Design, Analysis & optimization of non linear waveguide taper

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					                                       BANARAS HINDU UNIVERSITY
                                        INSTITUTE OF TECHNOLOGY
                                       DEPARTMENT OF ELECTRONICS
                                              ENGINEERING
                                                                                                                                                                                  

                                                                                                                                                            


                                                                                                                 Gram: Electronics Engineering
                                                                                                              Head of Dept.: Prof. P.Chakrabarti
                                                                                                                        Phone. : 0542-2307010
                                                                                                                         Fax N0:0542-2368925
                                                                                                                   e-mail:hd_electro@bhu.ac.in

Ref: IT/ECE/08-09                                                                                                                               Date:


                                                             CERTIFICATE

This is to certify that the dissertation entitled " Design, Analysis and Optimization of the
Cylindrical Waveguide Nonlinear Tapers " submitted by Mr. Deepak Singh Nagarkoti (Roll
No–07030506), to the Department of Electronics Engineering, Institute of Technology, Banaras
Hindu University, Varanasi, in partial fulfillment of the requirements for the award of the degree
"MASTER OF TECHNOLOGY" in Electronics Engineering (Microwave Engineering) is a
bonafide record of the work carried out by him under our supervision and guidance.

 

(Prof. P.K. Jain)                                                                                                                (Prof. P.Chakrabarti)
       Supervisor                                                                                                             Head of the Department




Deepak Singh                                                                          Digitally signed by Deepak Singh Nagarkoti
                                                                                      DN: cn=Deepak Singh Nagarkoti, c=IN,
                                                                                      o=CRMT, ou=BHU,
                                                                                      email=kecbackbencher@gmail.com
Nagarkoti                                                                             Reason: I am the author of this document
                                                                                      Date: 2009.08.18 19:14:31 +05'30'
Dedicated to My Mother
                           ACKNOWLEDGEMENT
Thesis work is an academic journey which is not possible without the contribution of people.
Here, I would like to thank all people who have helped and inspired me during my postgraduate
studies.

           First and foremost, I especially want to thank my supervisor, Prof. P.K. Jain for his
unflinching guidance, motivation, & encouragement from the very early stage of research upto
the writing of Thesis. His truly scientist intuition has made him as a constant oasis of ideas and
passions, which exceptionally inspire and enrich my growth as a student here at Centre of
Research in Microwave Tubes. I am indebted to him more than he knows.

           I am highly grateful to Prof. P.Chakrabarti, Head of the Department of Electronics
Engineering for providing me necessary facilities, effective management & valuable suggestions
for the success of this work. I would like to thank to all the faculty members, librarians & CRMT
staff members for their corporation during the course of this work.

           I am cordially thankful to Miss Smirity Dwivedi & Mr. Ashutosh, PhD scholars of
Centre of Research in Microwave Tubes, as it is difficult to complete the work without their
efforts.

           I would not forget to thank my friend Hitesh Mohan Trivedi (M.Tech. Part 2, Control
Systems, Electrical Engg.) & junior Chandrabhushan Pal (M.Tech.Part1,Microelectronics,
Electronics Engg.) for being always with me during the debugging of code.

           Above all, I thank my family for their love, cooperation and encouragement which was a
constant source of inspiration for me. Also  would like to give special thanks to my lab-mates
Abhay, Parikshit , Swapnil, Akram, & Kali who made an energetic & working environment in
the lab. I am also thankful to all my batchmates who made my stay at Varanasi so wonderful.

           Last but not least, I thank almighty my Family Isht dev Bhagwati & Laata along with
Lord Vishwanath for providing me strength and courage in completing the work.



Date:                                                                 (Deepak Singh Nagarkoti)
                                        Preface


There is always a need of tapered waveguide section for connection between two components
having different cross sections. The connection might be change the output of former part which
is input for tapered section to input of later part which is output of connection waveguide. Power
loss is the measure problem in using such connections.

Transmission lines having single mode propagation where part of total power incident get
transmitted & the remaining get reflected back. But in waveguide, Multimodal propagation takes
place where reflection is negligible except at mode cut-off frequencies. The measure problem in
it is transmission of power from incident mode to the spurious modes at the output. Hence,Loss
of power in terms of mode conversion has been take place. Here, we are developing a non-linear
taper giving minimum mode conversion to undesired mode at certain operating frequency.

In the development of gyrotron, non-linear taper is an important component at the output part of
the design. We are designing a general taper for matching of any cross section for any incident
mode and at any operating frequency. But here, we are taking 200kW,42 GHz, CW gyrotron is
an example for testing of our taper. TE0,3 mode is taken as incident mode at operating frequency
is 42 GHz . The input and output radius is taken as 13.99 mm & 42.5 mm and length is 350 mm
in our synthesis of non-linear taper. Power transmitted to output is desired in same TE0,3 mode
with very low mode conversion.
     Design, Analysis and Optimization of the Cylindrical
                Waveguide Nonlinear Tapers

                                    CONTENTS
            ACKNOWLEDGEMENT
            PREFACE

Chapter 1    Introduction                                                    1-4
      1.1 Background                                                             1
      1.2 Plan and Scope                                                         3

Chapter 2    RF Fundamentals and Electromagnetics                           5 - 17
      2.1 Cylindrical Waveguides                                                 5
               2.1.1 Propagating Modes                                           5
               2.1.2. Properties of Waveguide                                    8
      2.2 Quarter-Wave Transformer                                               9
      2.3 Tapered Transmission Lines                                            14
      2.4 Applications of Taper                                                 16

Chapter 3     Synthesis and Design of Cylindrical Waveguide Tapers         18 - 40
      3.1 Taper Synthesis                                                      18
               3.1.1 Coupling Coefficient                                      19
               3.1.2 Synthesis of circular but otherwise arbitrary taper       22
      3.2 Taper Design                                                         24
               3.2.1 Linear Taper                                              24
               3.2.2 Parabolic Taper                                           25
               3.2.3 Exponential Taper                                         26
               3.2.4 Arbitrary Profile Taper                                   27
               3.2.5 Raised Cosine Taper                                       28
               3.2.6 Modified Arbitrary Profile Taper                          30
      3.3 Cylindrical Taper Design for 200kW, 42GHz, CW Gyrotron               30
      Appendix 3.1: Derivation of Synthesis formula for Parabolic and
                     Raised Cosine Taper                                    32 -40

Chapter 4     Mode Conversion in Cylindrical Waveguide Tapers              41 - 47
       4.1 Physical Significance                                               41
       4.2 Low Mode Conversion Tapers                                          42
      4.3 Analytical Calculation of Mode Conversion                            43
      4.6 Graphical Analysis                                                   46
Chapter 5      Analysis of Cylindrical Waveguide Tapers                        48 - 69
       5.1 Fundamentals of Modal Analysis                                           48
       5.2 Mode Matching Technique(MMT)                                             49
                 5.2.1. Principle of Mode Matching                                  49
                 5.2.2.Scattering Matrix of a Uniform Section                       51
                 5.2.3. Scattering Matrix of a Junction                             52
                 5.2.4 Cascading of Scattering Matrix                               54
                 5.2.5 Calculation of Coupling Power Integrals                      55
       5.3. Numerical Computation                                                   56
       5.4 .Scattering Matrix Formulation of Cylindrical Waveguide Taper            57
       5.5. Analysis of Various Cylindrical Taper Designs                           62
       5.6 Analysis of Cylindrical Waveguide Taper for 200kW,42 GHz ,CW Gyrotron    64
       5.7 Mode Conversion in Various Cylindrical Taper Design                      66
       5.8 Mode Conversion in Cylindrical Waveguide Taper for 200kW,42 GHz, CW
       Gyrotron                                                                     68

Chapter 6   Particle Swarm Optimization of Cylindrical Waveguide Tapers 70 - 97
       6.1 Introduction                                                             70
       6.2 Parameter Selection in PSO                                               73
       6.3. Simplified PSO particle Trajectories                                    77
       6.4 Optimum Choice of Parameters for PSO                                     78
       6.5. Basic Algorithm of PSO                                                  78
       6.4. PSO of Cylindrical Waveguide Taper for 200kW,42 GHz ,CW Gyrotron        82
                6.6.1. Variation of Parameters                                      83 
                6.6.2. Optimization of Non-Linear Taper                             91

Chapter 7 Design Validation of Nonlinear Taper                                 98 -102
       7.1. Introduction                                                            98
       7.2. Design & Modes                                                          98
       7.3. Propagation of TE0,3 Mode                                               99
       7.4 Simulation Results                                                      101
       7.5 Comparison of Results                                                   102

Chapter 8 Conclusion                                                           103-104
BIBLIOGRAPHY                                                                   105-113
 
List of Figures
 

Figure 2.1: Variation of cut-off frequency with change in radius of taper………………....9

Figure 2.2: Variation in wave impedance with change in radius of taper …………………..9

Figure 2.3: Bandwidth Characteristic for a single quarter wave transformer in terms of
            Reflection coefficient……………………………………………………............11
Figure 2.4: N-section quarter wave transformer with a load at the end……………………..12

Figure 2.5: Improvement of bandwidth with increase in no. of quarter wave transformers

               in binomial transformer…………………………………………………………14

Figure 2.6: Impedance variation exponentially in tapered section………………………….16

Figure 2.7: Reflection coefficient calculations along the length in exponential taper………16

Figure 3.1: A non-linear taper with parabolic profile……………………………………….21
Figure 3.2: Conical Taper Profile ………………………………………………………….24
Figure 3.3: 3D conical taper design…………………………………………………………25
Figure 3.4: Parabolic Taper Profile…………………………………………………………25
Figure 3.5: 3D Parabolic Taper Design……………………………………………………..26
Figure 3.6: Exponential Taper Profile………………………………………………………26
Figure 3.7: 3D Exponential Taper Design…………………………………………………..27
Figure 3.8: Various profiles obtained with variation of shape factor in arbitrary equation...27
Figure 3.9: Raised Cosine Taper Profile……………………………………………………28
Figure 3.10: 3D Raised Cosine Taper Design ……………………………………………….29
Figure 3.11: Various profiles obtained with variation of shape factor in modified arbitrary
             equation………………………………………………………………………...30
    Figure 3.12: Random Raised Cosine profiles for a specific range of shape factor in modified
               arbitrary equation……………………………………………………………….31
Figure 4.1: Mode conversion in physical taper…………………………………………….42
Figure 4.2: Mode coupling along the length shown in linear(conical) taper………………..46

Figure 4.3: Mode coupling in various basic profiles………………………………………..47

Figure 4.4: Mode conversion dependent on length of taper………………………………....47
Figure 5.1: Block diagram showing scattering matrix with forward and reflection

            coefficients……………………………………………………………………...50

Figure 5.2: Junction between two sections of cylindrical waveguide……………………….52

Figure 5.3: Single Scattering Matrix divided into a number of cascaded S Matrix…………54
Figure 5.4: Step approximation of basic raised cosine function for analysis………………..61
Figure 5.5: Section of raised cosine step approximated taper……………………………….61

Figure 5.6: Transmission characteristics of basic profiles with frequency………………….63

Figure 5.7: Transmission coefficients of arbitrary profiles with frequency…………………65

Figure 5.8: Transmission coefficients in narrow band………………………………………65

Figure 5.9: The Mode conversion comparison at 42GHz in desired and spurious modes….69

Figure 6.1: Plot between ‘w’ and success percentage at a particular maximum velocity…..74
Figure 6.2: Average no. of iteration required plot with inertia weight……………………..74
Figure 6.3: Showing the effect of introduction of constriction factor………………………76
Figure 6.4: Basic Algorithm of PSO………………………………………………………..79
Figure 6.5: Plot between confidence factor and constriction factor………………………...81
Figure 6.6: Design of Non-Linear Taper with Variation of Output Radius………………...83

Figure 6.7: Variation of Transmission Coefficient with frequency for different output

             radius…………………………………………………………………………..84

 Figure 6.8: Variation of Transmission coefficient with frequency for different output radius

             around 42GHz………………………………………………………………….84
Figure 6.9: Design of Non-Linear Taper with Variation in Length………………………...85

Figure 6.10: Variation of Transmission Coefficient with Frequency for different length….86

Figure 6.11: Variation of Transmission coefficient with frequency for different length around

              42 GHz………………………………………………………………………. 86

Figure 6.12: Design of Non-Linear Taper with Variation in shape factor………………….87

Figure 6.13: Variation of Transmission Coefficient with Frequency for different shape…..88

Figure 6.14: Variation of Transmission coefficient with frequency for different length around

             42GHz…………………………………………………………………………88

Figure 6.15: Design of Non-Linear Taper with Variation in No. of Sections……………….90
Figure 6.16: Variation of Transmission Coefficient with Frequency for different No. of

              Sections……………………………………………………………………….90

Figure 6.17: Variation of Transmission coefficient with frequency for different no. of

              sections around 42 GHz……………………………………………………….91

Figure 6.18: Basic overview of PSO in Non-Linear Taper…………………………………92

Figure 6.19: PSO of Non-Linear taper with iteration ………………………………………95
Figure 6.20: Showing variation of shape factor from S=0.1 to S=10 & optimization at
              S=0.5331……………………………………………………………………...95
Figure 6.21: Variation of Transmission coefficient with frequency in PSO optimized

              design…………………………………………………………………………96

Figure 6.22: Variation of Transmission coefficients around desired 42 GHz……………....96

Figure 7.1:   The PSO optimized non-linear taper design in CST microwave studio………99

Figure 7.2:   TE0,3 mode at input port………………………………………………………99

Figure 7.3:   TE0,3 mode at output port……………………………………………………100

Figure 7.4:   TE0,3 mode propagation from the PSO optimized taper……………………..101

Figure 7.5:   The S parameter plots for PSO optimized taper in CST Microwave Studio...102

Figure 7.6:   Comparison of results with two different approaches……………………….103




 
List of Tables
Table 2.1: Table showing the eigen values in increasing order corresponding modes….……7

Table 5.1: Table showing data sheet for design dimensions of 35 sections taper……………62

Table 5.2: Table showing basic profile transmission coefficients & transmitted power at

           42GHz……………………………………………………………………………64

Table 5.3: Arbitrary taper transmission coefficients & transmitted power at 42GHz……….66

Table 5.4: Mode Conversion to spurious modes with different basic designs for
           (a) Linear Taper………………………………………………………………......67
           (b) Parabolic Taper…………………………………………………………….....67
           (c) Exponential Taper………………………………………………………….....68
           (d) Raised Cosine Taper……………………………………………………….....68
Table 5.5: Mode Conversion to spurious modes with different Shape Factors

          (a) For shape factor S=0.4………………………………………………………..68

          (b) For shape factor S=0.5………………………………………………………..68

          (c) For shape factor S=0.6………………………………………………………..69

          (d) For shape factor S=0.7………………………………………………………..69

Table 6.1: Table showing parameters variation for PSO………………………………….....78

Table 6.2: Table showing the optimum choice of parameters for unimodal and multimodal

           functions…………………………………………………………….. …………..82
Table 6.3: Parameters under variation for non-linear taper design…………………………..83
Table 6.4: Transmission coefficient with variation of output radius at 42 GHz……………..85
Table 6.5: Transmission coefficient with variation of length at 42 GHz………………….....87
Table 6.6: Transmission coefficient with shape factor of length at 42 GHz…………………89

Table 6.7: Transmission coefficient with no. of sections at 42 GHz………………………...91

Table 6.8: PSO parameters involve in PSO of nonlinear Taper……………………………...92

Table 6.9: Variables for PSO of nonlinear taper……………………………………………..93

Table 6.10: Mode conversion analysis of PSO optimized taper………………..……………97

Table 7.1: Comparison of results with two different approaches at 42 GHz ……………...103
                                                                        Chapter 1: Introduction     
 




                                                                      Chapter 1



Introduction



1.1    Background

In the past few decades, there has been considerable effort to provide coherent, high power
sources in the electromagnetic spectrum. Among many of these devices, gyrotrons have
proven to be efficient sources for RF generation at high power levels and upto very high
frequencies. The application for gyrotrons range from microwave sources for general science
& industry, medicine, high power radar, plasma diagnostics & material sintering to RF driver
for high gradient accelerator. The main motivation for the development of high frequency and
high average power gyrotrons, however, is the application in magnetic fusion devices for
plasma heating and for electron current drive, which require a frequency range above
100GHz with power levels in excess of several hundred kilowatts.
       The gyrotron basically consist of a resonant cavity in a strong, continuous magnetic
field. A beam of relativistic electrons enter the cavity and interacts with the RF (Microwave)
field present in the cavity. The signal growth mechanism consists of a transfer of energy
between the electron and the RF field. The electrons must be moving at relativistic speeds so
that their mass will change as their velocities changes. This leads to the formation of bunches
of electrons in areas, where the electrons lose energy to the RF field. The work done in the
present dissertation is on the waveguide section for the connection of the output section of
gyrotron cavity to the uniform output collector waveguide section. The waveguide section is

Institute of Technology, Banaras Hindu University                                               1 
 
                                                                         Chapter 1: Introduction     
 
a nonlinear taper for 42GHz, 200kW, CW gyrotron operating in the TE0,3 cavity mode with
axial output collection.The Electron beam after RF interaction in cavity need collection at the
collector. Direct connection of collector to cavity output leads to generation of heat and
backward flow of same leads to disturbance in RF interaction. To keep the heat away from
the cavity end, a connection waveguide is needed between cavity output & collector
waveguide. Gyrotron output systems consist of a tapered cylindrical waveguide section,
which connects the output of the RF interaction structure, cavity, with the main collector of
the gyrotron. In the present case, the mode of the RF signal output of gyrotron cavity is in the
TE0,3 mode. The requirement of this nonlinear taper for the gyrotron or any other system is to
connect the two sides with maximum power transmission and to provide very low mode
conversion. Hence, the transmission coefficient corresponding to desired mode should be
very high. Reflection is very less because of multimode propagation.
       In 1958, HG Unger described that raised cosine profile taper have very low mode
conversion or low spurious modes [1]. In next year, L. Solymer proved that by making the
non-uniform waveguide more and more gradual, the amplitude of all spurious modes tend to
zero [2] and hence, assuming a pure incident mode, a sufficiently gradual non-uniform
waveguide may be represented by a single non-uniform transmission line. CCH Tang (1961)
designed an optimum taper of minimum length with the assumption that the taper possess
perfect symmetry and its axis is perfectly straight, only TE0n modes will be excited in the
tapered region [3]. Tomiyasu Klyo in 1971 developed a waveguide taper of minimum length at
General Electric Company [4]. Rudolf P Hecken(1972) demonstrated that optimum taper has
maximum bandwidth for a given length or minimum length for a given bandwidth [5]. In
1973,Hecken introduced modification in taper design equation for the condition when ratio of
end diameter becomes too large as reconversion from the spurious mode to incident mode is
neglected [6].
       Detailed work on modes in cylindrical waveguides & waveguide discontinuities is
needed for analysis of complex waveguides & the propagating modes [7-9]. To overcome the
complexity & advancement in computer technology,H Flugel and E Kuhn (1988) developed
the rigorous analysis of circular waveguide tapers using the high speed digital computer in
1980s.Modified Dolph-Chebychev taper turns out to give the best performance with respect
to both minimum spurious mode           excitation with minimum taper length [10]. In the
meantime, Jeff M. Neilson (1989) working on development of cascaded scattering matrix
code for analysis of tapers with complicated electromagnetic equations [11]. WG Lawson at
the University of Maryland developed a similar code which require less memory, had rapid

Institute of Technology, Banaras Hindu University                                                2 
 
                                                                       Chapter 1: Introduction     
 
convergence for the backward modes, and could rapidly find results for the forward modes
[12]. Also, it was shown that under some conditions a raised cosine profile yields less mode
conversion than the modified Dolph-Chebyshev profile. A taper with raised cosine profile
with input radius 25.4mm, output radius 63.5mm, and length 500 mm keeping the forward
power in parasitic mode below -47dB and the reflected power below -64dB. The scattering
matrix code is based on mode matching technique rather than time and memory consuming
FEM based for circular and rectangular waveguide analysis invented by Silvester in late 60s
[13]. In 1996, Deitmer Wagner also developed Scattering matrix code and analysis of
Gyrotron FU IV A has been done [14,15]. In late 60s, mode matching technique has been
introduced by Wexler (1967)[16], PJB Clarricoats (1971) went for development of computer
code at University of Leeds [17]. The effort was appreciable but not much effective due to
lack of much advancement in computer technology at that time. Also, Clarricoats contributed
a lot in the analysis of tapered & corrugated horn & waveguide using numerical computation
[18-20]. G L James in early 80’s worked on application of mode matching technique in
waveguide problems [21, 22].
         Transmission line is single mode propagating waveguide and it can be analyzed by
reflection coefficient calculation [23-30]. Mode analysis is not an important issue while
dealing with transmission lines & hence planer microstrip tapered transmission
lines[21,32].Single mode tapered waveguide behaves as a transmission line [33,34].As it is
well understood that overmoded tapered cylindrical waveguide involves very little reflection
even for the worst possible profile is considered. Rather, it involves mode conversion to
spurious modes. Mode conversion issues of waveguides & mode convertor synthesis works
on the same principle involves transmission of power to desire different from the incident
mode[35-48]. Mode convertors [21, 49] are similar to tapers but in tapers desired mode in
output is same as the incident mode. This mode conversion analysis can be done step by step
field matching after no. of section division of the waveguide. Mode matching technique is a
very effective analysis method followed by huge crowd in waveguide & horn designs [50-
72]. Eberhart & shi in 1995 proposed an optimization technique for non-linear functions. This
particle swarm optimization technique [73-88] is used here for optimization of non-linear
taper.
         To overcome the complexity & advancement in computer technology, many
simulation software are available apart from theoretical and experimental work [89-99].
Commercial software Ansoft HFSS [89] and CST microwave studio [90] are popular and
they are developed on the principle of FEM and FDTD principle respectively. CASCADE

Institute of Technology, Banaras Hindu University                                              3 
 
                                                                         Chapter 1: Introduction     
 
[94]-96] & SCATTMAT [97] are the commercially available based Scattering matrix code
for analysis of passive circuits working on personal computers in a window environment.
These codes simplifies very complex calculations involves in analysis of any passive circuit.



