Analysis and Design of a Wideband Multibeam Array

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Lim Wai Yean

A THESIS SUBMITTED

FOR THE DEGREE OF PH.D. OF ENGINEERING

DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2010 Acknowledgements

I count it a blessing to be able to meet very special people who help me along the way in life. I am very fortunate to encounter some of these people in my research for this thesis. The thesis would never have been completed if not for them. I would like to specially thank my advisor, Professor Yeo Tat Soon, for his belief that I will be able to complete the research, as well as for his support and encouragement. I am also much indebted to my other advisor, Dr Chan Kwok Kee for his guidance and regular technical discussions. His strong technical background is of tremendous help and his strict adherence to high standards of research has left an indelible mark on me. I am grateful to Dr Chio Tan Huat, who …rst encouraged me to embark on this arduous journey of research, and for being a friend who is ever ready to o¤er his advice when I needed it. I would like to express my appreciation to my friends from DSO National Laboratories, for their ability to lighten my mood when I am down and for being such wonderful friends. It is not possible to list all people who have made a di¤erence in some way or another to this thesis but they’ll always stay in my heart. I would like to thank my family members, especially my parents, for their forbearance and support during these few years. Last but not least, I am eternally grateful to Lord Buddha for his immeasurable wisdom that I have the blessing to learn. His words of wisdom have taught me many lessons which helped me tide over many di¢ cult situations.

"All composite things Are like a dream, a phantasm, a bubble, and a shadow, Are like a dew-drop and a ‡ash of lightning; They are thus to be regarded." - Shakyamuni Buddha Abstract

The geodesic lens is a versatile device that is capable of radiating a beam with omni- directional coverage in azimuth or a directive beam that can be scanned 360o in azimuth. The lens takes the form of a conical coaxial cylinder where the larger end is used to excite a biconical antenna and the smaller end is fed by multiple sectoral waveguides equally distrib- uted around the periphery of the cylinder. These sectoral guides are in turn fed by coaxial probes. To provide matched input ports, the sectoral guides have ridged sections to act as impedance transformer. With the proper amplitude and phase excitations of these probes, power is combined within the lens and radiated out of the bicone antenna. The geodesic lens has …ve major components - the feed section consisting of sectoral waveguides, a junction between a ring of sectoral guides and a coaxial guide, a conical coaxial cylinder, a transition between a coaxial guide and a radial guide, and a series of stepped radial guides approximating a monocone or a ‡ared biconial antenna. This is a thesis on the theoretical developments carried out for the complete electromagnetic analysis of the geodesic lens antenna. The analytical precedure employed is the …eld matching method. Within any cross-section of the lens antenna, the electric and magnetic …elds are represented by a series of natural eigenmodes. For non-canonical cross-sections, the modes are determined by the transverse resonance technique (TRT). Each discontinuity in the lens is characterized by the Generalized Scattering Matrix (GSM). The GSMs of all the junctions are combined together to give an overall description of the performance of the lens.

i Contents

1 Introduction 1

1.1 Thesisobjectives ...... 5 1.2 Organization of thesis ...... 5

2 The geodesic cone 8

2.1 Mode coupling coe¢ cients for a coaxial waveguide step junction ...... 9 2.2 S-matrix of a coaxial waveguide junction ...... 14 2.3 Results...... 15

3 Modelling the Coaxial-Radial waveguide Junction 20

3.1 GAM of Coaxial-Radial Guide Junction: Y11 ...... 21

3.2 GAM of Coaxial-Radial Guide Junction: Y21 ...... 24

3.3 GAM of Coaxial - Radial Guide Junction: Y22 ...... 27

3.4 GAM of Coaxial - Radial Guide Junction: Y12 ...... 30 3.5 Results...... 34

4 Modelling the bicone/monocone antenna 39

4.1 Radial Waveguide Step Junction ...... 41 4.2 Radial Waveguide Step Mode Coupling Coe¢ cients ...... 52 4.3 Determining the S-parameters of a section of radial line ...... 55 4.4 Results...... 57

ii 5 Modes of the ridge sectoral waveguide 61

5.1 Ridge Sectoral Guide Symmetric TMz Mode...... 62

5.2 Ridge Sectoral Guide Asymmetric TMz Mode ...... 69

5.3 Ridge Sectoral Guide Symmetric TEz Mode ...... 72

5.4 Ridge Sectoral Guide Asymmetric TEz Mode...... 77 5.5 Results...... 82

6 Modelling the sectoral-coaxial junction 83

6.1 Coupling coe¢ cients between a sectoral waveguide - coaxial waveguide junction 85 6.2 Results...... 89

7 Geodesics of a cone 96

7.1 Derivation of the geodesic path ...... 97 7.2 DesignProcedure ...... 102 7.3 Results...... 106

8 Conclusions 113

8.1 Summary and conclusions ...... 113 8.2 Furtherwork ...... 115

A Circular Coaxial Waveguide Mode Functions 116

B Cylindrical Radial Waveguide Mode Functions 122

C Sectoral Waveguide Mode Functions 128

iii List of Figures

1.1 16-element circular array providing steered beam...... 2 1.2 Side view of the geodesic lens...... 6 1.3 Perspective view of the geodesic lens...... 7

2.1 Approximating the geodesic cone by a series of cascaded coaxial waveguides. 9 2.2 The coaxial junction to be modelled by mode matching...... 10 2.3 Flowchart showing steps to compute the GSM for a coaxial cone...... 11 2.4 The step-down coaxial junction modelled as a cascade of 3 waveguides. . . . 14 2.5 S11 amplitude. Cone height=101.6mm, bottom inner radius=42.291mm, bottom outer radius=50.621mm, top inner radius=63.4365mm, top outer radius=71.767mm...... 16 2.6 S11 phase of coaxial cone...... 17 2.7 S12 phase of coaxial cone...... 17 2.8 Comparing the phase results of the dominant mode in a shorted cone. . . . . 18 2.9 Comparing the phase results of the second higher order mode in a shorted cone. 18 2.10 Comparing the phase results of the third higher order mode in a shorted cone. 19

3.1 Flowchart showing the steps to computing the S-matrix of a coaxial-radial junction...... 22 3.2 Diagram of the coaxial to radial waveguide coupling ...... 23

iv 3.3 HFSS model to test the modelling of the coaxial-to-radial waveguide junction usingGAM...... 35 3.4 S11 magnitude for test case 1 (a = 0.635mm, b = 1mm, d = 2mm)...... 35 3.5 S11 phase for test case 1...... 36

3.6 Running test case 1 with kmax=3k, 5k and 20k, and with number of modes = 1,3and33respectively...... 36 3.7 S11 amplitude for test case 2 (a = 2mm, b = 4mm, d = 3mm)...... 37 3.8 S11 phase for test case 2...... 37 3.9 Running test case 2 with di¤erent number of modes...... 38 3.10 Return loss of a wideband coaxial to radial waveguide transition. d=0.5mm, b=1mm,a=0.635mm...... 38

4.1 Approximating a bicone with sections of radial waveguides...... 40 4.2 Modelling a radial waveguide as an equivalent transmission line...... 41 4.3 Flowchart showing steps to computing the GSM for a bicone/monocone. . . 42 4.4 Model tested with mode matching and HFSS...... 57 4.5 S11 amplitude comparison between mode matching and HFSS...... 58 4.6 S11 phase comparison between mode matching and HFSS...... 59 4.7 Fabricated monocone...... 59 4.8 Monocone Antenna with dimensions in mm...... 59 4.9 Comparison of measured results with simulation for a monocone antenna. In mode matching, 100 sections are used with each step height = 0.38mm. . . . 60

5.1 Diagram of the cross-section of geodesic cone showing the feeding of the cone by sectoral ridge waveguides...... 62 5.2 Ridge Sectoral Waveguide ...... 63

6.1 Flowchart showing steps to compute the GSM for a coaxial-sectoral waveguide junction...... 84

v 6.2 Rotation of the sectoral guide with respect to the coaxial guide ...... 85 6.3 Two sectoral-coaxial junctions placed back-to-back...... 90 6.4 S11 magnitude of sectoral-coaxial junction for test case 1. Inner radius = 45mm, outer radius = 50mm, coaxial waveguide length = 40mm, 10 sectoral

waveguides. In mode matching, kmax=6k is being used to limit the number of modes. At operating frequency of 10GHz, 470 sectoral modes and 477 coaxial modes are being used. Runtime for the mode matching method is 7 seconds, while that for Ansoft HFSS is 3 minutes 26 seconds...... 91 6.5 S21 amplitude of sectoral-coaxial junction for test case 1...... 92

6.6 Using di¤erent number of modes for test case 1. For kmax=0.2k, 1k, 3k and 8k, number of modes used are 3, 19, 63 and 665 repectively...... 93 6.7 S21(port5,port1) phase of sectoral-coaxial junction for test case 2. Inner radius = 41.275mm, outer radius = 51.275mm, 4 sectoral waveguides. . . . . 94 6.8 S21(port6,port1) phase of sectoral-coaxial junction for test case 2...... 95

7.1 Geodesic path of a ray travelling from input to output of cone...... 97 7.2 Top view of the geodesic path of the lens...... 100 7.3 Top view of geodesic lens showing the reference path travelled by the central ray and a path travelled by another ray...... 102 7.4 Plotting the variation of beamwidth with cone angle. Frequency of operation

= 13GHz, o=42mm; uniform line source is assumed...... 103 7.5 Plotting the variation of linear aperture length with sin( ) ...... 104 7.6 Plotting the variation of beamwidth with cone angle. Frequency of operation

= 13GHz, o=35mm; uniform line source is assumed...... 105 7.7 Top view showing the mapping of the input probe to the maximum output angle. Each probe is given a number...... 106 7.8 Flowchart showing the design procedure of the geodesic lens...... 107 7.9 Calculated radiation pattern at 10GHz for 24 input feeds...... 108

vi 7.10 Calculated radiation pattern at 13GHz for 24 input feeds...... 109 7.11 Calculated radiation pattern at 11GHz for a …xed set of phase excitations. . 110 7.12 Calculated radiation pattern at 12GHz for a …xed set of phase excitations . . 111 7.13 Calculated radiation pattern at 13GHz for a …xed set of phase excitations. . 112

vii List of Tables

2.1 Simulation time when di¤erent maximum number of modes are used . . . . . 16

5.1 Comparison of the …rst 14 cuto¤ propagation constants calculated using TRT and HFSS. Constants are in radians/meter. Dimensions of sectoral ridge are: a=10mm, c=15mm, d=20mm, b=25mm...... 82

viii Chapter 1

Introduction

Antennas are important components in communication systems. They act as the eyes, ears and voices in the wireless application, enabling two users to communicate with each other without the use of cables. Antenna can be passive or active, the di¤erence being the passive antenna does not have external amplifying devices connected to it while active antenna has external devices such as low noise and power ampli…ers which draw power from an external source and thus may face heating issues. To increase the working range of the antenna, the ampli…ers may be improved, or a larger aperture is used. As antenna performance is so critical to system performance, it is imperative that innovative antennas designs be conceived so as to achieve the highest e¢ ciency given a …xed aperture size and a carefully selected ampli…er. An omni-directional and high gain antenna is highly desirable in many applications. For example, in radar, 360o coverage is needed to detect threats all round and high gain is needed for long distance target tracking. However, it is not possible to have high gain together with broad beam coverage, since they are inversely proportional to each other. Instead, what is usually implemented is an electronically or mechanically steered high gain beam to cover a broad sector. The electronically scanned beam is preferable to that of the mechanical one as it is able to direct its beam almost instantaneously and thus improves reaction time of

1 the system. Moreover, mechanical scan requires the use of motor which is susceptible to breakdown. There have been many antenna con…gurations that are devised to provide electronically scanned beams for all round coverage. One such con…guration is by putting the antenna elements in a circular fashion [1][2][3] (…gure 1.1), with each element properly excited to form a beam in the desired direction. Unfortunately, this con…guration involves the excitation of only one half of the array since the other half is in the opposing side and hence is not part of the radiating aperture. This e¤ectively reduces the e¢ ciency of the array to at most 50% and the aperture size has to be increased when high e¤ective radiating power (ERP) [4] is required. Moreover, circular array usually makes use of a complex and lossy network to commutate the excitation around the cylinder [1][5], making implementation complicated and further adding loss.

Figure 1.1: 16-element circular array providing steered beam.

To improve the e¢ ciency of the array, the geodesic lens antenna is conceived [6][7][8]. This geodesic lens antenna is similar to the geodesic luneberg lens antenna which was devised in the early 1960s [9][10][11][12][13][2]. However, there is a main di¤erence between the two in that in the latter case, only one feed point can be used at any time to give a single beam in the required direction, hence the low e¢ ciency problem has not improved. For the lens

2 antenna reported in this thesis, although there is still only half the radiating aperture, this antenna is able to improve the e¢ ciency by exciting all the feeds at the same time, thereby increasing its ERP. The geodesic lens antenna (…gures 1.2 and 1.3) comprises a radiating element and a beamformer called the geodesic cone. The radiating element serves as a conduit to transfer energy from the beamformer to free space, while the beamformer takes care of phasing the electromagnetic waves so that the beam to form a focused beam. The input feeds to the beamformer are located at the bottom of the geodesic cone and they are excited with varying amplitude and phase so as to collimate in di¤erent angles. By making use of the fact that electromagnetic waves, like light waves, travel in the shortest possible path, ray from each feed is traced as it travels between a parallel plate to the output of the cone and is then summed spatially to give a beam in the desired direction. The geodesic antenna can also operate to give an omni-directional, low gain pattern by applying uniform phase and amplitude to each feed, or achieve other beam shapes by appropriately applying the correct signal distributions. Other advantages of the lens include low insertion loss of the parallel plate and its inherent light-weight since technically only sheets of conductor are needed to construct the lens. Forming high gain beams requires a large number of input excitations. To prevent high sidelobes or grating lobes, the input feeds surrounding the bottom coaxial cone should be spaced half a wavelength at the highest frequency. The number of input ports necessitates the development of e¢ cient and accurate modelling tools so that optimizations can be performed as fast as possible and simulation results will be close to that of measurements to avoid empirical tuning. Computational electromagnetics can be divided into two broad categories, namely the numerical and the analytical techniques [14]. Popular numerical techniques include the Finite Element Method (FEM) [15], Method of Moment (MoM) [16] and the Finite Di¤erence Time Domain (FDTD) [17] method. They share the basic principle of subdividing the

3 geometry into a number of small elements, and the determination of the unknown “by the application of the method to the patchwork of elements which approximates the original geometry” [18]. The discretization of the unknown electromagnetic property is known as meshing and it is generally true that the …ner the mesh, the better is the accuracy of the solution. For methods such as FEM and FDTD, the space around the radiating problem has to be meshed and an absorbing boundary has to be imposed, thus adding to the complexity of the problem. Numerical methods are however, very useful in solving general structures that do not have canonical shapes. The Mode Matching Method (MMM) [19][20] belongs to the category of analytical techniques. MMM uses equivalent circuits to represent waveguide discontinuities in terms of lumped circuit elements. It is an accurate method as opposed to single mode equivalent circuits because it takes care of the interaction between higher order modes that are excited by the junction discontinuity. This is especially important where distances between discontinuities are not large and higher modes that are excited at one junction are near enough to the next junction to a¤ect it drastically [21]. Provided that su¢ cient number of modes are being considered in MMM, it is very accurate and so it is frequently used to calculate the generalized scattering matrix for canonical waveguide discontinuities [22][23][24]. On a side note, MMM is fundamentally a moment method which is based on the approximation of unknowns by Fourier series [26]. The geodesic lens antenna can be subdivided into components in which di¤erent techniques are used to solve them. There are three main components, namely, the feed structure, the coaxial cone and the radiating element. A circular array of ridge sectoral feeds, each of whom is probe-fed, is used to propagate energy into the geodesic cone, which is a tapered coaxial waveguide. The sectoral to coaxial junction can be found by MMM using the mode functions for sectoral and coaxial waveguides [27]. MMM is used to model the geodesic cone which is approximated as a series of waveguide step discontinuities cascaded with short sections of coaxial waveguide [28][29][30][31][32][33]. The coupling of the energy out of the cone for radiation is through a coaxial-to- radial waveguide junction which is best modelled

4 using the Generalized Admittance Matrix (GAM) since it is the “natural” representation for cavities with ports realized on conducting walls” [34]. The GAM method is a way of evaluating multimode network representation in terms of admittance parameters [35][36][37]. The S-parameters for each component are then linked together using the Generalized Scat- tering Matrix (GSM) [38][39][40][41]. An antenna suitable as the radiating element is the bicone or monocone since they are well-known for their broad bandwidth [42]. Similar to the analysis for the coaxial cone, the radiating element can be approximated by a series of cascaded radial waveguide, with their discontinuities solved by MMM. Other kind of radiating elements such as slots or Vivaldi can be used but absorbing materials need to be placed in certain locations to prevent re‡ection back to the feed [43].

1.1 Thesis objectives

The motivation for this dissertation is to develop an e¢ cient and accurate technique to solve for the geodesic lens antenna. As the geodesic lens antenna is usually big in size and requires a large number of feeds, commercial software is usually not able to solve it or will take unduly long time to do so. To verify the code, individual parts of a small geodesic lens will be simulated with the developed code and compared with the commercial software Ansoft HFSS. A computer with a 2.8 GHz Intel Core 2 Duo processor is being used for the simulations.

1.2 Organization of thesis

Chapter 2 is devoted to the modelling of the geodesic cone that is essentially a tapered coaxial waveguide. The coupling equations are presented and results are compared with commercial software to demonstrate the accuracy of using MMM. Chapter 3 consists of the formulation for a coaxial-to-radial waveguide junction using the GAM method.

5 Figure 1.2: Side view of the geodesic lens.

Chapter 4 gives the formulation for the solving of a monocone antenna using MMM. Chapter 5 is devoted to …nding the modes of a sectoral ridge waveguide using the Trans- verse Resonance Technique (TRT). The sectoral ridge waveguide is used as the feed to the coaxial cone. However, the problem of using multiple sectoral ridge waveguides for impedance transformation will not be considered in this thesis. Chapter 6 comprises the formulation of the coaxial to sectoral waveguide discontinuity. Chapter 7 presents the equations that de…ne the geodesic path of rays that leave a given feed point. These equations will be used to generate the signal phase to be excited at each feed Chapter 8 concludes the thesis with a summary of what have been done and also suggests future work.

