … … … … .29

3.3.2 TM Wave … … … … … … … … … … … … … … … … … … … … … … … … … … … … .38

3.4 Artificial Rectangular Waveguide … … … … … … … … … … … … … … … … … … … … .44

3.4.1 TE Wave … … … … … … … … … … … … … … … … … … … … … … … … … … … … .44

3.4.2 TM Wave … … … … … … … … … … … … … … … … … … … … … … … … … … … … .51

3.5 PEC Rectangular Waveguide with PEC Fin … … … … … … … … … … … … … … … … 57

3.6 L-Shape Waveguide … … … … … … … … … … … … … … … … … … … … … … … … … … .64

4. DETERMINATION OF WAVENUMBERS OF THE SUPPORTED MODES BY

WAVEGUIDES USING METHOD II … … … … … … … … … … … … … … … … … … … … .68

4.1 PEC Rectangular Waveguide … … … … … … … … … … … … … … … … … … … … … … .69

4.1.1 TE Wave … … … … … … … … … … … … … … … … … … … … … … … … … … … … .69

4.1.2 TM Wave … … … … … … … … … … … … … … … … … … … … … … … … … … … … .71

4.2 Artificial Rectangular Waveguide … … … … … … … … … … … … … … … … … … … … .73

4.2.1 TE Wave … … … … … … … … … … … … … … … … … … … … … … … … … … … … .73

4.2.2 TM Wave … … … … … … … … … … … … … … … … … … … … … … … … … … … … 74

4.3 PEC Rectangular Waveguide … … … … … … … … … … … … … … … … … … … … … … .76

4.4 L-Shape Waveguide … … … … … … … … … … … … … … … … … … … … … … … … … … .77

5. CONCLUSION … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … .79

BIBLIOGRAPHY … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … .80

VITA … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . … … … … .83

vii

LIST OF TABLES

TABLE PAGE

3.1 Comparison of Solutions for PEC Rectangular Waveguide TE and TM Waves with an

Analytical Method … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . .35

3.2 Comparison of Solutions for Artificial Rectangular Waveguide TE and TM Waves with an

Analytical Method … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . .49

3.3a Cutoff frequencies of first few modes for h=0 … … … … … … … … … … … … … … … … … … . .58

3.3b Cutoff frequencies of first few modes for h=b/4 … … … … … … … … … … … … … … … … … … 58

3.3c Cutoff frequencies of first few modes for h=b/2 … … … … … … … … … … … … … … … … … … 59

3.3d Cutoff frequencies of first few modes for h=3b/4 … … … … … … … … … … … … … … … … … .59

3.3e Cutoff frequencies of first few modes for h=b … … … … … … … … … … … … … … … … … … . .59

3.4 Cutoff frequencies of first few modes of an L-Shape Waveguide … … … … … … … … … … . . .65

viii

LIST OF FIGURES

FIGURE PAGE

1.1 Staggered mesh … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … 6

1.2 Original mesh … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … . .7

2.1 End view of two wire line … … … … … … … … … … … … … … … … … … … … … … … … … … … .13

2.2 End view of coaxial cable … … … … … … … … … … … … … … … … … … … … … … … … … … . .14

2.3 Waveguide shapes … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … .14

2.4 Rectangular Waveguide … … … … … … … … … … … … … … … … … … … … … … … … … … … . .16

3.1 Classification of different cases of nodes … … … … … … … … … … … … … … … … … … … … … 24

3.2 Interior nodes of any Cartesian problem domain … … … … … … … … … … … … … … … … … . .25

3.3 TE case and PEC boundary … … … … … … … … … … … … … … … … … … … … … … … … … … .27

3.4 TM case and PEC boundary … … … … … … … … … … … … … … … … … … … … … … … … … … 27

3.5 PEC Rectangular Waveguide cross section … … … … … … … … … … … … … … … … … … … .28

3.6 Field distribution of the axial component of the magnetic field for mode of PEC

20 Rectangular Waveguide … … … … … … … … … … … … … … … … … … … … …푇퐸… … … … … … … .36

3.7 Field distribution of the axial component of the magnetic field for mode of PEC

21 Rectangular Waveguide … … … … … … … … … … … … … … … … … … … … …푇퐸… … … … … … … .36

3.8 Field distribution of the axial component of the magnetic field for mode of PEC

30 Rectangular Waveguide … … … … … … … … … … … … … … … … … … … … …푇퐸… … … … … … … .37

3.9 Field distribution of the axial component of the magnetic field for mode of PEC

02 Rectangular Waveguide … … … … … … … … … … … … … … … … … … … … …푇퐸… … … … … … … .37

3.10 Field distribution of the axial component of the electric field for mode of PEC

11 Rectangular Waveguide … … … … … … … … … … … … … … … … … … … … …푇푀… … … … … … … .42

3.11 Field distribution of the axial component of the electric field for mode of PEC

21 Rectangular Waveguide … … … … … … … … … … … … … … … … … … … … …푇푀… … … … … … … .43

3.12 Artificial Rectangular Waveguide … … … … … … … … … … … … … … … … … … … … … … .44

3.13 Field distribution of the axial component of the magnetic field for

mode of Artificial Rectangular Waveguide … … … … … … … … … … … … … … … . .50

21 3.14 Field푇퐸 distribution of the axial component of the magnetic field for

mode of Artificial Rectangular Waveguide … … … … … … … … … … … … … … … . .50

12 3.15 Field푇퐸 distribution of the axial component of the electric field for

mode of Artificial Rectangular Waveguide … … … … … … … … … … … … … … … .55

31 3.16 Field푇푀 distribution of the axial component of the electric field for

mode of Artificial Rectangular Waveguide … … … … … … … … … … … … … … … .56

22 3.17 PEC푇푀 Rectangular Waveguide with PEC Fin … … … … … … … … … … … … … … … … … … 57

3.18a Field distribution of the axial component of the electric field of for “h=0” … … . .60

11 3.18b Field distribution of the axial component of the electric field of 푇푀 for “h=b/4” … . .61

11 3.18c Field distribution of the axial component of the electric field of 푇푀 for “h=b/2” … . .61

11 3.18d Field distribution of the axial component of the electric field of 푇푀 for “h=3b/4” … .62

푇푀11

3.18e Field distribution of the axial component of the electric field of for “h=b”

11 (degenerate mode) … … … … … … … … … … … … … … … … … … … … … … … …푇푀… … … … … .62

3.18f Field distribution of the axial component of the electric field of for “h=b”

11 (degenerate mode) … … … … … … … … … … … … … … … … … … … … … … … …푇푀… … … … … .63

3.19a An L-Shaped Waveguide type 1 … … … … … … … … … … … … … … … … … … … … … … … . .64

3.19b An L-Shaped Waveguide type 2 … … … … … … … … … … … … … … … … … … … … … … … . .64

3.20 Field distribution of the axial component of the electric field of for an L-Shape

11 Waveguide … … … … … … … … … … … … … … … … … … … … … … … … …푇푀… … … … … … … . .66

3.21 Field distribution of the axial component of the electric field of for an L-Shape

21 Waveguide … … … … … … … … … … … … … … … … … … … … … … … … …푇푀… … … … … … … . .66

3.20 Field distribution of the axial component of the electric field of for an L-Shape

22 Waveguide … … … … … … … … … … … … … … … … … … … … … … … … …푇푀… … … … … … … . .67

3.20 Field distribution of the axial component of the electric field of for an L-Shape

31 Waveguide … … … … … … … … … … … … … … … … … … … … … … … … …푇푀… … … … … … … . .67

4.1a Determination of cutoff frequencies of TE wave for PEC Rectangular waveguide ( =

16 , = 11) … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … …푛푥.69

4.1b Determination푛푦 of cutoff frequencies of TE wave for PEC Rectangular waveguide ( =

31 , = 21) … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … …푛푥.70

4.2a Determination푛푦 of cutoff frequencies of TM wave for PEC Rectangular waveguide ( =

16 , = 11) … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … …푛푥.71

푛푦

4.2b Determination of cutoff frequencies of TM wave for PEC Rectangular waveguide ( =

31 , = 21) … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … …푛푥.71

4.3a Determination푛푦 of cutoff frequencies of TE wave for Artificial Rectangular waveguide

( = 16 , = 11) … … … … … … … … … … … … … … … … … … … … … … … … … … … … .73

4.3b Determination푛푥 푛푦 of cutoff frequencies of TE wave for Artificial Rectangular waveguide

( = 31 , = 21) … … … … … … … … … … … … … … … … … … … … … … … … … … … … .74

4.4a Determination푛푥 푛푦 of cutoff frequencies of TM wave for Artificial Rectangular waveguide

( = 16 , = 11) … … … … … … … … … … … … … … … … … … … … … … … … … … … … .74

4.4b Determination푛푥 푛푦 of cutoff frequencies of TM wave for Artificial Rectangular waveguide

( = 31 , = 21) … … … … … … … … … … … … … … … … … … … … … … … … … … … … .75

4.5 Determination푛푥 푛푦 of cutoff frequencies of TM wave for PEC Rectangular waveguide with

PEC Fin ( = 21 , = 21) … … … … … … … … … … … … … … … … … … … … … … … … .76

4.6 Determination푛푥 of cutoff푛푦 frequencies of TM wave for an L-Shape waveguide ( =

21 , = 21) … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … …푛푥.77

푛푦

xii

CHAPTER I

FINITE DIFFERENCE METHOD

Analysis of hollow conducting waveguides is an area that has been in the limelight lately.

Various methods to find the cutoff wavenumbers are available in the literature. For example,

finite difference time domain approach by using the integral form of Maxwell’s equations [1],

finite element method [2], method of moments by using the surface integral equation [3, 4], and

spectrum of two dimensional solutions with an application of asymptotic waveform evaluation

[5].

Treatment of waveguides with arbitrary cross section is a challenging task to solve by an

analytical method. Though numerous approaches are available in the literature to solve this

problem, they may be deficient in the areas, viz. simplicity, accuracy and speed.

A finite difference approach with an application of eigenvalue method has been used here

because of its simplicity, accuracy and speed. This method to solve for the waveguide’s cutoff

wavenumbers, cutoff frequencies and mode shapes has been explained.

1.1 Introduction to Finite Difference Method

The finite difference method (FDM) was first developed by A. Thom in the 1920s under the title “the method of squares” to solve nonlinear hydrodynamic equations. Finite

difference methods have dominated computational science since its inception and were the

method of choice in the 1960s and 1970s [6].

From the study of [7], while integral equation based methods have been employed occasionally [8,9], partial differential equation (PDE) based methods have been proven to be more practical and have seen broader use.

