Refinements to the design of waveguideslot arrays

Soumya Sheel

This Thesis is submitted in fulfilment of the requirement for the degree ofIF49 Doctor of Philosophy

Science and Engineering FacultyQueensland University of Technology

2019

I would like to dedicate this thesis to my loving parents . . .

Statement of Original Authorship

The work contained in this thesis has not been previously submitted to meet requirements for an

award at this or any other higher education institution. To the best of my knowledge and belief, the

thesis contains no material previously published or written by another person except where due

reference is made.

Soumya Sheel

2- ?/01/ Zoi cl

QUT Verified Signature

Acknowledgements

I would like to thank my supervisor, Dr. Jacob Coetzee, for the guidance and encouragementthroughout my time as his student. I would also like to acknowledge Prof. David Lovell,Prof. Bronwyn Harch and Prof. Clinton Fookes for providing the funding which allowed meto undertake this research and for the continuous support during my PhD candidature. I amalso particularly grateful to Queensland University of Technology for awarding me the QUTPostgraduate Research Award and facilitating my development as a researcher.

Finally, I would like to thank SAIVT lab for all the awesome Fridays, and providing themuch needed break from work.

Abstract

Waveguide slot arrays are widely used in radar and telecommunication applications, especiallywhere high gain, increased power-handling capacity and compact dimensions are critical.Radar applications demand radiation patterns with sharp narrow main beams and very specificsidelobe levels for detection and tracking, while other applications require main beams withvery specific shapes.

Slot array design was formalised in the early 1980s. Fundamental design equations weredeveloped to achieve desired slot excitations and an impedance match at the centre frequency.The design for slot arrays is a mature field, but suffers from limitations that in some cases mayresult in reduced performance. This thesis focuses on extending the theory and developingnew design methods that compensate for five known shortcomings of the conventional designmethodology.

Slots arrays are often fed via a main line with centred inclined longitudinal slots, placedat right angles below the branch lines. Mutual coupling between slots in an array is known tohave a great impact on individual slot excitations. The existing formulation does not accountfor higher order mode coupling effects between coupling slots in the main line and the twoadjacent radiating slots. To date, there is no known design method that compensates forthese effects without the use of dedicated analysis software. This thesis presents a detailedmathematical derivation and a step-by-step procedure to account for higher-order modecoupling between the coupling slots and the radiating slots straddling them. The technique isvalidated using illustrative design examples. The new technique is able to achieve stringentdesign specifications.

The conventional formulation for the calculation of the inclination angles and lengthsof the coupling slots in the main line of planar slot arrays relies on the assumption thatall scattering parameters can be expressed in terms of the reflection coefficient and that allscattering parameters are either in-phase or out-of-phase at resonance. This assumption isvalid for very small waveguide wall thickness or large inclination angles, but for realisticvalues of wall thickness and smaller inclination angles, substantial phase differences betweenscattering parameters have been observed. In an array environment, coupling slots aresubject to internal higher-order mode mutual coupling, with the TE20 and TE01 modes being

viii

the major contributors to the coupling. An alternative model for the coupling junction ispresented. Expressions for the equivalent impedance and the coupling coefficient that do notrely on the aforementioned assumptions, and which compensate for the higher-order modecoupling effects are derived.

There is a growing tendency to construct slotted waveguide arrays using light-weight,non-metallic materials like Carbon Fibre Reinforced Plastic. For high frequency applications,small dimensions make implementation of slot arrays in Substrate-Integrated Waveguide(SIW) with etched slots a viable option. In both these cases, the waveguide exhibit inherentlosses that cause some slots to be under-illuminated. This can affect the overall performanceof the antenna. A mathematical derivation of the modified non-linear equations that compen-sate for these effects is presented. The method accounts for arbitrary levels of waveguideloss, and the only additional input required is accurate data for the phase and attenuationconstants of the lossy waveguide.

Although the viability of slot arrays in SIW has been widely reported, high-precisiondesign of arrays to stringent specifications is limited. For narrow-beam low-sidelobe orshaped-beam radiation patterns, strict control over the excitation of individual elements isvital. Conventional design principles fall short in this regard, largely due to failure of thesimplified equivalent circuit for slots. All existing design methods assume a symmetric fielddistribution in the slot aperture. The distribution is known to become asymmetrical for largeoffsets, especially in reduced height or dielectric-filled waveguide. An improved designtechnique that takes asymmetry in slot field into consideration, is presented.

A new feed structure for SIW slot arrays is also proposed. The technique accommodatesfor waveguide losses, and is capable of providing arbitrary complex power split ratios tovariable output port impedances. Shaped-beam arrays in SIW and quarter-height metallicwaveguide are implemented and meet the desired specifications. Antennas designed usingthe new technique offer superior performance compared to the conventional designs.