1.2    Plan and Scope

This Thesis presents the synthesis, design, analysis, optimization and EDA simulation of
nonlinear tapered waveguide section connecting the output of the gyrotron cavity having
smaller diameter with the uniform collector waveguide of larger diameter. This taper has a
significant role to play in extracting the RF power developed within the device to the out
waveguide with minimum loss and mode conversion. The results obtained by analytical
techniques are compared with simulation software results as well as predicted results.

       The Thesis is organized in the following manner. An overview of RF fundamentals
and electromagnetic analysis necessary for the present dissertation will be presented in
Chapter 2. In Chapter 3, synthesis and design of different types tapers will be studied
followed by the most important issue mode conversion analysis will be studied in Chapter 4.
In Chapter 5, the taper analysis will be described using mode matching technique for various
designs. Optimization of taper will be undergone using a modern technique, particle swarm
optimization in Chapter 6. In the last Chapter 7, the optimum parameters obtained from last
chapter will be used in simulation tools and the results will be compared with the analytically
obtained results. The conclusion of work is briefly explained in chapter 8.




Institute of Technology, Banaras Hindu University                                                4 
 
                                                     Chapter 2: RF Fundamentals & Electromagnetics 
 




                                                                         Chapter 2



RF Fundamentals and Electromagnetics


This chapter includes circular waveguide description followed by the modes propagation in
it. It is followed by cut-off frequencies and wave impedance calculation for nonlinear circular
waveguide. Quarter wave transformer will be discusses for impedance matching in
transmission lines [23] and synthesis of nonlinear tapered transmission line will be discussed.
The Chapter is terminated with application of tapers.



2.1 Cylindrical Waveguides
As the taper have circular cross section, here, we discuss some basic description of only
circular waveguide. The waveguide must be metal structures for the propagation of
microwave frequencies. Microwaves ranges from 1GHz to 300GHz or correspondingly 30cm
to 1mm in terms of wavelength. A part of microwave range having wavelength in millimeters
or in other words having frequencies more than 30 GHz are well known as millimeter waves.
The impedance of waveguide is not frequency independent like transmission line. At the
mean time, cut off frequency of a circular waveguide (off course, rectangular waveguide too)
is dependent on the propagating mode and geometry of waveguide. For a uniform waveguide,
cut off frequency for any particular mode under propagation is constant. So, the wave
impedance also remains same throughout the length of uniform cylindrical waveguide [24].




Institute of Technology, Banaras Hindu University                                                5 
 
                                                      Chapter 2: RF Fundamentals & Electromagnetics 
 
2.1.1 Propagating Modes

In circular waveguides, mode of propagation is either TEmn or TMmn mode, where the
subscripts have a different meaning. The first subscript indicates the number of full-wave
patterns around the circumference of the waveguide. The second subscript indicates the
number of half-wave patterns across the diameter. If the E field is perpendicular to the length
of the waveguide, no electric field lines parallel to the direction of propagation. Thus, it must
be classified as operating in the TE mode. The dominant mode in circular waveguide is TE1,1
.The TE and TM modes propagating in waveguide can be calculated with the Bessel function
of first kind and second kind, respectively. Cylindrical waves can be used to study the
propagation characteristics of horns with cross sections which are rectangular, circular,
elliptical or arbitrary. The horns/tapers can have metal or dielectric boundaries and can be
homogeneous (e.g., hollow) or inhomogeneous (e.g., coaxial, corrugated, dielectric loaded).
The same general principles apply in all cases. The electric and magnetic field in the
homogeneous waveguide can be divided into axial components Ez and Hz and the transverse
components Et and Ht. The forward travelling fields are
                ,                  ,                                                       (2.1)

                 ,                     ,                                                   (2.2)

where     and        refer to the transverse co-ordinates of a general co-ordinate system. The
axial field components satisfy the 2-dimensional scalar electromagnetic wave equations.
     E     k H        0                                                                    (2.3)
     H     k H         0                                                                   (2.4)
For homogeneous waveguides, one of the axial field components will be zero (Hz=0 for TM
modes and Ez=0 for TE modes), but for inhomogeneous waveguides both axial components
exist. In this latter case, either the Ez or the Hz wave equation can be used. The wave number
k is given by

                                                                                           (2.5)

where for the most general medium filling the waveguide,

                                             1,             1

Then,




Institute of Technology, Banaras Hindu University                                                   6 
 
                                                     Chapter 2: RF Fundamentals & Electromagnetics 
 
                                           .

The transverse laplacian operator     depends on the co-ordinate system.

As we are dealing with circular coordinates, so the Laplacian operator for circular coordinates
is


                             .                                                            (2.6)


The solutions are products of Bessel functions and trigonometric functions.


     Table 2.1: Showing the eigenvalues in increasing order and corresponding modes.
        Modes             Eigenvalues               Modes               Eigenvalues
         TE11                     1.841                    TE42                9.282
         TM01                     2.405                   TM32                  9.761
         TE21                     3.054                    TE23                9.969
         TE01                     3.832                    TE03                10.173
         TM11                     3.832                   TM13                 10.173
         TE31                     4.201                    TE52                 10.52
         TM21                     5.136                   TM42                 11.065
         TE41                     5.317                    TE33                11.346
         TE12                     5.331                   TM23                 11.62
         TM02                      5.52                    TE14                11.706
         TM31                      6.38                   TM04                 11.792
         TE51                     6.416                   TM52                 12.339
         TE22                     6.706                    TE43                12.682
         TE02                     7.016                   TM33                 13.015
         TM12                     7.106                    TE24                13.17
         TM41                     7.588                    TE04                13.324
         TE32                     8.015                   TM14                 13.324
         TM22                     8.417                    TE53                13.987
         TE13                     8.536                   TM43                 14.372
         TM03                     8.645                   TM24                 14.796
         TM51                     8.771

The wavenumber for the circular waveguide is different for TM & TE modes. For TMnm
modes, the axial electric field and wavenumber are given by

        ,                                                                                 (2.7)

and                    0                                                                    (2.8)




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                                                         Chapter 2: RF Fundamentals & Electromagnetics 
 
where a is the radius of the cylindrical waveguide. The wavenumber is equal to the roots of
Bessel function of order n. For TEnm modes axial magnetic field and wavenumber are given
by:

               ,                                                                              (2.9)

and                            0                                                               (2.10)

The wavenumber is now equal to the roots of the derivative of the Bessel function of the
order n. The lowest order mode in the circular waveguide is the TE11 mode.

2.1.2 Properties of Waveguide
The cut off frequency at any cross section of waveguide is given by

                                                                                             (2.11)
                                           .

Where,  =permeability of medium
                    =permittivity of medium
                   =radius of waveguide
                    =eigenvalue of TEmn mode .
For our design parameters of taper, the waveguide has input and output radii are 13.99 mm
and 42.5mm, respectively. The propagating mode is TE03, and has eigenvalue     =10.174.

The graph above is showing that cutoff frequency varies from 34.7229 GHz at the input to
11.4229 GHz at the output.

The wave impedance of non-uniform waveguide section for TEmn modes can be calculated by
the well known expression


                                                                                               (2.12)



Where              is the operating frequency i.e. 42GHz in our problem.

The figure 2.2 showing that the input & output wave impedance of tapered section is given
by 669.4 ohms and 391.74 ohms, respectively

The matching of two waveguide sections having impedance

                                                 Z    669.4 Ω

                                                 Z    391.74 Ω


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                                                                                                       Chapter 2: RF Fundamentals & Electromagnetics 
 



                                                                                   Cut off Frequency of Non Line ar taper
                                                                      35




                                                                      30




                                           cut off frequency in GHz
                                                                      25




                                                                      20




                                                                      15




                                                                      10
                                                                        10    15      20        25       30        35       40        45
                                                                                           Radius of Taper in mm



      Figure 2.1: Variation of cut-off frequency with change in radius of taper

                                                                                   W ave impedance of Non linear taper
                                          700


                                          650


                                          600
                  Wave impedance in ohm




                                          550


                                          500


                                          450


                                          400


                                          350
                                             10                              15      20         25       30          35          40        45
                                                                                           Radius of Taper in mm



       Figure 2.2: Variation in wave impedance with change in radius of taper



2.2 Quarter Wave Transformer
Impedance transformation is the ability to change impedance by adding a length of
transmission line. If the line attenuation is neglected, the line impedance for a lossless line
can be given as

                                                                                                                                                (2.13)

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                                                                           Chapter 2: RF Fundamentals & Electromagnetics 
 

For a quarter wave transformer,                        ,

                       .                                                                                                              (2.14)

If two unmatched transmission lines with output impedance of first is                                                                  and input
impedance of second is                   . Then, Impedance matching can be achieved by using a quarter
wavelength section in between having characteristic impedance                                                               .
If    be the electrical length of the transformer at the frequency .The phase constant                                                          is
function of frequency.
As,

For a TEM wave in an air filled line,

At any frequency the input impedance presented to the main line is



Where,
Consequently, the reflection coefficient is

                                                                                                                                        2.15

gives,




The magnitude of reflection coefficient is calculated as
                                                                    |              |
                                                                               4
                                 1
                                                                                                                                        2.16
                  2
             1                              

for   near   2,
the above equation can be approximated as
             |             |
                               |cos |                                                                                           2.17
             2
It has periodic nature because input impedance has periodic variation with frequency or
wavelength, i.e., the impedance repeats its value every time the electrical length of the
transformer changed by               .


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                                                                                                 Chapter 2: RF Fundamentals & Electromagnetics 
 
                                                   Bandw idth characterstic for a single quarter w ave transformer
                                        0.35


                                         0.3


                                        0.25

                Reflection coeffcient
                                         0.2


                                        0.15


                                         0.1


                                        0.05


                                          0
                                               0                pi/2                   pi            3.pi/2                        2.pi
                                                                           electrical length of transforme r


Figure 2.3: Bandwidth Characteristic for a single quarter wave transformer in terms of
                              reflection coefficient.

If      is the maximum value of reflection coefficient that can be tolerated, the useful
bandwidth provided by the transformer is corresponding to the range ∆ . As the increase is
very fast, so useful bandwidth is very small . One can calculate the                                                                      at the edge of useful
passband by putting   to                                .
Thus,
                                                   2
                                                                                                                                                              2.18
                                                            1           

For TEM wave, =                                                            2

Where      is the frequency for which                                                      2
In this case bandwidth is given by
                                                                               ∆           2
                                                                                                   2
                                                                           ∆           2

The fractional bandwidth is given by
      ∆                          4                          2
            2                                                                                           .                                                      2.19
                                                                            1       
In the above discussion of bandwidth, it is assumed that the characteristic impedances                                                                                and
     were independent of frequency. This is good approximation for transmission line but

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                                                     Chapter 2: RF Fundamentals & Electromagnetics 
 
wave impedance varies with frequency in waveguides. Also, in transmission line and
waveguides, there are reactive fields excited at the junctions of the different sections, brought
about because of the change in geometrical cross section necessary to achieve the required
characteristic impedances. These junction effects can often be represented by a pure shunt
susceptance at each junction. Even inclusion of susceptive elements will also vary the
performance of any practical transformer from the predicted performance based on an ideal
model where junction effects are neglected. Hence, junction effects and the frequency
dependency of the equivalent characteristic impedances are neglected. Also, one can increase
the bandwidth by using multisection quarter wave transformers [25].




          Figure 2.4: N-section quarter wave transformer with a load at the end

The figure 2.4 shown is an N-section quarter wave transformer. If the first junction the
reflection coefficient is




Similarly, at the th junction, the reflection coefficient is

                                                  




The last reflection coefficient is




Z0 is a characteristic impedance and not necessarily equal to               .

       Each section has same length                  , & will be a quarter wave long at the
matching frequency .The load ZL is assumed to be a pure resistance, and may be greater or
smaller than Z0. In this analysis, it is chosen greater, so that all        , where  is the
magnitude of . If ZL is smaller than Z0, all            are negative real numbers and only
modification required in the theory is by replacing all   by - .


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                                                                 Chapter 2: RF Fundamentals & Electromagnetics 
 
        For a First approximation the total reflection coefficient is the sum of the first order
reflected waves only and is given by

                                                                           … … . .

where         accounts for the phase retardation introduced because of the different distances
the various partial waves must travel.
Consider the transformer is symmetrical,




                                                                                etc.
It becomes,
                                                                                                                   …]

Last term is                             when N is odd &                              when N is even.

Using Fourier cosine series,

                   cos          cos             2           ……..            cos             2         +……..]                  (2.20)

Last term is             cos  when N is odd &                            when N is even.

It should now be apparent that by proper choice of the reflection coefficients      , and hence
the Zn , a variety of passband characteristics can be obtained. As series has cosine nature, it is
periodic over the interval corresponding to the frequency range over which the length of
each transformer section changes by half wavelength.

In Binomial transformer, we choose

                     1                    .                                                                            2.21

Magnitude of reflection coefficient in binomial transformer made up of N sections of quarter
wave transformers is given by

                           . cos                                                                                      2.22

Considering the wave impedance of our taper section, the reflection coefficient is calculated
through binomial transformer. The figure 2.5 showing that increase in number of section
introduced, bandwidth is enhanced.




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                                                                             Chapter 2: RF Fundamentals & Electromagnetics 
 
                                              Ba ndw idth cha ra cte rstic for a Binom ia l tra nsform e r
                                   0.35
                                                                                                    N=2
                                                                                                    N=4
                                    0.3
                                                                                                    N=10
                                                                                                    N=20
                                   0.25

           Reflection coeffcient
                                    0.2


                                   0.15


                                    0.1


                                   0.05


                                     0
                                          0        pi/2                  pi              3.pi/2             2.pi
                                                          e le ctrica l le ngth of tra nsform e r



        Figure 2.5: Improvement of bandwidth with increase in no. of quarter wave
                         transformers in binomial transformer.


2.3 Tapered Transmission Lines
Klopfenstein [26] and Collin [27] improve the impedance matching issues of transmission
lines. Reflection coefficient calculation is sufficient for getting perfect match or optimum
design [26]-[31].. Any arbitrary real load impedance could be matched to a line over a
desired bandwidth by using multi-section matching transformers. As the number of sections,
N, increases, the step changes in characteristic impedance between the sections become
smaller. Thus, in the limit of an infinite number of sections, we approach a continuously
tapered line. In practical, a matching transformer must be of finite length, and no more than a
few sections long. If the taper length is short and step length is larger, the insertion loss of a
linear taper is lower. However, if the taper length is longer and step length is larger, the
insertion loss of raised cosine taper is smaller [32]. But instead of discrete sections, the line
can be continuously tapered. Then by changing the type of taper, we can obtain different
passband characteristics. Based on the theory of small reflections, the reflection coefficient
Z(z).
        Consider the continuously tapered transmission line made up of a number of
increment sections of length ∆ , with the impedance change ∆                                               from one section to the
next. The increment reflection coefficient from the step at z is given by




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                                                                       Chapter 2: RF Fundamentals & Electromagnetics 
 
                Z   ∆Z        Z    ∆Z
        ∆                                                                                                                              2.23
                Z   ∆Z        Z    2Z

In the limit as ∆            0,we have an exact differential

                         Z
            Z                Z
    Γ                             dz
            Z

Since,




Then, by using the theory of small reflections, the total reflection coefficient at z=0 can be
found by summing all the partial reflections with their appropriate phase shifts.

                1                             
                                                                                                                                         2.24
                2

where           2   ,

So, If Z(z) is known,        can be found as a function of frequency. Alternatively, if                                                          is
specified, then in principle Z(z) can be found.

        A tapered impedance matching network is defined by two characteristics. First is its
length L and second is its taper function Z(z).If Z1 and Z2 are the input and output impedance
of taper, let us discuss exponential taper here.

            Consider an exponential taper shown in figure 2.6, where

                                                      exp            ,               0              ,

As at z = 0, Z(0) = Z1

and at z = L, Z(L) = Z2 = Z1exp(aL)

or ,

The reflection coefficient is calculated as



                                                                      ln                  

                                                  
                                                                                 

                                                  
                                                                                                                                     (2.25)

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                                                                                                                         Chapter 2: RF Fundamentals & Electromagnetics 
 
Figure 2.7 gives the variation of reflection coefficient. Thus, in transmission lines where
single mode is propagating, reflection coefficient calculation is sufficient for analysis.

                                                                                              Exponential variation of impedance w ith length
                                                              700


                                                              650




                   Wave impedance of waveguide in ohm
                                                              600


                                                              550


                                                              500


                                                              450


                                                              400


                                                              350
                                                                                 0           50        100       150       200        250         300         350
                                                                                                             Length of Taper in mm



             Figure 2.6 Impedance variation exponentially in tapered section



                                                                                0.35


                                                                                 0.3


                                                                                0.25
                                                        Reflection coeffcient




                                                                                 0.2


                                                                                0.15


                                                                                 0.1


                                                                                0.05


                                                                                     0
                                                                                         0        pi           2pi            3pi           4pi         5pi
                                                                                                        electrical length of transformer



    Figure 2.7 Reflection coefficient calculations along the length in exponential taper



2.4 Applications of Taper
Taper has wide application in matching of two sections of transmission lines as we have
discussed above. In circular or rectangular waveguide system, two different cross sections are
connected using tapered waveguide section. The two different structures undergo matching


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                                                     Chapter 2: RF Fundamentals & Electromagnetics 
 
might be both circular with different radii, both rectangular [53,54] with different cross
section, or might be rectangular to circular[65-67] & vice versa. Horn antennas are also
rectangular or circularly tapered either in linear or non-linear way. In planer circuits, tapered
micro-strip lines are used for matching of two lines of different width. Most tapered lines are
implemented in strip-line or micro-strip. We can modify the characteristic impedance of the
transmission line by simply tapering the width of the conductor. In other words, we can
continuously increase or decrease the width of the microstrip or stripline to create the desired
impedance taper [31, 32].




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                                        Chapter 3: Synthesis & Design of Cylindrical Waveguide Tapers 
 




                                                                           Chapter 3



Synthesis                    and             Design              of        Cylindrical
Waveguide Tapers


3.1 Taper Synthesis:
Two ideal uniform lossless transmission lines are coupled together, that it is to be expected
that system can support four normal modes; two in forward and two backward directions. The
two backward modes can be neglected as including them not much affect the physical insight
[1].
    The amplitude for the two coupled lines can be given by coupled line equations as
                  d I1/dz = -j(β1(z)+c(z)) I1 + j c(z) I2                                      (3.1)


                 d I2/dz = jc(z) I1 – j(β2(z) + c(z)) I2                                      (3.2)
where,
          I1, I2 = Amplitude of desired and undesired modes, respectively.
          β1, β2 = Propagation constants for desired and undesired modes, respectively.
          c(z) = Coupling coefficient
For the modes reasonably far from cut off of both modes, then reflection caused by a taper
are ordinarily negligible.      These self coupling terms can be neglected from the above
equations.
The equations becomes,


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                                             Chapter 3: Synthesis & Design of Cylindrical Waveguide Tapers 
 
            d I1/dz = - jβ1(z) I1 + j c(z) I2                                                                                    (3.3)
            d I2/dz = - jβ2(z) I2 + j c(z) I1                                                                                     (3.4)
Coupling coefficient c(z) may be written in terms of the tapered dimension ‘a’ as
                                     c(z)= K(a,f)da/dz                    (K0/a) da/dz                                    (3.5)
    where, a = local radius for cylindrical taper and local length for rectangular length.
K is weakly dependent on f and can be neglected [33], hence second expression is
surprisingly better approximated even modes are near the cut off. In the second expression,
K0 is independent of ‘a’ and ‘f’ and depends upon only coupled modes.