6 Figure 1.3: Perspective view of the geodesic lens.

7 Chapter 2

The geodesic cone

The geodesic cone is a tapered coaxial waveguide that can be approximated by a series of partially overlap step sections of coaxial waveguide (…gure 2.1) . The blown up view of the discontinuity between two o¤set coaxial waveguides is shown in …gure 2.2. The mode matching method is employed to derive the generalized scattering matrix of the cone as it is computationally fast and accurate. The modal content of the coaxial cone can also be easily determined and thus may be controlled by changing the dimensions of the inner and outer conductors. The ‡owchart in …gure 2.3 illustrates the sequence in which the steps are performed to achieve the generalized scattering matrix for the coaxial cone. The steps start with divid- ing the cone into tractable sections, followed by …nding the modal content of the sections under consideration, after which the coupling coe¢ cient is calculated for the discontinuity in-between the two sections. The S-matrix can be easily computed given the coupling coef- …cient. The S-matrix of the discontinuity is then cascaded with short sections of intercon- necting transmission lines, and the process repeats for the next section of coaxial waveguide. The following sections discussed more on the computation of the coupling matrix.

8 Figure 2.1: Approximating the geodesic cone by a series of cascaded coaxial waveguides.

2.1 Mode coupling coe¢ cients for a coaxial waveguide

step junction

In the following evaluation of the mode coupling integral, the i-th mode refers to a mode of guide 1 with mode indices (m; n) and the j-th mode refers to a mode of guide 2 with mode indices (p; q). There are two orientations that a mode can be in. They are the vertical orientation (named as the ‘V ’group) and the horizontal orientation (named as the ‘H’group). The coupling coe¢ cient of a coaxial waveguide step junction is computed from the following expression.

(2) b2 2 n;m j (2) (1) P = ej !hi zdrdr^ (2.1) ij v (1) ! ui u aZ2 Z0 h i t where is the mode impedance of the coaxial waveguide, n; m represent either TE, TM or TEM modes, indices, and 1,2 represent 2 adjoining waveguide regions. ej and hi are derived from the electric and magnetic mode functions [44]. A permutation of the TE, TM and TEM modes therefore requires nine coupling coe¢ cient integrals [45] although some integrals are zero and need not be evaluated [46].

TMmn mode circular coaxial waveguide 1 - TMpq mode circular coaxial waveguide 2

9 Figure 2.2: The coaxial junction to be modelled by mode matching.

(Group ‘V ’)

(2) ea 2 ea j i j (2) ea 2 (1 + 0m) a1 a2 ea for m = p = 0 i j i 8 r a2 a1 6 ea ea j i > e e a e b e b and = > 0 ea ea 2 0 ea 2 2 a2 a1 > a2m j m i b2m j m 6 > a1 a2 a1 > > h i > 0 for m = p > 6 V > Pij = > (2) for m = p = 0 (2.2) > j > (1) 6 <> ea ea i and j = i r a2 a1 > (2) ea 2 ea > j i j 2 > (2) ea 2 > a1 a2 ea for m = p = 0 > i j i > r a2 a1 > ea ea > j > ea ea eaa i > e j e i e ea e i 2 and a = a > b21 b2 0 b2 a21 j 0 2 6 1 > a2 a1 a1 > > n o :> TEM mode circular coaxial waveguide 1 - TEM mode circular coaxial waveguide 2 (Group ‘V ’)

(2) b2 b2 ln a ln a P V = j 2 = 2 ij (1) v v (2.3) v 2 b1 3 2 h b1 i3 ui u ln u ln u a1 u a1 u u u t t4 5 t4 h i5 TM0n mode circular coaxial waveguide 1 - TEM mode circular coaxial waveguide 2

10 Figure 2.3: Flowchart showing steps to compute the GSM for a coaxial cone.

11 (Group ‘V ’) (2) V j 2 e ea b2 e ea a2 Pij = (1) 0 i 0 i (2.4) v a1 a1 u i b2 u 2 ln a t 2 r TEM mode circular coaxial waveguide 1 - TM0q mode circular coaxial waveguide 2 (Group ‘V ’)

V Pij = 0

TEmn mode circular coaxial waveguide 1 - TEpq mode circular coaxial waveguide 2 (Group ‘V ’)

(2) ha ha 2 j i j (1) ha 2 2 a1 a2 ha i j i for m = p = 1; 2; 3; ::: 8 r a2 a1 ! ha ha > j > and = i > h ha h0 ha a2 h ha b2 h0 ha b2 a2 a1 > a2 + b2 6 > m j m i a1 m j a2 m i a1 V > Pij = > h i > 0 m = p <> 6 (2) ha ha j i j (1) for m = p = 1; 2; 3; ::: a1 a2 2 > i > r ha ha > j > 2 h ha h00 ha 2 h ha b2 h00 ha b2 i > a b and a = a > 2 m j m j 2 m j a2 m j a2 2 1 > > h i (2.5) :> TEmn mode circular coaxial waveguide 1 - TMpq mode circular coaxial waveguide 2 (Group ‘V ’) There is no coupling between the TE modes of guide 1 and TM modes of guide 2, i.e.

V Pij = 0

TEM mode circular coaxial waveguide 1 - TEpq mode circular coaxial waveguide 2 (Group ‘V ’) There is no coupling between the TEM modes of guide 1 and TE modes of guide 2.

V Pij = 0

12 TEmn mode circular coaxial waveguide 1 - TEM mode circular coaxial waveguide 2 (Group ‘V ’) There is no coupling between the TE modes of guide 1 and TEM modes of guide 2.

V Pij = 0

TMmn mode circular coaxial waveguide 1 - TEpq mode circular coaxial waveguide 2 (Group ‘V ’)

(2) b b a P V = j e ea 2 h ha 2 h ha e ea 2 (2.6) ij v (1) m i a m j a m j m i a ui 1 2 1 u t

TMmn mode circular coaxial waveguide 1 - TMpq mode circular coaxial waveguide 2 (Group ‘H’)

H V Pij = Pij , m = p = 1; 2; ::: (2.7)

TMmn mode circular coaxial waveguide 1 - TEpq mode circular coaxial waveguide 2 (Group ‘H’)

P H = P V , m = p = 1; 2; ::: (2.8) ij ij

TEmn mode circular coaxial waveguide 1 - TEpq mode circular coaxial waveguide 2 (Group ‘H’)

H V Pij = Pij (1 + 0m) , m = p = 0; 1; 2; ::: (2.9)

TEmn mode circular coaxial waveguide 1 - TMpq mode circular coaxial waveguide 2 (Group ‘H’) There is no coupling between TE modes of guide 1 and TM modes of guide 2, i.e.

H Pij = 0

13 2.2 S-matrix of a coaxial waveguide junction

Figure 2.4: The step-down coaxial junction modelled as a cascade of 3 waveguides.

The conical geodesic lens cylinder is represented by a series of partially overlap step sections of coaxial waveguide, as shown in …gure 2.4. An e¢ cient way of treating this partially overlap step junction, which consists of an input waveguide 1, an output waveguide 2 and the in-between common aperture represented as waveguide 0, is to consider all three waveguides together to give a S-matrix characterization of the junction.

(0;1) 1 (0;1) T [S11] = 2 P [W ] P [I] (2.10)

(0;1) 1 (0;2) T [S12] = 2 P [W ] P (2.11)

(0;2) 1 (0;1) T T [S21] = 2 P [W ] P = [S12] (2.12)

(0;2) 1 (0;2 T [S22] = 2 P [W ] P [U] (2.13) 14 where T T [W ] = P (0;1) P (0;1) + P (0;2) P (0;2)

[U] and [I] are identity matrices P (0;1) = mode coupling matrix between coaxial waveguide 0 and 1 P (0;2) = mode coupling matrix between coaxial waveguide 0 and 2 Both mode coupling matrices can be found in the previous section.

2.3 Results

As the coaxial cone is a critical portion of the geodesic structure, it is important to ensure the accuracy given by the mode matching method. Two software packages Ansoft HFSS and FEKO are used to simulate a test case of the cone. HFSS uses FEM while FEKO uses MoM/MLFMM. For waveguide structures, HFSS is able to solve the problem relatively quickly while FEKO could only solve it at the lower frequencies. In simulating the cone using MMM, the number of higher order modes with cuto¤ frequency smaller than kmax

will be taken into account. kmax is usually set to four times or more that of the operating frequency, but it may be a matter of trial and error to ensure convergence . A total of thirty

sections is being used so that the step at each junction is less than 20 wavelength. Figures 2.5 and 2.6 show the S11 of the dominant mode for a matched coaxial cone, simulated using the mode matching method and HFSS . Figure 2.7 shows the S12 phase results for the same cone. Results from the mode matching method are in excellent agreement with that of HFSS. To check the higher order mode results for the cone, the larger end of the cone is terminated with a short-circuit and the phase parameter of the higher order mode is compared. The dimensions of the cone remains the same as the un-shorted case above. Figures 2.8, 2.9 and 2.10 show the comparison for the …rst three modes computed using mode matching and Ansoft HFSS. The results are in excellent agreement and this shows

15 Figure 2.5: S11 amplitude. Cone height=101.6mm, bottom inner radius=42.291mm, bottom outer radius=50.621mm, top inner radius=63.4365mm, top outer radius=71.767mm.

kmax=0.2k kmax=1k kmax=3k Number of modes used 5 29 221 Time to compute 52sec 1min 58sec 17min 55sec

Table 2.1: Simulation time when di¤erent maximum number of modes are used the mode matching method as an accurate tool. Figure 2.3 compares the S-matrices for the coaxial cone when di¤erent number of modes are being used. Table 2.1 lists the maximum number of modes used for di¤erent kmax and the time taken to complete the simulations. As a comparison, HFSS takes 1 minute 55 seconds to …nish the simulation. This shows that although more modes can be used to give a higher accuracy, a balance has to be struck between simulation time and accuracy. Simulation results show that using a smaller kmax give fairly accurate answers without long computational time, and for a designer who needs …rst-cut results quickly, this may be a viable option.

16 Figure 2.6: S11 phase of coaxial cone.

Figure 2.7: S12 phase of coaxial cone.

17 Figure 2.8: Comparing the phase results of the dominant mode in a shorted cone.

Figure 2.9: Comparing the phase results of the second higher order mode in a shorted cone.

18 Figure 2.10: Comparing the phase results of the third higher order mode in a shorted cone.

19 Chapter 3

Modelling the Coaxial-Radial waveguide Junction

Electromagnetic wave from the geodesic cone needs to be coupled out into free space for radiation. Radiation may be achieved by attaching antenna elements to the circumferential slots at the output of the cone, or by coupling energy from the slots to a radial waveguide. The transition from a coaxial to radial waveguide is being modelled here for the possibility of using a monocone or bicone as the antenna element. The transition from coaxial to radial line has been studied by a number of researchers because of their wide usage in many microwave devices [55][56][57]. Applications include power dividers/combiners as well as in antenna feed structures. Previous works are exper- imental or empirical [58], or based on the mode expansion method [59]. While the latter method o¤ers an accurate way to analyze the junction, it has been used only for the case where there is rotational symmetry. This is not applicable to the analysis of the geodesic lens antenna where the beam is scanned and there is azimuthal variation. The generalized admittance matrix (GAM) approach is used in this thesis to model the junction between the coaxial cone and the radial waveguide. In this approach, GAM is used instead of generalized scattering matrix (GSM) as it is mathematically simpler and able to analyze complicated

20 multiport networks without large matrix inversion [34]. Even with its advantages, the GAM method should be carefully used because of singularity issues that may arise [34]. This singularity issue can be avoided in the analysis of the coaxial-radial junction by using only a small admittance matrix and a short section of waveguide between the discontinuities. Figure 3.1 shows the ‡owchart broadly illustrating the steps to calculating the S-matrix of the coaxial-radial waveguide junction. The steps start with calculating the modal content for the coaxial and radial waveguides, iterating through each mode in the two waveguides when calculating for the admittance matrix of the junction and converting the admittance matrix to S-matrix. The calculation of the admittance matrix of the discontinuity is detailed in the next few sections.

3.1 GAM of Coaxial-Radial Guide Junction: Y11

A PEC is placed across port 2, the radial port, while the coaxial waveguide port is excited by

mode i. The tangential electric Et and magnetic …elds Ht across the guide cross-section may be represented in terms of modal voltage V , current I, wave amplitudes and mode vectors as follows.

iz iz E1t = Vieit = Aie + Bie eit (3.1)

iz iz [Aie Bie ] hit H1t = Iihit = (3.2) Zi

At port 1, we are looking into a coaxial waveguide that is shorted by the top wall of the radial guide. A is the amplitude of the incident wave; B is that of the re‡ected wave; i is the propagation constant and Zi is the mode impedance. Since the tangential electric …eld vanished at z=d, we …nd that

21 Figure 3.1: Flowchart showing the steps to computing the S-matrix of a coaxial-radial junction.

22 Figure 3.2: Diagram of the coaxial to radial waveguide coupling

2 id Bi = Aie (3.3)

The voltage and current coe¢ cients of mode i then become

id Vi (z) = 2Aie sinh [ (d z)] (3.4) i

2 id Ii (z) = Aie cosh [ i (d z)] (3.5) Zi

The self admittance at port 1 is given as the ratio of the current and voltage coe¢ cients.

Ii (z = 0) 1 Y11;ii = = coth ( id) (3.6) Vi (z = 0) Zi

Since there is no discontinuity inside the shorted coaxial guide, there is no mode conver- sion. Yii is therefore a diagonal matrix with the element given by the above expression. It applies to the TE, TM and TEM modes.

23 3.2 GAM of Coaxial-Radial Guide Junction: Y21

With mode i incident at port 1, the tangential magnetic …eld across port 2, which is short- circuited, is given by

V H = I h + i h z^ (3.7) 2t;i i it j iz

However, the tangential magnetic …eld in port 2 (r=b1) may also be expanded in terms of the modes of the radial waveguide connected to the port i.e.

H2t = Ilhlt (3.8) l X where

d 2

Il = elt H2t r^ b1 d dz (3.9) Z0 Z0 In equation 3.9, the current coe¢ cient is obtained through mode orthogonality. Substi- tuting the tangential magnetic …eld H2t;i in equation 3.7 for H2t in equation 3.9 we get the induced current in port 2 due to voltage excitation in port 1.

d 2

Il;i = e1t H2t;i r^ b1 d dz (3.10) Z0 Z0

From equations 3.4 and 3.10, the mutual admittance Y21 between port 2 and port 1 is found to be

d 2 elt H2t;i r^ b1 d dz 0 0 Y21;li = d (3.11) R 2RAie i sinh [ (d z)] i In the circular coaxial waveguide of port 1, there are TE and TM modes, belonging to two di¤erent groups or orientations, and the TEM mode. In the radial waveguide of port

24 2, there are TE and TM modes also with two di¤erent orientations. The properties of these modes are found in the Appendix B. The mutual admittance of eqn 3.11 has been evaluated for the various combinations of modes types and orientations for the two ports. The equations are given below. The nomenclatures used are as follows. For port 1 or guide 1, i = mode number = (m; n), m = 1st mode index, n = 2nd mode index For port 2 or guide 2, l = mode number = (u; v), u = 1st mode index, v = 2nd mode index

Calculation of Y21: TM Mode port 1 - TM Mode port 2

ee For modes of di¤erent orientation, i.e. ‘V’- ‘H’, Y21;li = 0 For modes of the same orientations and m = u = 0, 6

d " b e eb vz ee v 1 i e0 i 1 e Y21;li = e e m cos cosh [ i (d z)] dz (3.12) dZi sinh ( i d) a1 a1 d Z0

For modes of the same ‘V ’orientations and m = u = 0,

d 2b " e eb vz ee 1 v i e0 i 1 e Y21;li = e e m cos cosh [ i (d z)] dz (3.13) dZi sinh ( i d) a1 a1 d Z0

ee For modes of the same ‘H’orientations and m = u = 0, Y21;li = 0

Calculation of Y21: TEM Mode port 1 - TM Mode port 2

The TEM mode is considered as part of the TM mode with ‘V’orientation. For u = 0, Y ee = 0 6 21;li ee For modes of ‘H’orientation and u = 0, Y21;li = 0 For modes of ‘V’orientation and u = 0,

25 j 2 " Y ee = v i (3.14) 21;li e 2 v 2 Zi d 2 ln b1 i d a1 r i = j i = propagation constant of TEM mode

Calculation of Y21: TE Mode port 1 - TM port 2

he For modes of di¤erent orientations, Y21;li = 0

he For modes of the same orientation and m = u = 0, Y21;li = 0 For modes of the same orientation and m = u = 0, 6

" m hb +1 Y he = v h i 1 21;li dZh sinh hd m a 8 9 i i 1 > 1 > < = d kh v ci vz > h > hd k sin d sinh: i (d; z) + i v 8 9 dz (3.15) vz h Z0 > cos cosh i (d z) > < d = :> ;> Calculation of Y21: TM Mode port 1 - TE Mode port 2

eh There is no coupling between TM modes of port 1 and TE modes of port 2, i.e. Y21;li = 0

Calculation of Y21: TEM Mode port 1 - TE Mode port 2

eh There is no coupling between the TEM mode of port 1 and TE modes of port 2, i.e. Y21;li = 0

Calculation of Y21: TE Mode port 1 - TE Mode port 2

hh For modes of di¤erent orientation, Y21;li = 0 For modes of the same orientation and m = u = 0, 6

h h h hh 1 kci n i 1 h i b1 Y21;li = 2 m (3.16) j k d a1 h + n a1 i d 26 is the impedance of the medium.

hh For modes of the same ‘V’orientation and m = u = 0, Y21;li = 0 For modes of the same ‘H’orientation and m = u = 0;

h h h hh 1 kci n i 1 h i b1 Y21;li = 2 m (3.17) j k d a1 h + n a1 i d 3.3 GAM of Coaxial - Radial Guide Junction: Y22

A PEC is placed across port 1, the circular coaxial waveguide port. Port 2, the radial

waveguide port, is excited by mode l. The radial waveguide is shorted at r=a1 by the centre conductor of the coaxial guide of port 1. The incident and re‡ected waves at port 2 are used to compute the self admittance as shown below. Depending on the location of port 2 and the mode type, the equation for the self admittance changes with the radial functions used.