Contents

List of Figures xiii

List of Tables xvii

1 Introduction 11.1 Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Modes of a waveguide . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Different types of slot radiators in a waveguide . . . . . . . . . . . . . . . 3

1.2.1 Coupling slot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Waveguide-fed slot arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Slot data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.2 Design of slot arrays and representation of slots . . . . . . . . . . . 81.3.3 Analysis of slot arrays . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Waveguide arrays implemented in non-metallic materials . . . . . . . . . . 101.5 Shortcomings in existing design procedures . . . . . . . . . . . . . . . . . 11

1.5.1 Higher-order mutual coupling between the coupling slot and neigh-bouring radiating slots. . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5.2 Phase discrepancy in coupling junctions . . . . . . . . . . . . . . . 111.5.3 Waveguide material-losses in the design of slot arrays . . . . . . . 131.5.4 Asymmetrical fields in the design of waveguide slot arrays . . . . . 13

1.6 Aims and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.7 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Higher order mode compensation between the coupling slots and the radiatingslots straddling them 192.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 Fundamental mode analysis of the coupling junction . . . . . . . . 22

x Contents

2.2.2 Higher-order mode scattering in a coupling junction . . . . . . . . 282.3 Compensation for higher-order coupling . . . . . . . . . . . . . . . . . . . 37

2.3.1 TE20 mode compensation . . . . . . . . . . . . . . . . . . . . . . 392.3.2 TE01 mode compensation . . . . . . . . . . . . . . . . . . . . . . 42

2.4 Design equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.4.1 Radiating slots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.4.2 Coupling slots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.5 Design procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.6 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.6.1 Standard-height array . . . . . . . . . . . . . . . . . . . . . . . . . 512.6.2 Half-height array . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3 Design refinements for the feed of a planar slot array 633.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.2.1 Conventional formulation . . . . . . . . . . . . . . . . . . . . . . 643.2.2 Coupling junction . . . . . . . . . . . . . . . . . . . . . . . . . . 673.2.3 Amended formulation . . . . . . . . . . . . . . . . . . . . . . . . 68

3.3 Design procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.4 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.4.1 Manufactured prototype . . . . . . . . . . . . . . . . . . . . . . . 80

4 Compensation of waveguide losses in the design of slot arrays 874.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.2 Slot characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.2.1 Carbon fibre reinforced plastic slot arrays . . . . . . . . . . . . . . 884.3 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.3.1 Design equations for radiating slots . . . . . . . . . . . . . . . . . 914.3.2 Equivalent network analysis . . . . . . . . . . . . . . . . . . . . . 944.3.3 Design procedure for arrays fed via coupling slots . . . . . . . . . 984.3.4 Design procedure for a single-layer array . . . . . . . . . . . . . . 1004.3.5 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5 Compensation for asymmetrical fields in the design of waveguide slot arrays 1115.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.2 Characterisation of a single slot . . . . . . . . . . . . . . . . . . . . . . . . 112

Contents xi

5.2.1 Analysis of slot fields in SIW . . . . . . . . . . . . . . . . . . . . 1135.3 Design relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.3.1 Calculation of slot voltage components . . . . . . . . . . . . . . . 1155.3.2 Equivalent network analysis . . . . . . . . . . . . . . . . . . . . . 118

5.4 Design procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.5 Single-layer feeding structure for slot arrays . . . . . . . . . . . . . . . . . 123

5.5.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.5.2 Design example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.6 Coaxial line to SIW transition . . . . . . . . . . . . . . . . . . . . . . . . 1345.6.1 Transition design . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355.6.2 Design example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.7 Validation of proposed design method . . . . . . . . . . . . . . . . . . . . 1365.8 An 8×8 element SIW array with circular beam radiation pattern . . . . . . 1385.9 An 8×8 element array in quarter-height PEC waveguide with circular beam

radiation pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6 Conclusion 1516.1 Limitations in the proposed design procedures . . . . . . . . . . . . . . . . 1526.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Bibliography 155

List of Figures

1.1 Waveguide characteristics. . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 TE modes in a waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Surface currents in a waveguide . . . . . . . . . . . . . . . . . . . . . . . 51.4 A planar slotted waveguide array fed via inclined coupling slots. . . . . . . 61.5 Normalised susceptance and conductance vs normalised slot length [1] . . . 71.6 Dispersion curve for reduced-height waveguide with a = 22.86 mm and

b = 5.08 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.7 Dispersion curve for standard-height waveguide with a = 22.86 mm and

b = 10.16 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1 A planar slotted waveguide array (centre) with equivalent circuits for a branchof radiating slots (above) and the main line with inclined coupling slots (below). 21