3.1.1 Coupling Coefficient:
The normalized propagation constant Pm , for mode mn is given by
                                  Pmn = βmn / k = [1 – (fc,mn /f)2]1/2                                                            (3.6)
Where,
            fc,mn = cut off frequency of mode ‘mn’

                                             ,          .                                                                             3.7

            βm = propagation constant of mode ‘m’
            k = free space propagation constant.
If the two modes far from cut off , then one can approximate for a TE0n / TM0n mode for n=0
as
                                                 P0     ∆β a2 / λ           ,                                                   (3.8)
here, λ = free space wavelength
                                                      ∆β = (β1 - β2)
Taper synthesis parameter, P0 can be calculated using (3.8)
For smooth cylindrical waveguide, using equation
           TE01-TE02 coupled modes                              P0 = 2.75
           TE02-TE03 coupled modes                              P0 = 4.32
For TE0m -TE0n coupling

                                             ,                    /                                                                  3.9

For TE1m –TM1n coupling
                                         ,                            /              /                                                3.10

For TM1m –TM1n coupling

                                     ,                      /                                                                          3.11


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                                              Chapter 3: Synthesis & Design of Cylindrical Waveguide Tapers 
 
where, Xm, Xn are Bessel functions roots satisfying J0’(Xm) =0
    K0 is obtained by putting Pm and Pn in the above equation (3.11) unity .
Taper synthesis parameter, K0
         For smooth cylindrical waveguide,
                           TE01-TE02 coupled modes                          K0 = 1.557
                           TE02-TE03 coupled modes                          K0 = 2.630
Coupled line equations can be expressed in terms of linear combination of local normal
modes as
                                          ξ                  ξ
                                   cos w z            sin        (z)]                            (3.12)
                                          ξ                  ξ
                                   sin w z            cos        (z)]   ,                         (3.13)

where, W1 and W2 are wanted and unwanted local normal modes respectively. W1 is
corresponding to TE01 and W2 is corresponding to TE0n ( n ≠ 1 ) ,that’s unwanted mode.
                           β = (β1 + β2) / 2                                                        (3.14)
                           Φ = (β2 – β1) / 2                                                        (3.15)
                       2          2 1/2
               Г = ( Φ + c(z) )           =∆β / (2 Cos ξ)                                           (3.16)

                            Г Φ
            Cos ξ /2                                                                                (3.17)
                             Г

                           Г Φ
            Sin ξ /2 =                                                                              (3.18)
                             Г

where, ξ is taper angle. After solving above sine and cosine half angle terms
                     Cot ξ = Φ /c(z)                                                               (3.19)
     tan ξ = c(z) / Φ = 2c(z) /(β2 – β1)                                                           (3.20)
                    tan ξ = 2c(z) / ∆β                                                             (3.21)
Where, from (3.8)
                                          ∆β     P0 λ / a2
As P0 = constant, independent of ‘a’ and ‘f’ and only mode dependent when ‘f’ is very high
to the cut-off. Hence, ∆β = constant,
         As we are well aware that if there is no mutual coupling between modes then taper
angle ξ must be constant. Therefore, from (3.19) ⇒ c(z) is also constant, i.e. , (K0/a) da/dz =
Constant.
Clearly, da /dz = Constant
Now, For designing a taper of parabolic profile,
            da /dz = F / a , F = constant                                                      (3.22)

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                                       Chapter 3: Synthesis & Design of Cylindrical Waveguide Tapers 
 
It is so because by integration we find the axial length ‘z’ dependent on the square of the
radius ‘a’ as
                      z = [a2 (z) – a2(0) ] / 2F




                        Figure 3.1: A non-linear taper with parabolic profile


Writing a (0) = a1 and a(L) = a2 ,
We have ,
                             F = ( a22 – a12) /2L and a(z) = (a12 + 2Fz)2
Subtituting, Eq. (3.12), (3.13) in couple line equation pair (3.3) (3.4), One can get the normal
modes ,wanted or unwanted both must satisfy the equation
          dw1/dz + jГ w1 = (1/2) w2 .d ξ /dz                                            (3.23)

          dw2/dz - jГ w2 = (- 1/2) w1 .d ξ /dz                                          (3.24)

These are the linear differential equation pair. This equation pair is coupled and directly
proportional to d ξ / dz .
                                  If ξ = constant,      then d ξ / dz =0
Means the normal modes are decoupled and the above equations becomes
       dw1/dz + jГw1 = 0 ,                                                              (3.25)
and d w2 /dz - jГw2 = 0           .                                                     (3.26)
The normal mode solutions may write down immediately as
        w1 (z) = w1 (0) exp (jГz)                                                       (3.27)
        w2 (z) = w2 (0) exp (-jГz)                                                      (3.28)
remember, w1 (z) is wanted mode and also called as fast normal mode and w2 (z) is unwanted
mode and also called as slow normal mode.
        For gentle change in taper angle. so that d ξ /dz changes very slowly, that means
coupling is there but its extent is very little. This condition is called as hyper coupling




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                                                   Chapter 3: Synthesis & Design of Cylindrical Waveguide Tapers 
 
between the normal modes. In such case, one can prefer very long taper for achieving the
same. The hyperbolic coupling coefficient is given by
            1 dξ
                        1                                                                                                         3.29
           2Г  dz


3.1.2 Synthesis of Circular but Otherwise Arbitrary Taper:
         Suppose all the power initially at input of taper, at z = 0 is in mode 1 , TE01 mode
                Then,            at z = 0,                 I1(0) = 1 , I2(0) = 0
Then, putting in Eq. (3.12) and (3.13),
                       w1 (0) Cos ξ /2 - w2 (0) Sin ξ /2 = 1                                                                       (3.30)
                      w1 (0) Cos ξ /2 + w2 (0) Sin ξ /2 = 0                                                                        (3.31)
on solving the above equations,                     w1 (0) = Cos ξ /2                 ,
                                                    w2 (0) = - Sin ξ /2
Let the taper angle initially and at end is zero, ξ (0) = 0, ξ (L) = 0
                                                      w1(0) = 1 , w2(0) = 0
Since, the magnitude of I1 and I1 are both linear combination of w1 and w2 , the phasing of
w1 and w2 at the end of the taper z = L will be correct for making
                    I1( L) =1                 I2( L) = 0
As desired if
                                   ,                    [ I2( L) = 0] ,                                                            (3.32)

where,                 Г z dz                  .                                                                                   (3.33)
The expression to mode conversion to unwanted mode is
                     |I  L |   = sin ξ. sin                                                                                        (3.34)
As from the equations. (3.5), (3.8), and (3.21)
                       c(z)        (K0/a) da/dz
                    tan ξ = 2c(z) / ∆β
                         ∆β       P0 λ / a2
From above three equations
             tan ξ =          K da/dz                                                                                              (3.35)
                       λ P

for parabolic profile , eq. (3.22) gives
                                                                   da/dz = F / a
Substituting in above expression. we get
                                                        tan ξ = 2K0F/ λP0

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                                                     Chapter 3: Synthesis & Design of Cylindrical Waveguide Tapers 
 
                                                                             = 2K0(a22 – a12) /λP0L             (3.36)
Using the relation ,
                                                   Sin  ξ                                .                       (3.37)

    Implies
                                           Sin  ξ                                             1/                 (3.38)
                                                                             λL
                                                                      

Now, from (3.33) :
                                                                 Г z dz                   ,
and also from (3.21):                          Г(z) =∆β / (2 Cos ξ)                                              (3.39)
Therefore,
                                                                                    λ 
                                                                                                                 (3.40)
                                                                                         ξ

For calculating the value of cos ξ , one can start from Eq. (3.35)
                                                                             tanξ               K da/dz
                                                                                         λ P

we get ,
                                                                             √τ K
                                                                 ξ                                               (3.41)
                                                                                 λ

where,                     R= cosec2 ξ                                                                    (from eq.(3.38))
    Putting the above values in                     expression, we get:
                                                                         √           ln                          (3.42)

Hence, the expression for mode conversion, i.e., eq. (22)
                                                             |I  L |   = sin ξ. sin                              (3.43)
Substituting, sin ξ  &                     , from eq. (3.38) and (3.40), we get

                                                   √                      
                                        
                           |I  L |                                                       ,                       (3.44)

                                                           λL
    Where                                  1                                                       .             (3.45)
                                                        

These formulas prove a means of synthesizing a taper. For a given a1 and a2 , L can be chosen
to make   |I  L |   vanish.
           The minimum length required to produce a null in the mode conversion

For n = 1, from eq. (3.32)
                                                           [I2 ( L) = 0] .                                       (3.46)



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                                                              Chapter 3: Synthesis & Design of Cylindrical Waveguide Tapers 
 
Using equation (3.42), (3.44) and (3.45), one can easily get the expression for taper length Ln
for which the mode conversion vanishes
                                                                                                     /
                                                                                               1         .               (3.47)
                                                          λ

Tapers with different profiles are synthesized using different profile functions.




3.2 Taper Design:
       The taper design technique used above involves first analysis of taper and then design
of taper having very low mode conversion. Doing the analysis along with design is quite
cumbersome work even we discussed only parabolic case in the last section. A smart
approach is to design tapers as per the basic profiles [34] and analyze them later.

3.2.1 Linear Taper:
A linear taper has conical shape and can be generated a ramp function not passing through
origin around the abscissa. Linear taper can be synthesized with given input radius (a1),
output radius (a2) and Length (L) is given by the function
                                                         a(z) = a1 + (a2 - a1).(z/L)                                     (3.48)
One can obtain the design profile as shown in figure 3.2 and the corresponding 3D taper
obtained by sweeping around the axis as shown in figure 3.3


                                                                    Conica l Ta per Profile
                                           45


                                           40


                                           35
                   Radius of Taper in mm




                                           30


                                           25


                                           20


                                           15


                                           10
                                                0   50        100        150        200            250       300   350
                                                                    Le ngth of Ta pe r in mm


                                                    Figure 3.2 Conical Taper Profile
For our design dimensions, i.e., a1 = 13.99mm, a2 = 42.5mm, L=350 mm

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                                                               Chapter 3: Synthesis & Design of Cylindrical Waveguide Tapers 
 




                                                       Figure 3.3: 3D conical taper design
3.2.2 Parabolic Taper:
       Parabolic taper is obtained by using a parabolic function sweep around the abscissa.
Parabolic taper can be synthesis with given input radius (a1), output radius (a2) and Length
(L) is given by the function
                                a(z) = a1 + (a2 - a1).(z/L)2                             .                         (3.49)
For our design dimensions, i.e., a1 = 13.99mm, a2 = 42.5mm, L = 350mm
One can obtain the design profile as shown in figure 3.4 &3.5


                                                                       Parabolic Taper Profile
                                              45


                                              40


                                              35
                      Radius of Taper in mm




                                              30


                                              25


                                              20


                                              15


                                              10
                                                   0      50     100       150       200         250   300   350
                                                                       Length of Taper in mm


                                                       Figure 3.4: Parabolic Taper Profile

Institute of Technology, Banaras Hindu University                                                                            25 
 
                                                              Chapter 3: Synthesis & Design of Cylindrical Waveguide Tapers 
 




                                                    Figure 3.5: 3D Parabolic Taper Design


3.2.3 Exponential Taper:
       Exponential taper can be synthesis using an exponential function with given input
radius (a1),output radius (a2) and Length (L) is given by the function
                                                    a(z) = a1 + (a2 - a1) .exp[8.(z-L)/L]                            (3.50)
For our design dimensions, i.e., a1 = 13.99mm, a2 = 42.5mm, L = 350mm.
One can obtain the design profile as shown in figure 3.6
                                                                     Expone ntial Tape r Profile
                                           45


                                           40


                                           35
                   Radius of Taper in mm




                                           30


                                           25


                                           20


                                           15


                                           10
                                                0       50     100        150        200           250   300   350
                                                                      Length of Ta per in m m


                                                     Figure 3.6: Exponential Taper Profile

Institute of Technology, Banaras Hindu University                                                                              26 
 
                                                                   Chapter 3: Synthesis & Design of Cylindrical Waveguide Tapers 
 


                                                                     Exponential 3D Taper Design




                   1

                 0.8

                 0.6

                 0.4

                 0.2

                   0
                  50
                                                                                                                                     50
                                                        0
                                                                                                               0

                                                                            -50     -50



                                                Figure 3.7: 3D Exponential Taper Design




3.2.4 Arbitrary Profile Taper:
       An arbitrary tapered waveguide profile [35], with given input radius (a1), output
radius (a2) and Length (L) is given by the function
                                                             a(z) = a2 + (a1 - a2) . (1-z/L)S                              ,              (3.51)


                                                                                Arbitrary Profile
                                               45


                                               40           S=20
                                                                   S=10
                                                                          S=5
                                               35
                       Radius of Taper in mm




                                                                                   S=2
                                               30
                                                                                          S=1

                                               25                                                   S=0.5
                                                                                                       S=0.3
                                               20
                                                                                                                         S=0.1


                                               15


                                               10
                                                    0       50        100       150       200           250        300         350
                                                                            Length of Taper in mm


    Figure 3.8: Various profiles obtained with variation of shape factor in arbitrary
                                        equation

Institute of Technology, Banaras Hindu University                                                                                                   27 
 
                                                                Chapter 3: Synthesis & Design of Cylindrical Waveguide Tapers 
 


Where , S is the shape factor.
       S is always greater than zero and for S=1 and S=2, one could get linear and parabolic
profile respectively
       The taper so obtained have drawbacks that, it is not uniform at the ends. A taper
should have local uniformity at both the ends. It is necessary for better feeding to the next
device and best feeded by former device.


3.2.5 Raised Cosine Taper:
       A raised cosine taper can be designed by using raised cosine function. The peculiarity
of this function is that it has smoothness at the ends. Raised cosine taper [36] can be synthesis
with given input radius (a1), output radius (a2) and Length (L) is given by the function


                                                  a(z) = 0.5 (a1 + a2) + 0.5 (a2 - a1). cos[(z/L -1).π]          (3.52)


                                                                  Raised cosine Taper Profile
                                         45


                                         40


                                         35
                 Radius of Taper in mm




                                         30


                                         25


                                         20


                                         15


                                         10
                                              0         50      100       150       200       250   300   350
                                                                      Length of Taper in mm


                                                     Figure 3.9: Raised Cosine Taper Profile


For our design dimensions, i.e., a1= 13.99mm, a2 = 42.5mm & L=350 mm
One can obtain the design profile as shown in figure 3.9.




Institute of Technology, Banaras Hindu University                                                                          28 
 
                                             Chapter 3: Synthesis & Design of Cylindrical Waveguide Tapers 
 
                                             Raised cosine 3D Taper Design




                      1

                  0.8

                  0.6

                  0.4

                  0.2

                   0
                  50
                                                                                        50
                                     0
                                                                             0

                                                      -50       -50



                           Figure 3.10: 3D Raised Cosine Taper Design


We assume a step size of Δa, so that the number of steps (M) in the step tapered waveguide
section is given by

                                                                                              (3.53)

The width of the ith waveguide section in the taper is given by
                                                      ai = a1 + i. Δa                         (3.54)
                               th
While the location of j             discontinuity is determined by solving the profile equations
mentioned above. To find the value of z corresponding to
                                         a (zi) = a1 + (2.j -1). Δa/2                         (3.55)
where j = 1, 2, …. N.



3.2.6 Modified Arbitrary Profile Taper:
       Keeping the basic raised cosine nature and arbitrary profile equation, Using(3.51) and
(3.52), a modified arbitrary profile taper[37]with given input radius (a1), output radius (a2)
and Length (L) is given by the function:

              .   1       2.                 . sin          .     1   2.                      (3.56)

Again, S is the shape factor.



Institute of Technology, Banaras Hindu University                                                       29 
 
                                                                Chapter 3: Synthesis & Design of Cylindrical Waveguide Tapers 
 
Now, keeping all parameters constant except the shape factor ‘S’, one can generate any
possible shape from S=0 to S=100 or more.
                                                                      Modified Arbitrary Profile
                                          45


                                          40            S=0.1

                                                                S=0.3
                                          35                                S=0.5
                  Radius of Taper in mm


                                                                                 S=0.7
                                          30                                             S=1
                                                                                                     S=2

                                          25                                                                S=5


                                          20


                                          15                                                               S=10
                                                                                                                  S=20

                                          10
                                               0          50    100        150       200           250     300       350
                                                                       Length of Taper in mm


    Figure 3.11: Various profiles obtained with variation of shape factor in modified
                                   arbitrary equation


3.3 Design of a Raised Cosine Taper for 42GHz 200kW, CW Gyrotron
         In the gyrotron development, taper is connecting the output section of a gyrotron
cavity to the uniform output waveguide section. The design of non linear taper operating in
TE03 cavity mode provide a good match between input and output sections of the taper with
very low spurious mode content. The cavity output radius is 13.99mm and the uniform
collector waveguide radius is 42.5mm in our problem. Keeping the dimensions constant,
different exponential, triangular, Chebyshev, parabolic etc. taper profile can be used. But we
use raised cosine profile as it yields very low mode conversion [1].Equation (3.56) can be
modified as for raised cosine nature

              .                           1        2.            . sin       .      1    2.                                (3.57)

Where,
K=0.3+0.7. rand(0,1)
Modified Arbitrary profile for certain range of shape factor provides raised cosine profile,
having performance better than the basic raised cosine profile discussed above. The range of
shape factor S is taken in a narrow range form S=0.3 to S=1 as shown in figure 3.12 as the


Institute of Technology, Banaras Hindu University                                                                                30 
 
                                                             Chapter 3: Synthesis & Design of Cylindrical Waveguide Tapers 
 
random raised cosine profile for uptaper for 200kW, 42GHz, CW Gyrotron. This range of S
provides large no. of tapers, all are almost raised cosine nature. Analysis of these designs
provide a pathway for exactly optimum taper suitable for our gyrotron design problem


                                                Random raised cosine profiles for gyrotron output uptaper
                                       45


                                       40
                                                         S=0.3

                                       35
               Radius of Taper in mm




                                       30
                                                                               S=1


                                       25


                                       20


                                       15


                                       10
                                            0      50       100       150       200       250     300       350
                                                                  Length of Taper in mm


    Figure 3.12 Random Raised Cosine profiles for a specific range of shape factor in
                            modified arbitrary equation




Institute of Technology, Banaras Hindu University                                                                       31 
 
                                                                                        Appendix ­3.1 
   
                                           APPENDIX – 3.1


Derivation of Synthesis Formula for Parabolic and Raised Cosine Taper


                                                                                    (A1)


                                                 ,                                 (A2)

where      and     are amplitudes of desired modes and ′ and ′ are the amplitudes of undesired
modes and ,         are the respective propagation constant. Reasonably far from cutoff of both the
modes, reflection caused by a taper are ordinarily negligible, and they have been neglected in equation
in eq(A1).

       Coupling coefficient     and     are calculated by using telegraphist’s equation and both phase
constant and coupling coefficient are the function of the distance along the taper. Power conversion
requires,
                 |    |          |   |                                           (A3)

 Now for tapers in circular or rectangular waveguide, the coupling coefficient               or    can be
written in terms of tapered dimension a as

                           ,                     .                               (A4)

  Here      term is deciding factor means which taper we have applied we can put the distribution factor
  of that taper here. Here a is the local radius of circular waveguide. is depending only on coupled
  modes and not on a and f.
         After solving the equation (A1) and (A2), we get local normal modes. Normal modes are
  waveforms in a uniform waveguide which propagate without change of field configuration or in
  terms of coupled line description, do not couple mutually. Analogously, local normal modes in a non-
  uniform waveguide are waveforms of a local cross section which propagate without change of field
  configuration in a waveguide which uniform with respect to that local cross section.

                               cos         sin                                   (A5)

                                     cos             sin                 .        (A6)

   Here     and     are the magnitudes of local normal modes of the taper and equation A1 and A2
  shows the coupling between these modes.
  Now we substitute the equation A5 and A6 in eq.A1 and A3, we get,

  Institute of Technology, Banaras Hindu University                                          Page 32 
   
                                                                                                        Appendix ­3.1 
 

                        Г                                                                  (A7)

                        Г                                                                  (A8)

    Where

and             tan     2                    2                                             (A9)
                                                 ∆

                                                             ∆
                 Г           ∆               4                   .                        (A1.1)

Here one thing is important that the coupling between                      and      will be zero in the taper if ξ is
constant or, from eqn. (A9) is constant.
                                 ∆


                                     ∆                                                       (A1.2)

 Here is the free space wavelength. For modes reasonably far from cutoff eq. A1.2 is a good
approximation with depending only on the modes and on a or frequency.

Equation A7 and A8 are coupled only through the terms proportional to     . If ξ is constant, they
reduce to uncoupled equations for and    . For gentle change in taper angle, with

                                                         1                                   (A1.3)
                                         Г

After solving equation A7 and A8, we get the value of                         and        using differential equation of
integrating factor method, in this solution we get,
                                                     ′       ′
                                             Г                                                (A1.4)

The initial conditions in the taper are    0                     1 and    0      0. The taper begins with zero cone
angle; hence, from (7) and (8),     0   1 and                      0   0,

At the end      of the taper, the unwanted mode amplitude is,

            |    |                                                                             (A1.5)

At the taper end the cone angle is again zero,                          0 .therefore |            | equals|        |. Eqn.
(A1.5) integrated by parts becomes,

            |    |       Г                                                                     (A1.6)

The mode conversion in a smooth but otherwise arbitrary taper can be calculated with eqn. (A1.5)
and (A1.6).