Several key attributes combine to make the finite difference method, a useful and powerful tool. First is the method’s simplicity; Maxwell’s equations in differential form are discretized in space and time in a straight forward manner. Second, since the method tracks the time-varying fields through out a volume of space, finite difference results lend themselves well to scientific visualization methods. These, in turn, provide the user with excellent physical insights on the behavior of electromagnetic fields.

Since then, the method has found applications in solving different field problems.

While other methods, such as the finite element method and boundary element have enjoyed recent popularity, finite difference methods are still utilized for a wide array of computational engineering and science problems.

1.2 General Concepts of Finite Difference Method

Before proceeding to discuss the mathematical formalities of the finite difference method,

we will discuss about the Taylor’s series. From elementary calculus, we remember that the

Taylor series formula is used to expand a function, ( ) as a power series,

푓 푥 ( ) ( ) ( ) = ( ) + ( ) + ( ) + 1! 2! 2 푥 − 푎 ′ 푥 − 푎 ′′ 푓 푥 푓 푎 푓 푎 푓 푎 ⋯

∞ ( ) ( ) = ( )( ) (1.1) ! 푛 푥 − 푎 푛 푓 푥 � 푓 푎 푛=0 푛 We replace and by + and respectively, the Taylor series now becomes

푥 푎 푥 ∆푥 푥 ( ) ( + ) = ( ) + ′( ) + ′′( ) + 1! 2! 2 ∆푥 ∆푥 푓 푥 ∆푥 푓 푥 푓 푥 푓 푥 ⋯ ∞ ( ) ( + ) = ( )( ) (1.2) ! 푛 ∆푥 푛 푓 푥 ∆푥 � 푓 푥 푛=0 푛 It is convenient to take only the first two terms of the right hand side of the previous equation, upon solving for the first order, forward difference approximant to ( ), ′ 푓 푥 ( + ) ( ) ( ) = (1.3) ′ 푓 푥 ∆푥 − 푓 푥 푓 푥 ∆푥 Likewise, we can define the first order, backward difference approximant to ′( ), (just substitute instead in the expansion) 푓 푥

−∆푥 ∆푥 ( ) ( ) ( ) = (1.4) ′ 푓 푥 − 푓 푥 − ∆푥 푓 푥 ∆푥 And, the first order, central difference approximant to ′( ), is obtained by the above two equations, and is given by 푓 푥

( + ) ( ) ( ) = (1.5) 2 ′ 푓 푥 ∆푥 − 푓 푥 − ∆푥 푓 푥 ∆푥

Similarly, we can also obtain the second order approximations, which include the simple

calculations, and they are given as: The forward difference formula for the second order

derivative

( + 2 ) 2 ( + ) + ( ) ′′( ) = (1.6) ( ) 푓 푥 ∆푥 − 푓 푥 ∆푥 푓 푥 푓 푥 2 ∆푥 The backward difference formula for the second order derivative

( ) 2 ( ) + ( 2 ) ′′( ) = (1.7) ( ) 푓 푥 − 푓 푥 − ∆푥 푓 푥 − ∆푥 푓 푥 2 ∆푥 The central difference formula for the second order derivative

( + ) 2 ( ) + ( ) "( ) = (1.8) ( ) 푓 푥 ∆푥 − 푓 푥 푓 푥 − ∆푥 푓 푥 2 ∆푥 To apply the difference method to find the solution of a function ( , ), we divide the solution region in the – plane into equal rectangles or mesh of sides 푈 푥 and푦 y .We let the coordinates ( , ) of a 푥typical푦 grid point or node be Δ푥 Δ

푥 푦 = = 0,1,2,3 … …

푥 푖∆푥 푖 = = 0,1,2,3 … …

푦 푗∆푦 푗 and the value of U at P be

U = U( , )

푃 푖 푗

We use the second order central difference method in this thesis from now. With the above notation , the second order central difference approximations of the derivatives of U at the (i,j)th node are obtaıned as follows,

U( 1, ) 2U( , ) + U( + 1, ) U = (1.9a) |( , ) ( ) 푖 − 푗 − 푖 푗 푖 푗 푥푥 푖 푗 2 ∆푥 U( , 1) 2U( , ) + U( , + 1) U = (1.9b) |( , ) ( ) 푖 푗 − − 푖 푗 푖 푗 푦푦 푖 푗 2 ∆푦 Where, U |( , ) U |( , ), represents the relationship of the neighboring nodes in the

푥푥 푖 푗 푦푦 푖 푗 and 푎푛푑 respectively.

푥 − 푑푖푟푒푐푡푖표푛 푦 − 푑푖푟푒푐푡푖표푛 From equations (1.9a) and (1.9b),

( ) + ( ) 2 ( ) ( ) + ( ) 2 ( ) ( ) = + (1.10) ( ) ( ) 2 푢 푅 푢 퐿 − 푢 푃 푢 푇 푢 퐵 − 푢 푃 ∇ 푡 푢 푃 2 2 ∆푥 ∆푦 1.3 Types of Meshing

Before discussing the formulation of the TE and TM modes, it is necessary to briefly

discuss the two meshing techniques that were adopted for solving the different problem domains

considered in this thesis.

Here, nodes are created by considering the two conditions. The first condition is to create

the nodes with constant spacing along the ( . . , ), and constant spacing along the ( . . , ). This is flexible enough푥 − 푎푥푖푠to consider푖 푒 푑푒푙푥 different spacing between nodes in

푦 − 푎푥푖푠 푖 푒 푑푒푙푦 ( . . , ). In the second condition, it is important the wave

푥 − 푎푥푖푠 푎푛푑 푦 − 푎푥푖푠 푖 푒 푑푒푙푥 ≠ 푑푒푙푦 5

equation being solved is applicable in free space. From this, it is important to avoid the creation of nodes on boundary of the problem domain.

First, the staggered mesh which is shown in the figure 1.1, it was used by T. K. Sarkar in

[10]. In this type of meshing, the spacing between the nodes is constant in their

respective ( . . , , ). And the exterior nodes (i.e.,

outer most푎푥푖푠 nodes)푖 푒are푑푒푙푥 placed푖푛 푥with− 푎푥푖푠 half 푎푛푑of the푑푒푙푦 actual푖푛 푦 spacing− 푎푥푖푠 away from the boundary in their

respective axis ( . . , , ). 푑푒푙푥 푑푒푙푦 푖 푒 2 푖푛 푥 − 푎푥푖푠 푎푛푑 2 푖푛 푦 − 푎푥푖푠 Second, the original mesh which is shown in the figure 1.2, we consider the constant

spacing in the creation of every node with respect to the other node in their respective

( . . , , ). But in this case, the exterior nodes are placed푎푥푖푠

away푖 푒 from푑푒푙푥 the푖푛 boundary푥 − 푎푥푖푠 with푎푛푑 a푑푒푙푦 very푖푛 small푦 − 푎푥푖푠fraction of the actual spacing.

Staggered Mesh 1

0.9 delx/2 , 0.8 dely/2

0.7 delx , 0.6 dely 0.5 Y-axis 0.4

0.3

0.2 n

0.1 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X-ax i s

Figure 1.1: Staggered mesh

Original Mesh 1

0.9

0.8

0.7 delx dely 0.6

0.5 Y-axis 0.4

0.3 n 0.2

0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X-ax i s

Figure 1.2: Original mesh

Here we place the exterior nodes away from the boundary (towards inside the problem domain) with a very small fraction of the actual spacing in their respective instead of considering on the boundary, due to consideration of free space wave equation.푎푥푖푠 We refer these nodes (i.e., exterior nodes) as “nodes on the boundary” in this thesis for our convenience.

From the above two types of meshing, the original meshing, gives us the better solution for the domain closer to the boundaries of the problem when compared to the staggered meshing, this is due to the nodes near to the boundary. Based on this, we continue with the original meshing in solving the problems in the thesis.

1.4 FD approach for solving PDE

The finite difference techniques are based upon approximations, which permit replacing differential equations by finite difference equations. These finite difference approximations are algebraic in form; they relate the value of the dependent variable at a point in the solution region to the values at some neighboring points.

Thus a finite difference solution basically involves three steps:

(1) Dividing the solution region into a grid of nodes.

(2) Approximating the given differential equation by finite difference equivalent that relates the

dependent variable at a point in the solution region to its values at the neighboring points.

(3) Solving the difference equations subject to the prescribed boundary conditions and/or initial

conditions.

We consider the Helmholtz or wave equation in free space, + k = 0 (1.11) 2 2 ∇ 푢 푢 Where, for modes = = ( , ) (1.12 ) −푗훽푧 푇푀 푢 퐸푧 푒푧 푥 푦 푒 푎 and, for modes = = ( , ) (1.12 ) −푗훽푧 푇퐸 푢 퐻푧 ℎ푧 푥 푦 푒 푏 = = = (1.13) where, k is the wavenumber given by 2 2 2 2 휔 2 4휋 푓 2 2 푘 푐 휔 휇휀 푐 The are the permeability and permittivity of the free space respectively, and is the speed of휇 the푎푛푑 light.휀 푐

We solve these problems numerically by two methods using FD approach, and refer to them as (Eigenvalue Method) [11], and (Frequency Scanning Method) from now.푀푒푡 ℎ표푑 퐼 푀푒푡ℎ표푑 퐼퐼

1.4.1

Consider퐌퐞퐭퐡퐨퐝 퐈the wave equation (1.11),

+ k = 0 (1.14) 2 2 ∇ 푢 푢 + + + k = 0 (1.15) 2 2 2 휕 휕 휕 2 � 2 2 2 � 푢 휕푥 휕푦 휕푧 and from the equations (1.12a-b),

+ + k = 0 (1.16) 2 2 휕 휕 2 2 � 2 2 − β � 푢 휕푥 휕푦 + = ( k ) (1.17) 2 2 휕 휕 2 2 � 2 2� 푢 β − 푢 휕푥 휕푦 = + = k , then equation (1.17) becomes Let 2 2 and let 2 휕 휕 2 2 2 2 2 ∇푡 휕푥 휕푦 푘푡 − β = (1.18) 2 2 ∇푡 푢 − 푘푡 푢 At cutoff frequency, = 0. So,

훽 = k (1.19) 2 2 푡 Here we use the central difference∇ 푢 formula− 푢 for the second order derivative for the filling of matrix , this will be discussed in chapter III. If is a non-zero vector and

푀� 푉� 9

= (1.20)

then is said to be an eigenvector of 푀� 푉with� 휆associated푉� eigenvalue λ. On solving this, we get

various푉� eigenvalues λ and corresponding푀� eigenvectors . From the equations (1.19) and

(1.20) we can observe the eigenvalues (λ) are nothing but푉� the wavenumbers ( k ) and their 2 corresponding eigenvectors are the solution of the particular mode or −the associated

wavenumber. This method is used in detail to solve the different types of waveguides in

chapter III.