2.2 Geometry of a radiating slot. . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 Geometry of a coupling junction. . . . . . . . . . . . . . . . . . . . . . . . 222.4 Equivalent circuit for coupling slot t. . . . . . . . . . . . . . . . . . . . . . 232.5 (a) Magnitude and (b) phase of S11 for a slot coupler calculated using CST

Microwave Studio and the approximate dominant mode approach. . . . . . 292.6 (a) Magnitude and (b) phase of S31 for a slot coupler calculated using the

CST Microwave Studio and the approximate dominant mode approach. . . 302.7 Geometry for calculation of TE20 mode coupling between radiating slot (t,kt)

and coupling slot t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.8 Offset of slot (1,3,3) as computed during each iterative step. . . . . . . . . 512.9 Phase errors in the excitations of the radiating slots in a 5×4 element array. 542.10 Amplitude errors in the excitations of the radiating slots in a 5×4 element

array. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.11 Phase errors in the excitations of the coupling slots in a 5×4 element array. 552.12 Amplitude errors in the excitations of the coupling slots in a 5×4 element

array. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

xiv List of Figures

2.13 H-plane radiation pattern of the 5×4 element array. . . . . . . . . . . . . . 562.14 E-plane radiation pattern of the 5×4 element array. . . . . . . . . . . . . . 572.15 Return loss of the 5×4 element array. . . . . . . . . . . . . . . . . . . . . 572.16 H-plane radiation pattern ofthe 8×8 element array. . . . . . . . . . . . . . 612.17 E-plane radiation pattern ofthe 8×8 element array. . . . . . . . . . . . . . 62

3.1 A planar slotted waveguide array (center) with equivalent circuits for a branchof radiating slots (above) and the main line with inclined coupling slots (below). 65

3.2 Geometry of coupling junctions. . . . . . . . . . . . . . . . . . . . . . . . 663.3 Phase variation between the reflected and scattared wave in a coupling

junction in standard-height waveguide. . . . . . . . . . . . . . . . . . . . . 683.4 Phase variation between the reflected and scattared wave in a coupling

junction in half-height waveguide. . . . . . . . . . . . . . . . . . . . . . . 693.5 Geometry of the prototype feed structure. . . . . . . . . . . . . . . . . . . 793.6 Amplitude distribution for the 5 branches. . . . . . . . . . . . . . . . . . . 823.7 Phase spread for the 5 branches. . . . . . . . . . . . . . . . . . . . . . . . 823.8 Reflection coefficient for the conventional and proposed designs. . . . . . . 833.9 Manufactured sections of the model. . . . . . . . . . . . . . . . . . . . . . 833.10 Prototype assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.11 Assembled prototype. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.12 Measurement set-up with two N-type transitions and matched terminations

at other ports. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.13 Measured reflection of the prototype. . . . . . . . . . . . . . . . . . . . . . 86

4.1 Resonant slot length as a function of slot offset in PEC and CFRP half-heightwaveguide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.2 Normalized resonant slot conductance as a function of slot offset in PEC andCFRP half-height waveguide. . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.3 Normalized resonant slot conductance as a function of slot inclination anglein PEC and CFRP half-height waveguide. . . . . . . . . . . . . . . . . . . 90

4.4 Resonant slot length as a function of slot offset in PEC and CFRP standard-height waveguide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.5 Normalized resonant slot conductance as a function of slot offset in PEC andCFRP standard-height waveguide. . . . . . . . . . . . . . . . . . . . . . . 90

4.6 Normalized resonant slot conductance as a function of slot inclination anglein PEC and CFRP standard-height waveguide. . . . . . . . . . . . . . . . . 91

List of Figures xv

4.7 A planar slotted waveguide array (centre) with equivalent circuits for a branchof radiating slots (above) and the main line with inclined coupling slots (below). 92

4.8 Geometry for the nth slot in branch t. . . . . . . . . . . . . . . . . . . . . . 934.9 Geometry of t th coupling junction. The coupling slot of length lt is rotated

through an angle θt with respect to the axis of the main line, with θt positivefor clockwise rotation. The ports are numbered as indicated. . . . . . . . . 96