Institute of Technology, Banaras Hindu University                                                             Page 33 
 
                                                                                                        Appendix ­3.1 
    
Design of Parabolic Taper:
Now we can apply the distribution function for parabolic taper in place of                               , F is constant.
Integrate this function. We get,

                                            ,

                                    0           2

                                    0           2                                  (A2.1)

 Writing            0 and                       for a taper of overall length L, we have,

                                                2                                   (A2.2)

 Subject to the above approximation, the normal modes                      and      propagate without coupling in a
parabolic taper:

                                        0                                          (A2.3)

                                        0                                          (A2.5)

 If all of the power is initially in mode 1 at z=0, then     0     1,    0     0, and from eq. A5 and A6 ,
we get        0    cos ,         0      sin . Since | | and | | are both linear combinations of         and
     in eq. A5 and A6, the phasing of       and      at the end of the taper z=L will be correct for making
| | 1 and | | 0 as desired, if

                            ,                              0                                (A2.6)
 The explicit expression for the mode conversion to the undesired mode is,
           |        |                                                                        (A2.7)
 In terms of constants in eq. (A4) and eq. (A1.2),

                                √
           |        |                                                                        (A2.8)

                1           /                                                                (A2.9)

 These formulas provide a means of synthesizing a taper, since given     and , L can be chosen to make
|      | vanish. The minimum length required to produce a null in the mode conversion                 is
shorter than that required for other nonlinear tapers. The taper lengths  for which the mode conversion
vanishes can be found explicitly from eqn. (A2.8) and eq. (A2.9), we get,


                                                          1                                   (A 3.1)



   Institute of Technology, Banaras Hindu University                                                         Page 34 
    
                                                                                             Appendix ­3.1 
    
 According to equation (A4) the coupling coefficient K (a,f) for various waveguide tapers and modes
can be calculated as, the normalized propagation constant  for mode m is,


                                          1                                        (A3.2)

where      the cutoff frequency and k is is the free space propagation constant.



Smooth Circular Waveguide:
For                 coupling

                            ,                                                       (A3.3)

                                                                           ′
 Where      and                 are the Bessel function roots satisfying       0 and take            and
everywhere unity.

For                 coupling

                ,                                                                  (A3.4)

For                     coupling

                    ,                                                               (A3.5)

For                         coupling

                        ,                                .                          (A3.6)


Design of Raised Cosine Taper
Here before going to detailed analysis, we have to find the value of                               is used for
stipulation of the             modes using telegraphist’s equation, we get,

                                                                               (A4.1)
where
Now here the distribution function for raised cosine taper is 2                     is used in
                                                                                (A4.2),
We get,


   Institute of Technology, Banaras Hindu University                                              Page 35 
    
                                                                                              Appendix ­3.1 
    

           log                                                                      (A4.3)

   Applying boundary condition, we get,

                      1. . … … . .                for n is odd                       (A4.4)


                   . . …..                     for n is even                         (A4.5)

Substituting the eqn. (A4.4) and eqn. (A4.5) in eqn. (A4.3), we get,

For n=1, and a=

                      1                 log                                          (A4.6)

 For n=2, and a=

                                    2    log                                         (A4.7)

For n=3, and a=

                  1                                 log          .                  (A4.8)

The actual length of taper in terms of the parameters          is obtained from eqn. (A1.4), we get,

                      4                                                                 (A4.9)

Where      is the phase constant in free space. The last term in eqn. (A4.9) is negligibly small, in general.
Substitution of (A4.6), (A4.7) and (A4.8) in eqn. (A4.9), we get:
For n=1
                          1                                                    2


                                                                                    (A5.1)
where                         log

For n=2,

                                                                                         2    2    2



                                                                                     (A5.2)

   Institute of Technology, Banaras Hindu University                                               Page 36 
    
                                                                                        Appendix ­3.1 
   

where    2                     log           .

        For moderate values of , the axial distance z can be calculated by term by term integration. The
total length of the taper is approximately,

                                                                                    (A5.3)



For example, we will design a taper whose conversion distribution follows the raised cosine function

                 2                                   for n =1                      (A5.4)

Integration of eqn. (A1.5) for the mode conversion in the waveguide taper is extended over parameter ,

                           |             |                                         (A5.5)

Put the eqn. (A5.4) in eqn. (A5.5), and integrate this, we get,




                                                                2

                                                                                   (A5.6)

                                         2                                          (A5.7)

  After solving the eqn. (A5.7), as using integration by part method and put the limit, after that
substitute the solution in eqn.(A5.6), we get,
                       |         |
                                                                                    (A5.8)


We assume the                        modes to be far enough from cutoff so that,

                                                 1                                 (A5.9)

For both modes throughout the taper. Furthermore, we assume the taper to be gentle enough so that

                                     1                                                 (A6.1)




  Institute of Technology, Banaras Hindu University                                          Page 37 
   
                                                                                           Appendix ­3.1 
      
Generalized Telegraphist’s Equations of the circular Electric Waveguide Taper
  Taking symmetric structure,

                        =0,                                                     (A7.1)

                                                                               (A7.2)

  Put eqn. (A7.1) in eqn. (A7.2), we get,
                                                                                (A7.3)
  Similarly,
                                                                                (A7.4)
  and
                                                                                (A7.5)
  after substituting the eqn. (A7.1), we get,

                                                                                (A7.6)

   Where      ,    , and  , are the only no vanishing components of the field,     is the dielectric
  permittivity, the magnetic permeability and the angular pulsation. The exponential dependence of
  time is understood.
  The boundary conditions of the waveguide taper are, at r equal to a:

                               0                                                (A7.7)

                                            .                                    (A7.8)

  The field at any cross section of the paper is represented as a superposition of the fields of the normal
  modes in a cylindrical guide of the same cross section:

                           ∑                                                     (A7.9)
                                   √


                               ∑                                                 (A8.1)
                                       √


                           ∑                                                      (A8.2)
                                   √

  Where     and     are Bessel functions of the first kind and   is the mth zero of .

            The          and      have the dimensions of voltages and currents. The factors of  and    are
  normalized so that                    is the complex power flow in each normal mode. It has to be kept in
  mind that a is a function of z.

     Institute of Technology, Banaras Hindu University                                          Page 38 
      
                                                                                          Appendix ­3.1 
    
            The boundary condition (A7.7) is satisfied by the individual terms of the series for . Hence,
this series converges uniformly. Not so the series for : (31) is a representation for      only in the open
interval 0         , since the individual terms vanish at r=a but, according to (A7.8),    does not.

       The relationship between    and     is found by substituting in (A7.6) the series (A7.9) for
and the series (A8.2) for , and comparing coefficients

                                                                                      (A8.3)

        To convert Maxwell’s equation into generalized telegraphist’s equation, we introduce (A7.9) and
(A8.1) into (A7.2) and (A7.3), multiply both sides of both equations by           and integrate over the
cross-section. Since the series for    does not converge uniformly, we write for the left hand side of
(A7.3),

                                                                                        (A8.4)

 and invert integration and differentiation in the second term of this expression.
          The generalized telegraphist’s equations have the following form:

                                                   ∑                                   (A8.5)

                                                    ∑                 .                 (A8.6)

 The summations are extended over all m except m=n. the quantity        is the phase constant of the phase
constant of the nth mode in a cylindrical guide of the particular cross section;    is a function of a and
therefore of z.
           The generalized telegraphist’s equation represents an infinite set of coupled no uniform
transmission lines. For our purpose, it is convenient to write the transmission lines equations not in
terms of currents and voltages, but in terms of the amplitudes of forward and backward travelling waves.
Thus, let A and B be the amplitudes of the forward and backward waves of a typical mode at a certain
cross section. The mode current and voltages are related to the amplitudes by,

                       √                   ,                                         (A8.7)

                                       ,                                             (A8.8)
                       √

where K is the wave impedance,

                                   .                                                 (A8.9)

 If the currents and voltages in the generalized telegraphist’s equations (A8.5) and (A8.6) are represent
in terms of the travelling-wave amplitudes, after some obvious additions and subtractions, the following
equations for coupled travelling waves are obtained:


   Institute of Technology, Banaras Hindu University                                             Page 39 
    
                                                                                       Appendix ­3.1 
   
                                ′
                                       ∑                                      (A9.1)

                                ′
                                      ∑                             .         (A9.2)

The k’s are coupling coefficients defined by:

                                                                             (A9.3)

            ′
                                                                             (A9.4)

   For a cylindrical guide,   is equal to zero and (9.2) and (9.3) reduce to uncoupled transmission- line
  equations.




  Institute of Technology, Banaras Hindu University                                         Page 40 
   
                                          Chapter 4: Mode Conversion in Cylindrical Waveguide Tapers 
 




                                                                             Chapter 4




Mode Conversion in Cylindrical Waveguide
Tapers



4.1 Physical Significance:


An overmoded cylindrical waveguide taper transmit the entire power incident on it with very
little reflection for any profile (even the worst one taken) and almost all the power incident to it
is transmitted to the output section. But, in most cases, the power at output is not in the same
incident mode rather mode conversion occur to spurious modes. One can always be built the
profile to have as low as mode conversion as is wanted in a certain frequency band merely by
making it long enough. However, an optimally designed taper has the smallest possible length
for a given difference in diameters as its two ends and for a specific unwanted mode level in a
given frequency.
       The transmission line taper is different than waveguide taper for matching impedances. It
is nothing but a tapered waveguide in which only one mode is propagating. Power can only be
converted into reflected waves, and it is this reflected power which is kept small in a properly
designed transmission line taper. If more than one mode is propagating, power will be scattered

Institute of Technology, Banaras Hindu University                                                 41 
 
                                             Chapter 4: Mode Conversion in Cylindrical Waveguide Tapers 
 
not only into the reflected wave but also into the other propagating modes. The power scattered
into backward travelling waves is quite small and one can neglect the same in a multimode
waveguide taper. Therefore, the mode conversion in the waveguide transition corresponds to the
reflection in the transmission line taper.
In the figure 4.1, one can see that taper transfer and share the RF energy to other modes. At the
beginning of taper, it is maximum power transfer. The goal of optimization is to terminate the
taper at that particular length at which the mode conversion is minimum or largest energy
remains in the desired mode.




                               Figure 4.1: Mode conversion in physical taper.

       Waveguide tapers in which axial position is proportional to the tapered direction squared
can be synthesized from virtually exact solutions of the coupled mode equations. Such tapers
have much less mode conversion over narrow bands than equal linear conversion over narrow
bands than equal length linear tapers. The minimum length require to produce a null in mode
conversion is also shorter than that for other non linear tapers.


4.2 Low Mode Conversion Tapers:
       Mode conversion is minimized by keeping attention on the fact that the profile variation
should be very slow with axis. A perfect taper is one free of mode conversion. Conical tapers
when connected to the cylindrical waveguide matched at both ends with uniform waveguides of
different radii, a cylindrical guide excites a series of spherical waves in the conical guide. An
incident TE01 wave will excite all the TE0m waves, results a high spurious mode level. There are
three possible ways to sort out this undesired mode conversion
       1) S. P. Morgan [38] has suggested and worked out the design of dielectric inserts placed
near the junction, which, acting as quasi optical lenses, transform the cylindrical waves into the


Institute of Technology, Banaras Hindu University                                                    42 
 
                                          Chapter 4: Mode Conversion in Cylindrical Waveguide Tapers 
 
spherical waves. However, because of the dispersive character of the lenses and of the
waveguide, good broadband performance is difficult to achieve.
       2) When there were transition from cylindrical waves to spherical waves (approximately),
one has to taper the profile angle from zero at the cylindrical guide to the finite value of the
conical waveguide [8]. This is an approximate solution and has good broadband characteristics.
If this is gradually enough, nearly all the power incident in the cylindrical wave will be
transformed into the spherical wave, with very low spurious modes.
       3) The length of conical taper is kept very long generally not excites high level of modes
and hence, introduce very little mode conversion to higher electric modes.
       The smooth taper with very little mode conversion have zero first derivative at both ends.
The fields excited in any cross section by an incident TE01 wave can be expressed as the sum of
the TEo,m waves of a cylindrical guide having the same radius as that of the cross section. Taper
appears to be an infinite set of mutually coupled transmission lines, each line representing one of
the cylindrical TE0,m waves. Also, relative power coupled with the forward waves to backward
waves is quiet small. So, one can consider only forward travelling waves.
       For parabolic tapers, output end connected with the cylindrical waveguide involves slow
variation of profile as seen in figure 3.3, but at input it shows abrupt discontinuity leads to high
mode conversion. The opposite situation is one in exponential taper where high discontinuities
occur at output junction as shown in figure 3.5.
       Raised cosine profile in figure 3.8 involves better continuity & slow smooth variation
throughout the length as well as at the end junctions.

4.3 Analytical Calculation of Mode Conversion:
       Let the uniform waveguide G1 extend from z = -∞ to z = 0, and the uniform waveguide
G2 from z = L to z = ∞. Let us connect them by a non-uniform waveguide section having surface
equation F(x, y, z) = 0, which is differentiable as a function of z. A plane perpendicular to the z
axis cuts this surface in a single closed curve, the cross section of the non-uniform waveguide.
The interior of any cross section is denoted by S(z), and its boundary by C(z).The cross sections
at z = 0 and z = L corresponds to those of the uniform waveguides G1 and G2 respectively. Our
purpose is to determine the spurious modes in the G2, if waveguide G1 is fed by a pure mode, i.e.,
TE03 mode in our case.


Institute of Technology, Banaras Hindu University                                                 43 
 
                                            Chapter 4: Mode Conversion in Cylindrical Waveguide Tapers 
 
         The non-uniform waveguide may be regarded as a system of coupled transmission lines
where the coupling coefficient is function of z. The field intensities in the non-uniform
waveguide may be represented by equivalent voltage and currents. The differential equation
system for these voltages and currents is known as the generalized Telegraphist’s equations
                                                     ∑                                            (4.1)



                                                    ∑              ,                              (4.2)

where i & p denote arbitrary modes (no need to discriminate the E & H modes).Vi & Ii are the
equivalent voltages and currents for the mode i.            is the propagation coefficient, and     is the
wave impedance.          and     represent the voltage and current transfer coefficients respectively.
There is no mutual impedance between the voltages and currents of different modes, and there is
a simple connection between the current and voltage transfer coefficients.
         The transfer coefficient     may be expressed as

                                                                   ,                              (4.3)

where,       &       are the mode vector functions[9] of the corresponding modes satisfying the
normalized conditions

                                                        1          .                              (4.4)

As we are dealing with TE modes or H modes, so for TE modes
                                                    Ψ              ,                              (4.5)
where,
           = the gradient operator transverse to the z axis.
           = the unit vector in the direction if z axis.
         Ψ function satisfy the differential equations.

                                        Ψ       k     β       .Ψ       0                          (4.6)

where, k =       .

         The representation in terms of forward wave and backward travelling waves is more
suitable for description of a wave phenomenon. Assuming that wave impedance (K) neither be




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                                              Chapter 4: Mode Conversion in Cylindrical Waveguide Tapers 
 
zero nor be infinite at any point of the waveguide, we introduce as new variables the amplitude
of the forward and backward travelling waves,              &            , by the relations

                                                                                                         (4.7)

                                                                                                         (4.8)
Substituting this into above, we get,

                                                                ∑                                        (4.9)

                                                                ∑                                    ,   (4.10)

where     is forward coupling coefficient and              is the backward coupling coefficient. Both may
expressed in terms of the transfer functions as follows:

                                                                                 .                       (4.11)

If the waveguide is fed by mode m, the boundary conditions for the differential equation system
are as follows:
                                  0           ,                0                                         (4.12)
                                  0      0,                0             ( i m)          .               (4.13)
In our problem, we neglect the backward wave coupling and consider that only forward
travelling waves are propagating in the waveguide system.
        Substituting the mode vector functions into (4.3), using green’s theorem and stroke’s
theorems, the relation

                                          tan     on the boundary C(z) .                                 (4.14)

The transfer coefficient for TE modes can be obtained as:

                                       tan .           .                                                 (4.15)

                                      tan . Ψ .             .            ,                   .           (4.16)

The coupling coefficient for TE modes[39] can be obtained as
                      0                                                                                  (4.17)

                              .   .       .                         .        .       .
                                                                                                 .       (4.18)


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                                             Chapter 4: Mode Conversion in Cylindrical Waveguide Tapers 
 

where,                      k     β      ,           ,                                         (4.19)




                         Angle between the outward normal to C(z) and the normal to the non
                    uniform waveguide
                    ds=the element of C(z) curve
From the above expressions, one can conclude that coupling coefficient change very little with
frequency; it is too near to the cut off frequency. So, one can consider coupling coefficient
independent of frequency. The coupling of any one TE mode to other TE mode is larger the
nearer their corresponding cut off numbers are. This is only a rough rule as the effect of line
integrals in expressions is not taken into account.



4.4 Graphical Analysis:
         Coupling coefficient Analysis for conical tapers having input radius 13.99 mm and output
radius 42.5mm with length 350 mm:




           Figure 4.2: Mode coupling along the length shown in linear(conical) taper

From the figure 4.2, it is clear that length 350 mm is not sufficient to reduce the coupling upto
zero. One has to increase the length of taper to avoid coupling in conical taper.
The above figure shows the coupling coefficient for TE03-TE02 modes for different taper profile
for our desired input and output radius i.e.13.99 mm & 42.5mm respectively and desired length

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                                                                        Chapter 4: Mode Conversion in Cylindrical Waveguide Tapers 
 
350mm. Raised cosine profile shows no coupling at the ends of taper. So, Raised cosine profile
is our point of interest rather than other profile.The figure 4.3 shows the coupling coefficient for
TE03-TE02 modes for different taper profile for our desired input and output radius i.e.13.99 mm
& 42.5mm respectively and desired length 350mm.
                                                            -3
                                                     x 10               Coupling of TE03-TE02 mode in different tapers
                                                2

                                                                                                                            conica l
                                                                                                                           Para bolic
                                               1.8
                                                                                                                           Exponential
                                                                                                                           Raise d cosine

                                               1.6



                                               1.4



                                               1.2
                        Coupling coefficient




                                                1



                                               0.8



                                               0.6



                                               0.4



                                               0.2



                                                0
                                                     0           50   100            150               200           250         300        350
                                                                                   Le ngth of Tape r in mm




                      Figure 4.3: Mode coupling in various basic profiles

Raised cosine profile shows no coupling at the ends of taper. So, Raised cosine profile is our
point of interest rather than other profile. Kiyo Tomiyasu proposed a waveguide taper of
minimum length in 1971[4] and shows the mode coupling coefficient for TE01-TE02 obtained is
similar to that obtained by Doane[33] for the same raised cosine profile. As shown in figure 4.4,
that the mode conversion is within control (under 10%) when the taper length is around 340-
360mm or more.




                  Figure 4.4:Mode conversion dependent on length of taper

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                                                        Chapter 5: Analysis of Cylindrical Waveguide Tapers 
 




                                                                                 Chapter 5



Analysis of Cylindrical Waveguide Tapers



5.1. Fundamentals of Mode Analysis
Modal Analysis is done in the horn or taper waveguide to predict the electromagnetic fields. This can be
summarized in series of steps [50]:
       1. The form of the field components and the relationship between the field components is derived
          from Maxwell’s equations.

       2. The form of the propagating waves is found from solving the Helmholtz wave equation for the
          co-ordinate system appropriate to the geometry of the structure.

       3. The boundary conditions are used to find expressions for the propagation coefficients and
          amplitude coefficients which apply to the specific cross section of the horn. For hollow
          rectangular and circular cross sections, the propagation coefficients are closed form
          expressions. For other cases, the propagation coefficient must usually be derived by solving
          an equation by iterative techniques.

       4. The field components are obtained .It is convenient to divide these into the component along
          the axis of the horn (the z direction) and the components transverse to the axis of the horn.


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                                                       Chapter 5: Analysis of Cylindrical Waveguide Tapers 
 
          The transverse components make up the aperture fields and are used to predict the radiation
          characteristics.


       The horn or tapered waveguide will support two orthogonal mode sets. For waveguides, which
have perfectly conducting smooth walls and are filled with a homogenous material (i.e., hollow or
completely filled with dielectric), the two mode sets consist of ‘transverse magnetic’(TM) modes with
Hz = 0 and ‘transverse electric’ (TE) modes with Ez = 0. In general, there are an infinite number of
possible propagating modes. The number of finite propagating modes will depend on the relative size of
the cross section of the waveguide. These are designated TMnm and TEnm modes where n and m are
integrals and refer to solution to the wave equation in the two coordinate directions. When the cross
section of the waveguide is inhomogeneous (for example, two dielectrics or corrugated), the TM and TE
mode are coupled together and in principle all field components exists. The modes are now called hybrid
modes. The modal characteristics of a horn can be found either by using ‘cylindrical waves’ or by using
‘spherical waves’ [8].