1.4.2

퐌퐞퐭퐡퐨퐝From the 퐈퐈equation (1.16),

= ( k ) (1.21) 2 2 2 ∇푡 푢 β − 푢 At cutoff frequency, = 0. Thus,

β ( + k ) = 0 (1.22) 2 2 ∇푡 푢 This means that when = , for some mode, the matrix + k is singular. Thus, its 2 2 푐 푡 determinant is ‘0’. The matrix푓 푓 + k is composed of the addition∇ of the matrix (as 2 2 푡 mentioned in the Method I), and k∇, where k is given by the equation (1.13). This can be푀� done 2 2 for any range of frequency, so that we can obtain the modes of operation in that particular range

frequency. We will obtain this by plotting the range of frequency and the determinant of the

matrix. The value of the determinant of the matrix either changes from the positive value to the

negative value or vice versa, the cutoff frequencies of the particular modes which are in the

given frequency range are obtained, as the curve intersects the zero-value at each and every

cutoff frequency. This method is used in detail to solve the different types of waveguides in the chapter IV.

The filling of the matrix will be similar in both and except for ( , ) in Method II, in the method,푀� we add the term k for 푀푒푡the ℎ표푑( ,퐼 ) element푀푒푡ℎ in표푑 the퐼퐼 matrix 2 over푀 an푖 operating푖 range of frequency. It will be discussed in detail for푀 the푖 푖 both mode and mode while solving the different types of waveguides in the chapters III and IV.푇퐸 푇푀

CHAPTER II

WAVEGUIDES

2.1 Introduction

The portion of the electromagnetic spectrum which falls between 1000 megahertz and

100,000 megahertz is referred to as the microwave region. Before discussing the principles and applications of microwave frequencies, the meaning of the term microwave as it is used in this module must be established. On the surface, the definition of a microwave would appear to be simple because, in electronics, the prefix “micro” normally means a millionth part of a unit.

Micro also means small, which is a relative term, and it is used in that sense in this module.

Microwave is a term loosely applied to identify electromagnetic waves above 1000 megahertz in frequency because of the short physical wavelengths of these frequencies.

Short wavelength energy offers distinct advantages in many applications. For instance, excellent directivity can be obtained using relatively small antennas and low-power transmitters.

These features are ideal for use in both military and civilian radar and communication applications. Small antennas and other small components are made possible by microwave frequency applications. This is an important consideration in shipboard equipment planning where space and weight are major problems. Microwave frequency usage is especially important in the design of shipboard radar because it makes possible the detection of smaller targets.

Microwave frequencies present special problems in transmission, generation, and circuit design that are not encountered at lower frequencies. Conventional circuit theory is based on voltages and currents while microwave theory is based on electromagnetic fields.

The concept of electromagnetic field interaction is not entirely new, since electromagnetic fields form the basis of all antenna theory. However, many students of electronics find electromagnetic field theory very difficult to visualize and understand [12].

2.2 Waveguide theory

The two-wire transmission line used in conventional circuits is inefficient for transferring electromagnetic energy at microwave frequencies. At these frequencies, energy escapes by radiation because the fields are not confined in all directions, as illustrated in figure 2.1.

Figure 2.1: End view of two wire line [12].

Coaxial lines are more efficient than two-wire lines for transferring electromagnetic

energy because the fields are completely confined by the conductors, as illustrated in figure 2.2.

Figure 2.2: End view of coaxial cable [12].

Waveguides are the most efficient way to transfer electromagnetic energy. Waveguides are essentially coaxial lines without center conductors. They are constructed from conductive material and may be rectangular, circular, or elliptical in shape, as shown in figure 2.3.

Figure 2.3: Waveguide shapes [12].

2.2.1 Waveguide Advantages

Waveguides have several advantages over two-wire and coaxial transmission lines. For example, since the copper losses ( ) are inversely proportional to the surface area of the 2 conductor. Two-wire transmission lines퐼 푅 have large copper losses because they have a relatively small surface area. The surface area of the outer conductor of a coaxial cable is large, but the surface area of the inner conductor is relatively small, thus the copper losses are also large in the coaxial cable compared to the waveguides, because the large surface area of waveguides greatly reduces copper losses.

2.2.2 Waveguide Disadvantages

Physical size is the primary lower-frequency limitation of waveguides. The width of a waveguide must be approximately a half wave length at the frequency of the wave to be transported. For example, a waveguide for use at 1 megahertz would be about 500 feet wide (i.e., approximately 166 meters). This makes the use of waveguides at frequencies below 1000 megahertz increasingly impractical. The lower frequency range of any system using waveguides is limited by the physical dimensions of the waveguides.

2.3 Rectangular Waveguides

The rectangular waveguide with a cross section as illustrated in figure 2.4 is an example

of a waveguiding device that will not support a transverse electromagnetic (TEM) wave.

Consequently, it turns out that unique voltage and current waves do not exist, and the analysis of

the waveguide properties has to be carried out as a field problem rather than as a distributed-

parameter-circuit problem.

Figure 2.4: Rectangular Waveguide.

Since a TEM wave does not have any axial field components and there is no center

conductor on which conduction current can exist, a TEM wave cannot be propagated in a hollow rectangular or cylindrical waveguide. The types of waves that can be supported (propagated) in a hollow waveguide are the transverse electric (TE) and transverse magnetic (TM) modes.

Specifically, for transmission lines, the solution of interest is a transverse electromagnetic wave with transverse components only, that is = = 0, whereas for waveguides, solutions

푧 푧 with = 0 and = 0 are not possible. Because퐸 of퐻 the widespread occurrence of such field

푧 푧 solutions,퐸 the following퐻 classification of solutions is of particular interest [13].

1. TEM waves. These solutions have = = 0. This is a function of the

푧 푧 transverse coordinates only and is a solution퐸 퐻 of the two-dimensional Laplace’s

equation.

2. TE or H, modes. These solutions have = 0 and 0. All the field

푧 푧 components may be derived from the axial component퐸 of퐻 magnetic≠ field.

푧 3. TM or , modes. These solutions have = 0 and 퐻 0. The field components

푧 푧 may be퐸 derived from . 퐻 퐸 ≠

퐸푧 As discussed, the hollow rectangular waveguide can support TE and TM Waves but not

TEM waves. We will discuss about this in detail from [14],

On assuming the waveguide region is source free, Maxwell’s equation can be written as

x = µ (2.1)

∇ 퐸� −푗휔 퐻� x = (2.2)

∇ 퐻� 푗휔휀퐸� With an dependence, the three components of each of the above vector equation −푗훽푧 can be reduced as푒 following:

+ = µ , (2.3 ) 휕퐸푧 푗훽퐸푦 −푗휔 퐻푥 푎 휕푦 = µ , (2.3 ) 휕퐸푧 −푗훽퐸푥 − −푗휔 퐻푦 푏 휕푥 = µ , (2.3 ) 휕퐸푦 휕퐸푥 − −푗휔 퐻푧 푐 휕푥 휕푦

+ = , (2.3 ) 휕퐻푧 푗훽퐻푦 푗휔휀퐸푥 푑 휕푦 = , (2.3 ) 휕퐻푧 − − 푗훽퐻푥 푗휔휀퐸푦 푒 휕푥 = , (2.3 ) 휕퐻푦 휕퐻푥 − 푗휔휀퐸푧 푓 휕푥 휕푦 The above equations (2.3) can be solved for four transverse field components in terms of

and as follows

퐸푧 퐻푧 = , (2.4 ) 푗 휕퐸푧 휕퐻푧 퐻푥 2 �휔휀 − 훽 � 푎 푘푐 휕푦 휕푥 = + , (2.4 ) −푗 휕퐸푧 휕퐻푧 퐻푦 2 �휔휀 훽 � 푏 푘푐 휕푥 휕푦 = + µ , (2.4 ) −푗 휕퐸푧 휕퐻푧 퐸푥 2 �훽 휔 � 푐 푘푐 휕푥 휕푦 = + µ , (2.4 ) 푗 휕퐸푧 휕퐻푧 퐸푦 2 �−훽 휔 � 푑 푘푐 휕푦 휕푥 where, = , (2.5 ) 2 2 2 푘푐 푘 − 훽 푎 has been defined as the cutoff wavenumber;

2 = = (2.5 ) 휋 푘 휔�휇휀 푏 휆 is the wavenumber of the material filling the waveguide region.

2.3.1 TE Wave

As we discussed earlier, for TE wave = 0 and 0. All the field components may

푧 푧 be derived from the axial component of magnetic퐸 field.퐻 From≠ the equations (2.4) we get the

푧 transverse electric and transverse magnetic퐻 fields by considering the axial field components

( = 0 and 0) and can be written in terms of the component as

퐸푧 퐻푧 ≠ 퐻푧 = , (2.6 ) −푗훽 휕퐻푧 퐻푥 2 푎 푘푐 휕푥 = , (2.6 ) −푗훽 휕퐻푧 퐻푦 2 푏 푘푐 휕푦 µ = , (2.6 ) −푗휔 휕퐻푧 퐸푥 2 푐 푘푐 휕푦 µ = , (2.6 ) 푗휔 휕퐻푧 퐸푦 2 푑 푘푐 휕푥 In this case, 0 and the propagation constant = is generally a function 2 2 푐 푐 of frequency and the푘 geometry≠ of the guide. To apply equations훽 �푘 (2.6),− 푘 one must first find

푧 from the Helmholtz wave equation. This can be solved by adopting a method of separation퐻

variables followed by the tangential field boundary conditions. From which, we will be able to to

apply the field distributions (2.6).

And from [14], the propagation constant is

β = = ( ) ( ) (2.7) 2 2 2 푚휋 2 푛휋 2 �푘 − 푘푐 �푘 − − 푎 푏

Which is seen to be real, corresponding to a propagation mode, when

> = ( ) + ( ) 푚휋 2 푛휋 2 푘 푘푐 � 푎 푏 Each mode (combination of m and n) thus has a cutoff frequency given by

푓푐푚푛 1 = = ( ) + ( ) (2.8) 2 µ 2 µ 푘푐 푚휋 2 푛휋 2 푓푐푚푛 � 휋� 휀 휋� 휀 푎 푏 The mode with the lowest cutoff frequency is called the dominant mode. Since we have assumed

a>b, the lowest cutoff frequency occurs for the ( = 1, = 0) mode:

푇퐸10 푚 푛 1 = (2.9) 2 µ 푓푐10 푎� 휀 Thus, the is the dominant TE mode and, as we will see, the overall dominant mode

10 of the rectangular 푇퐸waveguide.