4.10 Slot voltage amplitude errors for the 8×8 element slot array. . . . . . . . . 1054.11 Slot voltage phase errors for the 8×8 element slot array. . . . . . . . . . . 1054.12 H-plane pattern for the 8×8 element slot array. . . . . . . . . . . . . . . . 1064.13 E-plane pattern for the 8×8 element slot array. . . . . . . . . . . . . . . . 1074.14 Reflection of the 8×8 element slot array. . . . . . . . . . . . . . . . . . . 1074.15 Slot voltage amplitude errors for the 1×8 element liner slot array. . . . . . 1094.16 H-plane pattern for the 1×8 element slot array. . . . . . . . . . . . . . . . 109

5.1 (a) Geometry of longitudinal slot in SIW and (b) its equivalent circuit. . . . 1125.2 (a) Slot field distribution for (a) slot with 1 mm offset illustrating the nor-

malised amplitude and (b) the phase variation along the length of the slot fordifferent slot lengths, (c) slot with 2 mm offset illustrating the normalised am-plitude and (d) the phase variation, and (e) slot with 3 mm offset illustratingthe normalised amplitude and (f) the phase variation. . . . . . . . . . . . . 114

5.3 (a) Equivalent circuit for the t th branch and (b) voltages, currents and ad-mittances associated with the T-network representing the nth slot of brancht. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.4 Geometry of single layer planar arrays with (a) a differential and (b) analternating feed structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.5 (a) Waveguide feed structure with a cascaded series of T-junctions, and (b)geometries of a single T-junction (left: multiple pins, center: 1 pin and right:rectangular plate). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.6 Two port equivalent network of the a single junction. . . . . . . . . . . . . 1275.7 Equivalent network of the entire feed structure. . . . . . . . . . . . . . . . 1285.8 Geometry of the single layer SIW feed structure with 8 branches fed from

alternate ends. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1325.9 (a) The amplitude in ports 2, 3, 4 and 5, and (b) phase distribution in ports 2,

3, 8 and 9 of the feed structure versus frequency. . . . . . . . . . . . . . . 1335.10 Reflection of the feed structure. . . . . . . . . . . . . . . . . . . . . . . . . 1335.11 Microstrip to SIW transition. . . . . . . . . . . . . . . . . . . . . . . . . . 1345.12 Geometry of the proposed feed structure. . . . . . . . . . . . . . . . . . . . 136

xvi List of Figures

5.13 Implementation of a back-to-back coaxial line to SIW transition. . . . . . . 1375.14 Scattering parameters of a back-to-back coaxial line to SIW transition. . . . 1375.15 8×8 element SIW slot array fed using a single layer feed, with branches fed

from alternate sides. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.16 (a) 3-D radiation pattern using conventional design, (b) proposed design. . . 1425.17 (a) H-plane radiation pattern of the 8×8 element array with with a circular

main beam, and (b) zoomed-in comparison of the main lobe. . . . . . . . . 1435.18 (a) E-plane radiation pattern of the 8×8 element array with with a circular

main beam, and (b) zoomed-in comparison of the main lobe. . . . . . . . . 1445.19 Comparison of the input reflection coefficients between the conventional and

the proposed design techniques. . . . . . . . . . . . . . . . . . . . . . . . 1455.20 (a) H-plane radiation pattern of the 8×8 element array with with a circular

main beam, and (b) zoomed-in comparison of the main lobe. . . . . . . . . 1485.21 (a) E-plane radiation pattern of the 8×8 element array with with a circular

main beam, and (b) zoomed-in comparison of the main lobe. . . . . . . . . 1495.22 Comparison of the input reflection coefficients between the conventional and

the proposed design techniques of the 8×8 element quarter-height array. . . 150

List of Tables

2.1 Dimensions of the radiating slots of the 5×4 planar array. . . . . . . . . . 532.2 Dimensions of the coupling slots of the 5×4 planar array. . . . . . . . . . 542.3 Dimensions of the radiating slots of the 8× 8 planar array in half-height

waveguide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582.4 Dimensions of the coupling slots of the 8×8 planar array. . . . . . . . . . 60

3.1 Slot dimensions of the feed network. . . . . . . . . . . . . . . . . . . . . . 813.2 Comparison of the simulated and measured scattering parameters of the

prototype. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.1 Dimensions of the radiating slots of the 8th branch of the 8×8 planar array. 1024.2 Dimensions of the coupling slots of the 8×8 planar CFRP array. . . . . . . 1044.3 Dimensions of the radiating slots of the 1×8 linear array. . . . . . . . . . . 108

5.1 Validation of the feed structure. . . . . . . . . . . . . . . . . . . . . . . . . 1325.2 Dimensions of the implemented SMA to SIW transition. . . . . . . . . . . 1365.3 Dimensions of the radiating slots of the 8×8 SIW array with a circular main

beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.4 Dimensions of the radiating slots of the 8× 8 quarter-height array with

circular beam radiation pattern. . . . . . . . . . . . . . . . . . . . . . . . . 145