Calculation of Y22: TMz Modes

In the above cut-o¤ region, the tangential electric Et and magnetic …elds Ht across the guide cross-section may be represented in terms of modal voltage V , current I, wave amplitudes and mode vectors as follows.

e e e (1) e (2) E2t = Vi elt = Al Hu (kvr) + Bl Hu (kvr) elt (3.18)

j kv ee e (1)0 e (2)0 H2t = Il hlt = e Al Hu (kvr) + Bl Hu (kvr) hlt (3.19) Zl kl h i At port 2, we are looking into a radial waveguide that is shorted by the centre conductor

of the coaxial guide. Al is the amplitude of the incident wave; Bl is that of the re‡ected

e wave; kl is the propagation constant and Zl is the mode impedance. Since the tangential

electric …eld vanishes at r=a1, we …nd that

27 (1) e e Hu (kva1) Bl = Al (2) (3.20) Hu (kva1) The voltage and current coe¢ cients of mode l then become

e e Al (1) (2) (2) (1) Vl (r) = (2) Hu (kvr) Hu (kva1) Hu (kvr) Hu (kva1) (3.21) Hu (kva1)

j k Ae e v l (1)0 (2) (2)0 (1) Il (r) = e (2) Hu (kvr) Hu (kva1) Hu (kvr) Hu (kva1) (3.22) Zl kl Hu (k a ) v 1 h i

The self admittance at port 2 is given as the ratio of the current and voltage coe¢ cients.

e (1)0 (2) (2)0 (1) ee Il (r = b1) j kv Hu (kvb1) Hu (kva1) Hu (kvb1) Hu (kva1) Y22;ll = e = e (1) (2) (2) (1) (3.23) Vl (r = b1) Zl kl " Hu (kvb1) Hu (kva1) Hu (kvb1) Hu (kva1) #

Since there is no discontinuity inside the shorted coaxial guide, there is no mode conver-

sion. Y22 is therefore a diagonal matrix with the element given by the above equation, It applies to all the TE and TM modes.

st nd In the transition region, Bessel functions of the 1 kind and 2 kind, Yu are used for the radial functions instead of the Hankel functions, so that the self admittance becomes

e 0 0 ee Il (r = b1) j kv Ju (kvb1) Yu (kva1) Yu (kvb1) Ju (kva1) Y22;ll = e = e (3.24) V (r = b1) Z kl Ju (kvb1) Yu (kva1) Yu (kvb1) Ju (kva1) l l

In the cut-o¤ region, the Bessel functions are replaced by the modi…ed Bessel function of

st nd the 1 kind, Iu, and 2 kind, Ku, so that the self admittance is rewritten as

e 0 0 ee Il (r = b1) j kv Iu (kvb1) Ku (kva1) Ku (kvb1) Iu (kva1) Y22;ll = e = e (3.25) V (r = b1) Z kl Iu (kvb1) Ku (kva1) Ku (kvb1) Iu (kva1) l l

28 Calculation of Y22: TEz Modes

In the above cut-o¤ region, the tangential electric Et and magnetic Ht across the guide cross-section may be represented in terms of modal voltage V , current I, wave amplitudes and mode vectors as follows.

kv h h h h (1)0 h (2)0 h E2t = Vi elt = jZl Al Hu (kvr) + Bl Hu (kvr) elt (3.26) kl h i hh h (1) h (2) h H2t = Il hlt = Al Hu (kvr) + Bl Hu (kvr) hlt (3.27) At port 2, we are looking into a radial waveguide that is shorted by the centre conductor of the coaxial guide. Al is the amplitude of the incident wave; Bl is that of the re‡ected

h wave; kl is the propagation constant and Zl is the mode impedance. Since the tangential electric …eld vanishes at r=a1, we …nd that

(1)0 h h Hu (kva1) Bl = Al (3.28) (2)0 Hu (kva1)

The voltage and current coe¢ cients of mode i then become

k Ah h v h l (1)0 (2)0 (2)0 (1)0 Vl (r) = j Zl Hu (kvr) Hu (kva1) Hu (kvr) Hu (kva1) (3.29) (2)0 kl Hu (k a ) v 1 h i

Ah h l (1) (2)0 (2) (1)0 Il (r) = Hu (kvr) Hu (kva1) Hu (kvr) Hu (kva1) (3.30) (2)0 Hu (k a ) v 1 h i The self admittance at port 2 is given the ratio of the current and voltage coe¢ cients.

h (1) (2)0 (2) (1)0 hh Il (r = b1) 1 kl Hu (kvb1) Hu (kva1) Hu (kvb1) Hu (kva1) Y22;ll = = (3.31) h h (1)0 (2)0 (2)0 (1)0 Vl (r = b1) jZl kv "Hu (kvb1) Hu (kva1) Hu (kvb1) Hu (kva1)#

Since there is no discontinuity inside the shorted coaxial guide, there is no mode conver- sion. Y22 is therefore a diagonal matrix with the element given by the above equation. It

29 applies to all the TE and TM modes.

st nd In the transition region, Bessel functions of the 1 kind,Ju , and 2 kind, Yu , are used for the radial functions instead of the Hankel functions. The self admittance then becomes

0 0 hh 1 kl Ju (kvb1) Yu (kva1) Yu (kvb1) Ju (kva1) Y22;ll = h (3.32) jZ kv J 0 (kvb1) Y 0 (kva1) Y 0 (kvb1) J 0 (kva1) l u u u u In the cut-o¤ region, the Bessel functions are replaced by the modi…ed Bessel function of

st nd the 1 kind, Iu, and 2 kind, Ku, so that the self admittance is rewritten as

0 0 hh 1 kl Iu (kvb1) Ku (kva1) Ku (kvb1) Iu (kva1) Y22;ll = h (3.33) jZ kv I 0 (kvb1) K 0 (kva1) K 0 (kvb1) I 0 (kva1) l u u u u

3.4 GAM of Coaxial - Radial Guide Junction: Y12

With mode l incident at port 2, the tangential magnetic …eld across port 1, which is short circuited, is given by V H = I h ^ + l h r^ (3.34) 1t;l l l j lr

The tangential magnetic …eld in port 1 (z=0) may also be expanded in terms of the modes of the circular coaxial waveguide connected to the port, i.e.

H1t = Iihit (3.35) i X where

b1 2

Ii = eit H1t z^ r d dr (3.36) aZ Z0 1 Equation 3.36 is obtained by taking the inner product of equation 3.35 with the electric …eld mode vector. The current coe¢ cient results from mode orthogonality. Substituting the tangential magnetic …eld H1t;l in equation 3.34 for H1t in equation 3.36 we get the induced

30 current in port 1 due to voltage excitation in port 2.

b1 2

Ii;l = eit H1t;l z^ r d dr (3.37) aZ Z0 1

The mutual admittance Y12 between port 1 and port 2 is found to be

b1 2 eit H1t;l z^ r d dr a1 0 Y12;il = (3.38) R R Vl (r =b1)

In the circular coaxial waveguide of port 1, there are TE and TM modes, belonging to two di¤erent groups or orientations, and the TEM mode. In the radial waveguide of port 2, there are TE and TM modes also with two di¤erent orientations. The mutual admittance of equation 3.38 has been evaluated for the various combinations of modes types and orientations for the two ports. The results are summarized below. The nomenclatures used are as follows. For port 1 or guide 1, i = mode number = (m; n), m = 1st mode index, n = 2nd mode index For port 1 or guide 1, l = mode number = (u; v), u = 1st mode index, v = 2nd mode index

Calculation of Y12: TM Mode port 1 - TM Mode port 2

The voltage and current coe¢ cients of mode l in port 2 are

e e Al Vl (r) = [Ju (kvr) Yu (kva1) Yu (kvr) Ju (kva1)] (3.39) Yu (kvr)

j k Ae e v l 0 0 Il = e Ju (kvr) Yu (kva1) Yu (kvr) Ju (kva1) (3.40) Zl kl Yu (kva1) h i Substituting equations 3.34, 3.39 and 3.40 into 3.38, we arrive at the following expressions

31 for the mutual admittance.

ee For modes of di¤erent orientation, i.e. ‘V’- ‘H’, Y12;li = 0

For modes of the same orientation and m = u = 0, 6

ee Ii;l (z = 0) jk"v Y12;il = = Vl (r = b1) kvd [Ju (kvb1) Yu (kva1) Yu (kvb1) Ju (kva1)] b1 e er i e0 i 0 0 m Ju (kvr) Yu (kva1) Yu (kvr) Ju (kva1) + a1 a1 r dr (3.41) 8 2 er 9 1 m e i aZ > m [Ju (kvr) Yu (kva1) Yu (kvr) Ju (kva1)] > 1 < kv r a1 = :> ;> This radial integral has to be evaluated numerically.

ee For modes of the same ‘H’orientation and m = u = 0, Y12;li = 0 For modes of the same ‘V’orientation and m = u = 0,

ee Ii;l (z = 0) jk2"v Y12;il = = Vl (r = b1) kvd [J0 (kvb1) Y0 (kva1) Y0 (kvb1) J0 (kva1)] b1 e er i e0 i 0 0 0 J0 (kvr) Y0 (kva1) Y1 (kvr) J0 (kva1) r dr (3.42) a1 a1 aZ1 h i

where

er er er e0 i p i e i e 0 = J1 Y0 ( i ) + Yl J0 ( i ) (3.43) 2 e a1 Yo ( ) a1 a1 2 i 1 Y 2 c e 0 ( i ) r

Calculation of Y12: TEM Mode port 1 - TM port 2

The TEM mode is considered as part of the TM mode with ‘V’orientation. For u = 0, Y ee = 0 6 12;il ee For modes of ‘H’orientation in port 2 and u = 0, Y12;il = 0 For modes of ‘V’orientation in port 2 and u = 0,

32 jk2" Y ee = v (3.44) 12;il dk2 2 ln b1 v a1 r

Calculation of Y12: TE Mode port 1 - TM Mode port 2

he For modes of di¤erent orientations, Y12;il = 0

he For modes of the same orientation and m = u = 0, Y12;il = 0 For modes of the same orientation and m = u = 0, 6

H +1 ee Ii;l (z = 0) jk"vm Y12;il = = 8 9 Vl (r = b1) kvd [Ju (kvb1) Yu (kva1) Yu (kvb1) Ju (kva1)] > 1 > < =V

b1 er h i 0 0 > > m Ju (kvr) Yu (kva1) Yu (kvr) Ju (kva1) + : ; a1 dr (3.45) 8 h hr 9 1 i h0 i aZ > m [Ju (kvr) Yu (kva1) Yu (kvr) Ju (kva1)] > 1 < kv a1 a1 = :> ;> Calculation of Y12: TE Mode port 1 - TE Mode port 2

hh For modes of di¤erent orientation, Y12;il = 0

hh For modes of the same ‘V’orientation and m = u = 0, Y12;il = 0 For modes of the same ‘H’orientation and m = u = 0,

h hh j i v Y12;il = kb1 J 0 (kvb1) Y 0 (kva1) Y 0 (kvb1) J 0 (kva1) a1 d 0 0 0 0 b1 hr h0 i 0 0 0 0 0 J0 (kvr) Y0 (kva1) Y0 (kvr) J0 (kva1) r dr (3.46) a1 aZ1 h i

For modes of the same orientation and m = u = 0, 6

hh j v Y12;il = kb1 [J 0 (kvb1) Y 0 (kva1) Y 0 (kvb1) J 0 (kva1)] d u u u u b1 2 hr m 1 h i 0 0 m Ju (kvr) Yu (kva1) Yu (kvr) Ju (kva1) r kv a1 r dr(3.47) 8 h h hr 9 i 0 i 0 0 0 0 aZ > + m Ju (kvr) Yu (kva1) Yu (kvr) Ju (kva1) > 1 < a1 a1 = > > : 33 ; Calculation of Y12: TM Mode port 1 - TE Mode port 2

eh For modes of di¤erent orientation, Y12;il = 0 For modes of the same orientation and m = u, Y eh = 0 6 12;il For modes of the same orientation and m = u = 0, 6

H +1 eh m v Y12;il = 8 9 jkb1 [J 0 (kvb1) Y 0 (kva1) Y 0 (kvb1) J 0 (kva1)] d u u u u > 1 > < =V

b1 e er i 1 e0 i 0 > 0 > m Ju (kvr) Yu (kva1) Yu (kvr:) Ju (k;va1) a1 kv a1 dr (3.48) 8 e er 9 i 0 0 0 0 aZ > +m Ju (kvr) Yu (kva1) Yu (kvr) Ju (kva1) > 1 < a1 = > > : eh ; For modes of the same orientation and m = u = 0, Y12;il = 0

Calculation of Y12: TEM Mode port 1 - TE Mode port 2

There is no coupling between the TEM mode of port and the TE mode of port 2.

eh Thus Y12;il = 0

3.5 Results

Results computed from the GAM are compared with HFSS. In HFSS, the radial waveguide port is terminated with a radiating/absorbing boundary so that only the return loss at the input coaxial port is being tested. The HFSS model is shown in …gure 3.3 and results are shown in …gures 3.4, 3.5, 3.7 and 3.8. Both results agree well with each other. Figure

3.6 compares the results for test case one when di¤erent kmax are being used. For kmax=3k and 5k, there is no di¤erence in their results while for kmax=20k, the result is only slightly di¤erent. Figure 3.9 shows the comparison for test case two and corroborates the postulation that results for kmax=3k and 5k are the same while that for kmax=20k is similar. GAM is also proven to be extremely fast in computing this junction, needing only 1 second when kmax

34 is set to 3k in both test case one and two, while the commerical software Ansoft HFSS takes about 12 seconds and 17 seconds for case one and case two respectively. The …eld matching method is much faster because the distance between the parallel plate is small, and this will always be the case when good transition is needed. The return loss of a design example of a wideband coaxial to radial waveguide transition which will be used in the monocone antenna in chapter 4 is shown in …gure 3.10.

Figure 3.3: HFSS model to test the modelling of the coaxial-to-radial waveguide junction using GAM.

Figure 3.4: S11 magnitude for test case 1 (a = 0.635mm, b = 1mm, d = 2mm).

35 Figure 3.5: S11 phase for test case 1.

Figure 3.6: Running test case 1 with kmax=3k, 5k and 20k, and with number of modes = 1, 3 and 33 respectively.

36 Figure 3.7: S11 amplitude for test case 2 (a = 2mm, b = 4mm, d = 3mm).

Figure 3.8: S11 phase for test case 2.

37 Figure 3.9: Running test case 2 with di¤erent number of modes.

Figure 3.10: Return loss of a wideband coaxial to radial waveguide transition. d=0.5mm, b=1mm, a=0.635mm.

38 Chapter 4

Modelling the bicone/monocone antenna

A candidate for the radiating element of the geodesic lens is the bicone or monocone antenna[47][49]. While there are many ways to analyze the bicone, they are either sim- plifying the problem by assuming that the antenna is circularly symmetric [48][49], or that they make use of complicated spherical modes to model them [50]. In this work, the bicone antenna is being analyzed by cascading sections of radial waveguides (…gure 4.1), with the discontinuity between each section computed by MMM [51]. In this way, the generalized scattering matrix of the antenna can be obtained. Extensive research in radial lines has been done in the past [27][52][53]. The radial waveguide is being modelled as an equivalent transmission line which introduces the modal voltage V and modal current I as measures of the transverse electric …eld Et and magnetic

…eld Ht (…gure 4.2). Each transmission line corresponds to a propagating or evanescent mode (TEM, TE or TM), depending on whether the wave amplitudes remain constant or decrease rapidly with the distance travelled. By making use of a multiple transmission line description, higher order modes can be taken into account. However, unlike in other uniform waveguide, in a radial transmission line representation, the characteristic mode impedance

39 varies with radial distance and hence has to be speci…ed at each reference plane [53]. The ‡owchart for computing the radiating bicone is given in …gure 4.3. As is done in the usual mode matching methodology, the modal content of each section of radial waveguide is computed and ordered according to its cuto¤ frequency. The coupling coe¢ cient for the radial step junction is then computed by doing a numerical integration of the electric mode functions of radial waveguide 1 and the magnetic mode functions of radial waveguide 2. There are two points to take note when mode matching between two radial waveguides. One, the form as represented by Hankel function denoting incoming or outgoing spherical waves should be converted to the more familiar exponential form. Secondly, as in the aforementioned, because the radial transmission line is not uniform, it is represented by the ABCD matrix before converting to S-matrix.

Figure 4.1: Approximating a bicone with sections of radial waveguides.