From [15], mode exists in the source plane ′, but is not allowed to propagate in

00 the wave guide (vanishes푇퐸 at any point ′ ). This mode푧 − is푧 associated with the “power sorted” in the vicinity of the source plane. Note푧 ≠ 푧also that it has no variation with respect to x or y variables. Still, this does not violate the boundary conditions, as the longitudinal magnetic field should have the maximum value on the perfectly conducting walls of the waveguides, including the special case of a constant value all over the cross section. It is worth mentioning that this

term will also occur in the dual problem: the electric field produced by a longitudinal electric

current source in a waveguide with perfectly magnetic walls.

2.3.2 TM Wave

Similarly, for TM wave = 0 and 0. All the field components may be derived

푧 푧 from the axial component of퐻 electric field.퐸 From≠ the equations (2.4) we get the transverse

푧 electric and transverse magnetic퐸 fields by considering the axial field components ( = 0 and

푧 0) and can be written in terms of the component as 퐻

퐸푧 ≠ 퐸푧 = , (2.10 ) 푗휔휀 휕퐸푧 퐻푥 2 푎 푘푐 휕푦 = , (2.10 ) −푗휔휀 휕퐸푧 퐻푦 2 푏 푘푐 휕푥 = , (2.10 ) −푗훽 휕퐸푧 퐸푥 2 푐 푘푐 휕푥 = , (2.10 ) −푗훽 휕퐸푧 퐸푦 2 푑 푘푐 휕푦 As in the TE case , 0 and the propagation constant β = is a function of 2 2 푐 푐 frequency and the geometry푘 of ≠the line or guide , is found from the� Helmholtz푘 − 푘 wave equation,

푧 and the solving procedure is same as the TE case. 퐸And from [14], the propagation constant is

β = = ( ) ( ) (2.11) 2 2 2 푚휋 2 푛휋 2 �푘 − 푘푐 �푘 − − 푎 푏 Which is seen to be real, corresponding to a propagation mode, when > = ( ) + ( ) 푚휋 2 푛휋 2 푘 푘푐 � 푎 푏 Each mode (combination of m and n) thus has a cutoff frequency for TM case is same as

푐푚푛 that of the TE mode, and is given by 푓

1 = = ( ) + ( ) (2.12) 2 µ 2 µ 푘푐 푚휋 2 푛휋 2 푓푐푚푛 � 휋� 휀 휋� 휀 푎 푏 and is real for propagating modes, and imaginary for evanescent modes. The cutoff frequency for the modes is also same as that of the modes. The guide wavelength and phase

푚푛 푚푛 velocity푇푀 for TM modes are also the same as those푇퐸 for TE modes.

The field expressions are zero for either = 0 = 0, thus there are no

, modes, and the lowest order 푚 mode표푟 푛to propagate (lowest cutoff

00 10 01 frequency)푇푀 푇푀 is 표푟the푇푀 mode. The lowest cutoff frequency푇푀 for mode is given by

푇푀11 푇푀11 1 1 1 = ( ) + ( ) (2.13) 2 µ 2 2 푓푐11 � � 휀 푎 푏

At a given operating frequency , only those modes having < will propagate; modes 푓 푓푐 푓 with > will lead to an imaginary β (or real α), meaning that all field components will decay

푐 exponentially푓 푓 away from the source of excitation. Such modes are referred to as cutoff, or evanescent modes. If more than one mode is propagating, the waveguide is said to be overmoded.

Degenerate modes: Two or more modes having the same cutoff frequencies with the different field distributions are known as the degenerate modes.

CHAPTER III

DETRMINATION OF WAVENUMBERS OF THE SUPPORTED

MODES BY WAVEGUIDES USING METHOD I

3.1 Introduction

The finite difference method is a commonly employed numerical method in solving the electromagnetic problems. We will discuss finite difference method usage to determine the

different possible modes and their field distributions in the different types of waveguides. We

can obtain any number of possible mode’s wavenumbers and the cutoff frequencies in both the cases (i.e., TE and TM cases) in that particular

r waveguide. Here we adopt two methods using finite difference approach, and we will compare them with the analytical solutions for the first two waveguides.

In this chapter, we determine the wavenumbers and cutoff frequencies of the modes, that are supported by the waveguides, by adopting the finite difference Method I, and we are familiar

to use the Method I, as we have already discussed in chapter I.

Mainly we will discuss the four different types of waveguides:

(i) Perfect Electric Conductor Rectangular Waveguide: A rectangular wave guide

with PEC walls on all sides.

(ii) Artificial Rectangular Waveguide: A rectangular waveguide with PEC boundary

on bottom and top walls, and perfect magnetic conductor (PMC) boundary on side

walls. We use the terminology Artificial Rectangular Waveguide in this thesis

from now.

(iii) PEC Rectangular Waveguide with PEC Fin.

(iv) L-Shape Waveguide

Original Mesh 1

0.9

0.8 7 5 8 6 0.7 0.6 9

0.5 Y-axis 0.4

0.3 2 4 0.2 1 3 0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X-ax i s

Figure 3.1: Classification of different cases of nodes.

3.2 Different cases of nodes used in Matrix filling

Here we use the central difference formula for the second order derivative (equations

1.9 and 1.10) for the filling of matrix , in this filling procedure we come across the nine different cases of nodes to be considered.푀� The nine different cases of nodes can be observed clearly from figure 3.1,

3.2.1 Interior nodes:

Figure 3.2: Interior nodes of any Cartesian problem domain.

We have,

( ) + ( ) 2 ( ) ( ) + ( ) 2 ( ) ( ) = + (3.1) ( ) ( ) 2 푢 푅 푢 퐿 − 푢 푃 푢 푇 푢 퐵 − 푢 푃 ∇ 푡 푢 푃 2 2 ∆푥 ∆푦 If we are at the interior node # " " , from the equations (1.9 - 1.10), ( , ) = 0 ,

푖 25 푀 푖 푗 푒푥푐푒푝푡

2 2 ( , ) = + (3.2 ) ( ) ( ) − − 푀 푖 푖 2 2 푎 ∆푥 ∆푦 1 ( , 1) = (3.2 ) ( ) 푀 푖 푖 − 2 푏 ∆푥 1 ( , + 1) = (3.2 ) ( ) 푀 푖 푖 2 푐 ∆푥 1 ( , ) = (3.2 ) ( ) 푀 푖 푖 − 푛푥 2 푑 ∆푦 1 ( , + ) = (3.2 ) ( ) 푀 푖 푖 푛푥 2 푒 ∆푦 Where, 1 is the node which is on the left to the node " ", + 1 is the node which is on the right to 푖the− node " ", is the node which is on the bottom푖 to푖 the node " ", + is the node which is on the top푖 to푖 the− 푛푥 node " ". 푖 푖 푛푥

푖 We discuss the two different cases of solving for the nodes on the left boundary, and can be used these principles in solving the other nodes of the problem domain.

3.2.2 TE case and PEC boundary:

We consider the TE case and PEC boundary. In this case, we do not have the left node.

The value of derivative of the field goes to zero at the boundary, and the value at the virtual node

(left node) is equal to the right node. Thus, the value of the right node with respect to the node on

the boundary " " from the equation (3.2c) is,

푖 2 ( , + 1) = ( ) 푀 푖 푖 2 ∆푥 26

Figure 3.3: TE case and PEC boundary.

3.2.3 TM case and PEC boundary:

Figure 3.4: TM case and PEC boundary.

We consider the TM case and PEC boundary. In this case, we do not have the left node.

The value of the field goes to zero at the boundary, and the value at the virtual node (left node) is opposite to the right node. Thus, the value of the right node with respect to the node on the boundary " " from the equation (3.2c) is,

푖 ( , + 1) = 0

3.3 PEC Rectangular Waveguide 푀 푖 푖

PEC rectangular waveguide is a hollow rectangular waveguide with the PEC walls. In this thesis, the dimensions of the waveguide are same in all the cases we consider.

Figure 3.5: PEC Rectangular Waveguide cross section.

We consider the dimensions as = 1.5 = 1.0 throughout this thesis, and the cross section of the waveguide in 푎 plane푐푚 is considered푎푛푑 푏 to 푐푚solve the problem.

푥 − 푦

We consider the mesh size as, = = , = 1 1 푑푒푙푥 �푛푥−1� 푎푛푑 푑푒푙푦 �푛푦−1� 푤ℎ푒푟푒 푛푥 31 , = 21

푛푦 ( ),

and (푛푥 − 푛푢푚푏푒푟 표푓 푛표푑푒푠 푖푛 푡ℎ푒 푑푖푟푒푐푡푖표푛 표푓 푥 − 푎푥푖푠).

푛푦 − 푛푢푚푏푒푟 표푓 푛표푑푒푠 푖푛 푡ℎ푒 푑푖푟푒푐푡푖표푛 표푓 푦 − 푎푥푖푠

3.3.1 TE Wave

From the equation (1.12b), we have = = ( , ) . Thus for TE waves, the axial or −푗훽푧 푧 푧 longitudinal magnetic field will be present and the푈 axial퐻 electricℎ 푥 field푦 푒 is zero.

We consider the original meshing in solving each and every problem from now, and it is better to recall regarding the nodes on the boundary from the chapter I, as they are not exactly on the boundary, but we address the particular nodes as “the nodes on the boundary” for our convenience.

Case 1: This includes the node which is on the left-bottom corner of the problem domain,

and it is numbered as “1” in figure 3.1. This node has no left and bottom nodes, the values of the right and top nodes corresponding to the left-bottom node can be calculated by using the image

theory from [16], and all the boundaries we have here in this problem are PEC, and the field we

are calculating is the magnetic field propagating in . Thus,

푧 − 푑푖푟푒푐푡푖표푛 2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦

2 ( , + 1) = ( ) 푀 푖 푖 2 ∆푥 2 ( , + ) = ( ) 푀 푖 푖 푛푥 2 ∆푦 Case 2: This includes the nodes which are on the bottom boundary of the problem domain, and it is numbered as “2” in figure 3.1. These nodes have no bottom nodes, the values of the left, right and top nodes corresponding to the particular bottom node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 1 ( , 1) = ( ) 푀 푖 푖 − 2 ∆푥 1 ( , + 1) = ( ) 푀 푖 푖 2 ∆푥 2 ( , + ) = ( ) 푀 푖 푖 푛푥 2 ∆푦 Case 3: This includes the nodes on the right-bottom corner of the problem domain, and it is numbered as “3” in figure 3.1. This node has no right and bottom nodes, the values of the left and top nodes corresponding to the right-bottom corner node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 2 ( , 1) = ( ) 푀 푖 푖 − 2 ∆푥 30