Chapter 1

Introduction

Geometric simplicity, efficiency, polarization purity together with an ability to radiate broad-side beams make slot arrays ideal solutions for many radar, communications and navigationapplications. They find wide application in systems that require narrow-beam or shaped-beampatterns, especially when high power, light weight and limited scan volume are priorities. Pla-nar arrays are implemented using a number of rectangular waveguides (branch line guides),arranged side-by-side with radiating slots cut in the walls of the branch lines. These branchescan be excited using various feed structures. A popular feeding mechanism compriseswaveguides (main lines) located behind and at right angles to the branch lines with centered-inclined coupling slots cut in the walls. Single layer feed structures can be employed tofeed branches from the side, especially in high frequency applications with small antennadimensions.

1.1 Waveguides

Waveguides are hollow metal pipes, usually rectangular or cylindrical in shape. They arecapable of guiding waves with minimal loss and have high power-handling capability. Awaveguide acts as a high pass filter in that waves at frequencies above the cut-off frequencycan propagate along it, while frequency components below the cut-off frequency will largelybe reflected or are rapidly attenuated along the waveguide.

The cut-off frequency of a waveguide depends on the shape and size of its cross-section.The content of this thesis deals with rectangular waveguides. These include conventionalair or dielectric-filled metallic waveguide, non-metallic waveguide and substrate-integratedwaveguide. A rectangular waveguide has a width a, and a height b, with a > b, as shown inFigure 1.1. The z-axis is conventionally aligned with the axis of the waveguide, and this isthe direction in which waves propagate. The cutoff frequency is determined by the larger

2 Introduction

value of the inner dimension of the waveguide cross section, a. Air-filled metallic waveguideshow very low loss, and in practice, losses can be neglected during the design of waveguideantennas. In contrast, dielectric-filled, non-metallic and substrate-integrated waveguide havehigher losses that will need to be taken into consideration in antenna design.

b

axz

y

Figure 1.1 Waveguide characteristics.

1.1.1 Modes of a waveguide

Maxwell’s equations are a set of four vector-differential equations that govern all of electro-magnetic field behaviour. Waveguide modes are discrete solutions to Maxwell’s equationsinside the waveguide that satisfy all the relevant boundary conditions. Waveguide modeshave unique field distributions, and collectively they form a complete basis for the fields thatcan exist inside the waveguide. The modes are classified as either TE or TM [2]. TE mode orTransverse Electric mode indicates that the E-Field is orthogonal to the axis of the waveguideand for TM mode or Transverse Magnetic mode, the H-field is orthogonal to the axis of thewaveguide.

A specific mode is denoted as TEmn or TMmn, where the m and n indicate the integernumber of half-cycle sinusoidal variations of a field component along the longer dimensiona and shorter dimension b, respectively.

The TEmn or TMmn mode can only propagate in the waveguide if the frequency f ≥ fcmn.In such cases, it will have a wavelength λmn = 2π/βmn [m], where βmn [rad/m] is the phaseconstant. If the TEmn or TMmn mode is excited in the waveguide with f < fcmn, it willexperience attenuation of e−αmnl over a distance l, where αmn [Np/m] is the attenuationconstant.

For TEmn modes, the smallest possible value of either m or n is 0, but m = n = 0 is notincluded. For TMmn modes, m = 0 and n = 0, meaning that the smallest possible value mand n is 1. Since a > b, the mode with the lowest cutoff frequency will be the TE10 mode.This mode is called the dominant waveguide mode. It has a cutoff frequency of

1.2 Different types of slot radiators in a waveguide 3

fc10 =1

2a√

µε(1.1)

where µ and ε are the permeability and the permittivity of the medium filling the waveguide.Since a > b, the first higher-order mode in the waveguide is the TE20 mode, with a cutofffrequency fc20 = 2 fc10. In most commercial waveguide, a > 2b. In those cases, the nexthigher-order mode will be TE01 with a cutoff frequency fc01 = 1/2b

√µε [Hz]. In most

applications, waveguide dimensions are chosen to only allow single mode operation, so thatfc10 < f < fc20 [3]. Only the dominant mode is then able to propagate in the waveguide, andall higher-order modes will be below cutoff. Higher-order modes are excited in the vicinity adiscontinuity in the waveguide, but get attenuated to virtually disappear at short distancesaway from the discontinuity.

Figure 1.2 shows electric field distribution of the the three lowest-order TE modes in arectangular waveguide.

