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This is M.Tech. dessertation thesis involves deep and detailed study of non linear cylindrical tapers
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:firstname.lastname@example.org 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@example.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 . 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  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 . Tomiyasu Klyo in 1971 developed a waveguide taper of minimum length at General Electric Company . Rudolf P Hecken(1972) demonstrated that optimum taper has maximum bandwidth for a given length or minimum length for a given bandwidth . 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 . 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 . In the meantime, Jeff M. Neilson (1989) working on development of cascaded scattering matrix code for analysis of tapers with complicated electromagnetic equations . 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 . 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 . 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), PJB Clarricoats (1971) went for development of computer code at University of Leeds . 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  and CST microwave studio  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 -96] & SCATTMAT  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  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 . 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) Institute of Technology, Banaras Hindu University 7 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 Institute of Technology, Banaras Hindu University 8 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) Institute of Technology, Banaras Hindu University 9 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 . Institute of Technology, Banaras Hindu University 10 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 Institute of Technology, Banaras Hindu University 11 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 . 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 - . Institute of Technology, Banaras Hindu University 12 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. Institute of Technology, Banaras Hindu University 13 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  and Collin  improve the impedance matching issues of transmission lines. Reflection coefficient calculation is sufficient for getting perfect match or optimum design -.. 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 . 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 Institute of Technology, Banaras Hindu University 14 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) Institute of Technology, Banaras Hindu University 15 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 Institute of Technology, Banaras Hindu University 16 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]. Institute of Technology, Banaras Hindu University 17 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 . 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, Institute of Technology, Banaras Hindu University 18 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 , 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 Institute of Technology, Banaras Hindu University 19 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) Institute of Technology, Banaras Hindu University 20 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 Institute of Technology, Banaras Hindu University 21 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 Institute of Technology, Banaras Hindu University 22 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) Institute of Technology, Banaras Hindu University 23 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  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 Institute of Technology, Banaras Hindu University 24 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 , 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  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 taperwith 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 .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  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 . 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 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 Institute of Technology, Banaras Hindu University 44 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 can be obtained as 0 (4.17) . . . . . . . (4.18) Institute of Technology, Banaras Hindu University 45 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 Institute of Technology, Banaras Hindu University 46 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 and shows the mode coupling coefficient for TE01-TE02 obtained is similar to that obtained by Doane 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 Institute of Technology, Banaras Hindu University 47 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 : 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. Institute of Technology, Banaras Hindu University 48 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’ . 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 Institute of Technology, Banaras Hindu University 49 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 , Masterman and Clarricoats  and English . 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  & 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 Institute of Technology, Banaras Hindu University 50 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] =  , (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. 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) Institute of Technology, Banaras Hindu University 52 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 Institute of Technology, Banaras Hindu University 53 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 : 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 Institute of Technology, Banaras Hindu University 56 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. 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 . 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 Institute of Technology, Banaras Hindu University 58 Chapter 5: Analysis of Cylindrical Waveguide Tapers √ . . . (5.45) √ . . It is very difficult to calculate manually, one can use MATHEMATICA  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 . Institute of Technology, Banaras Hindu University 59 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  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 Institute of Technology, Banaras Hindu University 60 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 , a scattering Institute of Technology, Banaras Hindu University 61 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. Institute of Technology, Banaras Hindu University 62 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  & MATLAB 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. Institute of Technology, Banaras Hindu University 63 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 . 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 Institute of Technology, Banaras Hindu University 64 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 = Institute of Technology, Banaras Hindu University 65 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 , 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 Institute of Technology, Banaras Hindu University 66 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 .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 Institute of Technology, Banaras Hindu University 67 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 Institute of Technology, Banaras Hindu University 68 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. Institute of Technology, Banaras Hindu University 69 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. 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 . 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 Institute of Technology, Banaras Hindu University 70 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) Institute of Technology, Banaras Hindu University 71 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. Institute of Technology, Banaras Hindu University 72 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) 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. Institute of Technology, Banaras Hindu University 73 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. Institute of Technology, Banaras Hindu University 74 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) 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 Institute of Technology, Banaras Hindu University 75 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. Institute of Technology, Banaras Hindu University 76 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 . 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) Institute of Technology, Banaras Hindu University 77 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 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 Institute of Technology, Banaras Hindu University 78 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. Institute of Technology, Banaras Hindu University 79 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. Institute of Technology, Banaras Hindu University 80 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. Institute of Technology, Banaras Hindu University 81 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 .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. Institute of Technology, Banaras Hindu University 82 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 . 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. Institute of Technology, Banaras Hindu University 83 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 Institute of Technology, Banaras Hindu University 84 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. Institute of Technology, Banaras Hindu University 85 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. Institute of Technology, Banaras Hindu University 86 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. Institute of Technology, Banaras Hindu University 87 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. Institute of Technology, Banaras Hindu University 88 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. Institute of Technology, Banaras Hindu University 89 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. Institute of Technology, Banaras Hindu University 90 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 as well as mode converters. 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. Institute of Technology, Banaras Hindu University 91 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 Institute of Technology, Banaras Hindu University 92 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, 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% Institute of Technology, Banaras Hindu University 94 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 Institute of Technology, Banaras Hindu University 95 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 , it is found the PSO optimized non-linear taper has very low mode conversion. Institute of Technology, Banaras Hindu University 96 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 Institute of Technology, Banaras Hindu University 97 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.This optimum design can be validate using many simulation software available like High Frequency Structure Simulator (HFSS) & CST microwave studio . 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. Institute of Technology, Banaras Hindu University 99 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) Institute of Technology, Banaras Hindu University 100 Chapter 7: Design Validation of Nonlinear Taper (b) (c) (d) Figure 7.4: TE0,3 mode propagation in the PSO optimized non-linear taper Institute of Technology, Banaras Hindu University 101 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. Institute of Technology, Banaras Hindu University 102 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% Institute of Technology, Banaras Hindu University 103 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 Institute of Technology, Banaras Hindu University 103 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 Bibliography  Unger H G, “Circular Waveguide Taper of Improved Design” Belll System Technical Journel, Vol 37, pp. 899-912,1958.  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"Design, Analysis & optimization of non linear waveguide taper"