2 ( , + ) = ( ) 푀 푖 푖 푛푥 2 ∆푦 Case 4: This includes the nodes which are on the right boundary of the problem domain, and it is numbered as “4” in figure 3.1. These nodes have no right nodes, the values of the left, bottom and top nodes corresponding to the particular right node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 2 ( , 1) = ( ) 푀 푖 푖 − 2 ∆푥 1 ( , ) = ( ) 푀 푖 푖 − 푛푥 2 ∆푦 1 ( , + ) = ( ) 푀 푖 푖 푛푥 2 ∆푦 Case 5: This includes the node which is on the right-top corner of the problem domain, and it is numbered as “5” in figure 3.1. This node has no right and top nodes, the values of the left and bottom nodes corresponding to the right-top corner node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 2 ( , 1) = ( ) 푀 푖 푖 − 2 ∆푥

2 ( , ) = ( ) 푀 푖 푖 − 푛푥 2 ∆푦 Case 6: This includes the nodes which are on the top boundary of the problem domain,

and it is numbered as “6” in figure 3.1. These nodes have no top nodes, the values of the left,

right and bottom nodes corresponding to the particular top node can be calculated and are given

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 1 ( , 1) = ( ) 푀 푖 푖 − 2 ∆푥 1 ( , + 1) = ( ) 푀 푖 푖 2 ∆푥 2 ( , ) = ( ) 푀 푖 푖 − 푛푥 2 ∆푦 Case 7: This includes the node which is on the left-top corner of the problem domain, and it is numbered as “7” in figure 3.1. This node has no left and top nodes, the values of the right and bottom nodes corresponding to the left-top corner node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 2 ( , + 1) = ( ) 푀 푖 푖 2 ∆푥

2 ( , ) = ( ) 푀 푖 푖 − 푛푥 2 ∆푦 Case 8: This includes the nodes which are on the left boundary of the problem domain,

and it is numbered as “8” in figure 3.1. These nodes have no left nodes, the values of the right, top and bottom nodes corresponding to the particular top node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 2 ( , + 1) = ( ) 푀 푖 푖 2 ∆푥 1 ( , ) = ( ) 푀 푖 푖 − 푛푥 2 ∆푦 1 ( , + ) = ( ) 푀 푖 푖 푛푥 2 ∆푦 Case 9: This includes the nodes which are in the interior of the problem domain, and it is

numbered as “9” in figure 3.1. These nodes have left, right, bottom and top nodes, the values of

these nodes corresponding to the particular interior node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 1 ( , 1) = ( ) 푀 푖 푖 − 2 ∆푥 1 ( , + 1) = ( ) 푀 푖 푖 2 ∆푥

1 ( , ) = ( ) 푀 푖 푖 − 푛푥 2 ∆푦 1 ( , + ) = ( ) 푀 푖 푖 푛푥 2 ∆푦 Now we have completed the filling of the matrix , and by using the in-built functions of

the matlab from [17], we solve the above matrix for the푀� eigenvalues and their corresponding

eigenvectors, which are nothing but the wavenumbers and the field distribution of the axial

component of the magnetic field ( ) corresponding to the particular wavenumber respectively.

ℎ푧 The dominant mode in this case is , and we can see the comparison of the

10 wavenumbers and their cutoff frequencies for푇퐸 the first few modes obtained by this method

(Method I), with the analytical Method in table 3.1. The error increases as the mode increases

and we can decrease the error by increasing the node approximations (i.e., ).

푛푥 푎푛푑 푛푦 And also we see the field distributions of the axial component of the magnetic field for

two modes ( ) modes from figures 3.6, and 3.7.

푇퐸20 표푟 푇퐸21 푚표푑푒 We can observe the modes with the same cutoff frequency from table

02 30 3.1; we call these modes as the degenerate푇퐸 푎푛푑 modes푇퐸. As we discussed about these modes in chapter

II, the modes with the same cutoff frequencies but different field distributions are called the degenerate modes. We will see the field distributions of these modes from figures 3.8, and 3.9.

Table 3.1: Comparison of Solutions for PEC Rectangular Waveguide TE and TM Waves with an

Analytical Method.

PEC Rect. W. Analytical Method Numerical Method I

Modes Wavenumber Cutoff- Wavenumber Cutoff- ( , ) ( ) Frequency ( ) Frequency ퟐ ퟐ 풎풏 풎풏 ( )( ) ( )( 푻푬 푻푴 풌 풌 풇풄 풊풏 푮푯풛 풇풄 풊풏 푮푯풛 43864.908449 10.000000 43824.837054 9.995431

푻푬ퟏퟎ 98696.044011 15.000000 98493.275239 14.984583

푻푬ퟎퟏ , 142560.95246 18.027756 142318.112293 18.012395

푻푬ퟏퟏ 푻푴ퟏퟏ 175459.63379 20.000000 174819.194130 19.963466

푻푬ퟐퟎ , 274155.67781 25.000000 273312.469368 24.961525

푻푬ퟐퟏ 푻푴ퟐퟏ 394784.17604 30.000000 391547.869639 29.876782

푻푬ퟎퟐ 풂풏풅 푻푬ퟑퟎ , 438649.08449 31.622777 435372.70669 31.504456

푻푬ퟏퟐ 푻푴ퟏퟐ , 493480.22005 33.541020 490041.14487 33.423941

푻푬ퟑퟏ 푻푴ퟑퟏ , 570243.80984 36.055513 566367.06376 35.932744

푻푬ퟐퟐ 푻푴ퟐퟐ

Field distribution of the axial component of the magnetic field for TE20 mode

0.06

0.04

0.02

-0.02

-0.04

-0.06 0.01 0.008 0.006 0.015 0.004 0.01 0.002 0.005 0 0 y-axis x-axis

Figure 3.6: Field distribution of the axial component of the magnetic field for

mode of PEC Rectangular Waveguide.

푇퐸20 Field distribution of the axial component of the magnetic field for TE21 mode

0.08

0.06

0.04

0.02

-0.02

-0.04

-0.06

-0.08 0.01 0.015 0.008 0.006 0.01 0.004 0.005 0.002 0 0 y-axis x-axis

Figure 3.7: Field distribution of the axial component of the magnetic field for

mode of PEC Rectangular Waveguide.

푇퐸21 36

Field distribution of the axial component of the magnetic field for TE30 mode

0.05

-0.05 0.01 0.005 0.01 0.015 0 0 0.005 y-axis x-axis

Figure 3.8: Field distribution of the axial component of the magnetic field for

mode of PEC Rectangular Waveguide.

푇퐸30 Field distribution of the axial component of the magnetic field for TE02 mode

0.05

-0.05 0.01 0.015 0.008 0.006 0.01 0.004 0.005 0.002 0 0 y-axis x-axis

Figure 3.9: Field distribution of the axial component of the magnetic field for

mode of PEC Rectangular Waveguide.

02 푇퐸 37

3.3.2 TM Wave

From the equation (1.12b), we have = = ( , ) . Thus for TM waves, the axial −푗훽푧 푧 푧 or longitudinal electric field will be present and the푈 axial퐸 magnetic푒 푥 푦 field푒 is zero.

Case 1: This node has no left and bottom nodes, the values of the right and top nodes corresponding to the left-bottom node can be calculated by using the image theory from [16], and all the boundaries we have here in this problem are PEC, and the field we are calculating here in

TM Wave is the electric field propagating in . Thus,

푧 − 푑푖푟푒푐푡푖표푛 2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 ( , + 1) = 0

푀 푖 푖 ( , + ) = 0

푀 푖 푖 푛푥 Case 2: These nodes have no bottom nodes, the values of the left, right and top nodes corresponding to the particular bottom node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 1 ( , 1) = ( ) 푀 푖 푖 − 2 ∆푥 1 ( , + 1) = ( ) 푀 푖 푖 2 ∆푥 ( , + ) = 0

푀 푖 푖 푛푥

Case 3: This node has no right and bottom nodes, the values of the left and top nodes corresponding to the right-bottom corner node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 ( , 1) = 0

푀 푖 푖 − ( , + ) = 0

푀 푖 푖 푛푥 Case 4: These nodes have no right nodes, the values of the left, bottom and top nodes corresponding to the particular right node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 ( , 1) = 0

푀 푖 푖 − 1 ( , ) = ( ) 푀 푖 푖 − 푛푥 2 ∆푦 1 ( , + ) = ( ) 푀 푖 푖 푛푥 2 ∆푦 Case 5: This node has no right and top nodes, the values of the left and bottom nodes corresponding to the right-top corner node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 ( , 1) = 0

푀 푖 푖 −

( , ) = 0

푀 푖 푖 − 푛푥 Case 6: These nodes have no top nodes, the values of the left, right and bottom nodes corresponding to the particular top node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 1 ( , 1) = ( ) 푀 푖 푖 − 2 ∆푥 1 ( , + 1) = ( ) 푀 푖 푖 2 ∆푥 ( , ) = 0

푀 푖 푖 − 푛푥 Case 7: This node has no left and top nodes, the values of the right and bottom nodes corresponding to the left-top corner node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 ( , + 1) = 0

푀 푖 푖 ( , ) = 0

푀 푖 푖 − 푛푥 Case 8: These nodes have no left nodes, the values of the right, top and bottom nodes corresponding to the particular top node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦

( , + 1) = 0

푀 푖 푖 1 ( , ) = ( ) 푀 푖 푖 − 푛푥 2 ∆푦 1 ( , + ) = ( ) 푀 푖 푖 푛푥 2 ∆푦 Case 9: This includes the nodes which are in the interior of the problem domain, and it is

numbered as “9” in the figure 3.1. These nodes have left, right, bottom and top nodes, the values

of these nodes corresponding to the particular interior node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 1 ( , 1) = ( ) 푀 푖 푖 − 2 ∆푥 1 ( , + 1) = ( ) 푀 푖 푖 2 ∆푥 1 ( , ) = ( ) 푀 푖 푖 − 푛푥 2 ∆푦 1 ( , + ) = ( ) 푀 푖 푖 푛푥 2 ∆푦 Similar to previous case (i.e., TE Wave), we solve for the wavenumbers and the field distribution of the axial component of the electric field ( ) for possible number of supported

푧 modes, and the dominant mode in this case is . 푒

푇푀11

For this case (TM Wave), we can see the comparison of the wavenumbers and their cutoff frequencies for the first few modes obtained by this method (Method I), with the analytical

Method in the table 3.1. And also we see the field distributions of the axial component of the electric field for s in the figures 3.10 and 3.11.

푇푀11 푎푛푑 푇푀21 푚표푑푒 Field distribution of the axial component of the electric field for TM11 mode

0.02

-0.02

-0.04

-0.06

-0.08

-0.1 0.01 0.008 0.015 0.006 0.004 0.01 0.002 0 0 0.005 y-axis x-axis

Figure 3.10: Field distribution of the axial component of the electric field for

mode of PEC Rectangular Waveguide.