5.2 Mode Matching Technique
       The modal matching technique is a powerful computer method of analyzing horn antennas or
nonlinear waveguides in which the actual profile is replaced by a series of uniform waveguide sections.
The waveguides can have any cross section and can be propagating wither cylindrical or spherical
modes.Mode matching is possible not only between two similar circular cross section rather two
rectangular cross section matching[51,52] or circular to rectangular or elliptical sections is also
possible[53-56].


5.2.1 Principles of Mode Matching Technique
       The mode matching technique involves matching the total modal field at each junction between
uniform sections so that conservation of power is maintained. From this process, the amplitudes of the
separate modes at the output of a junction can be deduced in terms of the amplitudes of the mode
spectrum at the input to the junction. The number of propagating and evanescent modes which are
needed to represent the total power, must be found by trial and error. The power of the modal matching
technique stems from the fact that the amplitudes of the modes can be expressed as the components of a
scattering matrix. Each junction along the length has its own scattering matrix. The matrices for all


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                                                        Chapter 5: Analysis of Cylindrical Waveguide Tapers 
 
junctions can be cascaded and an overall scattering matrix derived from horn. The process of computing
the overall scattering matrix can be decoupled from the process of obtaining the elements of a particular
scattering matrix. The later will depend on the geometry of the waveguide, but the formal is quite
general. Thus, the basic technique can readily be extended to different geometries. The overall scattering
matrix for the taper contains the input reflection coefficient and the output transmission coefficient
which the aperture fields are computed.
        The concept of mode matching at the junction was first attempted in the late 1960s and early
1970s by Wexler [16], Masterman and Clarricoats [17] and English [57]. However, the process of
computing the coefficient is lengthy and because of limited computer power was available, it was not
possible to do more than simple computations. The computational emphasis was on reducing the number
of modes to the minimum so that a numerical solution could be obtained. It was the arrival of powerful
computers which enabled the concept to be applied to the analysis of complete tapers or horns. It was
firstly developed by Wexler [16] & James [17,18]. The mode matching technique involves a large
amount of computation because there will be large number of modes to be matched across each junction
and a large number of junctions along the waveguide/horn.


             [A]                                                       [C]



                                    S Matrix for
                                   waveguide/horn


               [B]                                                       [D]

    Figure 5.1: Block diagram showing scattering matrix with forward and reflection coefficients.

        The mode matching technique can be considered as a method of obtaining the overall
transmission and reflection properties of a waveguide/horn. The waveguide is represented as a box,
where [A] and [B] are column matrices containing the forward and reflection coefficients of all the
modes looking into the waveguide from the source side. Similarly [C] and [D] represent column
matrices containing the forward and reflections coefficient of all the modes looking into the aperture of


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                                                           Chapter 5: Analysis of Cylindrical Waveguide Tapers 
 
the waveguide/horn from the outside. The characteristics of the horn are then given by a scattering
matrix[S],

                                                                      ,                       (5.1)

where, the scattering matrix [S] is

                                                                                              (5.2)

       The elements of [S] are the square matrixes describing the power coupling between all the modes
at the input with all the modes at the output of the horn. The reflection coefficient for the horn is
                                                             .                                (5.3)
For many waveguide/horns, there will only be one mode at the input (for instance, TE11 mode in circular
waveguide), in this case [A] will be
                                                     1
                                                     0           .                            (5.4)

The length of column matrix depends on the number of modes at the output of waveguide. The
transmission coefficient of the waveguide, from which the aperture fields are determined, is
                                                                      .                       (5.5)


5.2.2 Scattering matrix of a uniform section
       The scattering matrix elements for a uniform section of guide are
                                           [S11] = [S22] = [0] ,                              (5.6)
                                           [S12]=[S21]=[V]       ,                            (5.7)
where, [V] is an N×N diagonal matrix with the elements Vnn = exp(-γn ), where is the length of the
section and γn is the propagation constant for the nth mode in the waveguide (1 < n < N). In principle,
the waveguide section could contain lost material so that γn is complex. However, this would lead to
extensive computation and is generally unnecessary since the influence of lossy materials can usually be
adequately accounted by perturbation approach. The propagation coefficient is normally either purely
imaginary (γn = jβn) for travelling modes or is purely real (γn = αn) for evanescent modes. In both cases,
the elements of matrices are real. A substantial number of evanescent modes must be included in the
analysis. This is because the uniform sections will be relatively short in length so that the amplitude of a
decaying wave may still be significant by the time the wave reaches the next junction.


Institute of Technology, Banaras Hindu University                                                          51 
 
                                                             Chapter 5: Analysis of Cylindrical Waveguide Tapers 
 
5.2.3 Scattering matrix of a junction
       The derivation of the scattering matrix at the junction is more complicated, since it involve
matching of total power in all the modes on both sides of the junctions[58]. The number of modes on
the left hand side of the junction and the number of modes on the right hand side of the junction can in
general be arbitrary. However, it simplifies the analysis and the computational procedure if the number
of modes N is the same on both sides of the junction. This will be assumes in the following analysis




                                Surface SL 
                                                      Surface SR




                 Figure 5.3: Junction between two sections of cylindrical waveguide.


       Each uniform waveguide section contains travelling waves in which the transverse electric fields
can be represented as a spectrum of N nodes. The transverse electric and magnetic modal functions[8,9]
on the left hand side of the junction are represented by the subscript L (i.e., enL and hnL), and those on the
right hand side of the junction by the subscript R.
       The electric and magnetic fields on the left hand side are

                       ∑        exp              exp                      ,                     (5.8)

                       ∑        exp              exp                      ,                     (5.9)
where An and B    n   are the forward and reflected coefficients of mode n on the left hand side of the
junction.
On the right hand side of the junction, the fields have the form
                           ∑      exp                exp              ,                         (5.10)
                           ∑      exp                 exp             ,                         (5.11)


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                                                             Chapter 5: Analysis of Cylindrical Waveguide Tapers 
 
where Cn and Dn are the forward and reflected amplitude coefficients of mode n on the right hand side of
the junction, looking into the junction.
The total transverse fields must match across the junction. If the junction is at z = 0, then
                 ∑                         ∑                         ,                           (5.12)
                 ∑                         ∑                             .                       (5.13)
If the cross section area of the waveguide on the left hand side of the junction is sL and that on the right
hand side of the junction is sR, the boundary conditions give that the transverse electric fields over the
area(sR – sL) will be zero. The fields over the area sL will be continuous. The continuity of fields and the
orthogonality relationship between modes leads to a pair of simultaneous matrix equations.
                       [P] [ [A] + [B] ] =[Q] [ [C] + [D] ]          ,                           (5.14)
                       [P]T [ [D] – [C] ] =[R] [ [A] – [B] ]         ,                           (5.15)
where [A] and [B] are N- element column matrices in the section on the left hand side of the junction
containing the unknown modal coefficients A1 to AN and B1 to BN. Similarly, [C] and [D] are N-
element column matrices for the right hand side of the junction containing the unknown modal
coefficients C1 to CN and D1 to DN.
       The matrix [P] is an N×N square matrix whose elements are integrals representing the mutual
coupled pair between mode i on the left hand side and mode j in the right hand side.
                                    
                                                 .       .                                       (5.16)

The matrix [P]T is the transpose of [P], i.e., the rows and columns are interchanged. The matrix [Q] is
an N×N diagonal matrix describing the self coupled power between modes on the right hand side of the
junction. The elements are integrals over the area sR
                                    
                                                     .           .                              (5.17)

Similarly, the matrix [R] is an N×N diagonal matrix describing the self coupled power between modes
on the left hand side of the junction. The elements are integral over the area sL,
                                        
                                                     .       .                                   (5.18)

       The power coupling integrals in above three mode matching equations contain information about
the type of waveguide on either side of the junction. They must be evaluated for the appropriate
homogeneous or non homogeneous cross section. This can be done either analytically or numerically.
Numerical evaluation reduces the amount of mathematical effort but increases the computational time
since the integrals need to be evaluated for all modal combinations at each junction. Analytical


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                                                           Chapter 5: Analysis of Cylindrical Waveguide Tapers 
 
evaluation is only possible in some cases and may involve considerable mathematics, but where it is
possible the saving in computer time can be considerable.
        Equation (5.16), (5.17), and (5.18) needs to be rearranged into the scattering matrix form. This
gives the elements of [S] from equation ___
                               [S11]= [[R] + [P]T[Q]-1[P] ]-1 [ [R] – [P]T[Q]-1[P] ]          (5.19)
                               [S12] = 2[[R] + [P]T[Q]-1[P] ]-1[P]T                           (5.20)
                               [S21]=2[ [Q]+[P] [R]-1[P]T]-1 [P]T                             (5.21)
                               [S22] = -[ [Q] + [P] [R]-1[P]T]-1[ [Q] - [P] [R]-1[P]T] .      (5.22)
The analysis above assumes that the area sR is greater than the area sL. If this is not the case, the
elements of [S] in equation(5.2) become

                                                               .                              (5.23)

This completes the description of the general mode matching technique for analyzing any combination
of junctions and waveguide sections.


5.2.4 Cascading of Scattering Matrix
        The waveguide is divided into a number of sections and junctions. For instance, the Taper has 35
junctions and 34 uniform sections. Each junction or section can be represented by its own scattering
matrix with a box as shown in figure 5.2 and equation (5.2) similar to equation. [S] now represents the
scattering properties of the individual junction or section. The scattering matrix of the complete horn is
made up of a series of scattering matrices as shown in figure



      [Sa]             [Sb]
         Figure 5.2: Single Scattering Matrix divided into a number of cascaded S Matrix


Scattering matrices are particularly useful because it is straightforward to cascade two scattering matrix.
If these two scattering matrices have elements

                                                                                              (5.24)

and


Institute of Technology, Banaras Hindu University                                                          54 
 
                                                          Chapter 5: Analysis of Cylindrical Waveguide Tapers 
 

                                                      .                                      (5.25)

Then the cascaded scattering matrix is

                                                                                             (5.26)

where,
                                                                                             (5.27)

                                                                                            (5.28)

                                                                                             (5.29)

                                                                                             (5.30)

where [I] is a unit matrix and [ ]-1 represents the inverse matrix. The cascading process has the advantage
that the exact number of junctions and sections does not have to be known at the start of the analysis as
the process proceeds from the input to the output in a recursive fashion [59, 60].
         The next stage is to determine the scattering matrices of the separate sections. These can be of
two forms, either a uniform section or a junction between two uniform sections.


5.2.5. Calculation of Coupling Power Integrals
         The complex power technique is used for calculation of coupling power integrals [61,62]. The
power coupling integrals involve the evaluation of integrals of the form

                          
                                  .                               .   .      ,               (5.31)

where the fields are evaluated for each mode and for either the left hand side or the right hand side of the
junction. The field components are obtained from equation (2.7) and (2.9).            The integrals can be
evaluated numerically or analytically. In this case dorm solutions are standard and should therefore be
used. In more complicated cases, for instances of a coaxial waveguide, numerical evaluation may be
preferable; the penalty will be a considerable increase in the computational time.
         Inserting the field components into the self coupling power integrals Qjj and Rii and evaluating
the integrals of products of Bessel functions gives for TE modes.
                                                                                              , (5.32)



Institute of Technology, Banaras Hindu University                                                         55 
 
                                                               Chapter 5: Analysis of Cylindrical Waveguide Tapers 
 
where      is the impedance of free space,         = /   = /     and C is a constant.
For TM modes
                                 QTM = QTE/                                                       (5.33)

The mutual coupling power integrals Pij are as follows [50]:
      a) Left hand side TE mode-Right hand side TE mode
                                           2                                                      (5.34)
      b) Left hand side TE mode-right hand side TM mode
                                               2                                                           (5.35)
      c) Left hand side TM mode-right hand TE mode
                                                                    0                                      (5.36)
      d) Left hand side TM mode-right hand side TM mode

                                                   2                                                       (5.37)

where,



                                                                                                           (5.38)
and
                                       ,                                                                   (5.39)
here C1 ,C2 are the constants.


5.3. Numerical Computation
                The theory outlined above can be implemented numerically using the following strategy:
The main part of computation is the four section, which have to be calculated for each section for each
mode. For hollow circular waveguide m the wave number,                  is the roots of Bessel functions as given
in equation (2.8) and (2.10). Most conical feeds use input circular waveguides propagating the TE11
mode. If the structure contains no asymmetry, the modes will all have an azimuthally dependency of
unity. Junctions will excite only TE 1m and TM1n modes.
         The time taken to compute the input and output coefficients for a horn depends on the number of
modes and the number of sections. The time is proportional to the number of sections and approximately
proportional to the square of the number of modes. It is important to be able to choose the number of


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                                                           Chapter 5: Analysis of Cylindrical Waveguide Tapers 
 
sections and the number of modes correctly so that the desired accuracy is achieved without unnecessary
computation.

    For              For      Wave number       computed for each waveguide section
    Each             Each
                              Left hand side of junction: self coupling power coefficient (Rii)
    Section          Mode
                              Right hand side of junction: self coupling power coefficient(Qjj)

                              Cross coupling coefficient (Pij)
                     Scattering coefficient for junction
                     Scattering matrix of junction cascaded with matrix for section length
                     Scattering matrix cascaded with scattering matrix computed at previous junction
                     Aperture modal coefficient [D] and input reflection coefficient [B]


          The total number of modes depends on the relative diameters at each junction and must be
chosen by repeat tests[7]. The larger the change in diameter, the more modes will be excited locally and
the higher the level of mode conversion. The smooth wall and dielectric loaded horns require relatively
few modes, but the large number of sections. The corrugated profile can be made up of straight sections
between the corrugations so the number of sections is twice the number of corrugations, but the number
of modes is high. This can be seen by computing the cross polar patterns for different numbers of modes
and comparing the peak cross polar level which is a sensitive measure of accuracy since the cross
polarization is difference between two orthogonal copular patterns. In practical computations, it is easy
to increase the number of sections because of the low number of modes required.


5.4 Scattering Matrix Formulation of Circular Waveguide Taper
          Mode matching analysis applied on waveguide circuits, microstrip circuits and horn antennas.
The whole circular waveguide non linear taper will be divided into step discontinuity. Larger the
number of sections taken, higher the accuracy in the analysis results obtained. The usual choice of the
testing eigenmodes as being those of the smaller guide for enforcing continuity and as those of the
smaller guide for enforcing the magnetic continuity at each junction is justified rigorously. For smooth
changes in waveguide dimensions the change is approximated by a large number of steps. At this point,
the type of waveguide is arbitrary but the common area between the two guides must be identical to the

Institute of Technology, Banaras Hindu University                                                          57 
 
                                                                   Chapter 5: Analysis of Cylindrical Waveguide Tapers 
 
cross section of the smaller waveguide. Also, all guides are required to posses the same centre line. This
simplifies the analysis since only modes with zero azimuthal variation (TE0,n) need to be considered.
Keeping in mind, the scattering matrix discussion in section .As fields defined left of the junctions and
right of the junctions as the sum of normal modes of respective waveguides. The set of equation (5.10),
(5.11), (5.12) and (5.13) can be rewrite by considering M modes in left side of junction and N modes in
right side of junction [63].

For left of junction,
                              ∑         exp                       exp                                         (5.40)

                              ∑          exp                      exp                                         (5.41)

For right side,
                              ∑        exp                   exp                                              (5.42)


                              ∑        exp                   exp                        .                     (5.43)

Here, M is chosen large for convergence in left side of junction. In similar way, N is the number of
modes chosen for right side of junction.               and         are the normalized vector functions for the mth
mode. Similarly,        and       are the normalized vector functions for the nth mode.

        In the circular waveguide, TE eigenmodes for the transverse electric fields can be calculated on
either side of junction using the relation


                                                 sin          ̂                   cos                         (5.44)
       √


For our mode of interest TE0,3
                                      Substitute m = 0 and n = 3 for calculation of desired eigenmode.
For m = 0,     = 1 and for m 0,        =2
        For our mode,             1
                                         Also,           10.174 for TE0,3 mode.
The above equation for TE0,3 can be modified as



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                                                                                    Chapter 5: Analysis of Cylindrical Waveguide Tapers 
 

                                                       
                                   √


                                                                                      .                 .   .
                                                                                                                               (5.45)
                                   √       .              .


It is very difficult to calculate manually, one can use MATHEMATICA [91] for the further calculations.
Also,
                                                                                                                               (5.46)
Therefore,


For both side of junctions, the normalization of                              and             ,using (5.25)
                                    
                                                                  .                                                            (5.47)
and from the orthogonality of waveguide mode, when
                                        
                                                                      .              0 .                                       (5.48)
Matching the electric field and magnetic fields over the common apertures between the two regions are
matched.
                           EL=ER inside SL, EL                                 =ER                 inside SL                   (5.49)
Or , one can modify the equation (5.48) as
                                                                       
                                  EL           . ds                       R
                                                                               ER                . ds                          (5.50)
Since, E = 0 on the conductor making up the surface SR-SL, the integral limit on the right hand side may
be modified.
                                                                       
                                  EL           . ds                       L
                                                                              ER                 . ds                          (5.51)
Using the properties
                             ∑                                                                                                 (5.52)
where ,
                                                               

                                                                          eL                   . ds
                                                              L

and
                                                               
                                                                  R
                                                                          eR                   . ds .
The other boundary condition required is
                                                                                                                     .

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                                                                      Chapter 5: Analysis of Cylindrical Waveguide Tapers 
 
Following a similar line of reasoning
                                                                   

                                        eL       . ds                  eL           . ds
                                    L                             L

giving
                                                              ∑                                                  (5.53)
                                                      
                                                         eL                  . ds

Equation (5.52) and (5.53) may be recast into a more compact matrix form, giving
                                                                                                                 (5.54)
                                                                                                                 (5.55)
This equation is converted into a scattering matrix format relating the normalized output vector B and D
to the normalized input vector A and B.
         The submatrices [S11],[S12],[S21] and [S22] are derived from the [P],                 ,[R],and [Q] matrices by
simple matrix math and equations (5.54) and (5.55)

                             √                                                      √                            (5.56)

                      S      2√                                                                                  (5.57)
                                                               
                      S      2                                    √                                              (5.58)

                                                                                               .                 (5.59)
This is the calculation at single junction. As the cascading describe in section 3.2.1, one can calculate
the overall S matrix. But, we require only the transmitted power for a given incident at input to the
output in desired mode .So, calculation for transmission coefficient S21 is sufficient for our analysis. The
cascading equation (5.29) only for S21 foe two junctions a and b

                                                                                           .                     (5.60)

         Suppose, a general raised cosine taper discussed in Chapter 3 of length 350 mm undertaken for
analysis using mode matching technique. Let us divide it in 35 sections, each length 10mm, uniform
waveguide as a step approximation [64] as shown in figure 5.4 & 5.5.
         One can calculate the coupling coefficient P, Q, and R of all junction for TE0,3 incident mode and
TE0,3 output mode using mode matching technique[65-72] .Larger the number of section taken, more
accurate the analysis can be done. The above case, we have done using 35 sections for clear


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                                                                                    Chapter 5: Analysis of Cylindrical Waveguide Tapers 
 
visualization of sections to the reader. In original, we have used 350 sections and hence 351 junctions
for highly accurate analysis. The cascading of results calculated for one section to whole waveguide
.

                                                             Step approximation of Raised cosine taper
                                               45


                                               40


                                               35
                       Radius of Taper in mm




                                               30


                                               25


                                               20


                                               15


                                               10
                                                    0   50       100       150       200       250       300   350
                                                                       Length of Taper in mm


              Figure 5.4: Step approximation of basic raised cosine function for analysis.




                                                                                                                  

                     Figure 5.5: Section of raised cosine step approximated taper.

    section, i.e.351 junctions, is really a cumbersome work. Some commercially available codes are
generally preferred at this stage of work for computation purpose. CASCADE 3.0 [94], a scattering



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                                                         Chapter 5: Analysis of Cylindrical Waveguide Tapers 
 
matrix tool working with the mode matching philosophy described in this chapter is used to continue the
calculation of analytical work from this point onwards



                    Table 5.1: Data sheet for design dimensions of 35 sections taper.

         Radius(a)                 Length                     Radius                    Length
                                    0.0000                                             180.0000
           13.99                                              28.8845
                                   10.0000                                             190.0000
          14.0474                                             30.1585
                                   20.0000                                             200.0000
          14.2191                                             31.417
                                   30.0000                                             210.0000
          14.5037                                              32.65
                                   40.0000                                             220.0000
           14.899                                             33.8476
                                   50.0000                                             230.0000
          15.4017                                                35
                                   60.0000                                             240.0000
          16.0078                                             36.098
                                   70.0000                                             250.0000
          16.7125                                             37.1328
                                   80.0000                                             260.0000
           17.51                                              38.0961
                                   90.0000                                             270.0000
          18.3939                                              38.98
                                  100.0000                                             280.0000
          19.3572                                             39.7775
                                  110.0000                                             290.0000
           20.392                                             40.4822
                                  120.0000                                             300.0000
           21.49                                              41.0883
                                  130.0000                                             310.0000
          22.6424                                             41.591
                                  140.0000                                             320.0000
           23.84                                             41.9863
                                  150.0000                                             330.0000
           25.073                                            42.2709
                                  160.0000                                             340.0000
          26.3315                                            42.4426
                                  170.0000                                             350.0000
          27.6055                                              42.5




5.5 Analysis of Various Taper Designs
       As discussed in previous chapters about different taper profiles, here analysis of each design will
be taken into account. Analysis of taper in respect of S parameters tells the performance of the every
design profile. Mode matching technique discussed above in very detail is used for this part of work.