40 Figure 4.2: Modelling a radial waveguide as an equivalent transmission line.

4.1 Radial Waveguide Step Junction

Tangential electric …eld of waveguide 1 (see labels in …gure 4.1) at junction:

M h h h (2)0 h (1)0 h E1t = Z1i A1iHm1 (kn1;r) + B1iHm1 (kn1r) e1i + i=1 q h i XM e e e (2) e (1) e Z1l A1lHu1 (kv1r) + B1lHu1 (kv1r) e1l (4.1) l=1 X p h i In general, the Hankel function is a complex number which is made up of amplitude and phase named in the following manner:

(2)0 H (kn1r) has amplitude G0 and phase 1 (negative sign is used to denote outward m1 travelling wave)

(1)0 Hm1 (kn1r) has amplitude G0 and phase 1 (positive sign is used to denote inward travelling wave)

(2) H (kv1r) has amplitude G1 and phase u1 1 (1) Hu1 (kv1r) has amplitude G1 and phase 1

41 Figure 4.3: Flowchart showing steps to computing the GSM for a bicone/monocone.

42 Zh = jZh kn 1 o ki Ze = jZe ki 1 o kn

(2) j 1 H (kv1r) = Ju1 (kv1r) jYu1 (kv1r) = G1 (kv1r) e (4.2) u1

(1) j 1 Hu1 (kv1r) = Ju1 (kv1r) + jYu1 (kv1r) = G1 (kv1r) e (4.3)

(2)0 (2) m1 (2) Hm1 (kn1r) = Hm1+1 (kn1r) + Hm1 (kn1r) (4.4) kn1r

(1)0 (1) m1 (1) Hm1 (kn1r) = Hm1+1 (kn1r) + Hm1 (kn1r) (4.5) kn1r where

2 2 G1 (kv1r) = Ju1 (kv1r) + Yu1 (kv1r) (4.6) q Hence, the tangential electric …eld of waveguide 1 can be written as:

M h h h j1 h j1 h E1t = G0 Z1i A1ie + B1ie e1i + i=1 q XM e e e j 1 e j 1 e G1 Z1l A1le + B1le e1l (4.7) l=1 X p

43 Tangential magnetic …eld of waveguide 1 at junction:

M h 1 h (2) h (1) h H1t(r; ; z) = A1iHm1(kn1r) + B1iHm1(kn1r) h1i + Zh i=1 1i h i XM e p1 e Ae H(2)0 (k r) + Be H(1)0 (k r) h (4.8) e 1l u1 v1 1l u1 v1 1l l=1 Z1l X h i p (2) H (kn1r) has amplitude T0 and phase 2 (negative sign is used to denote outward m1 travelling wave)

(1) Hm1(kn1r) has amplitude T0 and phase 2 (positive sign is used to denote inward travelling wave)

(2)0 H (kv1r) has amplitude T1 and phase u1 2 (1)0 Hu1 (kv1r) has amplitude T1 and phase 2 Similarly, the tangential magnetic …eld of waveguide 1 can be rewritten as follows:

M h h T0 h j2 h j2 H1t(r; ; z) = A e + B e h + h 1i 1i 1i i=1 Z1i XM e pT1 e j e j e A e 2 + B e 2 h (4.9) e 1l 1l 1l l=1 Z1l X p In the above equation, M h and M e represent the number of TE and TM modes in waveguide 1, and N h and N e represent the number of TE and TM modes in waveguide 2. A and B are the coe¢ cients of the incident and re‡ected waves at the junction; e and h are the modal vectors. The subscripts m1 and n1 are the 1st and 2nd mode indices of the TE modes while the subscripts u1 and v1 are 1st and 2nd mode indices of the TM modes. For guide 2, the mode numbering for TE modes is p and the 1st and 2nd mode indices of the TE modes are m2 and n2. For TM modes the mode numbering is q, with u2 and v2 as the 1st and 2nd mode indices.

44 Tangential electric …eld of waveguide 2 at junction:

N h h h (1)0 h (2)0 h E2t(r; ; z) = Z2p A2pHm2 (kn2r) + B2pHm2 (kn2r) e2p + p=1 X q h i N e e e (1) e (2) e Z2q A2qHu2 (kv2r) + B2qHu2 (kv2r) e2q (4.10) q=1 X q h i

(1)0 Hm2 (kn2r) has amplitude R0 and phase 1

(2)0 H (kn2r) has amplitude R0 and phase 1 m2 (1) Hu2 (kv2r) has amplitude R1 and phase 1 (2) H (kv2r) has amplitude R1 and phase u2 1 Zh = jZh kn 2 o ki Ze = jZe ki 2 o kn

(2) j 1 H (kv2r) = Ju2 (kv2r) jYu2 (kv2r) = R1 (kv2r) e (4.11) u2

(1) j 1 Hu2 (kv2r) = Ju2 (kv2r) + jYu2 (kv2r) = R1 (kv2r) e (4.12)

(1)0 (1) m2 (1) Hm2 (kn2r) = Hm2+1 (kn2r) + Hm2 (kn2r) (4.13) kn2r

(2)0 (2) m2 (2) Hm2 (kn2r) = Hm2+1 (kn2r) + Hm2 (kn2r) (4.14) kn2r where

2 2 R1 (kv1r) = Ju2 (kv2r) + Yu2 (kv2r) (4.15) q

45 The tangential electric …eld of waveguide 2 is rewritten as:

N h h h j 1 h j 1 h E2t(r; ; z) = R0 Z2p A2pe + B2pe e2p + p=1 X q N e e e j 1 e j 1 e R1 Z2q A2qe + B2qe e2q (4.16) q=1 X q Tangential magnetic …eld of waveguide 2 at junction:

N h 1 h (1) h (2) h H2t(r; ; z) = A2pHm2(kn2r) + B2pHm2(kn2r) h2p + h p=1 Z X 2p h i N e q 1 e Ae H(1)0 (k r) + Be H(2)0 (k r) h (4.17) Ze 2q u2 v2 2q u2 v2 2q q=1 2q X h i p (2) H (kn2r) has amplitude U0 and phase 2 m2 (1) Hm2(kn2r) has amplitude U0 and phase 2

(2)0 H (kv2r) has amplitude U1 and phase u2 2 (1)0 Hu2 (kv2r) has amplitude U1 and phase 2 The tangential magnetic …eld of waveguide 2 is rewritten as:

N h h U0 h j 2 h j 2 H2t(r; ; z) = A2pe + B2pe h2p + h p=1 Z2p X N e q U1 e j e j e A e 2 + B e 2 ) h (4.18) Ze 2q 2q 2q q=1 2q X p Matching the tangential electric …elds, E2t = E1t, across surface S1 of the junction located at radial distance r, leads to

M h M e h h j1 h j1 h e e j 1 e j 1 e G0 Z1i A1ie + B1ie e1i + G1 Z1l A1le + B1le e1l = i=1 l=1 X q X N h N e p h h j 1 h j 1 h e e j 1 e j 1 e R0 Z2p A2pe + B2pe e2p + R1 Z2q A2qe + B2qe e2q (4.19) p=1 q=1 X q X q 46 Taking the inner product of equation 4.19 with the TE magnetic …eld mode vectors of guide 2, gives

M e e e j 1 e j 1 e h G1 Z A e + B e e h r r d dz + 1l 1l 1l 1l 2j ZZ l=1 S1 X p M h b h h j1 h j1 h h G0 Z A e + B e e h r r d dz = 1i 1i 1i 1i 2j ZZ i=1 S1 X q N h b h h j 1 h j 1 h h R0 Z A e + B e e h r r d dz = 2p 2p 2p 2p 2j ZZ p=1 S1 X q N e b e e j 1 e j 1 e h R1 Z A e + B e e h r r d dz (4.20) 2q 2q 2q 2q 2j ZZ q=1 S1 X q b This results in

M e e e j 1 e j 1 e h G1 Z A e + B e e h r r d dz + 1l 1l 1l 1l 2j ZZ l=1 S1 X p M h b h h j1 h j1 h h G0 Z A e + B e e h r r d dz = 1i 1i 1i 1i 2j ZZ i=1 S1 X q h h j 1 h j 1 b R0 Z2p A2pe + B2pe (4.21) q Taking the inner product of eqn 4.19 with the TM magnetic …eld mode vectors of guide 2 and making use of mode orthogonality yields

M e e e j 1 e j 1 e e G1 Z A e + B e e h r r d dz + 1l 1l 1l 1l 2q ZZ l=1 S1 X p M h b h h j1 h j1 h e G0 Z A e + B e e h r r d dz = 1i 1i 1i 1i 2q ZZ i=1 S1 X q N e b e e j 1 e j 1 R1 Z2q A2qe + B2qe (4.22) q=1 X q

47 Equation 4.21 and 4.22 can be combined to give the following matrix equation.

ee eh j 1 e [F ] F e [0] [A1] 2 3 2 3 2 3 + he hh j1 h F F [0] e A1 6 7 6 7 6 7 4 5 4 5 4 5 ee eh j 1 e [F ] F e [0] [B1] 2 3 2 3 2 3 = he hh j1 h F F [0] e B1 6 7 6 7 6 7 4 5 4 5 4 5 j 1 e j 1 e e [0] [A2] e [0] [B2] 2 3 2 3 + 2 3 2 3 (4.23) j 1 h j 1 h [0] [e ] A2 [0] [e ] B2 6 7 6 7 6 7 6 7 4 5 4 5 4 5 4 5 The [F ] submatrix is de…ned by the following

Ze0 ee p 1l e e e e Fql = e1l h2q r r d dz , q = 1; 2; ::::N , l = 1; 2; ::::M (4.24) Ze0 S1 2q RR q b

Ze0 he p 1l e h h e Fpl = e1l h2p r r d dz , p = 1; 2; ::::N , l = 1; 2; ::::M (4.25) Zh0 S1 2p RR q b

Zh eh p 1i h e e h Fqi = e e1i h2q r r d dz , q = 1; 2; ::::N , i = 1; 2; ::::M (4.26) pZ2q S1 RR b

Zh hh p 1i h h h h Fpi = h e1i h2p r r d dz , p = 1; 2; ::::N , i = 1; 2; ::::M (4.27) pZ2p S1 RR b The next step is the matching of the tangential magnetic …elds, H2t = H1t, across surface

48 S1 of the junction located at radial distance r. This leads to

M h M e h e T0 h j2 h j2 T1 e j e j A e + B e h + A e 2 + B e 2 h = h 1i 1i 1i e 1l 1l 1l i=1 Z1i l=1 Z1l X X N h N e pU h p U e 0 h j 2 h j 2 1 e j 2 e j 2 A2pe + B2pe h2p + e A2qe + B2qe ) h2q (4.28) h Z2q p=1 Z2p q=1 X X q p Taking the inner product of equation 4.28 with the electric …eld TE mode vectors of guide 1, gives

M h h T0 h j2 h j2 h A1ie + B1ie e1j h1i r r d dz = Zh ZZ i=l 1i S1 X N h p b h U0 h j 2 h j 2 h A2pe + B2pe e1j h2p r r d dz + h ZZ p=1 Z2p S1 X N e q b U1 e j e j h e A e 2 + B e 2 ) e h r r d dz (4.29) Ze 2q 2q 1j 2q ZZ q=1 2q S1 X p b Taking the inner product of equation 4.28 with the electric …eld TM mode vectors of guide 1, yields

M e T1 e j e j e e A e 2 + B e 2 e h r r d dz = Ze 1l 1l 1j 1l ZZ l=1 1l S1 X N h p b h U0 h j 2 h j 2 e A2pe + B2pe e1j h2p r r d dz + h ZZ p=1 Z2p S1 X N e q b U1 e j e j e e A e 2 + B e 2 ) e h r r d dz (4.30) Ze 2q 2q 1j 2q ZZ q=1 2q S1 X p b Making use of mode orthogonality, equations 4.29 and 4.30 are combined to give the

49 following matrix equation:

T T ee he j 2 e [F ] F e [0] [A2] + 2 T T 3 2 3 2 3 eh hh j 2 h F F [0] [e ] A2 6 7 6 7 6 7 4 T T 5 4 5 4 5 ee he j 2 e [F ] F e [0] [B2] = 2 T T 3 2 3 2 3 eh hh j 2 h F F [0] [e ] B2 6 7 6 7 6 7 4 5 4 5 4 5 j 2 e j 2 e e [0] [A1] e [0] [B1] 2 3 2 3 + 2 3 2 3 (4.31) j2 h j2 h [0] e A1 [0] e B1 6 7 6 7 6 7 6 7 4 5 4 5 4 5 4 5 For compactness, equation 4.23 is rewritten as

21 11 12 22 [F ] D [A1] + [F ] D [B1] = D [A2] + D [B2] (4.32) where

[F ee] F eh e j 1 [0] ej 1 [0] 21 11 [F ] = 2 3 ; D = 2 3 ; D = 2 3 he hh j1 j1 F F [0] e [0] e 6 7 6 7 6 7 4 5 4 5 4 5

ej 1 [0] e j 1 [0] 12 22 D = 2 3 ; D = 2 3 j 1 j 1 [0] [e ] [0] [e ] 6 7 6 7 4 5 4 5

e e e e [A1] [B1] [A2] [B2] [A1] = ; [B1] = ; [A2] = ; [B2] = 2 h 3 2 h 3 2 h 3 2 h 3 A1 B1 A2 B2 6 7 6 7 6 7 6 7 4 5 4 5 4 5 4 5 Also equation 4:31 can be represented as

T 12 T 22 21 11 [F ] G [A2] + [F ] G [B2] = G [A1] + G [B1] (4.33)

50 where

ej 2 [0] e j 2 [0] 12 22 G = 2 3 ; G = 2 3 j 2 j 2 [0] [e ] [0] [e ] 6 7 6 7 4 5 4 5 e j 2 [0] ej 2 [0] 21 11 G = 2 3 ; G = 2 3 j2 j2 [0] e [0] e 6 7 6 7 4 5 4 5 Equations 4:32 and 4:33 are manipulated to give the following GSMs of the radial guide junction.

1 11 T 22 22 1 11 T 22 22 1 21 21 [S11] = G [F ] G D [F ] D [F ] G D [F ] D G n o n (4.34)o

11 T 22 22 1 11 T 12 22 22 1 12 [S12] = G [F ] G D [F ] D [F ] G G D D n o n (4.35)o

22 11 11 1 T 22 21 11 11 1 21 [S21] = D [F ] D G [F ] G [F ] D D G G (4.36) n o n o

1 22 11 11 1 T 22 11 11 1 T 12 12 [S22] = D [F ] D G [F ] G [F ] D G [F ] G D n o n (4.37)o

51 4.2 Radial Waveguide Step Mode Coupling Coe¢ cients

Calculation of coupling matrix [F ]: TM Mode Guide 1 - TM Mode Guide 2

The coupling between the TM mode of Guide 1 and the TM mode of Guide 2 is given by

Ze0 ee p 1l e e e e Fql = e1l h2q r r d dz , q = 1; 2; ::::N , l = 1; 2; ::::M (4.38) Ze0 S1 2q RR q b where the new mode characteristic impedances are de…ned as

G Ze e0 1 1l Z1l = (4.39) T1

and e R1Z e0 2q Z2q = (4.40) U1

st nd Let the 1 and 2 mode indices of mode l in guide 1 be (ul; vl) and mode q in guide 2 be (uq; vq).

0 if ul = uq 6 8 H > > Ze0 " 0 ee > p 1l vl > ICC [vl; d1; vq; d2; ] if ul = uq = 0 Fql = > e d1 8 9 (4.41) > Z2q0 > > 1 > < q < =V e0 pZ1l "vl > d ICC [vl; d1; vq; d2; ] > > if ul = uq = 0 > Ze0 1 : ; > 2q 6 > q > :>

"v = 1 if v = 0

"v = 2 if v = 0 6

52 d1 vz w (z + ) ICC [vl; d1;w; d2; ] = cos cos dz d1 dz Z0 sin v + w d1 cos v + w d1 + w d1 d2 2 d1 d2 2 d2 + h vi + wh d1 i d1 d1 d2 2 v w 8 8 d d 9 for = 2 sin v w 1 cos v w 1 w d1 d2 d d 2 d d 2 d 6 > > 1 2 1 2 2 > > < h vi wh d1 i = = > d d 2 (4.42) > 1 2 > <> d1 >cos w > for v = w = 0 2 : d2 ; d1 d2 6 > h i v w > d1 for = = 0 > d1 d2 > > :> Calculation of coupling matrix [F ]: TM Mode Guide 1 - TE Mode Guide 2

The coupling between the TM mode of Guide 1 and the TE mode of Guide 2 is given by

Ze0 he p 1l e h h e Fpl = e1l h2p r r d dz , p = 1; 2; ::::N , l = 1; 2; ::::M (4.43) Zh0 S1 2p RR q b where the new mode characteristic impedances are de…ned as

G Ze e0 1 1l Z1l = (4.44) T1

and h R0Z h0 2p Z2p = (4.45) U0

st nd Let the 1 and 2 mode indices of mode l in guide 1 be (ul; vl) and mode p in guide 2 be (mp; np).

53 0 for ul = mp 6 8 H > +1 > v1 > 8 2 2 ISS [vl; d1; np; d2; ] + 9 > k (vl=d1) d1 8 9 > > 1 > > Ze0 " > > > > > p 1l "v ul np > < =V > > > > for ul = mp = 0 > h d1 r d2 > H > > Z2p0 > > 6 > > > > > > q < : 1; = > np > 2 2 ICC [vl; d1; np; d2; ] > k (np=d2) d2 8 9 he > Fpl = > > > +1 > > > > < =V > > > H > < > > :> :>1 ;> ;> v1 > 8 2 2 ISS [vl; d1; np; d2; ] + 9 > k (vl=d1) d1 8 9 > > e > 0 > > Z 0 "n > > > > > p 1l "v ul p > < =V > > d r d > H > for ul = mp = 0 > Zh0 1 2 > > > 2p > > > > > q < : 0; = > np > 2 2 ICC [vl; d1; np; d2; ] > k (np=d2) d2 8 9 > > > 1 > > > > < =V > > > > > > > (4.46) :> :> :> ;> ;>

d1 vz w (z + ) ISS [v; d1; w; d2; ] = sin sin dz d1 d2 Z0 sin v w d1 cos v w d1 w d1 d1 d2 2 d1 d2 2 d2 2 h vi wh d1 i d1 d2 2 v w 8 d for = 0 sin v + w 1 cos v + w 1 d1 d2 d d 2 d d 6 > 1 2 2 2 > h v iw hd1 i = > B d + d 2 C (4.47) > B 1 2 C > @ A <> d1 cos w v = w = 0 2 d2 d1 d2 6 > 0 h i v = w = 0 > d1 d2 > > :> Calculation of coupling matrix [F ]: TE Mode Guide 1 - TM Mode Guide 2

Zh F eh = 1i eh he r r d dz , q = 1; 2; ::::N e i = 1; 2; ::::M h (4.48) qi Ze 1i 2q ZZ p 2q S1 = 0 p b

54 Calculation of coupling matrix [F ]: TE Mode Guide 1 - TE Mode Guide 2

The coupling between the TE mode of Guide 1 and the TE mode of Guide 2 is given by

h0 hh Z1i h h h h Fpi = e1i h2p r r d dz , p = 1; 2; ::::N i = 1; 2; ::::M (4.49) h0 ZZ pZ2p S1 q b

0 for mi = mp 6 8 H > > h0 " 1 hh > pZ1i np > ISS [n1; d1; np; d2; ] for mi = mp = 0 Fpi = > h d2 8 9 (4.50) > Z2p0 > > 0 > < q < =V h pZ1i0 "np > d ISS [n1; d1; np; d2; ] > > for mi = mp = 0 > Zh0 2 : ; > 2p 6 > q > where the new:> mode characteristic impedances are de…ned as

G Zh h0 0 1i Z1i = (4.51) T0

and h R0Z h0 2p Z2p = (4.52) U0

4.3 Determining the S-parameters of a section of radial

line

After …nding the S-parameters of the radial step discontinuity, the S-matrix for the sections of radial waveguide that connects the junction discontinuities need to be found. The S- matrices are then cascaded together using GSM. The radial waveguide holds certain properties unlike that of uniform sections of rectangular or circular waveguides, such that even in a perfectly matched lossless radial waveguide, the return loss is not zero. The section of radial waveguide is best modelled as an ABCD

55 matrix which will then be transformed to the S-matrix with the relevant characteristic impedance [54].