푇푀11

Figure 3.3: Field distribution of the axial component of the electric field for TM21 mode

0.1

0.05

-0.05

-0.1 0.01 0.008 0.015 0.006 0.004 0.01 0.002 0 0 0.005 y-axis x-axis

Figure 3.11: Field distribution of the axial component of the electric field for

mode of PEC Rectangular Waveguide.

푇푀21

3.4 Artificial Rectangular Waveguide

PEC rectangular waveguide is a hollow rectangular waveguide with the PEC walls. In

this thesis, the dimensions of the waveguide are similar to the section 3.3, in all the cases we

consider.

Figure 3.12: Artificial Rectangular Waveguide

The bottom and top boundaries are PEC, and the left and right boundaries are PMC in

Artificial rectangular waveguide.

3.4.1 TE Wave

From the equation (1.12b), we have = = ( , ) . Thus for TE waves, the axial −푗훽푧 푧 푧 or longitudinal magnetic field will be present 푈and the퐻 axialℎ 푥electric푦 푒 field is zero. Here we have two different conductors at the boundaries, and the field we need to calculate the magnetic field

propagating in . Thus, from [16], we can calculate for different cases as,

푧 − 푑푖푟푒푐푡푖표푛

Case 1: This node has no left and bottom nodes, the values of the right and top nodes corresponding to the left-bottom node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 ( , + 1) = 0

푀 푖 푖 2 ( , + ) = ( ) 푀 푖 푖 푛푥 2 ∆푦 Case 2: These nodes have no bottom nodes, the values of the left, right and top nodes corresponding to the particular bottom node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 1 ( , 1) = ( ) 푀 푖 푖 − 2 ∆푥 1 ( , + 1) = ( ) 푀 푖 푖 2 ∆푥 2 ( , + ) = ( ) 푀 푖 푖 푛푥 2 ∆푦 Case 3: This node has no right and bottom nodes, the values of the left and top nodes corresponding to the right-bottom corner node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 ( , 1) = 0

푀 푖 푖 −45

2 ( , + ) = ( ) 푀 푖 푖 푛푥 2 ∆푦 Case 4: These nodes have no right nodes, the values of the left, bottom and top nodes corresponding to the particular right node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 ( , 1) = 0

푀 푖 푖 − 2 ( , ) = ( ) 푀 푖 푖 − 푛푥 2 ∆푦 Case 6: These nodes have no top nodes, the values of the left, right and bottom nodes corresponding to the particular top node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆46푥 ∆푦

1 ( , 1) = ( ) 푀 푖 푖 − 2 ∆푥 1 ( , + 1) = ( ) 푀 푖 푖 2 ∆푥 2 ( , ) = ( ) 푀 푖 푖 − 푛푥 2 ∆푦 Case 7: This node has no left and top nodes, the values of the right and bottom nodes corresponding to the left-top corner node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 ( , + 1) = 0

푀 푖 푖 2 ( , ) = ( ) 푀 푖 푖 − 푛푥 2 ∆푦 Case 8: These nodes have no left nodes, the values of the right, top and bottom nodes corresponding to the particular top node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 ( , + 1) = 0

푀 푖 푖 1 ( , ) = ( ) 푀 푖 푖 − 푛푥 2 ∆푦 1 ( , + ) = ( ) 푀 푖 푖 푛푥 2 ∆푦 47

Case 9: This includes the nodes which are in the interior of the problem domain, and it is numbered as “9” in the figure 3.1. These nodes have left, right, bottom and top nodes, the values of these nodes corresponding to the particular interior node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 1 ( , 1) = ( ) 푀 푖 푖 − 2 ∆푥 1 ( , + 1) = ( ) 푀 푖 푖 2 ∆푥 1 ( , ) = ( ) 푀 푖 푖 − 푛푥 2 ∆푦 1 ( , + ) = ( ) 푀 푖 푖 푛푥 2 ∆푦 Similar to previous cases, we solve for the wavenumbers and the field distribution of the axial component of the magnetic field ( ) for possible number of supported modes, and the

푧 dominant mode in this case is . ℎ

푇퐸10 For this case (TE Wave), we can see the comparison of the wavenumbers and their cutoff frequencies for the first few modes obtained by this method (Method I), with the analytical

Method in the table 3.2. And also we see the field distributions of the axial component of the magnetic field for in figures 3.13 and 3.14.

푇퐸21 푎푛푑 푇퐸12 푚표푑푒푠

Table 3.2: Comparison of Solutions for Artificial Rectangular Waveguide TE and TM Waves

with an Analytical Method.

Art. Rect. Wav. Analytical Method Numerical Method I

Modes Wavenumber Cutoff- Wavenumber Cutoff- ( , ) (k) Frequency (k) Frequenc

풎풏 풎풏 ( )( ) 푻푬 푻푴 y ( )( 풇풄 풊풏 푮푯풛 풇풄 풊풏 푮푯 43864.908449 10.000000 43824.837054 9.995431

푻푬ퟏퟎ 98696.044011 15.000000 98493.275239 14.984583

푻푴ퟎퟏ , 142560.95246 18.027756 142318.112293 18.012395

푻푬ퟏퟏ 푻푴ퟏퟏ 175459.63379 20.000000 174819.194130 19.963466

푻푬ퟐퟎ , 274155.67781 25.000000 273312.469368 24.961525

푻푬ퟐퟏ 푻푴ퟐퟏ , 394784.17604 30.000000 391547.869639 29.876782

푻푬ퟑퟎ 푻푴ퟎퟐ , 438649.08449 31.622777 435372.70669 31.504456

푻푬ퟏퟐ 푻푴ퟏퟐ , 493480.22005 33.541020 490041.14487 33.423941

푻푬ퟑퟏ 푻푴ퟑퟏ , 570243.80984 36.055513 566367.06376 35.932744

푻푬ퟐퟐ 푻푴ퟐퟐ

Field distribution of the axial component of the magnetic field for TE21 mode

0.08

0.06

0.04

0.02

-0.02

-0.04

-0.06

-0.08 0.01 0.015 0.008 0.01 0.006 0.004 0.005 0.002 0 0 y-axis x-axis

Figure 3.13: Field distribution of the axial component of the magnetic field for

mode of Artificial Rectangular Waveguide.

푇퐸21 Field distribution of the axial component of the magnetic field for TE12

0.08

0.06

0.04

0.02

-0.02

-0.04

-0.06

-0.08 0.01 0.015 0.008 0.01 0.006 0.004 0.005 0.002 0 0 y-axis x-axis

Figure 3.14: Field distribution of the axial component of the magnetic field for

mode of Artificial Rectangular Waveguide.

푇퐸12 50

3.4.2 TM Wave

From the equation (1.12b), we have = = ( , ) . Thus for TM waves, the axial −푗훽푧 푧 푧 or longitudinal electric field will be present and푈 the퐸 axial푒 magnetic푥 푦 푒 field is zero. Here we have two different conductors at the boundaries, and the field we need to calculate the electric field propagating in . Thus, from [16], we can calculate for different cases as,

푧 − 푑푖푟푒푐푡푖표푛 Case 1: This node has no left and bottom nodes, the values of the right and top nodes corresponding to the left-bottom node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 2 ( , + 1) = ( ) 푀 푖 푖 2 ∆푥 ( , + ) = 0

푀 푖 푖 푛푥 Case 2: These nodes have no bottom nodes, the values of the left, right and top nodes corresponding to the particular bottom node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 1 ( , 1) = ( ) 푀 푖 푖 − 2 ∆푥 1 ( , + 1) = ( ) 푀 푖 푖 2 ∆푥 ( , + ) = 0

푀 푖 푖 푛푥 51

Case 3: This node has no right and bottom nodes, the values of the left and top nodes corresponding to the right-bottom corner node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 2 ( , 1) = ( ) 푀 푖 푖 − 2 ∆푥 ( , + ) = 0

푀 푖 푖 푛푥 Case 4: These nodes have no right nodes, the values of the left, bottom and top nodes corresponding to the particular right node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 2 ( , 1) = ( ) 푀 푖 푖 − 2 ∆푥 1 ( , ) = ( ) 푀 푖 푖 − 푛푥 2 ∆푦 1 ( , + ) = ( ) 푀 푖 푖 푛푥 2 ∆푦 Case 5: This node has no right and top nodes, the values of the left and bottom nodes corresponding to the right-top corner node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦

2 ( , 1) = ( ) 푀 푖 푖 − 2 ∆푥 ( , ) = 0

푀 푖 푖 − 푛푥 Case 6: These nodes have no top nodes, the values of the left, right and bottom nodes corresponding to the particular top node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 1 ( , 1) = ( ) 푀 푖 푖 − 2 ∆푥 1 ( , + 1) = ( ) 푀 푖 푖 2 ∆푥 ( , ) = 0

푀 푖 푖 − 푛푥 Case 7: This node has no left and top nodes, the values of the right and bottom nodes corresponding to the left-top corner node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 2 ( , + 1) = ( ) 푀 푖 푖 2 ∆푥 ( , ) = 0

푀 푖 푖 − 푛푥 Case 8: These nodes have no left nodes, the values of the right, top and bottom nodes corresponding to the particular top node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 2 ( , + 1) = ( ) 푀 푖 푖 2 ∆푥 1 ( , ) = ( ) 푀 푖 푖 − 푛푥 2 ∆푦 1 ( , + ) = ( ) 푀 푖 푖 푛푥 2 ∆푦 Case 9: This includes the nodes which are in the interior of the problem domain, and it is numbered as “9” in the figure 3.1. These nodes have left, right, bottom and top nodes, the values of these nodes corresponding to the particular interior node can be calculated and are given by

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 1 ( , 1) = ( ) 푀 푖 푖 − 2 ∆푥 1 ( , + 1) = ( ) 푀 푖 푖 2 ∆푥 1 ( , ) = ( ) 푀 푖 푖 − 푛푥 2 ∆푦 1 ( , + ) = ( ) 푀 푖 푖 푛푥 2 ∆푦

Similar to previous cases, we solve for the wavenumbers and the field distribution of the axial component of the electric field ( ) for possible number of supported modes, and the

푧 dominant mode in this case is . 푒

푇푀01 For this case (TM Wave), we can see the comparison of the wavenumbers and their cutoff frequencies for the first few modes obtained by this method (Method I), with the analytical

Method in the table 3.2. And also we see the field distributions of the axial component of the electric field for in figures 3.15, and 3.16.

푇푀31 푎푛푑 푇푀22 푚표푑푒푠

Field distribution of the axial component of the electric field for TM31 mode

0.08

0.06

0.04

0.02

-0.02

-0.04

-0.06

-0.08 0.01 0.005 0.015 0.005 0.01 0 0 y-axis x-axis

Figure 3.15: Field distribution of the axial component of the electric field for

mode of Artificial Rectangular Waveguide.