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                                                                                            Chapter 5: Analysis of Cylindrical Waveguide Tapers 
 
The mathematics involves in calculation of S parameters for each junction and cascading the scattering
matrix is very difficult task and is not possible manually. MATHEMATICA [92] & MATLAB[93] are
good software available for the calculations. Dedicated scattering matrix codes are also very good
alternative for the mode matching computation [94-97].
       The number of sections taken is important issue. More number of steps taken, less the
approximation, and better the transmission of power in every profile. Total power transmission in all
modes is dependent on profile along with its dependency on the no. of sections. Profile of taper is
responsible for mode conversion issue and also somewhat affected to the total power arrived at the
output in all modes. After certain large enough sections, the mode conversion and transmission is
saturated to certain value. Even increase in sections further remain those quantities unaffected.The
transmission power is almost approaches to 1 while reflected power is very less as the waveguide is
overmoded.
       Tapers designed in chapter 3 are analyzed here using mode matching technique. Analysis
frequency range is taken from 40GHz to 45GHz. Transmission coefficient (S21) is plotted with
frequency range. Our concern is calculation and observation of S11 and S21 at 42GHz for each of basic
design. Incident mode at input of taper is TE0,3 .S21 is total power transmitted at the output of taper
retained in TE0,3 mode. Transmission power calculation in desired mode is our objective.

                                                                       incident mode =TE03,Output mode=TE03
                                                          100

                                                           90

                                                           80
                      Transmission coefficient S21 in %




                                                           70

                                                           60

                                                           50

                                                           40

                                                           30
                                                                                                            Parabolic
                                                           20                                               Linear
                                                                                                            Exponential
                                                           10
                                                                                                            Raised Cosine

                                                            0
                                                             32   34    36      38     40       42     44        46         48
                                                                                 Frequency in GHz



             Figure 5.6: Transmission characteristics of basic profiles with frequency.


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                                                            Chapter 5: Analysis of Cylindrical Waveguide Tapers 
 
One can calculate the total transmitted and reflected power irrespective of modes in different taper
designs at 42GHz .Incident power in TE0,3 mode and the total power at output in TE0,3
       As the reflected power is negligible in overmoded waveguide, then, total power transmitted in a
desired mode is calculated with the corresponding S21.
Transmitted power in desired mode for given incident (T)
                                  = [0.01 * S21(desired output, incident mode)]2 x 100                  (5.62)

Table showing the calculated transmitted power in desired TE0,3 mode at output along with the
transmission coefficient. The incident power in TE0,3 mode is consider as unity.


        Table 5.2: Basic profile transmission coefficients and transmitted power at 42GHz.

                           Profile           |   |      Transmitted power at output inTE0,3
                      Linear               89.88%                   80.82%
                      Parabolic           71.484%                   51.10%
                      Exponential          32.48%                   10.55%
                      Raised Cosine       93.861%                   88.10%

The calculation of transmission coefficient for all basic profiles gives that for the raised cosine taper
have maximum transmission for desired mode. This is the analytical proof of our theory explained
earlier regarding smooth terminals of tapers [1].


5.6 Analysis of Cylindrical waveguide taper for 200kW, 42GHz, CW Gyrotron
       As we have discussed different design of taper in Chapter 3 for 200kW, 42GHz, CW gyrotron.
One can see in above section the performance of raised cosine taper (S21 = 93.861% or transmitted
power = 88.10%) is best among the profile samples taken. Thus, we take few shapes from arbitrary
profile resembling with raised cosine profile for analysis by varying shape factor S in equation (3.52).
The number of sections we have taken 350 for our analysis problem and as we have done the frequency
sweep transmission coefficient calculation for other profiles in previous section. For our design we can
calculate the same here.
       Using the arbitrary equation(3.56) described in chapter 3,one can obtained the profiles of taper
by varying Shape factor. Here we are dividing each profile in 350 sections or 351 junctions using the


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                                                                                                                                     Chapter 5: Analysis of Cylindrical Waveguide Tapers 
 
design equations. And as explained in section 5.5, one can calculate the S parameters. Our interest is
mainly on transmission coefficient, so we focus on S21 calculation in different frequency sweep.
                                                                                                            incident mode =TE03,Output mode=TE03
                                                                   100

                                                                                        90

                                                                                        80
                    Transmission coefficient S21 in %




                                                                                        70

                                                                                        60

                                                                                        50

                                                                                        40                                                                       S=0.3
                                                                                                                                                                 S=0.4
                                                                                        30                                                                       S=0.5
                                                                                                                                                                 S=0.6
                                                                                        20                                                                       S=0.7
                                                                                                                                                                 S=0.8
                                                                                        10
                                                                                                                                                                 S=1

                                                                                            0
                                                                                             30      32    34       36     38    40    42      44      46    48          50
                                                                                                                          Frequency in GHz

             Figure 5.7: Transmission coefficients of arbitrary profiles with frequency.

                                                                                                                incident mode =TE03,Output mode=TE03
                                                                                             100

                                                                                            99.5

                                                                                              99
                                                        Transmission coefficient S21 in %




                                                                                            98.5

                                                                                              98

                                                                                            97.5

                                                                                              97

                                                                                            96.5
                                                                                                                 S=0.4
                                                                                              96                 S=0.5
                                                                                                                 S=0.6
                                                                                            95.5                 S=0.7

                                                                                              95
                                                                                                35    36   37      38    39    40   41    42   43      44   45      46
                                                                                                                           Frequency in GHz



                                                                                            Figure 5.8: Transmission coefficients in narrow band.

       The variation of shape factor varies the transmission characteristics. Taking the shape factor in
steps from S = 0.3 to S = 1 only because in this range only the taper has raised cosine smooth nature. S =

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                                                          Chapter 5: Analysis of Cylindrical Waveguide Tapers 
 
0.5 gives the maximum power transmission of 96.90% in desired mode at output with transmission
coefficient 98.437%. Still we don’t know for which shape factor, we get maximum transmission in
desired mode. In the very next chapter, we use an optimization technique namely particle swarm
optimization for getting optimum design with maximum transmission in desired mode.
        Table 5.3: Arbitrary taper transmission coefficients and transmitted power at 42GHz.

                         Profile            |   |         Transmitted power at output inTE0,3

                        S=0.3            66.865%                        44.71%
                        S=0.4            98.427%                        96.88%
                        S=0.5            98.437%                        96.90%
                        S=0.6            97.718%                        95.49%
                        S=0.7            96.093%                        92.34%
                        S=0.8            94.3451%                       89.01%
                         S=1             89.3308%                       79.80%


5.7 Mode Conversion Analysis for Various Taper Designs
        In the above discussion, we have applied mode matching and concern with only incident mode
TE0,3 and transmitted mode TE0,3. We calculate the transmitted power in desired mode but don’t bother
for the transmission of power in spurious modes.
        As given by Neilson [11], the coupling coefficients for a system in which all waveguides share
the same axis of symmetry for mode TEmn1 at input to mode TEmn2at output is given by the relation.

                                                                                                     (5.61)
                                                            /



where     and       are two section under consideration having radius      and     , respectively.
                         incident mode and output mode eigenvalues, respectively
         m= azimuthal mode coupling.
As for TE0,n modes, azimuthal mode coupling is zero as no variation occur in azimuthal direction,
substituting m = 0 in above equation




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                                                         Chapter 5: Analysis of Cylindrical Waveguide Tapers 
 


                                                                                     .                                    (5.62)
                                                      



Now, for each section, one can calculate the coupling of TE0,m and TE0,n using this relation. Two
different modes having different eigen values are considered. For whole taper, i.e., 350 sections (351
junctions), calculation is quite cumbersome for calculation of power transmitted in spurious mode.
CASCADE 3.0 is used for avoiding the calculation complexity. The mode conversion to different
modes are shown in normalized form. Also, the power transmission to only symmetrical (degenerated)
modes [1].Here, Transmitted mode content normalized to total transmitted power and reflected mode
content normalized to reflected power. When incident mode is TE03 is considered and operating
frequency is 42GHz.Let the power incident in TE03 is 1 at the input of taper and total power transmitted
and total power reflected including desired mode as well as spurious mode, then
                                                                                                                             
     Normalized Transmitted power in a mode
                                                                                                             
                                                                                                                     (5.63) 
                                                                                                                       
        Normalized Reflected power in a mode
                                                                                                     
                                                                                                                          (5.64) 
Sum of Normalized Transmitted powers in all modes =1 (100 in %)
Table 5.4: Mode conversion to spurious modes with different basic designs
(a)Linear Taper                                                      (b)Parabolic Taper

Normalized transmitted power
                                                              Normalized transmitted power
Mode No.          % Power
                                                              Mode No.         % Power
TE 0,1             0.67
                                                              TE 0,1              2.93
TE 0,2            10.36
                                                               TE 0,2             20.17
TE 0,3            80.82
                                                               TE 0,3             51.10
TE 0,4             7.41
                                                               TE 0,4             22.24
TE 0,5             0.60
                                                               TE 0,5              2.97
TE 0,6             0.10
                                                               TE 0,6              0.44
TE 0,7             0.02
                                                               TE 0,7              0.10
TE 0,8             0.01
                                                               TE 0,8              0.03
TE 0,9             0.00
                                                               TE 0,9              0.01
TE 0,10             0.00
                                                               TE 0,10             0.00
TE 0,11             0.00
                                                               TE 0,11             0.00


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                                                     Chapter 5: Analysis of Cylindrical Waveguide Tapers 
 
(c)Exponential Taper                                                    (d)Raised cosine taper


Normalized transmitted power                                           Normalized transmitted power
Mode No.         % Power                                               Mode No.        % Power
TE 0,1            17.99                                                 TE 0,1           0.15
 TE 0,2             8.15                                                TE 0,2             7.75
 TE 0,3            10.55                                                TE 0,3           88.10
 TE 0,4            17.59                                                TE 0,4             3.89
 TE 0,5            31.43                                                TE 0,5             0.11
 TE 0,6            11.46                                                TE 0,6             0.00
 TE 0,7             2.23                                                TE 0,7             0.00
 TE 0,8             0.43
 TE 0,9             0.12
 TE 0,10             0.03
 TE 0,11             0.01
 TE 0,12             0.00
 TE 0,13             0.00
 TE 0,14             0.00


Again, the table showing that raised cosine profile showing low mode conversion to spurious
modes, and also the exponential profile is the worst one showing maximum mode conversion.
Reflection of power is very small and negligible in each case as the waveguide is overmoded.

5.8 Mode Conversion in Taper for 200kW, 42GHz, CW Gyrotron
       Modified raised cosine taper can be obtained using basic non linear taper design equation.
The input radius, output radius and length is kept constant and for different shape factors, one
can calculate the mode conversion.

        Table 5.5: Mode Conversion to spurious modes with different Shape Factors
    (a) Shape factor S=0.4                                (b) Shape factor S=0.5

Normalized transmitted power
Mode No.        % Power                                 Normalized transmitted power
   TE 0,1         0.26                                   Mode No.      % Power
   TE 0,2          2.85                                   TE 0,1      0.13
   TE 0,3         96.88                                   TE 0,2      2.70
   TE 0,4          0.01                                   TE 0,3      96.90
   TE 0,5          0.00                                   TE 0,4      0.28
   TE 0,6          0.00                                   TE 0,5      0.00
   TE 0,7          0.00                                   TE 0,6      0.00
   TE 0,8          0.00                                  TE 0,7       0.00

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                                                                                       Chapter 5: Analysis of Cylindrical Waveguide Tapers 
 



    (c) Shape Factor S=0.6                                                                           (d) Shape Factor =0.7

Normalized transmitted power                                                                Normalized transmitted power
  Mode No.         % Power                                                                   Mode No.         % Power
   TE 0,1            0.05                                                                      TE 0,1            0.09
   TE 0,2            3.64                                                                      TE 0,2            5.86
   TE 0,3           95.49                                                                      TE 0,3           92.34
   TE 0,4            0.82                                                                      TE 0,4            1.71
   TE 0,5            0.00                                                                      TE 0,5            0.00
   TE 0,6            0.00                                                                      TE 0,6            0.00


As shown in the figure 5.9, different raised cosine nature profiles showing very little mode
conversion as normalized transmission coefficient is very high in these cases. Raised cosine
profile with S=0.5 gives 96.3% power in incident TE0,3 mode only, and also having mode
conversion into 3 more spurious modes. Rather, in conical and basic raised cosine profile, mode
conversion occur upto 6 and 4 more degenerated modes.



                                                              120
     Normalized Transmission coefficent




                                                                              Mode conversion  analysis
                                                              100

                                                               80

                                                               60

                                                               40

                                                               20

                                          TE 0 n, n=1 to 14     0
                                                                     1    2   3    4    5   6    7     8    9   10   11   12   13   14
                                             Conical                0.67 10.3 80.8 7.41 0.6 0.1 0.02 0.01 0     0    0    0    0    0
                                             raised cosine          0.15 7.75 88.1 3.89 0.11 0   0     0    0   0    0    0    0    0
                                             modified raised cosine 0.14 3.3 96.3 0.24 0    0    0     0    0   0    0    0    0    0

    Figure 5.9: The Mode conversion comparison at 42GHz in desired and spurious modes.




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                          Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 




                                                                        Chapter 6



Particle Swarm Optimization of a
Cylindrical Waveguide Taper


6.1 Introduction:
Particle Swarm Optimization (PSO) is a method for optimization of continuous nonlinear
functions. This is a powerful tool when at the mean time we have to optimize multi-variables
simultaneously. It is a simple concept and paradigms can be implemented in a few lines of
computer code. It takes very less memory and at high speed. It is effective with several kinds
of problems[73].
       A satisfying simulation could be written keeping two ancillary variables in mind, i.e.,
nearest neighbor velocity matching and craziness. A population of birds (particles) was
randomly initialized with X and Y velocities. At every position in grid plan, the function has
fixed value. For every iteration, a loop in the program determined for each agent (particle),
which other agent was its nearest neighbour, then assigned that agent’s X and Y velocities to
the agent in focus. But unfortunately, the flock quickly settled on a unanimous, unchanging
direction. Therefore, a stochastic variable called Craziness was introduced.
       Heppner’s bird simulation involves birds(particles) flocked around a “roost”, a
position on the 2D space that attracted them until they finally landed there. This eliminates
the need for the variable like craziness [74]. But in Heppner’s bird, they knew where their
roost was, but in real life birds landed anywhere that meets their immediate needs, i.e., where
the food was available. As one could find that a flock went for food search in a new place


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                             Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 
put their best individual efforts and after some time accumulated at a single global best place.
In this way capitalizes one another’s knowledge.
       Cornfield vector: It is a 2D vector of XY coordinates on the pixel plane. Consider that
the value of agent at position (       ,     is zero (reference).Each agent was programmed to
evaluate its present position in terms of the equation
                  Eval=                                                      .                 (6.1)
Each agent remembered two things, the best value (represented as pbest[ ]) and the position
coordinates (pbestx[ ], pbesty[ ]) corresponding to it. Brackets indicates that these are arrays,
with no. of elements = no. of agents). As each agent moved through the 2D space, then its
axial velocities (Vx, Vy) was adjusted according to following manner.
If present[x] > pbest[ ]
                  then       Vx[ ]= Vx [ ] - rand( )* p_increment ,                          (6.2.1)
If present[x] < pbest[ ]
                  then        Vx[ ]= Vx [ ]+rand( )* p_increment ,                           (6.2.2)
If present[y] > pbest[ ]
                   then        Vy[ ]= Vy [ ]- rand( )* p_increment ,                         (6.2.3)
If present[y] < pbest[ ]
                      then     Vy[ ]= Vy [ ]+ rand( )* p_increment ,                         (6.2.4)
In this way, we can say that X velocities or Y velocities adjust forward/ backward or
up/down whether the agent is lead/lag or above/below the pbestx or pbesty, respectively.
       Also, Each Agent knew the globally best position that one member of flock had
found. Let the best position for group is to a variable called gbest, so that pbestx[gbest],
pbesty[gbest] was the group’s best position, and this information was available to all flock
members. As each member try to migrate towards global best. So, by considering
g_increment as a system parameter.
If presentx[ ] > pbest[gbest]
               then       Vx[ ]= Vx [ ] - rand( )* g_increment ,                             (6.3.1)
If presentx[ ] < pbest[gbest]
               then      Vx[ ]= Vx [ ]+rand( )* g_increment ,                                (6.3.2)
If presenty[ ] > pbest[gbest]
               then      Vy[ ]= Vy [ ]- rand( )* g_increment ,                               (6.3.3)
If presenty[ ] < pbest[gbest]
               then      Vy[ ]= Vy [ ]+ rand( )*g_increment ,                                (6.3.4)


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                           Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 
When p_increment and g_increment set relatively high ,the flock of around 15-20 agents
rapidly sucked into the cornfield within few iterations. While, same set low, it takes good
number of iterations to confine to best position according to cornfield vector.
        Elimination ancillary variables: Algorithm works just as well, and looks just as
realistic, without craziness, so it was removed. Also, it was shown that optimization actually
occurs slightly faster when nearest neighbor matching is removed. Due to this, The flock is
now a swarm, but it is well able to find the cornfield.
        The variables pbest and gbest and their increments are both necessary. pbest
resembles as each individual remembers its own experience and the velocity adjustment
associated with pbest. It is just like that as an individual tends to return to the place that
satisfied him the most in the past.on the other hand, gbest is something one wish to go as per
the public knowledge or other’s experience but hadn’t visited earlier. In the simulations, a
high value of p_increment relative to g_increment results high wondering of flock and take
more iterations to settle down while g_increment is relatively high then flock is rapidly
settled to fixed point within few iterations only. So, one should go for equal increments for
effective search of the problem .
        For multi dimensional search, simply change the presentx and presenty and
corresponding velocities from one dimensional arrays to D*N matrices, where D is any
number of dimensions and N is the number of agents.
Velocities equations could be modified to avoid inequality test by

     Vx [ ][ ]= Vx [ ][ ] + rand( )* p_increment * (pbestx[ ][ ] – presentx[ ][ ])         (6.4.1)

      Vx [ ][ ]= Vx [ ][ ] + rand( )* g_increment * (gbestx[ ][ ] – presentx[ ][ ])        (6.4.2)

Two sets of brackets are there because they are now matrices of agents be dimensions.
        But it is realized that there is no good way to guess whether p or g increment should
be larger. So, these terms were fired from the algorithm. The simplified particle swarm
optimizer now adjusts velocities as per the formula.

Vx [ ][ ]= Vx [ ][ ] + 2 *rand( )* (pbestx[ ][ ] – presentx[ ][ ])
                         + 2 * rand( ) * (pbestx[ ][gbest] – presentx[ ][ ])                 (6.5)

Another version of PSO which consider two types of agents: Explorers and settlers. Explorers
use the inequality test while settlers use the difference term.




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                              Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 
6.2 Parameter Selection in PSO:
           In PSO, instead of using generic operators, each particle adjusts its flying according to
its own flying experience (local best) as well its companion’s flying experience. Each particle
is treated as a point in a D- dimensional space [75-77].
    The ith particle is represented as
             Xi=( xi1 ,xi2 ,xi3 ……….,xiD ) .
The best previous position giving best fitness value of ith particle is recorded as
             Pi=( pi1 ,pi2 ,pi3 ……….,piD ) .
The velocity of ith particle is represented as
            Vi=( vi1 ,vi2 ,vi3 ……….,viD ) .
The index of best particle among the population is denoted by g.


The particles are manipulated according to following equation

    vid =w * vid +c1* rand( )*              + c2* Rand(                                     (6.6)
                                      ∆                          ∆

    xid =xid + vid                                                                          (6.7)
where,
                     w = inertia factor
                       = self- confidence
                       = swarm confidence

                               = particle memory influence
                        ∆

                               = swarm influence
                        ∆

rand ( ) and Rand ( ) are two random function in the range [0,1].

Stopping Criteria:
           The iteration is going on continuously until maximum change in best fitness value is
smaller than specified tolerance for a specific number of moves (S)[76]
               f (p ) – f(p       )                  q=1,2,3……S         .                       (6.7)
The inertia weight w is employed to control the impact of the previous history of velocities
on the current velocity, thus to influence the trade-off between global and local exploration
abilities of flying points.
           A large inertia weight w facilitates global exploration ( searching new areas) while
small inertia weight facilities local exploration and hence fine tune to a particular zone.