1.2 Different types of slot radiators in a waveguide

A slot cut in a metallic wall of the waveguide can radiate EM waves when the slot interruptsthe electric currents flowing on the waveguide walls. The radiation pattern and the gainare determined by the shape and size of the slot. The surface currents flowing on the metalwalls for a waveguide excited with a TE10 mode field are shown in the Figure 1.3. The axialmagnetic field increases toward the edges of the wall, and consequently the electric currentsalso increase in magnitude as a function of the offset from the waveguide centreline.

Slots can be cut in different orientations and positions in order to interrupt the currentson the walls of the waveguide. This sets up an oscillating field in the slot with a certainamplitude and phase depending on the slot dimensions.

One such slot is the broad-wall compound slot, which exhibits interesting properties. Itsamplitude and phase can be independently controlled by its offset and tilt with respect to thewaveguide axis while maintaining resonance [4]. These slots are ideally suited to travellingwave applications such as fuze antennas for missiles that require a beam at an angle withrespect to the axis of the missile.

The inclined slot centred at the axis of the waveguide is another example of an excellentseries element [5]. These slots behave like series elements for small offsets and rely on thetilt angle to achieve resonance. For larger offsets, the slot starts to behave as a shunt element

4 Introduction

(a) TE10 mode

(b) TE20 mode

(c) TE01 mode

Figure 1.2 TE modes in a waveguide

insensitive to the tilt angle. This property makes it versatile in array applications, as allelements can be designed to be resonant.

The transverse slot is another example of a radiating element. Although the use ofthis element leads to impedance matching problems in arrays due to the lack of controlof excitation amplitude [6], transverse slots in partially dielectric-filled waveguide providegreater control of excitation and produce reduced mutual coupling between slots [7].

Narrow-wall tilted slots that do not extend in the broad wall have been investigatedin [8, 9, 10]. Arrays utilising these slots give rise to cross-ploarization behaviour in theoff-axis planes. These slots usually extend to a small region on both broadwalls, whichcomplicates analysis of these slots. A design which avoids the cross-polarization effect byutilisng non-tilted narrow-wall slots has been proposed [11].

Longitudinal slots were studied in detail in [12, 13, 14, 15, 16, 17], and are the focus ofthis thesis. Falk [16, 17] presented a solution to the longitudinal slot cut in the broad wall

1.2 Different types of slot radiators in a waveguide 5

(a) Top view

(b) Side view

Figure 1.3 Surface currents in a waveguide

of a single ridged waveguide. A single longitudinal shunt slot in a rectangular waveguideradiating into a parallel plate region was analysed by Farooraghi [18]. An expression forthe magnetic field in the parallel plate region was developed and the slot admittance wascomputed. The Green’s function for the parallel plate region was developed and the electricfield in the slot was estimated. It is found that the resonant length for a given offset increasesfor some plate spacing and decreases with others. For a given offset, the resonant length willthus oscillate as a function of plate spacing.

The spectral domain technique has been applied to longitudinal slots radiating into anarbitrary cylindrical outer structure [19]. Antennas utilizing these longitudinal slots havebeen extensively studied since by Ando et al. [20] including a waveguide-fed parallel platearray antenna investigated by Hirokawa et al. [21].

1.2.1 Coupling slot

Planar slot arrays consisting of multiple waveguide branches are often fed via a main linewith centred inclined coupling slots placed orthogonal to the main lines. For broadwalllongitudinal slot arrays, coupling slots cut in the common broadwall control the amount ofpower into the branches, as shown in Figure 1.4.

The key relations for a 4-port coupling junction have been developed in [22]. Compoundcoupling slots allow arbitrary control over the magnitude and phase achieved with differenttilt angles and slot lengths [23, 24].

6 Introduction

Figure 1.4 A planar slotted waveguide array fed via inclined coupling slots.

In the design of resonant slot arrays fed via coupling slots, each slot is modelled as aseries impedance in the equivalent circuit for the main line. The design equations for thecalculation of coupling slot dimensions comprise expressions for the series impedance andthe coupling coefficient that relates the current through the series impedance to the voltagesacross the shunt elements in the branch line equivalent circuit. The equations are based onanalysis results of coupling slot junctions [24, 25].