For a length of radial line from r=r1 to r = r2, its ABCD matrix is of the form

V (r1) A (r1; r2) B (r1; r2) V (r2) 2 3 = 2 3 2 3 (4.53) I (r1) C (r1; r2) D (r1; r2) I (r2) 6 7 6 7 6 7 4 5 4 5 4 5 with

k r c 2 0 0 A (r1; r2) = Y (knr2) Jm (knr1) J (knr2) Ym (knr1) (4.54) 2 m n h i

2 knb "0m B (r1; r2) = j [Jm (knr2) Ym (knr1) Ym (knr2) Jm (knr1)] (4.55) 4! "0n

2! " 0n 0 0 0 0 C (r1; r2) = j r1r2 Ym (knr2) Jm (knr1) Jm (knr2) Ym (knr1) (4.56) b "0m h i

k r n 1 0 0 D (r1; r2) = Jm (knr2) Y (knr1) Ym (knr2) J (knr1) (4.57) 2 m m h i We can obtain the admittance matrix by the following transformation.

D (r1; r2) Y11 (r1; r2) = (4.58) B (r1; r2)

[B (r1; r2) C (r1; r2) A (r1; r2) D (r1; r2)] Y12 (r1; r2) = (4.59) B (r1; r2)

1 Y21 (r1; r2) = (4.60) B (r1; r2)

56 A (r1; r2) Y22 (r1; r2) = (4.61) B (r1; r2)

Finally the S-matrix obtained by

1 S = E Y E + Y (4.62) where

Y = Z0Y Z0 (4.63) p p Z0 is a diagonal matrix of characteristic impedance and E is the identity matrix

4.4 Results

The step discontinuity between radial waveguides is simulated with both the mode matching method and HFSS. The model being tested is shown in …gure 4.4 and the results given in …gures 4.5 and 4.6.

Figure 4.4: Model tested with mode matching and HFSS.

57 Figure 4.5: S11 amplitude comparison between mode matching and HFSS.

A monocone antenna has been built (…gure 4.7) and its dimension is shown in …gure 4.8. The monocone antenna is fed by the centre pin of a SMA connector inserted into a hole that is drilled at its bottom. The ‡ange of the connector is soldered to a circular plate that acts as the ground plane. Return loss computed from the mode matching method is compared with the measured result. Figure 4.9 shows the comparison of the results. Generally, the simulated result compares well with the measured data. The discrepancy at the lower frequencies is likely due to the fact that in the mode matching formulation, a large output aperture is assumed so that no energy is re‡ected from the radial-air interface, but in actual case some amount of energy is re‡ected back. The ground plane in the formulation is also modelled as having the same size as the top plate but in reality the ground plane is built bigger than the radius of the monocone. In terms of computational time, mode matching takes more than an hour while Ansoft HFSS takes about 23 minutes. This can be attributed to the intensive computations needed to calculate the large number of radial sections and junction discontinuities. Nevertheless, results demonstrated that the mode matching method developed is able to predict the antenna fairly well and it also shows the wide bandwidth achieved by the monocone.

58 Figure 4.6: S11 phase comparison between mode matching and HFSS.

Figure 4.7: Fabricated monocone.

Figure 4.8: Monocone Antenna with dimensions in mm.

59 Figure 4.9: Comparison of measured results with simulation for a monocone antenna. In mode matching, 100 sections are used with each step height = 0.38mm.

60 Chapter 5

Modes of the ridge sectoral waveguide

The feed to the coaxial cone is an array of ridge sectoral waveguides that are spaced uniformly around the bottom of the coaxial guide (…gure 5.1). The input ridge waveguide is in turn excited by a probe which can be carefully tuned for impedance matching. A number of sections of ridge waveguides with di¤erent cross-section may then be cascaded to transform the input impedance at the probe to the appropriate impedance at the mouth of the coaxial cone. This procedure of impedance transformation requires knowledge of the modal content in the ridge waveguide, and may be determined by a number of methods. The generalized transverse resonance …eld-matching technique (or TRT) [60]-[67] is used here to evaluate the mode spectrum of the ridge sectoral waveguide as it is computationally e¢ cient and accurate. In the TRT formulation, the cross-sectional eigenfunctions with the appropriate boundary conditions are used to derive the electric and magnetic …elds at the cross-section, and the …elds are matched across the dimensional variations. Similar to MMM, the projection method [68] is used to form the relevant equations in which the eigenvalues can be solved. The physical parameters of a ridge sectoral waveguide are de…ned in …gure 5.2. The top, bottom and side walls are all perfect electrical conductors (PEC). There is a plane of symmetry about = s. For symmetric modes, the plane of symmetry is a magnetic wall and for asymmetric modes, the plane of symmetry is an electric wall.

61 Figure 5.1: Diagram of the cross-section of geodesic cone showing the feeding of the cone by sectoral ridge waveguides.

5.1 Ridge Sectoral Guide Symmetric TMz Mode

The symmetric half of the ridge sectoral guide is divided into three regions. The potential function in terms of the unknown coe¢ cients for each region is …rst given. The tangential

…elds H and Ez are derived using this potential. Imposing the requirement for tangential …eld continuity across the regions leads to the mode characteristic equation.

Region I 0 , a c 1 The potential function for this region is given by the following series.

M I jkzz = Am [J (kra) Y (kr) J (kr) Y (kra)] sin () e (5.1) m=1;2;:: X 62 Figure 5.2: Ridge Sectoral Waveguide where

m 2 2 2 = ; m = 1; 2; :::M ; kz = k kr 1

M 0 0 jkzz H = Am J (kra) Y (kr) J (kr) Y (kra) kr sin () e (5.2) m=1;2;:: X h i

M 1 jkzz H = A [J (k a) Y (k ) J (k ) Y (k a)] cos () e (5.3) m r r r r m=1;2;:: X

2 M kr jkzz E = A [J (k a) Y (k ) J (k ) Y (k a)] sin () e (5.4) z j!" m r r r r m=1;2;:: X

M kz jkzz E = A J (k a) Y 0 (k ) J 0 (k ) Y (k a) k sin () e (5.5) !" m r r r r r m=1;2;:: X h i

M kz jkzz E = A [J (k a) Y (k ) J (k ) Y (k a)] cos () e (5.6) !" m r r r r m=1;2;:: X

63 Region II 0 ; c d s The potential function for this region is given by the following series.

N II jkzz = [BnYv (kr) + CnJv (kr)] sin (v) e (5.7) m=1;2;:: X where

(2n 1) 2 2 2 v = ; n = 1; 2; :::N ; kz = k kr 2s

N 0 0 jkzz H = BnYv (kr) + CnJv (kr) kr sin (v) e (5.8) n=1;2;:: X h i

N 1 jkzz H = [B Y (k ) + C J (k )] v cos (v) e (5.9) n v r n v r n=1;2;:: X

2 N kr jkzz E = [B Y (k ) + C J (k )] sin (v) e (5.10) z j!" n v r n v r n=1;2;:: X

N k jkzz E = [B Y (k ) + C J (k )] v cos (v) e (5.11) !" n v r n v r n=1;2;:: X

N k jkzz E = B Y 0 (k ) + C J 0 (k ) k sin (v) e (5.12) !" n v r n v r r n=1;2;:: X h i

Region III 0 ; d b 1 The potential function for this region is given by the following series.

I III jkzz = Di [J (krb) Y (kr) J (kr) Y (krb)] sin () e (5.13) i=1;2;:: X where

i 2 2 2 = ; i = 1; 2; :::I ; kz = k kr 1

64 I 0 0 jkzz H = Di J (krb) Y (kr) J (kr) Y (krb) kr sin () e (5.14) i=1;2;:: X h i

I 1 jkzz H = D [J (k b) Y (k ) J (k ) Y (k b)] cos () e (5.15) i r r r r i=1;2;:: X

2 I kr jkzz E = D [J (k b) Y (k ) J (k ) Y (k b)] sin () e (5.16) z j!" i r r r r i=1;2;:: X

I kz jkzz E = D [J (k b) Y (k ) J (k ) Y (k b)] cos () e (5.17) !" i r r r r i=1;2;:: X

I kz jkzz E = D J (k b) Y 0 (k ) J 0 (k ) Y (k b) k sin () e (5.18) !" i r r r r r i=1;2;:: X h i

Because Regions I and III have the same subtended angle 1, one may set M = I.

Match H between regions I and II at = c, then take the product with sin 0 and …nally integrate over 0 , we get the following matrix equation. This is the same as 1 taking the inner product with E.

se 0 0 [F12 ] Y12 [B] + J12 [C] = [W12][A] (5.19) nh i h i o where 1 se 2 m (2n 1) F = sin sin d ; m = 1; 2; :::M ; n = 1; 2; :::N 12;mn 1 1 2s 0 m (2n 1) m (2n 1) Rsin 1 sin + 1 1 2s 1 2s m (2n 1) (2n 1) (2n 1) = mh i mh i 1 2s se 2 1 + 2 1 6 F12;mn = 1 s 1 s 8 h i h i m (2n 1) > 1 = < 1 2s Y 0 = diagonal matrix with element Y 0 = Y 0 (k c) 12 :> 12;nn v r 65 0 0 0 J12 = diagonal matrix with element J12;nn = Jv (krc)

[W12] = diagonal matrix with element W12;mm = J (kra) Y 0 (krc) J 0 (krc) Y (kra) Similarly, we match Ez between regions I and II at = c, we end up with the following

M 1 0 Am [J (kra) Y (kr) J (kr) Y (kra)] sin () sin v d = m=1;2;:: X Z0 N s 0 [BnY (kr) + CnJ (kr)] sin (v) sin v d (5.20) m=1;2;:: X Z0

Rewriting the equation in matrix form,

[Y ][B] + [J ][C] = 1 [F se]T [V ][A] (5.21) 12 12 12 12 s where T is the transpose operator and

[Y12] = diagonal matrix with element Y12;nn = Yv (krc)

[J12] = diagonal matrix with element J12;nn = Jv (krc)

[V12] = diagonal matrix with element V12;mm = [J (kra) Y (krc) J (krc) Y (kra)]

Matching H and Ez between regions II and III at = d result in

se 0 0 [F32 ] Y23 [B] + J23 [C] = [W32][D] (5.22) nh i h i o

[Y ][B] + [J ][C] = 1 [F se]T [V ][D] (5.23) 23 23 32 32 s where

[Y23] = diagonal matrix with element Y23;nn = Yv (krd)

[J23] = diagonal matrix with element J23;nn = Jv (krd)

[V23] = diagonal matrix with element V23;ii = [J (krb) Y (krd) J (krd) Y (krdb)] 0 0 0 Y23 = diagonal matrix with element Y23;nn = Yv (krd) 66 0 0 0 J23 = diagonal matrix with element J23;nm = Jv (krd)

W 0 = diagonal matrix with element W23;ii = J (krb) Y 0 (krd) J 0 (krd) Y (krb) 23 1 se 2 i (2n 1) F = sin sin d ; i = 1; 2; :::I ; n = 1; 2; :::N 32;in 1 1 2s 0 Using eqnsR 5.21, 5.22 and 5.23, the coe¢ cient matrix A can be eliminated to obtain

[G11][G12] [B] 2 3 2 3 = 0 (5.24) [G21][G22] [C] 6 7 6 7 4 5 4 5 where

se T 1 se 0 s [G11] = [F12 ] [V12][W12] [F12 ] Y12 [Y12] 1 se T 1 se 0 s [G12] = [F12 ] [V12][W12] [F12 ] J12 [J12] 1 se T 1 se 0 s [G21] = [F32 ] [V32][W32] [F32 ] Y23 [Y23] 1 se T 1 se 0 s [G22] = [F32 ] [V32][W32] [F32 ] J23 [J23] 1 The roots of the determinant of eqn 5.24 will give the TMz mode cuto¤ wavenumber kr.

Once kr is determined, the unknown coe¢ cients of the …eld representation for that mode can be readily found for the three regions [25]. These coe¢ cients have to be normalized by the square root of the mode power propagating through the guide cross-section. This will ensure that each mode will carry a unit of power [26]. The power P in each region is found by integrating the Poynting vector over the cross- section. The integration in the -variable is done analytically while the integration with respect to the -variable has to be carried out numerically. The results are summarized below.

67 c 1 P I = !E !H zdd^ Za Z0 c 1

= (EH EH) dd Za Z0 k = z 1 k2 !" 4 r M c 2 [J (kra) Y 1 (kr) Y (kra) J 1 (kr)] + (5.25) 8 2 9 m=1;2;:: Z0 > [J (kra) Y+1 (kr) Y (kra) J+1 (kr)] > X < = :> ;>

c s P II = !E !H zdd^ Za Z0 c s

= (EH EH) dd Za Z0

N d 2 k [BnYv 1 (kr) + CnJv 1 (kr)] + z s 2 = kr (5.26) !" 4 8 2 9 n=1;2;:: > [BnYv+1 (kr) + CnJv+1 (kr)] > X Zc < = :> ;>

b 1 P III = !E !H zdd^ Za Z0 b 1

= (EH EH) dd Za Z0 k = z 1 k2 !" 4 r I b 2 [J (krb) Y 1 (kr) Y (krb) J 1 (kr)] + 2 Di (5.27) 8 2 9 i=1;2;:: Z > [J (krb) Y+1 (kr) Y (krb) J+1 (kr)] > X d < = :> ;>

68 The normalization factor for the coe¢ cients is the square root of the sum of the powers across the entire guide.

N e = 2 (P I + P II + P III ) (5.28) p

5.2 Ridge Sectoral Guide Asymmetric TMz Mode

The symmetric half of the ridge sectoral guide is divided into three regions. The potential function in terms of the unknown coe¢ cients for each region is …rst given. The tangential

…elds Hz and Ez are derived using this potential. Imposing the requirement for tangential …eld continuity across the regions leads to the mode characteristic equation.

Region 1 0 , a c 1 The potential function for this region is given by the following series.

M I jkzz = Am [J (kra) Y (kr) J (kr) Y (kra)] sin () e (5.29) m=1;2;:: X where

m 2 2 2 = ; m = 1; 2; :::M ; kz = k kr 1

M 0 0 jkzz H = Am Jm (kra) Y (kr) J (kr) Y (kra) kr sin () e (5.30) m=1;2;:: X h i

2 M kr jkzz E = A [J (k a) Y (k ) J (k ) Y (k a)] sin () e (5.31) z j!" m r r r r m=1;2;:: X

Region II , 0 ; ; c d s

69 The potential function for this region is given by the following series.

N II jkzz = [BnYv (kr) + CnJv (kr)] sin (v) e (5.32) n=1;2;:: X where

n 2 2 2 v = ; n = 1; 2; :::N ; kz = k kr s

N 0 0 jkzz H = BnYv (kr) + CnJv (kr) kr sin (v) e (5.33) n=1;2;:: X h i

2 N kr jkzz E = [B Y (k ) + C J (k )] sin (v) e (5.34) z j!" n v r n v r n=1;2;:: X

Region III 0 ; d b 1 The potential function for this region is given by the following series.

I III jkzz = Di [J (krb) Y (kr) J (kr) Y (krb)] sin () e (5.35) i=1;2;:: X where

i 2 2 2 = ; i = 1; 2; :::I ; kz = k kr 1

I 0 0 jkzz H = Di J (krb) Y (kr) J (kr) Y (krb) kr sin () e (5.36) i=1;2;:: X h i

2 I kr jkzz E = D [J (k b) Y (k ) J (k ) Y (k b)] sin () e (5.37) z j!" i r r r r i=1;2;:: X

Because regions I and III have the same subtended angle 1, one may set M = I.

Matching H between regions I and II at = c results in

ae 0 0 [F12 ] Y12 [B] + J12 [C] = [W12][A] (5.38) nh i h i o 70 Matching H between regions II and II at = d results in

ae 0 0 [F32 ] Y23 [B] + J23 [C] = [W32][D] (5.39) nh i h i o where

[W32] = J (krb) Y 0 (krd) J 0 (krd) Y (krb) [V12] = [J (kra) Y (krc) J (krc) Y (kra)] [W12] = J (kra) Y 0 (krc) J 0 (krc) Y (kra) [V32] = [J (krb) Y (krd) J (krd) Y (krb)] Using eqns 5.38, and 5.39 the coe¢ cient matrix A can be eliminated to obtain

[G11][G12] [B] 2 3 2 3 = 0 (5.40) [G21][G22] [C] 6 7 6 7 4 5 4 5 where

ae T 1 ae 0 s [G11] = [F12 ] [V12][W12] [F12 ] Y12 [Y12] 1 ae T 1 ae 0 s [G12] = [F12 ] [V12][W12] [F12 ] J12 [J12] 1 ae T 1 ae 0 s [G21] = [F32 ] [V32][W32] [F32 ] Y23 [Y23] 1 ae T 1 ae 0 s [G22] = [F32 ] [V32][W32] [F32 ] J23 [J23] 1 1 F ae = 2 sin m sin n d ; m = 1; 2; :::M ; n = 1; 2; :::N 12;mn 1 1 s 0 m n m n Rsin 1 sin + 1 1 s 1 s m = n mh n i mh n i 1 s ae 1 + 1 6 F12;mn = 1 s 1 s 8 h i h i > 1 m = n < 1 s 1 F ae = 2:> sin i sin n d ; i = 1; 2; :::I ; n = 1; 2; :::N 32;in 1 1 s 0 Similar to thatR done for the symmetric TMz mode, the roots of the determinant 5.40 will give the asymmetric TMz mode cuto¤ wavenumber kr. The equations for calculating the power is also the same as that for the symmetric TMz mode case.

71 5.3 Ridge Sectoral Guide Symmetric TEz Mode

The symmetric half of the ridge sectoral guide is divided into three regions. The potential function in terms of the unknown coe¢ cients for each region is …rst given. The tangential

…elds E and Hz are derived using this potential. Imposing the requirement for tangential …eld continuity across the regions leads to the mode characteristic equation. Region I 0 ; a c 1 The potential function for this region is given by the following series.