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                             Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 
Hence, w is suitably selected to keep balance between local and global exploration abilities
and thus require less iteration to find the optimum.
          In all simulation, using the same parameter setting for the PSO except the inertia
weight and maximum velocity allowed.
                 Population size = 20
                 Dynamic range of a position of particle is defined as (-100,100)
                 Maximum no. of iterations = 4000
If PSO cannot find the acceptable solution within 4000 iterations, it is considered as failure.
For each selected ‘w’ and Vmax, 30 runs are performed and calculated the successful runs
(in which solution is acceptable within 4000 iterations).
                 120.00%

                 100.00%
                                                                             Vmax=3
                  80.00%
                                                                             Vmax=4
                  60.00%
                                                                             Vmax =5
                  40.00%
                                                                             Vmax.=1
                  20.00%                                                     0

                     0.00%
                             0             0.5              1               1.5

    Figure 6.1: Plot between ‘w’ and success percentage at a particular maximum velocity.
                                                   .
              3000

              2500

              2000                                                          Vmax=3
                                                                            Vmax=4
              1500
                                                                            Vmax =5
              1000                                                          Vmax.=10
                                                                            Vmax.=Xmax
               500

                 0
                     0               0.5               1              1.5

          Figure 6.2: Average number of iteration required plot with inertia weight.



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                           Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 
Plot Analysis:
         For lower value of maximum velocity consideration, inertial weight ‘w’=1 ,while if
maximum velocity is considered high (Vmax >=3) ,then ‘w’=0.8 is a optimum choice
         As per the given plot , optimum range of inertia weight is from 0.6 to 0.8 for getting
global optimum with minimum iterations for higher value of maximum velocity
consideration and ‘w’=0.9-1.05 is optimum for low velocity consideration.
         When ‘w’ <0.8, if PSO finds the global optimum, then finds it fast with very less
umber of iteration in the available range of iterations while when ‘w’>1.2, it require more
than iteration and hence less success chances to find global optimum in this range of inertia
weight. When ‘w’=0.8 to 1.2, the PSO will have the best chance to find the global optimum
but also takes a moderate number of iterations.
         The work done by Clerc(1999)tells that for the convergence of particle swarm
optimization, the use of constriction factor is necessary. This factor is function of      and     ,
and it is included in (6.6), the modified equation is

            vid =K *[ vid +c1* rand( )*               + c2* Rand(               ]             (6.8)
                                             ∆                              ∆

where,

         K =                    , where     = c1 + c2 , confidence factor           >4     (6.9.1)

             =                            otherwise                                        (6.9.2)
Experimental Approach:
         For comparison of PSO with or without constriction factor, one can take sphere
function as an example which can be represented as
                                             ∑                                               (6.10)
Population size was set to 30, and the maximum no. of iteration was set to10,000.
         Inertia weight (w)[78] was set 0.9 in beginning of run and linear decrease to 0.4

                         w(t) =                                                              (6.11)

where nt is maximum number of time steps for which the algorithm is executed
         w(0) = initial inertia weight (0.9 in this case)
         w(nt) = final inertia weight
         w(t) = inertia at time step t, w(0)>w(nt)
keeping the condition Vmax= Xmax=10,000 ( in our example) and our case we are considering
c1 =c2 =2.05


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                             Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 
Therefore,     =4.1
                           and hence, K = 0.729
Substituting above in equation (6.8), the equation (6.8) becomes

         vid =0.729 *[ vid +2.05* rand( )*                 + 2.05* Rand(                   ]   (6.12)
                                                  ∆                               ∆

or,

      vid =0.729* vid +1.49445* rand( )*                   + 1.49445* Rand(                    (6.13)
                                                ∆                                      ∆

one can compare above equation with equation (5.1) gives the values
                                  ‘w’=0.729
                                  c1 = c2= 1.49445 .
The spherical function is run with 30 dimensions. It undergoes for 20 runs for each iteration
of spherical function using inertia weight (for a value of Xmax=100) and using constriction
(with Vmax=Xmax=10,000). The run is going on until an error less than 0.01 was obtained.
Number of iteration needed for each run to achieve this error value for both cases can be
obtained from the graph below
                1800

                1600

                1400

                1200
                                                                     inertia weight 
                1000                                                 (Xmax=100)

                 800
                                                                     constriction 
                 600                                                 factor(Vmax=Xmax=
                                                                     10,000)
                 400

                 200

                   0
                       0      5      10    15         20     25


             Figure 6.3: Showing the effect of introduction of constriction factor.


The average no of iteration for inertia weight is 1537.8
Range of iteration =1615-1485=130, 8.5% of the average
The average number of iterations using the constriction factor is 552.05
Range of iteration =599-503=96, 17.4% of the average.
Thus, constriction method yield faster results, with range to average quotient.


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                            Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 
6.3 Simplified PSO Particle Trajectories
         Analysis of behavior of PSO is not an easy task because of collective influences of
multiple particles and the stochastic elements. Ozcan and Mohan considered a simplified
PSO system with one particle of one dimension [73]. It is also assumed that personal best
and global best positions were same and remained constant, i.e., pbest = gbest = p. PSO is
assumed with no inertia, velocity clamping or constriction. Based on these assumption
                v(t) = v(t-1) +    1+      2)(p   – x(t -1))                                          (6.14)
                x(t ) =x(t-1) +v(t)         ,                                                         (6.15)
where            1=c1r1    and    2=c2r2               .                                              (6.16)
Solving above equation (6.14) and (6.15), we get,
                x(t) =(2- )x(t-1) – x(t-2) + p                                                        (6.17)
where,                                                                                                (6.18)
with initial conditions,
                          x(0) = x0, x(1) =(1- ) + v0+ p                                              (6.19)
as for a characteristic polynomial,
                                                                       2
                                            (1- ) ( 1- (2- ) +             ) gives the solutions as

                                                                                                       6.20)

                                                                   .                                  (6.21)

Where ,                                               4            .                                  (6.22)
This gamma,      is geometric parameter. The close form of equation (6.17) can be obtained as
                                  x(t) =                                                              (6.23)
where , k1, k2, and k3 are constants determined by initial condition of system.
                                      0           1        1   1
                                      1         = 1                         .                         (6.24)
                                      2           1
Solving the above matrix
                                                                                                      (6.25)
                                                                                                      (6.26)

                                                                                  .                    (6.27)

Let x0 = p, the trajectory equation of the particle changes to
                                   x(t)= (                                            .               (6.28)


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                            Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 
6.4 Optimal Choice of Parameters for PSO:
             , confidence factor was considered as an important parameter. Clerc and Kennedy
(2002) gave some clues that constriction factor method exhibits almosr linear convergence
for   >4 ,      is varied from 4.0 to 4.4 with the increment of 0.01. In the general case
                            1 in equation (6.9) given by Eberhart and Shi (2000)
One can determine K [79]from equation (6.9).
                   K =                 , where       = c1 + c2 ,        >4

                      =                                            otherwise
         The second parameter is Vmax, which limits the maximum distance a particle can
travel in one generation. Earlier we considered
                  Vmax= Xmax. that is modified as Vmax= Xmax                                      (6.29)

                            Table 6.1: Parameters variation for PSO


                      Parameter         Range               Increment        Steps
                                        [4.00,4.40]         .01              40
                                        [.001,.01]          .001             10
                                        [.01,.1]            .01              10
                                        [.1.1]              0.1              10
                                        [1.10]              1                10
                                        [10,100]            10               10
                                        [100,1000]          100              10
                              N         [10,150]            10               15

Where,       is a proportional factor to be determined. The increment of          is varied for different
interval. Last parameter is population size N. This was varied from 10 to 150 in the steps of
10.


6.5 Basic Algorithm of PSO:
The Basic algorithm of particle swarm optimization technique is well explained in number of
published literature using the points discussed in the previous sections. PSO is widely used in
many engineering optimization problems [80-82] involving non linear functions. As other
optimization techniques,it involves iterative nature so code development is the better choice

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                            Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 
with the algorithm steps mentioned below Steps involve in the process are discussed here one
by one.
          Step 1: initialize a population of N particles. For the ith particle ,its location in the
search space is randomly placed. Its velocity vector is vi =(vi1,vi2,….vid,…viD), in which
velocity in dth dimension is vid=rand( ) * Vmax. Vmax is calculated by equation (11) for which
Xmax needed is given in the problem for optimization.


          Step 2: Start the loop execution for the range assigned for     and   according to table
1. Then calculate K from equation (6.9) and suppose c1=c2= /2. Then implement three series
of experiment by varying       ,   and N within the given interval taking one as a variable and
other two keep constant for each case.




                              Figure 6.4: Basic Algorithm of PSO.




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                           Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 
        Step 3: Start the inner loop, set the no. of iteration t=1 and evaluate the fitness
function for each particle given by pbest. Out of all particles, the particle gives best fitness
value that’s index is equal to gbest.


        Step4: Compare the evaluated fitness value of each particle with i its pbest. If the
present value is better than pbest, the set the current location as the pbest location.
Furthermore, if the present value is better than gbest , then reset gbest to the current index in
the particle array.


        Step 5: Change the velocity and location of the particle according to

                vid =K *[ vid +c1* rand( )*            + c2* Rand(                 ]       (6.30)
                                                 ∆                          ∆

                xid =xid + vid                                                             (6.31)
check if the velocity and position exceed the constraints.


        Step6: Set t=t+1 and repeat step 4 and step 5 until the no. of iteration is greater than
the maximum iteration allowed.


        Step 7: Go to outer loop and implement Step 2 to Step 6 until the condition get
unsatisfied.

Effect of Confidence Factor,
        For each of the above function, average fitness variation with           from 4.0 to 4.4
keeping other two parameters      =0.1 and N=30.The plot of K with         in equation (8) can be
observe in the plot below. Increasing     will decrease the value of constriction factor. When
is too large ,than K becomes very smaal which means that the search distance of every step
for each particle is very small that leads to dominance of local exploitation. Rather, for small
    and hence large K increases the search distance and leads to dominance of global
exploration. A PSO system tries to balance exploration and exploitation by combining local
and global search ability. So K factor should be kept moderate.




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                                               Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 

                                    1

                                  0.95

                                   0.9

                                  0.85

            constriction factor
                                   0.8

                                  0.75

                                   0.7

                                  0.65

                                   0.6

                                  0.55

                                   0.5
                                         4   4.05   4.1   4.15   4.2    4.25   4.3   4.35   4.4   4.45   4.5
                                                                 confidence factor


            Figure 6.5: Plot between confidence factor and constriction factor.

       It is found in analysis that                          =4.05 is appropriate for high multimodal functions and
     4.10 is for unimodal function. Confidence factor ( ) is taken smaller for multimodal
function because more exploration ability is needed to multimodal function as compare to
unimodal function for finding global optimum.

Effect on Velocity Coefficient,
       The average fitness varied with                            from 0.001 to 1000 when          = 4.1 and N=30. It is
observed from curves that for unimodal functions, if                             is larger than 0.005, then performance
is very less affected. For Shaffer function, it is very difficult to predict the exact range, while
in high dimension restrain function, [0.01, 0.1] range is optimum. If Vmax is too large,
particles might easily pass over, skip the local searching ability and explore more global
space and thus, good global optimum and poor local optimum is obtained. Vmax is set at
about 5% of the dynamic range of the variable on each dimension for unimodal function and
50% for multimodal function.

Effect on Swarm Size, N
       Average fitness varies with population size N when                                   = 4.1 and    = 0.1. As seen
from the curves, unimodal functions remain unaffected with population size, the multimode
functions are giving good performance on increase in population size but large population
causes more computational efforts, so population size is always kept below 50. Generally,
between 20 to 50 particle population is preferred.



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                          Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 
    Table 6.2: Optimum choice of parameters for unimodal and multimodal functions
           Parameters                  Unimodal functions      Multimodal functions
           Confidence factor(          4.1                     4.05
           Velocity coefficient (      0.05                    0.5
           Population Size (N)         independent             30




6.6 PSO of Nonlinear Taper for 42GHz, 200kW,CW Gyrotron:
       This optimization technique can be used in any field of research but here we are
interested in application in electromagnetics [83].We have done analysis of tapers starting
from basic profiles, i.e., linear, parabolic, exponential and raised cosine. It is found that
raised cosine profile is giving very low mode conversion. So, using the modified arbitrary
taper design equation, we try to choose certain range of shape factor just by idea which are
giving similar raised cosine profiles and analyzed. Shape factor from S=0.3 to S=1 range is
decided as the range under consideration. In previous chapter, we analyze few profiles in this
range and most of them are giving a good transmission coefficients. Rather we find that more
specifically S=0.4 to S=0.7 gives greater than 95% transmission coefficient for TE      0,3   mode
at output when incident power is only in TE0,3 mode. Still, we are curious about the optimum
taper, i.e., the taper giving best transmission coefficient for TE0,3 mode at output. Particle
swarm intelligence is introduced here for optimization of taper and hence obtaining optimum
design parameters.



Parameter Selection
       Input radius of taper is same as the radius of gyrotron cavity, i.e., 13.99 mm .So, one
cannot change the input radius and it cannot be taken as variable. One can vary length of
taper and output radius and taken as parameters. Shape factor is also a parameter under
consideration along with the number of sections. As the range of shape factor variation
discussed above (0.3-1) is the observation basis. While we are doing optimization using
MATLAB code developed specifically for our problem, so we can increase the width of
range of shape factor for self satisfaction.




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                                                     Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 
           Table 6.3: Parameters under variation for non-linear taper design.
                                                        Parameters             Range
                                                        Output Radius          35 mm-45 mm
                                                        Length of taper        300 mm -500 mm
                                                        Shape factor           0.01- 10
                                                        No. of sections        50-500



6.6.1 Variation Effect of Parameters:
       We analyze one by one the effect of parameters one the performance of taper. In this
regard, varying one parameter at a time keeping the remaining constant [84]. Calculation of
transmission coefficient in desired mode using mode matching technique described in
Chapter 5 for every design.

Variation of output radius
       Input radius = 13.99 mm, Length of taper = 350 mm, number of sections = 350 and
shape factor = 0.5. Keeping these parameters constant, output radius is changed in the given
                                                                         Variation of output radius
                                            45


                                            40


                                            35
                    Radius of Taper in mm




                                            30


                                            25

                                                                                                            L=35 mm
                                            20                                                              L=37.5 mm
                                                                                                            L=40 mm
                                                                                                            L=42.5 mm
                                            15
                                                                                                            L=45 mm


                                            10
                                                 0         50      100       150       200            250      300      350
                                                                         Length of Tape r in mm



        Figure 6.6: Design of Non-Linear Taper with Variation of Output Radius.




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                                                                                            Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 
range leads to obtain the plot shown in figure. Using dedicated scattering matrix code of
mode matching, one can calculate the transmission coefficient at 42GHz and TE0,3 mode .



                                                                                        Variation of Output radius:incident mode =TE03,Output mode=TE03
                                                                                100

                                                                                  90

                                                                                  80
                                            Transmission coefficient S21 in %




                                                                                  70

                                                                                  60

                                                                                  50

                                                                                  40
                                                                                                                                         a2=35 mm
                                                                                  30
                                                                                                                                         a2=37.5 mm
                                                                                  20                                                     a2=40 mm
                                                                                                                                         a2=42.5 mm
                                                                                  10                                                     a2=45 mm

                                                                                   0
                                                                                    30      32     34    36     38    40    42     44    46     48     50
                                                                                                               Frequency in GHz



Figure 6.7: Variation of Transmission Coefficient with frequency for different output radius

                                                                                       Variation of Output radius:incident mode =TE03,Output mode=TE03
                                                                                100

                                                                       99.9

                                                                       99.8
              Transmission coefficient S21 in %




                                                                       99.7

                                                                       99.6

                                                                       99.5                                                               a2=35 mm
                                                                                                                                          a2=37.5 mm
                                                                       99.4                                                               a2=40 mm
                                                                                                                                          a2=42.5 mm
                                                                       99.3

                                                                       99.2

                                                                       99.1

                                                                                 99
                                                                                   40      40.5   41    41.5    42   42.5    43   43.5    44    44.5      45
                                                                                                               Frequency in GHz


    Figure 6.8: Variation of Transmission coefficient with frequency for different output
                                          radius around 42GHz



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                                                Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 


      Table 6.4 Transmission coefficient with variation of output radius at 42 GHz
                                                     Variation in output         Transmission
                                                           radius                coefficient
                                                           35 mm                 99.90%
                                                           37.5 mm               99.99%
                                                            40 mm                99.69%
                                                           42.5 mm               98.44%
                                                           45 mm                 95.87%

       Mode matching analysis of all possible tapers shown in figure 6.6 is done and
transmission coefficient calculated with frequency sweep. For getting a look of transmission
coefficient around the desired 42GHz, we have shown the closer view of figure 6.6 in figure
6.7. The variation of output radius affects a little the transmission coefficient at 42GHz
operating frequency in the considered range. Range is considered around the desired output
radius (= 42.5 mm) as shown in Table 6.4.

Variation of length
       The effect of variation of length is also analyzed in the similar way. The parameters
kept unchanged are input radius = 13.99mm, output radius = 42.5mm, number of sections =
350 and shape factor = 0.5.

                                                                   Variation of Length

                                       45


                                       40
               Radius of Taper in mm




                                       35


                                       30


                                       25
                                                                                                   L=300
                                                                                                   L=350
                                       20
                                                                                                   L=400
                                                                                                   L=450
                                       15                                                          L=500


                                            0   50   100   150     200     250   300   350   400     450   500
                                                                 Length of Taper in mm



            Figure 6.9: Design of Non-Linear Taper with Variation in Length.
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                                                                                                  Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 

                                                                                                 Variation of Length:incident mode =TE03,Output mode=TE03
                                                                                      100

                                                                                        90

                                                                                        80

                                                  Transmission coefficient S21 in %     70

                                                                                        60

                                                                                        50

                                                                                        40

                                                                                        30                                                                L=300 mm
                                                                                                                                                          L=350 mm
                                                                                        20                                                                L=400 mm
                                                                                                                                                          L-450 mm
                                                                                        10                                                                L=500 mm

                                                                                         0
                                                                                          30       32        34      36      38   40     42         44        46          48        50
                                                                                                                            Frequency in GHz

    Figure 6.10: Variation of Transmission Coefficient with Frequency for different length.

                                                                                               Variation of Length:incident mode =TE03,Output mode=TE03
                                                                         100

                                                              99.9

                                                              99.8
              Transmission coefficient S21 in %




                                                              99.7

                                                              99.6

                                                              99.5

                                                              99.4

                                                              99.3                                                                                   L=300 mm
                                                                                                                                                     L=350 mm
                                                              99.2                                                                                   L=400 mm
                                                                                                                                                     L-450 mm
                                                              99.1                                                                                   L=500 mm

                                                                                      99
                                                                                        40      40.5    41        41.5     42   42.5   43    43.5        44        44.5        45
                                                                                                                          Frequency in GHz


    Figure 6.11: Variation of Transmission coefficient with frequency for different length.




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                                               Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 
          Table 6.5: Transmission coefficient with variation of length at 42GHz.
                                                     Variation in Length        Transmission
                                                                                coefficient
                                                           300 mm               94.39%
                                                           350 mm               98.44%
                                                           400 mm               99.87%
                                                           450 mm               99.68%
                                                           500 mm               99.77%

       At 42GHz, transmission coefficient for desired mode is shown in Table 6.5 and it is
found that it is getting improved (not linearly) with increase in length. No doubt, an infinite
long taper is in fact the best one if radiation loss is neglected but for practical purpose; the
length of taper should be small as much as possible. Also, with increase in length of taper
consideration of radiation and hear loss on the metal surface of taper results power loss and
hence degraded the performance of taper. It is also found that variation of length around the
length under consideration (= 350 mm) not much introduce the mode conversion.

Variation with shape Factor S
       The effect of variation of shape factor in design equation is also analyzed in the
similar way. The parameters kept unchanged are input radius = 13.99 mm, output radius =
42.5 mm, number of sections =350 and Length of taper=350 mm.

                                                                Variation of Shape factor
                                      45


                                      40


                                      35
              Radius of Taper in mm




                                      30


                                      25
                                                                                                        S=0.3
                                                                                                        S=0.4
                                      20                                                                S=0.5
                                                                                                        S=0.6
                                                                                                        S=0.7
                                      15
                                                                                                        S=0.8
                                                                                                        S=1
                                      10
                                           0    50        100       150       200           250   300           350
                                                                Length of Taper in mm


        Figure 6.12: Design of Non-Linear Taper with Variation in shape factor.

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                                                                                         Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 




                                                                                              incident mode =TE03,Output mode=TE03
                                                                     100

                                                                            90

                                                                            80
            Transmission coefficient S21 in %




                                                                            70

                                                                            60

                                                                            50

                                                                            40                                                                 S=0.3
                                                                                                                                               S=0.4
                                                                            30                                                                 S=0.5
                                                                                                                                               S=0.6
                                                                            20                                                                 S=0.7
                                                                                                                                               S=0.8
                                                                            10
                                                                                                                                               S=1

                                                                             0
                                                                              30    32       34       36     38   40      42     44      46   48       50
                                                                                                            Frequency in GHz



Figure 6.13: Variation of Transmission Coefficient with Frequency with the shape factor.