1.3 Waveguide-fed slot arrays

Rectangular waveguide with slot-like radiating elements were first designed during WorldWar II in Canada, based on pioneering work by Watson [26], Stevenson [27] and Booker[28]. The basic theory of slots in rectangular waveguide and first attempts to find an analogywith transmission lines were done by Stevenson [27]. He gave a sound theoretical basis tothe experimental work of Watson. He formulated an integral equation for the slot apertureelectric field and estimated the resonance lengths of shunt slots using the Green’s function[29, 30]. Subsequently, Oliner [31, 32] developed a variational solution for the slot aperture

1.3 Waveguide-fed slot arrays 7

electric field and described the slot characteristics away from the resonant frequency. Heproposed the equivalent circuits for the transverse, longitudinal and centred-inclined slots,and the mathematical expressions to calculate the admittances/impedances of these equivalentnetworks that takes the thickness of the wall into account.

Oliner’s formulation did not take the dependence of the slot’s resonant length on itsposition and orientation in the waveguide wall into account. Yee [33] extended Oliner’svariational procedure account for the slot offset in calculation of the resonant length of alongitudinal broad-wall slot. This facilitated the design of slotted waveguide arrays withspecific desired radiation patterns, since the excitation of a radiating slots could be accuratelycontrolled by their physical parameters.

Stegen [1]] established a correlation between the phase of the admittance and the phase ofthe radiated field of a slot. He produced universal curves for the susceptance and conductanceof longitudinal slots cut in the broad-wall of a rectangular waveguide as a function of slotlength normalized to its resonant length, shown in Figure 1.5.

Figure 1.5 Normalised susceptance and conductance vs normalised slot length [1]

1.3.1 Slot data

The input data for any design procedure is the self-admittance and the resonant lengthsof slots as a function of offset and length. In the early 1960s, this data was generatedexperimentally, and Stegen’s data served as a standard for developing these slotted waveguide

8 Introduction

arrays. Due to the cost and complexity of the experimental measurements, computationalprocesses were implemented in order to characterize slots. It spawned the development ofthe Moment Method solution of the pertinent integral equations for the slot electric fields.

Various other techniques were implemented to characterize the slot conductance andthe resonant length [34, 35, 36, 37, 38, 39, 40]. A particularly successful technique is thesinusoidal Galerkin technique [41]. It allows for accurate slot conductance approximation,but the estimation of resonant lengths was not adequate due to the use of pulse expansionfunctions. An accuracy of better than 0.5% for the resonant length required a minimum of 9expansion functions [38, 39]. The design procedure using synthetic data is described in [40].

Stern and Elliott’s investigation in [41] led to a new design procedure involving aniterative solution of the coupled integral equations of the entire array, using the MomentMethod [42]. This design utilises piece-wise sinusoidal Galerkin models and was shown toaccurately replicate measured experimental slot data, especially for the zero-thickness case.

1.3.2 Design of slot arrays and representation of slots

The design of slotted waveguides evolved since the 1970s and eventually Elliott and Kurtz[43] introduced an accurate procedure for designing longitudinal broad-wall slotted wave-guides for resonant or standing wave type arrays. A resonant array is characterized by a shortcircuit termination and a half-guide wavelength spacing between slots. In [43], the lengthand offset of each slot is defined in the presence of mutual coupling between radiating slotsfor a specified aperture distribution and an impedance match, expanding Stevenson’s method[27] and modifying Booker’s relation [28] to analyse non-resonant longitudinal shunt slots inthe broad wall of a rectangular waveguide.

Elliott and Kurtz developed a general relation between the slot voltage and the modevoltage, thus deriving equations for the active, self and mutual admittances. This resulted ina design procedure that takes the mutual coupling into account [43]. The reciprocity theoremwas invoked to determine mutual coupling expressions.

This design procedure was further improved in [44] with the assumption of a half-cosineequi-phase aperture distribution that is also valid for slots in dielectric-filled waveguide[44]. The slots in the broadwall are represented by an equivalent network of shunt-elements(instead of a full T-network) in order to simplify the design.

Multiple radiators in close vicinity to each other influence individual slot characteristics.Calculation of mutual coupling during design synthesis accounts for the majority of run time.This area has thus been studied extensively.

A Taylor series approximation around half-guide wavelength slots and series expansiontechniques are employed by Mazzarella [45]. Mutual coupling between slots is studied in [46]

1.3 Waveguide-fed slot arrays 9

and is expressed as a single integral for narrow slots. Rengarajan [47] investigated mutualcoupling between longitudinal broadwall slots, taking into account the mutual interactionbetween the slot edges. The mutual coupling effects in these longitudinal shunt slots weresignificant compared to inclined longitudinal series slots, as reported by Kim and Elliott [48].

Rengarajan and Derneryd [49] presented a design for shaped-beam slot arrays employinglongitudinal slots and compound coupling slots. Rengarajan et al. analysed the effects ofhigher-order mode coupling in slot arrays [50]. He also proposed designs to compensatefor higher-order coupling in the design of arrays. The method involves the application of afull-wave finite element or method of moments analysis to an existing design, and performingoptimization by perturbing the slot dimensions to improve the array performance [51, 52].