푇푀31

Field distribution of the axial component of the electric field for TM22 mode

0.08

0.06

0.04

0.02

-0.02

-0.04

-0.06

-0.08 0.01 0.005 0.015 0 0.005 0.01 y-axis x-axis

Figure 3.16: Field distribution of the axial component of the electric field for

mode of Artificial Rectangular Waveguide.

푇푀22

3.5 PEC Rectangular Waveguide with PEC Fin

We consider a PEC rectangular waveguide with introducing PEC fin at the center of the bottom PEC wall towards the top PEC wall of the problem domain as shown in figure 3.17.

Figure 3.17: PEC Rectangular Waveguide with PEC Fin.

The dimensions we consider here are same as the previous sections, and the height of the

PEC Fin is “h” which can be varied. We observe four different cases by varying the length of the PEC pin for TM Wave.

The matrix filling is similar to the procedure of the TM Wave for the PEC rectangular waveguide in section푀� 3.3.2, except for the nodes on the PEC fin. The node of the PEC fin on the bottom PEC wall is considered as the case 10 and the other nodes on the PEC fin fall under the case 11.

For the case 10 node, we use the same procedure adopted for the case 1 and case 3 nodes in section 3.3.2, and for the case 11 nodes, we use the similar procedure in combination, adopted for the case 4 and case 8 nodes in section 3.3.2.

From the equation (1.12b), we have = = ( , ) . Thus for TM waves, the axial −푗훽푧 푧 푧 or longitudinal magnetic field will be present and푈 the퐻 axialℎ electric푥 푦 푒field is zero. All the boundaries we

have here in this problem are PEC similar to the section 3.3.2, and the field we are calculating is

the magnetic field propagating in . The four different heights of the PEC fin we consider here are 푧 − 푑푖푟푒푐푡푖표푛

3 = , , , 4 2 4 푏 푏 푏 ℎ 푎푛푑 푏 We will see the variation of the cutoff frequencies for the different cases of “h” and

compare with the case “h=0” below,

Table3.3a: Cutoff frequencies of first few modes for h=0

Mode

푇푀11 푇푀21 푇푀12 푇푀31 푇푀22 푇푀32 푇푀41 Cutoff 18.012 24.962 31.504 33.424 35.933 42.252 42.441 frequency for = 0

ℎ Table3.3b: Cutoff frequencies of first few modes for = /4

Mode ℎ 푏

푇푀� 11 푇푀� 21 푇푀� 12 푇푀� 31 푇푀� 22 푇푀� 32 푇푀� 41

Cutoff 18.925 24.962 32.539 34.404 35.933 43.140 42.441 frequency for = 4 푏 ℎ

Table3.3c: Cutoff frequencies of first few modes for = /2

Mode ℎ 푏

푇푀� 11 푇푀� 21 푇푀� 12 푇푀� 31 푇푀� 22 푇푀� 32 푇푀� 41

Cutoff 21.938 24.962 32.766 37.900 35.933 45.710 42.441 frequency for = 2 푏 ℎ Table3.3d: Cutoff frequencies of first few modes for = 3 /4

Mode ℎ 푏

푇푀� 11 푇푀� 21 푇푀� 12 푇푀� 31 푇푀� 22 푇푀� 32 푇푀� 41

Cutoff 24.611 24.962 35.100 41.228 35.933 46.658 42.441 frequency for 3 = 4 푏 ℎ Table3.3e: Cutoff frequencies of first few modes for =

Mode ℎ 푏

푇푀� 21 푇푀� 21 푇푀� 22 푇푀� 22 푇푀� 41 푇푀� 41 푇푀� 32

Cutoff 24.962 24.962 35.932 35.932 42.441 42.441 49.666 frequency for =

ℎ 푏

From the tables 3.3a-3.3e, we can observe that the cutoff frequency of the particular mode increases as the “h” increases and forms the degenerate mode at

= .

ℎ 푏 59

From the table 3.3e, we can see that the degenerate modes as,

, , … …

21 22 41 푇푀 푇푀 푇푀

In general, The PEC fin of variable height affects the cutoff frequencies of the

, modes with as an odd number, and does not affects the cutoff frequencies of the

푚 푛 푇푀 , modes with ′푚′ as an even number. For the PEC fin’s height equal to the height of the

푚 푛 푇푀waveguide ( = ),′ 푚′ , modes (with as an odd number) forms the degenerate mode with

푚 푛 the , modeℎ 푏 (where푇푀 = + 1). Thus′푚′ all the permitted modes are the degenerate modes ∗ ∗ 푚 푛 of 푇푀 , with as an even푚 number.푚

푚 푛 푇푀 ′푚′

The field distributions of in all the cases can be seen in the figures 3.18a - 3.18f.

11 Field distribution푇푀 of the axial component of the electric field of TM11 for “h=0”.

0.1

0.08

0.06

0.04

0.02

-0.02 0.01

0.005 0.015 0.01 0.005 0 0 y-axis x-axis

Figure 3.18a: Field distribution of the axial component of the electric field of for “h=0”.

푇푀11 60

Field distribution of the axial component of the electric field of TM11 for “h=b/4”.

0.1

0.08

0.06

0.04

0.02

-0.02 0.01

0.005 0.015 0.01 0.005 0 0 y-axis x-axis

Figure 3.18b: Field distribution of the axial component of the electric field of for “h=b/4”.

11 푇푀

Field distribution of the axial component of the electric field of TM11 for “h=b/2”.

0.08

0.06

0.04

0.02

-0.02 0.01

0.005

0.015 0.01 0.005 y-axis 0 0 x-axis

Figure 3.18c: Field distribution of the axial component of the electric field of for “h=b/2”.

푇푀11 61

Field distribution of the axial component of the electric field of TM for “h=3b/4”. 0.1 11

0.08

0.06

0.04

0.02

-0.02 0.01

0.005

0 y-axis 0 0.005 0.01 0.015 x-axis

Figure 3.18d: Field distribution of the axial component of the electric field of for

11 “h=3b/4”. 푇푀

Field distribution of the axial component of the electric field of TM11 for “h=b”(degenerate mode). 0.12

0.1

0.08

0.06

0.04

0.02

-0.02 0.01

0.005 0.01 0.015 0 0 0.005 y-axis x-axis

Figure 3.18e: Field distribution of the axial component of the electric field of for “h=b”

11 (degenerate mode). 푇푀

Field distribution of the axial component of the electric field of TM11 for “h=b” (degenerate mode).

0.05

-0.05

-0.1

-0.15 0.01 0.008 0.015 0.006 0.01 0.004 0.002 0.005 0 0 y-axis x-axis

Figure 3.14f: Field distribution of the axial component of the electric field of for “h=b”

11 (degenerate mode). 푇푀

3.6 L-Shape Waveguide

An L-shape waveguide with PEC walls will be solved in this section. Here the dimensions of the waveguide’s cross section are shown in the figures 3.19.

Figure 3.19a: An L-Shape Waveguide type 1.

Figure 3.19b: An L-Shape Waveguide type 2.

Here, in figure 3.19a, a piece of rectangular PEC patch is placed at the top-right corner of the PEC rectangular waveguide (figure 3.2). We have calculated the solution in both the cases

(figure 3.19a and figure 3.19b) which are in an exact agreement.

We solve this problem (figure 3.19a), for TM Wave and the axial component of the electric field. While matrix filling, for the nodes on the patch we take the values mentioned below and for the remaining nodes, we use almost similar procedure.

2 2 ( , ) = + ( ) ( ) − − 푀 푖 푖 2 2 ∆푥 ∆푦 ( , 1) = 0

푀 푖 푖 − ( , + 1) = 0

푀 푖 푖 ( , ) = 0

푀 푖 푖 − 푛푥 ( , + ) = 0

푀 푖 푖 푛푥 Table 3.4: Cutoff frequencies of first few modes of an L-Shape Waveguide.

Modes

푇푀� 11 푇푀� 21 푇푀� 22 푇푀� 31

Cutoff 23.891 32.328 35.933 39.816 Frequencies

And the field distributions for the modes in table 3.4 are given by figures 3.20 – 3.23.

Field distribution of TM11 for an L-Shaped Waveguide

0.12

0.1

0.08

0.06

0.04

0.02

-0.02 0.01 0 0.005 0.005 0.01 0.015 0 y-axis x-axis

Figure 3.20: Field distribution of the axial component of the electric field of for an L-

11 Shape Waveguide. 푇푀

Field distribution of TM21 for an L-Shaped Waveguide

0.15

0.1

0.05

-0.05

-0.1 0.01

0.005 0.015 0.01 0.005 0 0 y-axis x-axis

Figure 3.21: Field distribution of the axial component of the electric field of for an L-

21 Shape Waveguide. 푇푀

Field distribution of TM22 for an L-Shaped Waveguide.

0.1

0.05

-0.05

0.01

-0.1 0.005 0 0.005 0.01 0.015 0 y-axis x-axis

Figure 3.22: Field distribution of the axial component of the electric field of for an L-

22 Shape Waveguide. 푇푀

Field distribution of TM31 for an L-Shaped Waveguide.

0.15

0.1

0.05

-0.05 0.01 0.005 -0.1 0 0 0.005 0.01 0.015 y-axis x-axis

Figure 3.23: Field distribution of the axial component of the electric field of for an L-

31 Shape Waveguide. 푇푀

CHAPTER IV

DETERMINATION OF CUTOFF FREQUENCIES OF THE

MODES SUPPORTED BY WAVEGUIDES USING METHOD II

As we discussed in Method II in the chapter I, the matrix filling is similar to that we have

adopted in the Method I in the previous chapter (i.e., chapter II) in solving the different types of

waveguides, except the diagonal elements of the matrix. In this method, the diagonal elements of

the matrix are associated with the extra term "k " as in the equation (1.22), and this term is given by 2 the equation푀 �(1.13) as

4 = = = 2 2 2 2 휔 2 휋 푓 푘 2 휔 휇휀 2 Where, , 푐 , 푐

We푘 푖푠 add푡ℎ푒 this푤푎푣푒푛푢푚푏푒푟 term (square푐 − of푠푝푒푒푑 the wavenumber)표푓 푙푖푔ℎ푡 푓 − 푓푟푒푞푢푒푛푐푦 for every diagonal element ( . . , ( , ))

for every frequency as we discussed in the section 1.4.2. We obtain the value of the푖 determinant푒 푀 푖 푖

of the matrix , and this is done for the range of frequency of interest. So, we can plot the range of �� frequency vs the푀 values of the determinant of the matrix at each and every frequency we calculated. A

point to remember is that, the increment in the frequency푀 �must be small in order to get the near-by value

of the zero to track the cutoff frequencies. We use an iterative method for range of frequencies, in

order to get the value of the determinant of the matrix at each and every frequency we consider.