                                                                                                  incident mode =TE03,Output mode=TE03
                                                                            100

                                                                            99.5

                                                                             99
                                        Transmission coefficient S21 in %




                                                                            98.5

                                                                             98

                                                                            97.5

                                                                             97

                                                                            96.5
                                                                                                   S=0.4
                                                                             96                    S=0.5
                                                                                                   S=0.6
                                                                            95.5                   S=0.7

                                                                             95
                                                                               35   36       37      38    39    40   41    42   43      44   45   46
                                                                                                             Frequency in GHz


Figure 6.14. Variation of Transmission coefficient with frequency for different length.




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                          Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 



        Table 6.6: Transmission coefficient with shape factor of length at 42 GHz.
                    Variation of shape factor Transmission coefficient

                             S=0.3                    66.865%
                             S=0.4                    98.427%
                             S=0.5                    98.437%
                             S=0.6                    97.718%
                             S=0.7                    96.093%
                             S=0.8                   94.3451%
                              S=1                    89.3308%


In table 6.6,variation of shape factor affects the transmission coefficient high enough around
those value of shape factor under consideration which provide raised cosine nature profile to
the taper.More sepcifically, it is found that S=0.4 to S=0.7 is the range where the
transmission coefficient is maximum and mode conversion is minimum.but it is difficult to
calculate the optimum value of shape facor for best transmission coefficient in desired mode
and minimum mode conversion without undergoing any optimization technique

Variation in the Number of Sections
       Keeping all design parameters constant and putting the attention on the analysis
parameter i.e.number of sections in which the whole taper is to be divided,one can check the
variation of transmission coefficient calculation.With input radius=13.99 mm, output
radius=42.5 mm, Length of taper=350 mm and shape factor =0.5.

       Using different no. of sections affect the transmission coefficient upto certain point as
shown in table 6.7 obtained from figure 6.16 and 6.17. For the length of taper considered 350
mm, more than 100 sections keep unchanged the transmission coefficient. But for lower
values of N, it leads high approximation and leads to high mode conversion.

In particle swarm optimization technique, we are using 350 sections but one can use 100
sections and will get almost same result. The reason behind 350 sections consideration is to
keep each section is of unit length.




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                                                                 Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 



                                                                         Variation of no. of sections for 10 mm length
                                                      14.9

                                                      14.8                   N=50
                                                                             N=100
                                                      14.7                   N=150
                                                                             N=200
                 Radius of Taper in mm                14.6                   N=350

                                                      14.5

                                                      14.4

                                                      14.3

                                                      14.2

                                                      14.1

                                                       14

                                                      13.9
                                                             0     1     2       3       4       5      6     7    8     9       10
                                                                                      Length of Taper in mm


       Figure 6.15: Design of Non-Linear Taper with Variation in No. of Sections.




                                                             Variation of no. of Sections:incident mode =TE03,Output mode=TE03
                                                      100

                                                       90
                  Transmission coefficient S21 in %




                                                       80

                                                       70

                                                       60

                                                       50

                                                       40

                                                       30                                                                N=50
                                                                                                                         N=100
                                                       20                                                                N=150
                                                                                                                         N=200
                                                       10                                                                N=350

                                                        0
                                                        30        32     34      36      38    40    42       44   46    48      50
                                                                                        Frequency in GHz


Figure 6.16 Variation of Transmission Coefficient with Frequency with No. of Sections.




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                                                                Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 
                                                             Variation of no. of sections:incident mode =TE03,Output mode=TE03
                                                      100

                                                      99.5                                                           N=50
                                                                                                                     N=100
                                                       99                                                            N=150




                  Transmission coefficient S21 in %
                                                                                                                     N=200
                                                      98.5                                                           N=350

                                                       98

                                                      97.5

                                                       97

                                                      96.5

                                                       96

                                                      95.5

                                                       95
                                                         40      40.5   41    41.5    42   42.5   43    43.5    44    44.5   45
                                                                                     Frequency in GHz


    Figure 6.17: Variation of Transmission coefficient with frequency with no. of sections.


              Table 6.7: Transmission coefficient with no. of sections at 42GHz.
                                                                No. of sections (N)        Transmission coefficient
                                                                         50                         95.88%
                                                                        100                         98.45%
                                                                        150                         98.44%
                                                                        200                         98.44%
                                                                        350                         98.44%


6.6.2 Optimization of Nonlinear Taper:
As taper and mode converter demand similar treatment for analysis.Mode conversion is
avoided in taper but desired in mode converter. Optimization technique can be appled for
tapers[84] as well as mode converters[85]. The taper design equation (3.56) consists of four
parameters, i.e., length of taper, Input radius, output radius and shape factor. Knowledge of
all four one can generate any profile. For analysis point of view, we need the fifth parameter,
i.e., no. of sections. The PSO parameters are defined in table 6.7 and the design parameter
range are defined in table 6.3.




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                           Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 
              Table 6.8: PSO parameters involve in PSO of nonlinear taper.

                       PSO Parameters                Value
                       Swarm Size                    20
                       Inertia weight factor, w      0.9-0.4
                       C1                            1.5
                       C2                            1.5
                       Tolerance                     0.00001%
                       (stopping criteria)


              Initialization of 
              parameters values in 
              range for each particle




                 Calculation of S21 for                     Update the parameters 
                     each particle                                with PSO 




                 Maximization of S21 




                 Best Optimum 
                parameters after 
               sufficient iteration 

                 Figure 6.18: Basic overview of PSO in Non-Linear Taper.

       Inertia factor is dependent on iteration with the equation (6.11) starting from 0.9 and
varies to 0.4 with iterations. C1 and C2 values remain constant throughout the optimization
and can be taken between 1.5 to 2. Here, we are kept them equal and value is 1.5.Tolerence
will come in action when S21 calculation in the two consecutive iterations has such less
difference that the process undergoes saturation and taken out of the loop. If stopping criteria
is not given, iteration should be given manually[86,87]. The advantage of tolerance is that we
need not to check again and the saturation point.

       Particle swarm optimization technique is described in very detail in first part of this
chapter. It involves a swarm size, each particle has provided the five parameters values or
position randomly in their range and each having variation or velocity (change in position in

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                           Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 
unit time) within the range using analysis technique transmission coefficient for desired mode
is calculated for whole swarm. Selection of maximum S21 out of the swarm size and
corresponding particle’s design parameters are considered as gbest. Each particle has its own
S21 and in consecutive iterations the design parameters corresponding to the higher S21 for
each particle gives pbest. This gbest and pbest along with the PSO parameters in table 6.8
gives the update in variation or velocity and thus update in value or position after each
iteration. This process is going on upto the desired iteration or some stopping criteria should
be there.
       In our technique, we are keeping fix the input radius to 13.99 mm and no. of sections
hardly affect the S21 when consider more than 100. So, considering only three parameters as
variables from Table 6.3 gives:


                     Table 6.9: Variables for PSO of non linear taper.
                                Parameters       Range
                              Output Radius      35 mm-45 mm
                              Length of taper    300 mm -500 mm
                              Shape factor       0.01- 10

As no. of variables are 3 hence the dimension of PSO becomes 3.the following steps gives the
clear cut idea about the technique.

       Step 1: Each of 20 particles having 3 variables in the given ranges generated
randomly develops a position matrix. Similary ,variation matrix is also calculatable from the
given range
Position(Variable)matrix
       [X]20x3 =[[a2=rand(35,45)]20x1 [L=rand(300,500)]20x1             [S=rand(0.1,10)]20x1 ]20x3
Velocity in unit time =Variation in variables in given range
Velocity(variation) matrix
        [V]20x3 =[ [rand(0,10)]20x1          [rand(0,200)]20x1         [ rand(0,9.9)]20x1 ]20x3

       Step 2: In Each Iteration, For each particle, A row combination in position matrix
gives Transmission coefficient matrix [S21]20x1


       Step 3: For First iteration,
                [gbest]1x3 = Row of Position matrix corresponding to Max.(S21 )

Institute of Technology, Banaras Hindu University                                                  93 
 
                             Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 
                   [pbest]20x3 =[X(k)]20x3
For higher iterations,
                  [gbest]1x3 =obtained by comparing previous Max.(S21 ) and current Max.(S21 )
                  [pbest]20x3 =Previous [S21]20x1 is one to one compared with new [S21]20x1and
the position matrix rows corresponding to higher [S21]
           Step 4: Velocity[Variation] update as equation (6.30) in any iteration k,
                       [V(k+1)]20x3=w*[V(k)]20x3+ C1.rand([pbest]20x3 - [X(k)]20x3)+
                                                       C2.rand([gbest]20x3 - [X(k)]20x3)
    Where, w = inertia weight varies from 0.9 to 0.4 linearly with iterations
            C1 =C2=constants=2
Position[Variables] update as equation (6.31)
                       [[X(k+1)]20x3 =[X]20x3+ [V(k+1)]20x3

           Step 5: Next iteration starts with updated values of variables. If after update, any
parameters cross the upper or lower range than corresponding highest possible value or
lowest possible value is assigned respectively.

           Step 6: Iteration going on continuously and maximum value of S21 is increasing
continuously. When the S21 calculation of two consecutive iterations is under tolerance, then
saturation occurs and position of all particles is almost same. Each column in position matrix
is very close to each other. At this time velocity matrix approaches to zero means no more
variation of particle is there or saturation occurs.
           It is found that for every time we are using same optimization steps, we get different
set of design parameters giving maximum S21 because of random nature involve in the
process. In section 5.2.1, It is found that variation of shape factor is the critical parameter
and affect the most the mode conversion issue. Also, for the 42GHz, 200kW, CW gyrotron,
all remaining design parameters are fixed. So, Applying particle swarm optimization for one
dimension only[89], a MATLAB code is designed which optimize the shape factor within the
range S=0.1 to S=10. Figure 6.19 is showing the improvement of S21 with increase in no. of
iterations. It is found that after 90-95 iteration, transmission coefficient is saturated and
further increase in iteration keep the performance of taper remain unchanged. The maximum
value of S21 obtained is 98.91%




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                                                                                Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 
                                                                                   Taper a nalysis w ith pa rticle sw arm optimization
                                                                    1

                                            0.98                                                                                    X: 100
                                                                                                                                    Y : 0.9891
                                            0.96




              Transimssion coefficent S21
                                            0.94

                                            0.92

                                                 0.9

                                            0.88

                                            0.86

                                            0.84

                                            0.82

                                                 0.8
                                                                        0         20       40          60          80         100           120    140
                                                                                                      no. of iteration


                                                                    Figure 6.19: PSO of Non-Linear taper with iteration.
       At that instant the position matrix of 20x1 has all values almost same which is the
optimum shape factor given by S=0.5331.Figure 6.20 is showing the variation range of shape
factor considered and optimum taper profile obtained by using particle swarm optimization.
                                                                                                  Variation of Shape factor
                                                                    45


                                                                    40


                                                                    35
                                            Radius of Taper in mm




                                                                    30


                                                                    25


                                                                    20


                                                                    15


                                                                    10
                                                                            0      50       100       150       200           250          300    350
                                                                                                  Length of Taper in mm


Figure 6.20: Variation of shape factor from S=0.1 to S=10 and optimization at S=0.5331



The optimum profile obtained with input radius = 13.99mm, output radius = 42.5mm, Length
= 350mm, No. of sections =350, Shape factor = 0.5331 undergone mode matching technique



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                                                                                               Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 
giving the S21=98.90% at 42GHz which is same off course as seen in figure 6.21 & 6.22,
because in analysis of taper in PSO technique we are using same mode matching technique.

                                                                                       Optim um design Ana lysis::incide nt mode =TE03,Output mode =TE03
                                                                                   1
                                                                                                                           X: 42
                                                            0.9                                                            Y : 0.9889

                Transmission coefficient S21                0.8

                                                            0.7

                                                            0.6

                                                            0.5

                                                            0.4

                                                            0.3

                                                            0.2

                                                            0.1

                                                                                   0
                                                                                    30         32       34    36      38   40     42      44    46      48        50
                                                                                                                     Frequency in GHz


Figure 6.21: Variation of Transmission coefficient with frequency in PSO optimized design

                                                                                             Optimum Design Analysis :incident mode =TE03,Output mode=TE03
                                                                                   1.01



                                                                                         1
                                               Transmission coefficient S21 in %




                                                                                                                        X: 42
                                                                                                                        Y : 0.9889
                                                                                   0.99



                                                                                   0.98



                                                                                   0.97



                                                                                   0.96



                                                                                   0.95
                                                                                      40         40.5    41   41.5    42   42.5    43   43.5   44    44.5    45
                                                                                                                     Frequency in GHz


       Figure 6.22: Variation of Transmission coefficients around desired 42GHz

Power transmitted to spurious modes for optimum design can also obtained as obtained in
Section 5.7 & 5.8. As already mentioned, total power transmitted in desired mode is given by
square of normalized transmission coefficient. Table 6.10 is obtained through dedicated
scattering matrix code [95], it is found the PSO optimized non-linear taper has very low mode
conversion.


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                          Chapter 6: Particle Swarm Optimization of a Cylindrical Waveguide Taper 
 




              Table 6.10: Mode conversion analysis of PSO optimized taper.
                                Normalized transmitted power
                                  Mode No.           % Power
                                   TE 0,1              0.0399
                                   TE 0,2                1.73
                                   TE 0,3               97.79
                                   TE 0,4              0.4199
                                   TE 0,5                0.00
                                   TE 0,6                0.00
                                   TE 0,7                0.00




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                                                     Chapter 7: Design Validation of Nonlinear Taper 
 




                                                                          Chapter 7




Design Validation of Nonlinear Taper



7.1 Introduction:

       In the previous chapter, optimum non linear taper design is obtained using particle
swarm optimization. Various different techniques of optimization also used for profile
optimization of tapers,mode converters & horns[89].This optimum design can be validate
using many simulation software available like High Frequency Structure Simulator
(HFSS)[90] & CST microwave studio[91] . CST Microwave studio work on finite difference
time domain (FDTD) technique & HFSS is based on finite element method (FEM).As these
approaches are totally different from the analytical approach we have used upto this point.
So, design is considered as valid if simulation showing same results as shown analytically
[98, 99].




7.2 Design and Modes:

       We prefer CST Microwave studio for our problem over HFSS because of difficulty in
generation of TE0,3 mode in HFSS. When we are dealing with higher modes or modal
analysis, HFSS become less effective compared to CST microwave studio. The reason behind


Institute of Technology, Banaras Hindu University                                                 98 
 
                                                     Chapter 7: Design Validation of Nonlinear Taper 
 
it is that only 25 modes one can excite in a wave port in HFSS. An integration line method is
proposed for generation of desired mode but it is very complicated process. The taper design
is shown in figure 7.1.




    Figure 7.1: The PSO optimized non-linear taper design in CST microwave studio.

       Number of modes considered at the input and output port of the taper, but we have to
deal with TE0,3 mode only. In input port, 14th mode is corresponding to desired TE0,3 mode.
Similarly at output port, 11th mode is the desired mode as shown in figure 7.2 and 7.3. TE0,3
mode number is calculate(14th incident,11th output) and corresponding S parameters (S21 and
S11) are obtained.




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                                                     Chapter 7: Design Validation of Nonlinear Taper 
 
                             Figure 7.2: TE0,3 mode at input port.




                            Figure 7.3: TE0,3 mode at output port.


7.3 Propagation of TE0,3 mode:

       Input port is excited by TE0,3 mode and propagation of mode can be seen at different
point along the length as shown in figure 7.4.It is found that the mode not undergo any
deterioration and almost same is obtained at time output. It proves that the mode conversion
is very less in the designed taper.




                                               (a)




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                                                     Chapter 7: Design Validation of Nonlinear Taper 
 




                                               (b)




                                               (c)




                                               (d)

       Figure 7.4: TE0,3 mode propagation in the PSO optimized non-linear taper

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                                                     Chapter 7: Design Validation of Nonlinear Taper 
 




7.4 Simulation Results:

         The S parameters calculated by the CST microwave studio are shown in figure 7.5.
14th mode at the input & 11th mode at the output is the desired TE0,3 mode. S21 plot is our
desired, here,S11 is also plotted in the same figure. S21 at 42 GHz is 0.9778 or 97.78% .




    Figure 7.5: The S parameter plots for PSO optimized taper in CST Microwave Studio.




         At the meantime reflection coefficient S11 for desired mode is 0.006665 or 0.6655% at
42 GHz. As already told reflection coefficient is very low in overmoded waveguide but
interesting point to be noted is that around 34-35 GHz ,there is drastic change in reflection
coefficient to the desired TE0,3 mode & it gives a peak (S11=50% approx.). This is because the
cut off frequency of input port for TE0,3 mode is 34.7229 GHz as calculated by equation
(2.11). After this cut off frequency, transmission becomes abruptly high & at the cut off, high
reflection of power in the desired mode is expected.




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                                                                               Chapter 7: Design Validation of Nonlinear Taper 
 




7.5 Comparison of Results:

       The CST microwave studio result is exported to MATLAB and plotted along with the
analytical result of optimum taper in the same frame for comparison as shown in figure 7.6.

                                                           incident mode =TE03,Output mode=TE03
                                              100

                                               90
           Transmission Coefficent S21 in %




                                               80

                                               70

                                               60

                                               50

                                               40

                                               30

                                               20                                              Analytical
                                                                                               CST Simulation
                                               10

                                                0
                                                30   32   34    36      38   40     42    44      46     48     50
                                                                       Frequency in GHz



                                 Figure 7.6: Comparison of results with two different approaches.

       The results obtained are analyzed at 42 GHz and it is found that S21 calculation for
desired mode through CST microwave studio is 97.78% and same thing already obtained is
98.89%

                                                          Table 7.1: Comparison of results at 42GHz

                    Approach                                         S21      Transmitted power in TE0,3 mode

                    Analytical                                       98.89%                 97.79%

                    CST MWS Simulation                               97.78%                 95.61%




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                                                                            Chapter 8: Conclusion 
 




                                                                        Chapter 8




Conclusion

 

Non-linear taper with input radius 13.99 mm ,output radius 42.5 mm & length 350 mm is giving
high transmission of power in desired mode & very less mode conversion is found. Mode
conversion is maximum from TE0,3-TE0,2 and very little in TE0,1 & TE0,4. No power is
transmitted at the output at 42 GHz in the remaining modes.

       An optimum nonlinear taper for 200kW, 42GHz, CW gyrotron is obtained in the work.
The optimum profile improves the transmission of power retain in desired incident mode. TE0,3
mode is the cavity mode in the gyrotron & same is the incident mode to the input of taper. In the
optimum non-linear taper, transmission coefficient, S21= 0.9890(98.9%) is obtain. Transmitted
power in desired TE0,3 mode at output port is 97.79% (|        |    0.9779 of the total power
incident to the taper input at 42 GHz. Transmission of power in TE0,3 mode is very small when
the operating frequency is less than 34 GHz & abruptly high transmission is found after 35 GHz
as the cut off TE0,3 mode frequency of input cross section is around 34.7 GHz.

       The collection of electron beam after RF interaction in gyrotron cavity is collected at
collector waveguide connected at the output of taper. The mode conversion reduction was our
aim and it is minimized throughout the length of taper. The remaining power (2.21%) is
transmitted to spurious modes. The power is transmitted to nearby degenerated modes only.

       Validation of the design in CST microwave studio has been carried out & transmission
coefficient obtained is S21= 0.9778 (97.78%) & total power transmitted at output port in desired

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                                                                             Chapter 8: Conclusion 
 
TE0,3 mode is 95.61% of the total incident power. This result is obtained on CST microwave
studio simulation of design in a 4 GB RAM dual core processor. Because of memory issues
during simulation, time domain analysis is carried out instead of more accurate frequency
domain analysis for our design. Deviation in transmission coefficient calculation is 1.132 %
while deviation in transmitted power in desired mode calculation is 2.23 % in both methods. This
variation in results is obvious as the simulation software working on finite difference time
domain method. Rather, simulation results can be improved and more closeness to analytical
calculation could be achieved.

       This optimum non-linear taper is connected at cavity output radius is 13.99 mm & input
of taper is connected to it. The output of taper is connected to the collector waveguide of radius
42.5 mm. The electron beam is propagating in TE0,3 mode through the taper and collected at the
collector waveguide. Smooth non-linear taper avoids the backward flow of heat, there is no
disturbance in RF interaction in cavity. The optimum non-linear taper design could be done with
700 or 1400 points along the length & corresponding profile variation. This design can be
fabricated & analysis could be done using vector network analyzer. One could get similar results
as obtained using analytical work as well as CST simulation.




Institute of Technology, Banaras Hindu University                                             104 
 
                                                                                Bibliography 
 




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Description: This is M.Tech. dessertation thesis involves deep and detailed study of non linear cylindrical tapers