M 1 I h 0 0 jkzz = Am J (kra) Y (kr) J (kr) Y (kra) cos () e (5.41) m=0;1;2;:: X h i where

m 2 2 2 = ; m = 0; 1; 2; :::M 1 ; kz = k kr 1

M 1 h 0 0 0 0 jkzz E = Am J (kra) Y (kr) J (kr) Y (kra) kr cos () e (5.42) m=0;1;2;:: X h i

M 1 1 h jkzz E = A J 0 (k a) Y (k ) J (k ) Y 0 (k a) sin () e (5.43) m r r r r m=0;1;2;:: X h i

M 1 2 kr h jkzz H = A J 0 (k a) Y (k ) J (k ) Y 0 (k a) cos () e (5.44) z jk m r r r r m=0;1;2;:: X h i

M 1 kz h jkzz H = A J 0 (k a) Y (k ) J (k ) Y 0 (k a) sin () e (5.45) k m r r r r m=0;1;2;:: X h i

72 M 1 kz h jkzz H = A J 0 (k a) Y 0 (k ) J 0 (k ) Y 0 (k a) k cos () e (5.46) k m r r r r r m=0;1;2;:: X h i

Region II 0 ; c d s The potential function for this region is given by the following series.

N II h h jkzz = BnYv (kr) + Cn Jv (kr) cos (v) e (5.47) n=1;2;:: X where

(2n 1) 2 2 2 v = ; n = 1; 2; :::; N ; kz = k kr 2s

N h 0 h 0 jkzz E = BnYv (kr) + Cn Jv (kr) kr cos (v) e (5.48) n=1;2;:: X h i

N 1 h h jkzz E = B Y (k ) + C J (k ) v sin (v) e (5.49) n v r n v r n=1;2;:: X

2 N kz h h jkzz H = B Y (k ) + C J (k ) cos (v) e (5.50) z jk n v r n v r n=1;2;:: X

N kz h h jkzz H = B Y 0 (k ) + C J 0 (k ) k cos (v) e (5.51) k n v r n v r r n=1;2;:: X h i

N kz h h jkzz H = B Y (k ) + C J (k ) v sin (v) e (5.52) k n v r n v r n=1;2;:: X Region III 0 ; d b 1 The potential function for this region is given by the following series.

73 I 1 III h 0 0 jkzz = Di J (krb) Y (kr) J (kr) Y (krb) cos () e (5.53) i=0;1;2;:: X h i where

i 2 2 2 = ; i = 0; 1; 2; :::I 1 ; kz = k kr 1

I 1 h 0 0 0 0 jkzz E = Di J (krb) Y (kr) J (kr) Y (krb) kr cos () e (5.54) i=0;1;2;:: X h i

I 1 1 h jkzz E = D J 0 (k b) Y (k ) J (k ) Y 0 (k b) sin () e (5.55) i r r r r i=0;1;2;:: X h i

I 1 2 kr h jkzz H = D J 0 (k b) Y (k ) J (k ) Y 0 (k b) cos () e (5.56) z jk i r r r r i=0;1;2;:: X h i

I 1 kz h jkzz H = D J 0 (k b) Y (k ) J (k ) Y 0 (k b) sin () e (5.57) k i r r r r i=0;1;2;:: X h i

I 1 kz h jkzz H = D J 0 (k b) Y 0 (k ) J 0 (k ) Y 0 (k b) k cos () e (5.58) k i r r r r r i=0;1;2;:: X h i

Because Regions I and III have the same subtended angle 1, one may set M = I.

Matching E and Hz at the boundary between region I and II at = c, and the boundary between region II and III at = d yield the following equation

h h G11 G12 [B] = [0] (5.59) 2 h h 3 2 3 G21 G22 [C] 6 7 6 7 4 5 4 5 74 where

T h sh 1 sh s 0 G12 = H12 [W12][V12] H12 [Y12] Y12 1 T h sh 1 sh s 0 G12 = H12 [W12][V12] H12 [J12] J12 1 T h sh 1 sh s 0 G21 = H32 [W32][V32] H32 [J23] Y23 1 T h sh 1 sh s 0 G22 = H32 [W32][V32] H32 [J23] J23 1 1 sh 2 m (2n 1) H = cos cos d ; m = 0; 1; 2; :::M 1 ; n = 1; 2; :::N 12;mn 1 1 2s 0 m (2n 1) m (2n 1) Rsin + 1 sin 1 1 2s 1 2s m (2n 1) (2n 1) + (2n 1) = hm i hm i 1 2s sh + 2 2 6 H12;mn = 1 s 1 s 8 h i h i m (2n 1) > 1 = < 1 2s 1 sh 2 i (2n 1) H = :> cos cos d ; i = 0; 1; 2; :::I 1 ; n = 1; 2; :::N 32;in 1 1 2s 0 0 0 0 Y12 = diagonalR matrix with element Y12;nn = Yv (krc)

0 0 0 J12 = diagonal matrix with element J12;nn = Jv (krc)

"m [W12] = diagonal matrix with element W12;mm = J 0 (kra) Y 0 (krc) J 0 (krc) Y 0 (kra) 2 1 m = 0 " = 8 > 2 m = 0 < 6 [Y12] = diagonal matrix with element Y12;nn = Yv (krc) :> [J12] = diagonal matrix with element J12;nn = Jv (krc)

[V12] = diagonal matrix with element V12;mm = J 0 (kra) Y (krc) J (krc) Y 0 (kra) [Y23] = diagonal matrix with element Y23;nn = Yv (krd)

[J23] = diagonal matrix with element J23;nn = Jv (krd)

[V32] = diagonal matrix with element V32;ii = J 0 (krb) Y (krd) J (krd) Y 0 (krb) 0 0 0 Y23 = diagonal matrix with element Y23;nn = Yv (krd)

0 0 0 J23 = diagonal matrix with element J23;nn = Jv (krd)

"i [W32] = diagonal matrix with element W32;ii = J 0 (krb) Y 0 (krd) J 0 (krd) Y 0 (krb) 2 1 i = 0 "i = 2 i = 0 6 The roots of the determinant of eqn 5.59 will give the symmetric TEz mode cuto¤ wavenumber kr.

75 The power in each region is given below.

c 1 P I = !E !H zdd^ Za Z0 c 1

= (EH EH) dd Za Z0 M 1 k 2 = z 1 Ah k 2 m m=0;1;2:: X 2 c 2 J 0 (kra) Y(kr) Y 0 (kra) J (kr) + 8 9 d (5.60) 2 2 2 Za > [J (kra) Y+1 (kr) Y (kra) J+1 (kr)] kr > < "m = :> ;>

d s P II = !E !H zdd^ Zc Z0 d s

= (EH EH) dd Zc Z0 d 2 N BhY (k ) + ChJ (k ) 2 v + kz s n v r n v r = 8 9 d (5.61) k 2 h h 2 2 m=1;2;:: > B Y 0 (kr) + C J 0 (kr) k > X Zc < n v n v r = :> ;>

76 b 1 P III = !E !H zdd^ Zd Z0 b 1

= (EH EH) dd Zd Z0 I 1 k 2 = z 1 Dh k 2 i i=0;1;2;:: X b 2 2 0 0 J (krb) Y (kr) Y (krb) J (kr) + d (5.62) 8 2 9 0 0 0 0 2 2 Z > J (krb) Y (krb) Y (krb) J (kr) kr > d < "i = :> ;> The normalization factor for the coe¢ cients is the square root of the sum of the powers across the entire guide.

N h = 2 (P I + P II + P III ) (5.63) p

5.4 Ridge Sectoral Guide Asymmetric TEz Mode

The symmetric half of the ridge sectoral guide is divided into three regions. The potential function in terms of the unknown coe¢ cients for each region is …rst given. The tangential

…elds E and Hz are derived using this potential. Imposing the requirement for tangential …eld continuity across the regions leads to the mode characteristic equation.

Region I 0 ; a c 1 The potential function for this region is given by the following series.

M 1 I h 0 0 jkzz = Am J (kra) Y (kr) J (kr) Y (kra) cos () e (5.64) m=0;1;2;:: X h i where

77 m 2 2 2 = ; m = 0; 1; 2; :::M 1 ; kz = k kr 1

M 1 h 0 0 0 0 jkzz E = Am J (kra) Y (kr) J (kr) Y (kr) cos () e (5.65) m=0;1;2;:: X h i

M 1 1 h jkzz E = A J 0 (k a) Y (k ) J (k ) Y 0 (k a) sin () e (5.66) m r r r r m=0;1;2;:: X h i

M 1 2 kr h jkzz H = A J 0 (k a) Y (k ) J (k ) Y 0 (k a) cos () e (5.67) z jk m r r r r m=0;1;2;:: X h i

M 1 kz h jkzz H = A J 0 (k a) Y (k ) J (k ) Y 0 (k a) sin () e (5.68) k m r r r r m=0;1;2;:: X h i

M 1 kz h jkzz H = A J 0 (k a) Y 0 (k ) J 0 (k ) Y 0 (k ) k cos () e (5.69) k m r r r r r m=0;1;2;:: X h i

Region II 0 ; c d s The potential function for this region is given by the following series.

N 1 II h h jkzz = BnYv (kr) + Cn Jv (kr) cos (v) e (5.70) n=0;1;2;:: X where

n 2 2 2 v = ; n = 0; 1; 2; :::N 1 ; kz = k kr s

78 N 1 h 0 h 0 jkzz E = BnYv (kr) + Cn Jv (kr) kr cos (v) e (5.71) n=0;1;2;:: X h i

N 1 1 h h jkzz E = B Y (k ) + C J (k ) v sin (v) e (5.72) n v r n v r n=0;1;2;:: X

N 1 2 kr h h jkzz H = B Y (k ) + C J (k ) cos (v) e (5.73) z jk n v r n v r n=0;1;2;:: X

N 1 kz h h jkzz H = B Y 0 (k ) + C J 0 (k ) k cos (v) e (5.74) k n v r n v r r n=0;1;2;:: X h i

N 1 kz h h jkzz H = B Y (k ) + C J (k ) v sin (v) e (5.75) k n v r n v r n=0;1;2;:: X

Region III 0 ; d b 1 The potential function for this region is given by the following series.

I 1 III h 0 0 jkzz = Di J (krb) Y (kr) J (kr) Y (krb) cos () e (5.76) i=0;1;2;:: X h i where

i 2 2 2 = ; i = 0; 1; 2:::I 1 ; kz = k kr 1

I 1 h 0 0 0 0 jkzz E = Di J (krb) Y (kr) J (kr) Y (krb) kr cos () e (5.77) i=0;1;2;:: X h i

I 1 1 h jkzz E = D J 0 (k b) Y (k ) J (k ) Y 0 (k b) sin () e (5.78) i r r r r i=0;1;2;:: X h i

79 I 1 2 kr h jkzz H = D J 0 (k b) Y (k ) J (k ) Y 0 (k b) cos () e (5.79) z jk i r r r r i=0;1;2;:: X h i

I 1 kz h jkzz H = D J 0 (k b) Y (k ) J (k ) Y 0 (k b) sin () e (5.80) k i r r r r i=0;1;2;:: X h i

I 1 kz h jkzz H = D J 0 (k b) Y 0 (k ) J 0 (k ) Y 0 (k b) k cos () e (5.81) k i r r r r r i=0;1;2;:: X h i

Because Regions I and III have the same subtended angle 1, one may set M = I.

Matching E and Hz …elds at the boundary between region I and II at = c, and the boundary between region II and III at = d.

h h G11 G12 [B] = [0] (5.82) 2 h h 3 2 3 G21 G22 [C] 6 7 6 7 4 5 4 5 where

h ah T 1 ah G = H [W12][V12] H [Y12] [] Y 0 11 12 12 12 h ah T 1 ah G = H [W12][V12] H [J12] [] J 0 12 12 12 12 h ah T 1 ah G = H [W32][V32] H [Y23] [] J 0 21 32 32 23 h ah T 1 ah G = H [W32][V32] H [Y23] [] J 0 22 32 32 23 1 H ah = 2 cos m cos n d ; m= 0; 1; 2; :::M 1 ; n = 0; 1; 2; :::N 1 12;mn 1 1 2 0 R1 Hsh = 2 cos i cos n d ; i = 0; 1; 2; :::I 1 ; n = 0; 1; 2; :::N 1 32;in 1 1 2 0 0 0 0 Y12 = diagonalR matrix with element Y12;nn = Yv (krc)

0 0 0 J12 = diagonal matrix with element J12;nn = Jv (krc) 80 "m W12 = diagonal matrix with element W12;mm = J 0 (kra) Y 0 (krc) J 0 (krc) Y 0 (kra) 2 1 m = 0 "m = 8 > 2 m = 0 < 6 [Y12] = diagonal matrix with element Y12;nn = Yv (krc) :> [J12] = diagonal matrix with element J12;nn = Jv (krc)

[V12] = diagonal matrix with element V12;mm = J 0 (kra) Y (krc) J (krc) Y 0 (kra) [Y23] = diagonal matrix with element Y23;nn = Yv (krd)

[J23] = diagonal matrix with element J23;nn = Jv (krd)

[V32] = diagonal matrix with element V32;ii = J 0 (krb) Y (krd) J (krd) Y 0 (krb) 0 0 0 Y23 = diagonal matrix with element Y23;nn = Yv (krd)

0 0 0 J23 = diagonal matrix with element J23;nn = Jv (krd)

"i [W32] = diagonal matrix with element W32;ii = J 0 (krb) Y 0 (krd) J 0 (krd) Y 0 (krb) 2 1 i = 0 "i = 8 > 2 i = 0 < 6 s 2 [] = diagonal matrix with element nn = :> 1 "n The roots of the determinant of eqn 5.82 will give the asymmetric TEz mode cuto¤ wavenumber kr. The power in each region is given below.

c 2 2 M 1 0 k 2 J (kra) Y (kr) + P I = z 1 Ah d m 8 2 9 k 2 0 0 0 0 2 2 m=0;1;2;::: Za > J (kra) Y (kr) J (kr) Y (kra) kr > X < "m = (5.83) :> ;>

2 N 1 c v h h 2 k BnYv (kr) + Cn Jv (kr) + P II = z s d (5.84) 8 2 9 k 2 h 0 h 0 2 2 n=0;1;2;:: > BnYv(kr) + Cn Jv (kr) kr > X Zc < "n = :> ;>

81 Mode type Ansoft HFSS (kr;in rad/m) Transverse Resonance Technique (kr;in rad/m) TE 123.8 124.9 TE 227.2 230.7 TE 234.0 236.6 TE 288.2 293.0 TE 405.1 408.0 TE 492.2 492.1 TE 630.7 630.5 TE 631.7 632.1 TM 648.3 645.1 TE 645.5 645.4 TM 714.6 709.5 TE 719.8 721.8 TE 796.0 796.8 TE 808.4 810.5

Table 5.1: Comparison of the …rst 14 cuto¤ propagation constants calculated using TRT and HFSS. Constants are in radians/meter. Dimensions of sectoral ridge are: a=10mm, c=15mm, d=20mm, b=25mm.

b 2 2 I 1 0 0 k 2 J (krb) Y (kr) J (kr) Y (krb) + P III = 1 z Dh d i 8 2 9 2 k 2 0 0 0 2 2 i=0;1;2;:: Z > J (krb) Y (kr) J (kr) Y (krb) kr > X d < "i = (5.85) :> ;>

5.5 Results

The accuracy of using TRT is being veri…ed by comparing kr obtained using TRT and Ansoft HFSS (Table 5.1). While HFSS cannot give information such as mode indices, this is not important as the comparison is only between ordered kr. From the table, we can draw the conclusion that the TRT is an accurate way to solve for the modes of the sectoral ridge waveguide.

82 Chapter 6

Modelling the sectoral-coaxial junction

An array of sectoral waveguides that are uniformly spaced are used as feeds to the coaxial cone. The …elds in each sectoral waveguide are assumed here to be excited by a probe. The matching of the probe to the sectoral waveguide is not being investigated here. The junction between the sectoral waveguides and the coaxial waveguide is being modelled by the mode matching method. As in typical mode matching procedure, the coupling between the coaxial waveguide and each sectoral waveguide is …rst being evaluated. Fig- ure 6.1 shows the ‡owchart with the steps to compute the S-matrix of the sectoral-coaxial junction. Firstly, modal content in both sectoral and coaxial waveguides are computed and ranked in order of their cuto¤ frequencies. The coupling coe¢ cient of the junction involves integration in both azimuth and rho (in the radial direction) coodinates, and it is wise to split them up so that for each sectoral waveguide with a di¤erent rotational position, integration in the rho coordinate needs not be computed again.

83 Figure 6.1: Flowchart showing steps to compute the GSM for a coaxial-sectoral waveguide junction.

84 6.1 Coupling coe¢ cients between a sectoral waveguide

- coaxial waveguide junction

Two numerical integrations appear in the calculation of the coupling coe¢ cients. The …rst is with respect to the radial variable and the second is with respect to the azimuthal variable. The latter variable is a function of the position of rotational angle s of the sectoral waveguide. For each sectoral waveguide, a new X’-Y’coordinate system is used. The following functions are evaluated for the integration.