We will see the results of the different types of waveguides, which we have solved by using Method I in chapter III. And we no more discuss about the procedure of solving in this chapter, as we have discussed in detail for the same waveguides in the previous chapter.

4.1 PEC Rectangular Waveguide

We consider the mesh size as,

1 1 = = , 1 1 푑푒푙푥 � � 푎푛푑 푑푒푙푦 � � ( 푛푥 − 푛푦 − ),

and (푛푥 − 푛푢푚푏푒푟 표푓 푛표푑푒푠 푖푛 푡 ℎ푒 푑푖푟푒푐푡푖표푛 표푓 푥 − 푎푥푖푠 )

We plot the curves푛푦 − 푛푢푚푏푒푟for two node표푓 푛표푑푒푠 approximations,푖푛 푡ℎ푒 푑푖푟푒푐푡푖표푛 one is표푓 푦 − 푎푥푖푠 = 16 , = 11 and the other is for = 31 , = 21. This is done for all the 푤problemsℎ푒푟푒 푛푥 in this chapter.푛푦

4.1.1 TE Wave푤ℎ 푒푟푒 푛푥 푛푦

PEC Rectangular Waveguide TE case nx=16, ny=11 1

0.8

0.6

0.4

0.2 X: 9.982 X: 14.94 X: 19.85 X: 29.51 Y: 0.008772 Y: 0.01462 Y: 0.002924 Y: 1.813e-05 0 X: 17.97 X: 24.85 -0.2 Y: 0.008772 Y: 0.008772 det(M/constant)

-0.4

-0.6

-0.8

-1 5 10 15 20 25 30 35 Frequency in GHz

Figure 4.1a: Determination of cutoff frequencies of TE wave for PEC Rectangular

Waveguide ( = 16 , = 11).

푛푥69 푛푦

PEC Rectangular Waveguide TE case nx=31, ny=21 1

0.8

0.6

0.4

0.2 X: 9.995 X: 18.01 X: 24.96 X: 29.88 Y: 0.00293 Y: 0.009317 Y: 0.003106 Y: 0.0142 0 X: 14.98 X: 19.96 -0.2 Y: -0.003109 Y: 0.009317 det(M/constant)

-0.4

-0.6

-0.8

-1 5 10 15 20 25 30 35 Frequency in GHz

Figure 4.1a: Determination of cutoff frequencies of TE wave for PEC Rectangular

Waveguide ( = 31 , = 21).

As the mode increases, the error in the푛푥 cutoff frequencies푛푦 increases. We can decrease the error by approximating the higher number of nodes, which is meant for the small mesh size.

And also we can observe little difference in the cutoff frequencies of Method I and

Method II, this is due to the increment in the frequency step in the iterative procedure.

In both the graphs above, we can see the curve at frequency =29.88 GHz is turning back without continuing the flow into the negative part of the graph. This is due to the degenerate modes .

02 30 푇퐸From푎푛푑 this 푇퐸we can also observe the degenerate modes very easily.

4.1.2 TM Wave

PEC Rectangular Wavegude TM case nx=16 ny=11 1

0.8

0.6

0.4

0.2 X: 24.85 X: 31.15 X: 35.57 Y: -0.002441 Y: 0.002924 Y: -0.002924 0 X: 33.07 -0.2 Y: 0.002924 det(M/constant)

-0.4

-0.6

-0.8

-1 15 20 25 30 35 Frequency in GHz

Figure 4.2a: Determination of cutoff frequencies of TM wave for PEC Rectangular

Waveguide ( = 16 , = 11).

PEC Rectangular푛푥 Wavegude푛푦 TM case nx=31 ny=21 1

0.8

0.6

0.4

0.2 X: 18.01 X: 24.96 X: 31.5 X: 35.93 Y: 0.03125 Y: 0.002929 Y: 0.01462 Y: 0.002924 0 X: 33.42 -0.2 Y: 0.01462 det(M/constant)

-0.4

-0.6

-0.8

-1 15 20 25 30 35 Frequency in GHz

Figure 4.2b: Determination of cutoff frequencies of TM wave for PEC Rectangular

Waveguide ( = 31 , = 21).

푛푥71 푛푦

For TM Wave also we have a good agreement with the analytical solutions and

Method I solutions, and we can compare them from table 3.1.

4.2 Artificial Rectangular Waveguide

We have all specifications, dimensions and procedure regarding the Artificial

Rectangular Waveguide in the section 3.4.

4.2.1 TE Wave

Artificial Wavegude TE case nx=16 ny=11 1

0.8

0.6

0.4

0.2 X: 9.982 X: 17.97 X: 24.85 X: 31.15 Y: 0.002925 Y: 0.002924 Y: 0.002924 Y: -0.007644 0 X: 19.85 Y: 0.002924 -0.2 X: 29.51 det(M/constant) Y: -0.1218 -0.4

-0.6

-0.8

-1 5 10 15 20 25 30 35 Frequency in GHz

Figure 4.3a: Determination of cutoff frequencies of TE wave for Artificial Rectangular

Waveguide ( = 16 , = 11).

푛푥 푛푦

Artificial Wavegude TE case nx=31 ny=21 1

0.8

0.6

0.4 X: 29.88 0.2 X: 9.995 X: 18.01 X: 24.96 Y: 0.09521 Y: 0.00293 Y: -0.002924 Y: 0.002924 0 X: 19.96 X: 31.51 -0.2 Y: 0.002924 Y: -0.004497 det(M/constant)

-0.4

-0.6

-0.8

-1 5 10 15 20 25 30 35 Frequency in GHz

Figure 4.3b: Determination of cutoff frequencies of TE wave for Artificial Rectangular

Waveguide ( = 31 , = 21).

4.2.2 TM Wave 푛푥 푛푦

Artificial Wavegude TM case nx=16 ny=11 1

0.8

0.6

0.4

0.2 X: 14.94 X: 17.97 X: 29.51 Y: 0.002924 Y: 0.008772 Y: 0.004583 0 X: 24.85 X: 31.14 -0.2 Y: -0.008772 Y: 0.002132 det(M/constant)

-0.4

-0.6

-0.8

-1 10 15 20 25 30 35 Frequency in GHz

Figure 4.4a: Determination of cutoff frequencies of TM wave for Artificial Rectangular

Waveguide ( = 16 , = 11).

푛푥 푛푦 74

Artificial Wavegude TM case nx=31 ny=21 1

0.8

0.6

0.4

0.2 X: 14.98 X: 24.96 X: 29.88 X: 33.42 Y: -0.002914 Y: 0.002924 Y: 0.002924 Y: 0.003023 0 X: 18.01 -0.2 Y: -0.002924 X: 31.5 det(M/constant) Y: -0.1073 -0.4

-0.6

-0.8

-1 10 15 20 25 30 35 Frequency in GHz

Figure 4.4b: Determination of cutoff frequencies of TM wave for Artificial Rectangular

Waveguide ( = 31 , = 21).

푛푥 푛푦 For the Artificial Rectangular Waveguide also, the solutions obtained by Method II are in good agreement with the analytical solutions and Method I solutions. We can compare this from table 3.2, for both TE and TM Waves.

4.3 PEC Rectangular Waveguide with PEC Fin

We have all specifications, dimensions and procedure regarding the PEC Rectangular

Waveguide with PEC Fin in the section 3.5.

Here we plot the solution curve only for the case = . From table 3.3e, we can see only the degenerate modes in this case, and compare with ℎthe cutoff푏 frequencies obtained from the solution curve obtained by using the Method II, which are very well agreed.

PEC Rectangular Waveguide with PEC Fin TM case nx=21 ny=21 0.5

0 X: 24.93 X: 35.91 -0.5 Y: -0.03278 Y: -0.002062

-1

-1.5

-2

-2.5

det(M/constant) -3

-3.5

-4

-4.5

-5 24 26 28 30 32 34 36 38 40 42 Frequency in GHz

Figure 4.5: Determination of cutoff frequencies of TM wave for PEC Rectangular Waveguide

with PEC Fin ( = 21 , = 21).

푛푥 푛푦

4.4 L-Shape Waveguide

We have all specifications, dimensions and procedure regarding the L-Shape Waveguide in the section 3.6.

On observing the table 3.4, we can say that the cutoff frequencies of an L-Shape

Waveguide in the Method II are very well agreed with the cutoff frequencies in the Method I.

L-Shape Waveguide TM case nx=21 ny=21 0.5

0.4

0.3

0.2

0.1 X: 23.89 X: 32.32 X: 39.67 Y: 0.006231 Y: 2.354e-10 Y: 0.002454 0 X: 35.91 -0.1 Y: 0.003115 det(M/constant)

-0.2

-0.3

-0.4

-0.5 22 24 26 28 30 32 34 36 38 40 42 Frequency in GHz

Figure 4.6: Determination of cutoff frequencies of TM wave for an L-Shape Waveguide

( = 21 , = 21).

In Method II, we calculate the 푛푥values of the푛푦 determinant of the matrix over a range of

frequencies, which are very high values and also at the same time very low values,푀� in order to

plot the values, we adopt a constant to divide the matrix before the value of determinant of the

matrix calculated. And also the scale size on the y-axis will푀� be decreased to the range where we

can track the cutoff frequencies easily.

When we compare the two methods we have adopted here in this thesis to determine the cutoff frequencies of the waveguides, Method I has an advantage over the Method II. The

advantage is that we can get the field distributions for the corresponding cutoff frequency

without any further procedure and calculations. Method I might require more computation, if the

matrix is very large.

Linear Interpolation:

Estimation of an unknown quantity between two known quantities

or drawing conclusions about missing information from the available information is known as

Interpolation. Interpolation is meant for estimation. There are few types of interpolations, and

Linear Interpolation is one among them.

By this we can calculate the exact cutoff frequencies from the above curve by making use

of the other data points in the curve. This makes easier to calculate the exact cutoff frequencies

from the curve, which is independent of the frequency step considered.

CHAPTER V

CONCLUSION

1. This method is known for simplicity and accuracy.

2. Determination of the cutoff frequencies and mode shapes of waveguides, which are

ill-suited for determination by an analytical method.

3. Determination of the supported modes in a given frequency very easily.

4. Extraction of the degenerate modes without any further effort.

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VITA

Sridhar Kolagani was born in Karimnagar, India, in 1988. He received his Bachelor of

Technology in Electronics and Communication Engineering from Jawaharlal Nehru

Technological University (JNTU), Hyderabad, India in 2010.

In 2010 he joined the Department of Electrical Engineering at the University of

Mississippi, and pursued his M.S. degree. From 2010 – 2012, he worked as a research / teaching assistant in the Department of Electrical Engineering. His research interest include numerical techniques in electromagnetics, metamaterials, periodic structures and graphene based cloaks.