Figure 6.2: Rotation of the sectoral guide with respect to the coaxial guide

(m+)o o sin 2 sin[ms+ 2 ] h (mi+) ; except m = = 0 SS1 (m; ; o; s) = 8 (6.1) <> 0; m = = 0

> (m )o o : sin sin[ms ] 2 2 h (mi ) ; m = SS2 (m; ; o; s) = 6 (6.2) 8 > o sin m o ; m = < 2 s 2 h i >(m+)o o sin : 2 cos[ms+ 2 ] h (mi+) ; except m = = 0 SC1 (m; ; o; s) = 8 (6.3) o ; m = = 0 <> 2

:> 85 (m )o o sin cos[ms ] 2 2 h (mi ) ; m = SC2 = 6 (6.4) 8 > o cos m o ; m = < 2 s 2 h i > Coupling between TM mode: in coaxial waveguide and TM mode in sectoral waveguide

If the TM mode in the coaxial waveguide has a vertical orientation, then

es es ea j j i T M;T M T M;T M Pij = ea SS1F1 + SS2F2 (6.5) si a a h i If the TM mode in coaxial waveguide has a horizontal orientation, then

es es ea j j i T M;T M T M;T M Pij = ea SC1F1 + SC2F2 (6.6) si a a h i where

b m r r r r T M;T M e ea e es e0 ea e0 es F1 = ea m i s Zu j m i Z j rdr (6.7) i r=a a jr=a a a a Za

b m r r r r T M;T M e ea e es e0 ea e0 es F2 = ea m i es Zu j + m i Z j rdr (6.8) i r=a a j r=a a a a Za

If the coaxial waveguide mode is TEM (TM00) then Pij = 0

Coupling between TM mode in coaxial waveguide and TE mode in sectoral waveguide

If the TM mode in the coaxial waveguide has a vertical orientation, then

86 hs hs ea j j i T M;T E T M;T E Pij = ea SS1F1 + SS2F2 (6.9) si a a h i If the TM mode in the coaxial waveguide has a horizontal orientation, then

hs hs ea j j i T M;T E T M;T E Pij = ea SC1F1 + SC2F2 (6.10) si a a h i

b m r r r r T M;T E e ea h0 hs e0 ea h hs F1 = ea r m i Z j m i es r Z j rdr (6.11) i a a a a j a a Za

b m r r r r T M;T E e ea h0 hs e0 ea h hs F2 = c r m i Z j + m i es r Z j rdr (6.12) i a a a a j a a Za

If the coaxial waveguide mode is TEM (TM00), then

hs b j 2 1 h hs r p P = ea Z dr; p = 1; 3; 5; :::; = (6.13) ij b r j a i 2 ln( ) 0 r a a q R

hs b j 2 1 h hs r p ea Z dr; p = 1; 3; 5; :::, = b r j a i 2 ln( ) 0 Pij = 8 r a a (6.14) > q R <> 0; p = 2; 4; 6; :::

> Coupling between: TE mode in coaxial waveguide and TM mode in sectoral waveguide

If the TE mode in the coaxial waveguide has a vertical orientation, then

es es ha j j i T E;T M T E;T M Pij = ha SS1F1 SS2F2 (6.15) si a a h i If the TE mode in the coaxial waveguide has a horizontal orientation, then

87 es es ha j j i T E;T M T E;T M Pij = ha SC1F1 SC2F2 (6.16) si a a h i where

b m r r r r T E;T M h ha e0 es h0 ha e es F1 = ha r m i Z j m i es r Z j rdr (6.17) i a a a a j a a Za

b m r r r r T E;T M h ha e0 es h0 ha e es F2 = ha r m i Z j + m i es r Z j rdr (6.18) i a a a a j a a Za

Coupling between TE mode in coaxial waveguide and TE mode in sectoral waveguide

If the TE mode in the coaxial waveguide has a vertical orientation, then

hs hs ha j j i T E;T E T E;T E Pij = ha SS2F1 SS1F2 (6.19) si a a h i

If the TE modes in the coaxial waveguide has a horizontal orientation, then

hs hs ha j j i T E;T E T E;T E Pij = ha SC2F1 SC1F2 (6.20) si a a h i

b m r r r r T E;T E h ha h hs h0 ha h0 hs F1 = ha r m i hs r Z j m i Z j rdr (6.21) " i a a j a a a a # Za

88 b m r r r r T E;T E h ha h hs h0 ha h0 hs F2 = c r m i hs r Z j + m i Z j rdr (6.22) " i a a j a a a a # Za

6.2 Results

The formulation for the sectoral-coaxial junction is being validated with HFSS. Two test cases are studied here. The …rst is a pure sectoral-coaxial junction and the second is two sectoral-coaxial junctions placed back-to-back. with the in-between section a uniform coaxial waveguide (…gure 6.3). For the …rst case, ten sectoral waveguides are used at the input of the coaxial waveguide, and for the second case, four sectoral waveguides are used at both the input and output of the coaxial guide. Figures 6.4-6.8 show that the formulation for the sectoral-coaxial junction is developed correctly, with excellent agreement between mode matching and HFSS. Mode matching also proves to be much faster than HFSS in simulating this junction. Figure 6.6 shows the results for test case one when di¤erent number of modes are used. For kmax=1k, 3k and 8k, the results remain unchanged, thus showing convergence while that for kmax=0.2k, result is only slightly di¤erent.

89 Figure 6.3: Two sectoral-coaxial junctions placed back-to-back.

90 Figure 6.4: S11 magnitude of sectoral-coaxial junction for test case 1. Inner radius = 45mm, outer radius = 50mm, coaxial waveguide length = 40mm, 10 sectoral waveguides. In mode matching, kmax=6k is being used to limit the number of modes. At operating frequency of 10GHz, 470 sectoral modes and 477 coaxial modes are being used. Runtime for the mode matching method is 7 seconds, while that for Ansoft HFSS is 3 minutes 26 seconds.

91 Figure 6.5: S21 amplitude of sectoral-coaxial junction for test case 1.

92 Figure 6.6: Using di¤erent number of modes for test case 1. For kmax=0.2k, 1k, 3k and 8k, number of modes used are 3, 19, 63 and 665 repectively.

93 Figure 6.7: S21(port5,port1) phase of sectoral-coaxial junction for test case 2. Inner radius = 41.275mm, outer radius = 51.275mm, 4 sectoral waveguides.

94 Figure 6.8: S21(port6,port1) phase of sectoral-coaxial junction for test case 2.

95 Chapter 7

Geodesics of a cone

The geodesic lens comprises a parallel plate waveguide in which waves travel within. The lens is excited by an array of probes that are spaced uniformly at the base of a coaxial cylinder and the parallel plate guides the waves from the feeds to the top of the cylinder. The base is made smaller than the top to minimize the number of spurious rays that can wrap around within the cylinder’sparallel plate, and so is essentially a tapered coaxial cone. Figure 7.1 shows the path of a ray in the cone travelling from the input feed to the an output point at the top of the cone. The probes are inserted into sectoral waveguides which are interfaced to the bottom of the cone. Each probe is excited with a certain phase and amplitude so that the rays emerging from the top of the cone are able to form a beam that is focused in the required azimuthal angle. By commutating the signal excitations, the directional beam can be made to steer for 360o coverage. The derivation of the geodesic path of the ray from the feed follows that of McFarland [6].

96 Figure 7.1: Geodesic path of a ray travelling from input to output of cone.

7.1 Derivation of the geodesic path

The derivation of the geodesic path follows that of classical geometrical optics (GO) theory. In GO, the electromagnetic wave is visualized as propagating along trajectories called rays that can be determined from simple geometrical rules and principles. One very important principle is that of Fermat’s, which states that the rays between any two points follow a path that makes the optical distance between them an extremum [69][70]. The di¤erential path length of a ray travelling from the input to the output of the cone is ds = (d)2 + d2 + dz2 (7.1) q and 2 2 S = 2_ + (1 +z _2)d = F ; ; _ d (7.2)

Z1 q Z1 where d dz _ = ; z_ = (7.3) d d

We want to …nd the shortest path between the input point and the output point located on a cone. The calculus of variation is employed to locate the extrema of functionals. In this

97 case, the functional is S, which has a function as its input and real numbers as its output. The typical way to start o¤ with …nding the extrema is by the use of the Euler-Lagrange equation [71], in which the integrand F must satisfy. The Euler-Lagrange equation is

@F d @F = 0 (7.4) @ d @ ()

Substituting F into the Euler-Lagrange equation, it can be shown that the geodesic passing from radial distance ;azimuthal angle to radial distance 0, azimuthal angle 0 satis…es the following equation.

p1 +z _2 () = c1 d + ( ) (7.5) 2 2 0 c1 Z0 p and i p1 +z _2 ( ) = c1 d + ( ) (7.6) i 2 2 0 c1 Z0 p An important theorem from that of Clairaut’s[72] stated that

sin ( ) = c1 (7.7)

where if "g is a geodesic curve on a surface of revolution, m is a meridian of the surface

intersecting the geodesic at a point P, and is the angle between g and m", [10]. c1 is the distance of closest approach of the ray measured from the z axis (…gure 7.1). Thus, making use of Clairaut’stheorem, and imposing the condition of focusing at in…nity,

sin ( ) = c1 (7.8) 0 0

where 0 and 0 are de…ned in …gure 7.1.

98 For a cone, its half cone angle is de…ned as

csc = p1 +z _2 (7.9)

Letting

= c1 csc (7.10)

where is an independent variable, and substituting into 7.5,

2 = p1 +z _ ( o) + ( ) (7.11) o

where = o for = o. Using eqn 7.7 and eqn 7.10,

1 o = sin sin (7.12) o

Substituting eqn 7.12 into eqn 7.11, the geodesic projection in the ; plane is

1 o = (csc ) sin sin ( ) + ( ) (7.13) o o o

The top view of the ray propagating from input feeds to the output of the cone is shown in …gure 7.2. For maximum e¢ ciency, there should be 1-to-1 mapping of each input feed and output point. This also ensures that there are no spurious rays that may distort the pattern. By setting (i) = and substituting this into eqn 7.13, the maximum output angle o(max) of a ray is derived. The maximum linear aperture length is ho. The azimuth beamwidth can be roughly calculated from the aperture length by the following formula [73], assuming a uniform aperture line source.

180 3dB Beamwidth = 0.866. . (7.14) ho

99 Hence the ratio o and the cone angle can be used as design parameters to control the i azimuth beamwidth of the radiation pattern.

Figure 7.2: Top view of the geodesic path of the lens.

The excitations at the feed have two components - amplitude and phase. The phase excitations at the feed are calculated to ensure that the rays coming from each feed sum together at the rim of the output cone to provide a wave front, while the amplitude excitations are needed to synthesize the required far …eld pattern. To derive the amplitude excitations for the feed, a relationship needs to be developed between the distribution at the output of the cone and the signal power distribution at the feed. The far …eld pattern from the cone is then determined from the aperture illumination given by the distribution at the output of the cone.

The relationship between the linear aperture variable Yo and the output ray angle is

Yo = o sin o (7.15)

The increment of output power into the linear aperture is P (yo) dyo, and the increment of input power from each input point is P (i) di: Assuming no loss within the lens, conservation of energy within the ‡ux tube dictates

100 that

P (yo) dyo = P (i) di (7.16)

By di¤erentiating eqn 7.15 and making use of eqn 7.13, the relationship between input and output distributions can be further developed into

P (yo) cos o P (i) = (7.17) csc cos o 1 csc + 2 i sin2 o o r From eqn 7.17, selecting a desired power distribution for the linear aperture will translate into a set of amplitude excitations. The phase excitations are calculated from the geodesic path of each ray coming out from the feed. As the focusing criteria is that all rays from the input feeds must have equal path lengths to a tangent line (…gure 7.3) at the point at which the reference (central) ray leaves the output rim of the cone, the distance traverse by the rays is a constant, i.e.

S + x = constant (7.18)

where S is as follows

o 1 +z _2 S = 2 2 d (7.19) s c1 Zi

with c1de…ned by eqn 7.8. The geodesic path length from the centre of each sectoral feed to a corresponding output point at the rim of the coaxial cone is calculated from eqn 7.19 and phase compensation added to it so as to have the same phase as the central (reference) ray.

101 Figure 7.3: Top view of geodesic lens showing the reference path travelled by the central ray and a path travelled by another ray.

7.2 Design Procedure

In the design of the geodesic lens, the antenna element and the geodesic cone should be considered separately. Each part is designed independently of each other except for the interface between the antenna and the cone (details in the GAM section). As a guide, the design of the cone should be done …rst since it determines the dimension of the output port which is cascaded with the antenna element. The following is the design methodology for designing the cone.

1. Set an initial value for the output radius o of the lens. Generate the graph (e.g. …gure 7.4) that shows the variation of beamwidth with cone angle for di¤erent i ratio. It is o assumed here that the line source at the output has a uniform distribution. If it is not, then …gure 7.5 can be used, and beamwidth can be estimated by multiplying the output aperture length by a constant. It can be seen from the graph that a high i ratio is required for high o gain, narrow beam width. As the ratio goes higher, the cone angle loses its importance as the beam width variation becomes smaller. Thus it can be conveniently assumed that for high gain requirements, which is usually the case, cone angle does not play a part in the shaping of the beam.

102 2. Vary the output radius o of the lens and generate the plot as above. Figure 7.6

shows what happens when o is reduced. As o decreases, there is a bigger region of small beamwidths that cannot be reached. The designer has to iterate the process of changing o so as to …nd the maximum i and the smallest that can achieve the required beamwidth. o o

o i From the relation h = tan( ) the shortest height of the cone (meaning smaller volume) can be designed.

Figure 7.4: Plotting the variation of beamwidth with cone angle. Frequency of operation =

13GHz, o=42mm; uniform line source is assumed.

i 3. For a particular set of cone angle, and the ratio , ho is determined. Many such o o sets can be generated sweeping both the cone angle and i over a range of values. One o

set that meets the requirement for ho is chosen.

4. is set from the chosen ratio i . The spacing of the probes around the input i o circular array is set so that it is roughly equal to the free space half wavelength at the highest operating frequency. For the n number of input feeds, check that not all modes

103 Figure 7.5: Plotting the variation of linear aperture length with sin( )

are cuto¤. If all are cuto¤, then re-choose the spacing between the input feeds so that the dimensions of the sectoral waveguide are bigger. However, bear in mind that the bigger the spacing, the higher the sidelobes and the possibility of grating lobes emerging. If there are too few sectoral waveguides, start from (1) and increase o. 5. Each sectoral waveguide is assigned a number (…gure 7.7). The amplitude distribution at the linear aperture is being projected back to each input sectoral waveguide using equation 7.17, and the amplitude needed to be excited at the sectoral waveguide is determined. 6. Calculate the phase of each geodesic path between the probe and output point at the circular rim. 7. Compensate the phase at each probe so that the phase delay from the probe to the designated wave front is equalized. 8. Run the mode matching code to obtain the full S-matrix of the cone. 9. Post-process the S-matrix, taking into account the calculated phase compensation in

104 Figure 7.6: Plotting the variation of beamwidth with cone angle. Frequency of operation =

13GHz, o=35mm; uniform line source is assumed.

(7) and the amplitude excitation computed in (5). 10. The amplitude and phase at the output of the cone due to each sectoral feed is being summed together to obtain the active transmission coe¢ cient. 11. The active transmission coe¢ cient is input into the formula stated below that is used to calculate the pattern for a circular array.

j[kR sin cos( n)+ n] E (; ) = In e (7.20) n j j X where I is the amplitude of the excitation, R is the radius of the circular array, = j j n n 2 N , N is the number of elements and is the phase of the excitation. The design procedure described above involves numerous iterations because of two con- siderations. The …rst is to have an antenna as small as possible while meeting the speci…ed requirements, and the second is the number of input sectoral waveguides that are needed to

105 Figure 7.7: Top view showing the mapping of the input probe to the maximum output angle. Each probe is given a number.

ensure no grating lobes and low sidelobes. The design procedure is outlined in a ‡owchart in …gure 7.8.

7.3 Results

An acid test for the calculated phase excitations at the feeds is by checking the beam pointing of the radiation pattern and its beamwidth. A wrongly calculated phase excitation will show up in the form of a skewed beam. A test case is being used to test the beam pointing accuracy as well as the beamwidth.. The dimensions of the cone have already been obtained from

steps (1) to (7). The dimensions are i = 42.291mm, o = 63.4365mm, h = 101.6mm, with 24 input feeds and operating frequency is 10GHz. Element spacing is 0.55 wavelength. A point to note is that all modes are cuto¤ at frequencies below 10GHz. Figure 7.9 shows the

106 Figure 7.8: Flowchart showing the design procedure of the geodesic lens.

107 Figure 7.9: Calculated radiation pattern at 10GHz for 24 input feeds. calculated normalized E-plane radiation pattern. The theoretical beamwidth is 19.154o and the simulated beamwidth is 20o. Another set of excitations is calculated at 13GHz and the computed radiation pattern is shown in …gure 7.10. Though the main beam points in the correct direction, grating lobes start to appear in the visible space owing to the large spacing (0.71 wavelength) between the elements. High sidelobes also come into place for the same reason. Next, an investigation is done to see the e¤ects of phase shift on radiation pattern. Phase excitations are …xed at 10GHz and radiation patterns are generated at di¤erent frequencies. Figures 7.11 to 7.13 show the radiation patterns for 11Ghz to 13GHz. At frequencies higher than 10GHz, sidelobes are visibly higher. This is attributed to two factors. One, the phase excitation is only exact at 10GHz and secondly, the increased spacing between elements.

108 Figure 7.10: Calculated radiation pattern at 13GHz for 24 input feeds.

The patterns, however, show that the beam does not squint with frequencies.

109 Figure 7.11: Calculated radiation pattern at 11GHz for a …xed set of phase excitations.

110 Figure 7.12: Calculated radiation pattern at 12GHz for a …xed set of phase excitations

111 Figure 7.13: Calculated radiation pattern at 13GHz for a …xed set of phase excitations.

112 Chapter 8

Conclusions

8.1 Summary and conclusions

The objective of this thesis is to use modal analysis to analyze the geodesic lens antenna. The mode matching method is chosen for its accuracy and e¢ ciency in solving for structures which have canonical cross-section. There are three main components to the geodesc lens antenna. They are the feed section, the geodesic cone which is the beamformer, as well as the radiating element. The geodesic cone is a tapered coaxial waveguide whose dimensions are selected to form a radiating beam of the required beamwidth. The radiating element being considered here is the monocone, while the waveguide that feeds to the coaxial cone is chosen to be the ridge sectoral waveguide. Using mode matching method the S-matrices for the di¤erent parts of the antenna are found and …nally cascaded using the generalized scattering matrix method. In this way, a complex problem that cannot be easily solved using commercial software is made less intractable. Chapter 2 of this thesis deals with the modelling of the coaxial cone which is approximated using short sections of coaxial waveguides that are o¤set from one another. The discontinuity between the coaxial waveguides is analyzed using mode matching and the overall S-matrix is obtained by cascading the individual S-matrices of the junction discontinuity and uniform

113 sections of waveguide. Results are compared with that of HFSS and found to be in excellent agreement. The simulation times using the mode matching method with di¤erent number of modes and HFSS are also being compared. Although higher number of modes will give a more accurate result, this comes at the expense of computation time. As the results in this chapter show, it is better to …nd a balance between accuracy and simulation time. Results also show that using a smaller number of modes needs not lead to drastically bad accuracy and more often than not, it gives a decent accuracy without using too much computation time. Chapter 3 presents the analysis of the coaxial to radial waveguide junction that serves to link the coaxial cone and the radial waveguide that is the radiator. The generalized admittance matrix approach is used in this case as it o¤ers simplicity and elegance in solving the problem and is also able to do it more e¢ ciently than the generalized scattering matrix way. The admittance matrix is however converted to S-matrix for linking up with the rest of the s-matrices of the other components. Results compared well with that of HFSS and there is much saving in computation time. Di¤erent number of modes are also used in mode matching to show convergence. Chapter 4 gives details for the analysis of the bicone/moncone antenna using mode matching method. Similar to the coaxial cone, the bicone/monocone is being subdivided into sections of radial waveguides o¤set from each other. Short sections of radial waveguides are used to join together these discontinuities. A wideband monocone antenna is built to verify the formulation. Generally, results agree well with HFSS. However, owing to the smaller steps between radial waveguides that is needed to achieve accuracy, the simulation time using mode matching is much longer than that of HFSS. It may be advisable to simulate the radiating element using HFSS for design purpose. In chapter 5, the transverse resonance technique is used to compute the modal content of a ridge sectoral waveguide. This technique is able to yield the cuto¤ frequency quickly and results are in agreement with that of HFSS.

114 The interface between an array of sectoral waveguides and the coaxial cone is studied in chapter 6. As in earlier chapters, the mode matching method is used. Results from two test cases show that it is an accurate and e¢ cient method compared with HFSS, which takes far too long to set up the geometry and thus is not suitable as a design tool. Using di¤erent number of modes in mode matching a¤ects the accuracy only slightly, and convergence is shown. Finally, in chapter 7, the geodesic nature of the lens is studied. The idea behind the lens is to …nd the shortest path length that a ray will travel as it enters the lens. The required phase and amplitude excitations are being computed by using geometric optics to trace the path of a ray and …nding out the compensation required so that the distances travelled by a group of rays in the focussed angle are a constant. The design methodology is also presented in this chapter and developed codes in chapter 2 and 6 are used to generate the far …eld pattern. Simulated radiation patterns show that beam pointing is accurate but sidelobes or grating lobes may come into the picture if the lens is not properly designed.

8.2 Further work

Although sectoral waveguides are used as feeds to the coaxial cone, it is di¢ cult to maintain propagating modes as the sectoral waveguides become smaller. Ideally, the ring of input feeds at the bottom of the cone should be able to pack closer together so as to approximate as much as possible a continuous source. Formulations need to be done for the case of sectoral ‘waveguides’without the PEC walls. The use of ridge sectoral waveguides in cascade to do impedance matching may also be included in the future.

115 Appendix A

Circular Coaxial Waveguide Mode Functions

TMz Modes

e sin (m) ee = i e0 ( er=a) ir a m i 8 9 <> cos (m) =>

:> cos (m;>) ee = m e ( er=a) i r m i 8 9 <> + sin (m) =>

:> + cos (m) ;> he = m e ( er=a) ir r m i 8 9 > sin (m) > < =

e :> sin (m) ;> he = i e0 ( er=a) i a m i 8 9 <> cos (m) => where :> ;>

2 e 1=2 e e p"m Ym ( i ) e e e e ( r=a) = 1 [Jm ( r=a) Ym ( ) Ym ( r=a) Jm ( )] m i 2 Y 2 (c e) i i i i m i

i = (m; n)

116 m = 0; 1; 2; ::: …rst mode index n = 1; 2; 3; ::: second mode index

b c = a b = radius of outer conductor a = radius of inner conductor

th Ym = m order Bessel function of the second kind

e th e i = n non-vanishing root of Bessel-Neumann function m (c ) 1 m = 0 "m = 8 > 2 m = 0 < 6 > Mode in horizontal orientation : 8 9 = 8 9 > > > Mode in vertical orientation > < = < = TM0n mode has no horizontal orientation. :> ;> :> ;> The mode propagation constant is 1=2 j k2 (ke )2 = jke propagating mode k > ke e ci i ci i = 8 2 1=2 > (ke ) k2 = ke non-propagating mode k < ke < ci i ci where :> k = operating wave number

e e kci = i =a, mode cuto¤ wave number The characteristic TM mode impedance is given by 1=2 k2 (ke )2 = (!") propagating mode e ci Zi = 8 2 1=2 > (ke ) k2 = (j!") non-propagating mode < ci "r = relative permittivity of medium inside the waveguide :> ! = operating angular frequency At a given cross-section of the coaxial waveguide, the transverse and longitudinal …elds may be expressed in terms of the modal voltage V , current I and the modal vectors.

e e e Ei = Vi ei

e e e Eir = Vi eir

e e e Hi = Ii hi

117 e e e Hir = Ii hir

e Hiz = 0 From Maxwell’sequation,

@rHe @He Ee = 1 i ir = Ieee iz j!"r @r @ i iz The z-componenth of thei modal vector is given by

2 e sin (m) ee = j i e e r iz k a m i a 8 > cos (m) < The modal voltage and current satisfy the Telegraphist or transmission line equation. > dV e : i = eZeIe dz i i i The modal voltage may be written in terms of the outgoing and incomng waves. The

e e corresponding coe¢ cients of these waves are Ai and Bi .

e e e e i z e + i z Vi = Ai e + Bi e Substituting into the transmission line equation yields the modal current

e e e 1 e i z e + i z Ii = Ze Ai e Bi e i TEM Mode

The lowest order mode is the TEM mode where cuto¤ wave number is ke = 0. It may

be considered as the TM00 mode with a vertical orientation. The …eld components of this mode are

e 1=2 1 e = [2 ln (b=a)] ir r e 1=2 1 h = [2 ln (b=a)] i r e The TEM mode propagation constant is i = jk and its mode impedance is i = 1=2 0 . ("0"r) h i

TEz Modes

Components of the transverse electric and magnetic …eld vector mode functions are given below.

118 ha cos (m) eh = i h0 har=a i a m i 8 > sin (m) < :>+ sin (m) eh = m h har=a ir r m i 8 > cos (m) <

ha :> cos (m) hh = i h0 har=a ir a m i 8 > sin (m) < :>+ sin (m) hh = m h har=a i r m i 8 > cos (m) < m = 1; 2; 3; :::; n = 1; 2; ::: :> where

2 1=2 h 2 2 h ha p"m Ym i m m m i r=a = h 1 h 1 h 2 8 Y 0 c c 9 " m i # " i # " i # < = ha 0 h ha 0 h Jm r=a Y Ym r=a J :i m i i m i ; h i 1 if m = 0 "m = 8 > 2 if m = 0 < 6 i = (m; n) :> m = 0,1,2,... …rst mode index n = 1,2,3,... second mode index

b c = a b = radius of outer conductor a = radius of inner conductor

Ym = mth order Bessel function of the second kind

ha h0 i = nth non-vanishing root of Bessel-Neumann function m (c ) i.e. the nth zero of either of the following characteristic functions:

119 J 0 c ha Y 0 ha Y 0 c ha J 0 ha = 0 m i m i m i m i J 0 ha Y 0 ha Y 0 ha J0 ha = 0 m i m i m i m i TE0n has no vertical orientation. The mode propagation constant is 1=2 j k2 kh 2 propagating mode h ci i = 1=2 8 h h 2 2 i > k k non-propagating mode < ci where h i :> k = wave number

h h ki = i =a, mode cuto¤ wave number The characteristic mode impedance is given by ! 2 1=2 propagating mode k2 (kh ) h ci Zi = 8 > h j! i > 2 1=2 non-propagating mode < (kh ) k2 ci At a given> h cross-sectioni of the coaxial waveguide, the transverse and longitudinal …elds :> may be expressed in terms of the modal voltage V , current I and the model vectors.

h h h Ei = Vi ei

h h h Eir = Vi eir

h Eiz = 0

h h h Hi = Ii hi

h h h Hir = Ii hir From Maxwell’sequation,

@Eh @rEh V h Hh = 1 ir i = i hh iz j!r @ @r iz The z-componenth of thei modal vector is given by

2 h cos (m) hh = j i h r r iz k a m i a 8 9 > sin (m) > < = The modal voltage and current satisfy the Telegraphist or transmission line equation. > > dIh : ; i = hY hV h dz i i i The modal current may be written in terms of the outgoing and incoming waves. The

h h corresponding coe¢ cients of these waves are Ai and Bi .

120 h h h h i z h + i z Ii = Ai e + Bi e Substituting into the transmission line equation yields the modal voltage

h h h hz h + hz V = Z A e i B e i i i i i h i

121 Appendix B

Cylindrical Radial Waveguide Mode Functions

The complete orthonormal set of modes of the cylindrical radial waveguide are represented

by the TEz and TMz modes. Because of the circular symmetry of the guide, each TE and TM mode is degenerate. They can be referred to as having a vertical orientation (called the ‘V ’group) or a horizontal orientation (called the ‘H’group).

122 TMz Modes

The transverse and longitudinal components of the electric and magnetic …eld modal vectors are written as follows. sin (m) ee = j "m n sin nz ir k 2r d d 8 9 > cos (m) > < = >+ cos (m)> e "n m n 1 nz : ; e = 2 sin i d r d kn d 8 9 > sin (m) > < = sin:> (m) ;> ee = "n cos nz iz d d 8 9 > cos (m) > < = > cos (>m) e "n m k nz: ; h = j 2 cos ir d r kn d 8 9 <> + sin (m) => e hiz = 0 :> ;> where k2 = k2 n 2 n d i = (m; n)

m = 0; 1; 2; ::: …rst mode index n = 0; 1; 2; ::: second mode index d = height between the lower and upper parallel plates 1 if m = 0 "m = 8 > 2 if m = 0 < 6 k = !op" :> Mode in horizontal orientation 8 9 = 8 9 > > > Mode in vertical orientation > < = < = The mode propagation constant ki is given by :> ;> :> ;> 2 n 2 m 2 2 2 ki = k = k k d r ci q The mode cut-o¤ wavenumber p is kci where

n 2 m 2 kci = d + r q

123 e The mode impedance Zi is de…ned as

2 Ze = d "m kn i "n 2r k ki where

= o = wave impedance of the medium "r"o At aq given cross-section of the radial waveguide, the transverse and longitudinal …elds may be expressed in terms of the modal voltages V and current I and the model vectors.

e e e Ei = Vi ei

e e e Eiz = Vi eiz

e e e Hi = Ii hi

e Hiz = 0 @He Ee = 1 i = Ieee ir j!" @z i ir @Ee @Ee V e He = 1 1 iz i = i he ir j! r @ @z ir The modal voltageh and currenti satisfy the Telegraphist or transmission line equation.

e dVi e e = jkiZ I dr i i The modal voltage may be written in terms if the incident and re‡ected waves. In the region where the wave is propagating, the modal voltage is expressed as

e e (2) e (1) 2 Vi = Ai Hm (knr) + Bi Hm (knr) ; kn > 0 and k > kci where

(2) (1) Hm is the Hankel function of the second kind and represents an outgoing wave. Hm is

the Hankel function of the …rst kind and has the property of an incoming wave. Ai and Bi are the corresponding wave amplitudes. In the transition region between the propagating region and the cut-o¤ region, the modal voltage is expressed as a combination of Bessel functions of the 1st and 2nd kind.

e e e 2 Vi = Ai Ym (knr) + Bi Jm (knr) ; kn > 0 and k < kci In the cut-o¤ region, the modal voltage is given in terms of the modi…ed Bessel functions of the 1st and the 2nd kind.

e e e 2 Vi = Ai Km ( nr) + Bi Im ( nr) ; kn < 0 and k < kci

124 where

n 2 2 n = k d Substitutingq the modal voltage into the transmission line equation will yield the following

e modal current Ii .

e j kn e (2)0 e (1)0 2 Ii = e Ai Hm (knr) + Bi Hm (knr) ; kn > 0 and k > kci Zi ki

e j kn h e 0 e 0 i 2 Ii = e Ai Ym (knr) + Bi Jm (knr) ; kn > 0 and k < kci Zi ki

e j n e 0 e 0 2 Ii = e Ai Km ( nr) + Bi Im ( nr) ; kn < 0 and k < kci Zi ki

TEz Modes

The transverse and longitudinal components of the electric and magnetic …eld modal vectors are written as follows. + sin (m) h "n m 1 nz e = jk 2 sin ir d r kn d 8 9 > cos (m) > < = cos (:>m) ;> eh = "m sin nz i 2r d 8 9 > sin (m) > < = h eiz = 0 :> ;> cos (m) hh = 1 "m n cos nz ir jk 2r d d 8 9 > sin (m) > < = > sin (m>) h "n n m 1 nz : ; h = 2 cos i d d r kn d 8 9 > + cos (m) > < = cos (:>m) ;> hh = "n sin nz iz d d 8 9 > sin (m) > < = where :> ;> k2 = k2 n 2 n d i = (m; n)

m = 0; 1; 2; ::: …rst mode index n = 1; 2; 3; ::: second mode index

125 d = height between the lower and upper parallel plates 1 if m = 0 "m = 8 9 > 2 if m = 0 > < 6 = k = !op" :> ;> Mode in horizontal orientation 8 9 = 8 9 > > > Mode in vertical orientation > < = < = The mode propagation constant ki is given by :> ;> :> ;> 2 n 2 m 2 2 2 ki = k = k k d r ci q The mode cut-o¤ wavenumber p is kci where

n 2 m 2 kci = d + r q h The mode impedance Zi is de…ned as

h 2r "n ki Z = k 2 i "m d kn where

h h h Ei = Vi ei

h Eiz = 0

h h h Hi = Ii hi

h h h Hiz = Ii hiz @Hh @Hh Eh = 1 1 iz i = Iheh ir j!" r @ @z i ir @Eh h h 1 h i Vi h i Hir = j! @z = hir The modal voltage and current satisfy the Telegraphist or transmission line equation.

h dIi h h = jkiY V dr i i The modal current may be written in terms of the incident and re‡ected waves. In the region where the mode is propagating, the modal current is expressed as

h h (2) h (1) 2 Ii = Ai Hm (knr) + Bi Hm (knr) ; kn > 0 and k > kci where

126 (2) (1) Hm is the Hankel function of the second kind and represents an outgoing wave. Hm is the Hankel function of the …rst kind and has the property of an incoming wave. Ai and Bi are the corresponding wave amplitudes. In the transition region between the propagating region and the cut-o¤ region, the modal current is expressed as a combination of Bessel functions of the 1st and the 2nd kind.

h h h 2 Ii = Ai Ym (knr) + Bi Jm (knr) ; kn > 0 and k < kci In the cut-o¤ region, the modal current is given in terms of the modi…ed Bessel functions

st nd of the 1 kind, Im, and 2 kind, Km.

h h h 2 Ii = Ai Km ( nr) + Bi Im ( nr) ; kn < 0 and k < kci where = n 2 k2 d Substitutingq the modal voltage into transmission line equation will yield the following

h modal voltage Vi .

Propagating Region:

h h kn h (2)0 h (1)0 2 V = jZ A Hm (knr) + B Hm (knr) ; k > 0 and k > kci i i ki i i n h i Transition Region:

h h kn h h 2 V = jZ A Y 0 (knr) + B J 0 (knr) ; k > 0 and k < kci i i ki i m i m n Cut-o¤ Region:

h h n h h 2 V = jZ A K 0 ( nr) + B I 0 ( nr) ; k < 0 and k < kci i i ki i m i m n

127 Appendix C

Sectoral Waveguide Mode Functions

The complete orthonormal set of modes of the sectoral waveguide consists of TE and TM mode types. There are no azimuthally degenerate modes.

128 TMz Modes

The transverse and longitudinal components of the electric and magnetic …eld TMmn modal vectors are

es ee = i Ze0 es r sin + o ir a i a 2 e e es r h o i ei = r Z i a cos + 2 2 kes e ( ci ) e esr h io eiz = k Z i a sin + 2

e e es r h o i hir = r Z i a cos + 2 es he = i Ze0 es r hsin +io ie a i a 2 where h i 1=2 2 es Y ( ) Ze es r = i 1 Y es r J ( es) J es r Y ( es) i a Y 2 c es i a i i a i po ( i ) i = (m; n)

m = 1; 2; 3::: …rst mode index n = 1; 2; 3::: second mode index = m o

o = included angle of sectoral waveguide

b c = a b = outer radius of guide a = inner radius of guide

st J = Bessel function of the 1 kind

nd Y = Bessel function of the 2 kind

es th i = n non-vanishing root of the following Bessel Neumann function

es b es es b es Y J ( ) J Y ( ) i a i i a i The mode propagation constant is given by j k2 kes2 propagating mode es ci i = 8 q 2 > kes k2 non-propagating mode < ci k = operatingq wave number > :es es i ki = a = mode cuto¤ wave number

129 The mode impedance is given by pk2 kes2 ci propagating mode = !" 8 pkes2 k2 ci non-propagating mode <> j!"

TEz Mode

The transverse and longitudinal components of the electric and magnetic …eld TEmn modal vectors are

h h hs r o eir = r Z i a sin + 2 hs h i h0 hs r h io ei = a Z i a cos + 2 2 khs h ( ci ) h hsr h io hiz = k Z i a cos + 2

h h hs r h o i hi = r Z i a sin + 2 hs hh = i Zh0 hs r hcos +i o ir a i a 2 where h i

1=2 2 0 hs 2 2 h hs r p"m J i Z i = 1 1 0 hs hs hs a 2o 8"J c i # " c i # " i #9 < = r r p hs 0 hs hs 0 hs = Y J J Y (C.1) i:a i i a i ; h i i = (m; n) 1 if m = 0 "m = 8 > 2 if m = 0 < 6 m = 0; 1; 2; 3::: …rst mode index :> n = 1; 2; 3::: second mode index = m o

o = included angle of sectoral waveguide

b c = a b = outer radius of guide

130 a = inner radius of guide

st J = Bessel function of the 1 kind

nd Y = Bessel function of the 2 kind

hs th i = n non-vanishing root of the following Bessel-Neumann function

Y 0 hs b J 0 ( ) J 0 hs b Y 0 ( ) = 0 i a i a The mode propagation constant is given by j k2 khs2 propagating mode hs ci i = 8 q 2 > khs k2 non-propagating mode < ci k = operatingq wave number > :hs hs i ki = a = mode cuto¤ wave number The mode impedance is given by ! propagating mode 2 hs2 hs pk kci i = 8 j! > 2 non-propagating mode < pkhs k2 ci :>

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137