Casulla et al. [53, 54] have also presented designs for shaped-beam slot arrays, includingthe feed structure of the array. A synthesis technique which takes excitation constraints inslot arrays has also been presented [55].

Accurate characterisation of coupling junctions plays a vital role in the design of shaped-beam slot arrays. Mazarella [56] developed a wideband equivalent circuit consisting of anon-ideal transformer with a reactance in parallel with the primary, or an ideal transformerwith a parallel reactance connected to the secondary. Rengarajan proposed an optimisationtechnique to account for the higher-order mode effects between the coupling slots [57].

Radial line and conventional parallel plate fed slot arrays find wide application for directbroadcast satellite and millimetre-wave applications. Takashi [58] proposed a design forsingle-layered radial line slot antennas. Hirokawa and Ando proposed a design for slotarrays excited with TEM waves utilising slot pairs in order to suppress reflections [21]. Theyalso developed single-layer slot arrays utilising various feed structures for direct broadcastsatellite applications [20, 59, 60, 61]. Single-layer arrays for high-efficiency millimetre-waveapplications have also been proposed [62].

1.3.3 Analysis of slot arrays

Hamadallah presented an off-centre frequency analysis of linear slot arrays [63]. This methodcan be used to study the performance of an array at various frequencies in the operationalband. Hamadallah’s work was further extended in [64] to analyse planar arrays at frequenciesother than the design frequency. An alternative direct approach for the analysis of planararrays was proposed in [65]. Their procedure predicts the frequency response of planararrays utilising a single linear system dependent on slot voltages and the feeding excitations.Rengarajan also reported the accurate Moment Method analysis of entire slot arrays [51, 52].The last two decades has seen rapid growth in development of modern analysis software basedon mature numerical techniques. Some commonly used software include CST Microwave

10 Introduction

Studio and HFSS. While these commercial software tools enable accurate analysis of slotarrays, they are computationally intensive, require extensive computational resources and arenot ideally suited for application as design tools for full arrays.

1.4 Waveguide arrays implemented in non-metallic mate-rials

With the growing demand for satellite applications, there is a tendency to implement slotarrays using lightweight, non-metallic material such as Carbon Fibre Reinforced Plastic(CFRP). Metal plating within the waveguide is sometimes used to achieve the desired levelsof efficiency [66, 67]. Antenna patterns for single slots in CFRP was first reported in [68].

The performance of CFRP slot arrays is compared to designs in aluminium in [69] wherea 2 dB ohmic loss is reported. Implementation of slot arrays in CFRP was reported in[70, 71, 72]. CFRP feed structures for slot arrays were also implemented utilising finite-element-method in HFSS [71]. Operating procedures for the production of slot arrays inCFRP are described in [73].

Application in high-frequency telecommunication systems often demands the use ofSubstrate Integrated Waveguide (SIW) with etched slots instead of conventional waveguidewith machined slots. SIW employs rows of narrowly spaced metallic vias between two platesof a substrate thus forming virtual side walls of a rectangular dielectric-filled waveguide[74, 75]. The use of SIW in antenna design was first introduced in [76], where slot antennaswere designed using Elliott’s conventional technique. Other designs were presented in[77, 78].

SIW and non-metallic waveguides exhibit inherent losses due to reduced conductivity ofthe waveguide walls, radiation loss due to leakage between vias and dielectric losses [79].Physical specifications for SIW like the size and spacing between vias, and the transversedisplacement between virtual sidewalls to emulate a dielectric-filled waveguide have alsobeen presented [75, 80].

Shaped-beam SIW slot arrays have also been proposed in [81]. This design uses thetechnique proposed by Elliott to determine the initial slot dimensions and then utilises afull-wave analysis software to optimize slot dimensions to achieve the desired pattern and animpedance match at the input. This procedure is computationally intensive and is prone tonon-convergence.

1.5 Shortcomings in existing design procedures 11

1.5 Shortcomings in existing design procedures

1.5.1 Higher-order mutual coupling between the coupling slot and neigh-bouring radiating slots.

A design technique that accounts for TE20 mode coupling between adjacent longitudinalslots was presented in [82]. It was found that the coupling was significant for slots withsmaller offsets and tilt angles, especially for reduced height waveguide arrays.

In a coupling junction consisting of a coupling slot with a pair of straddling radiatingslots, higher-order mode coupling effects are more significant [83, 84, 85, 23, 86, 87, 88].Researchers analysed examples to highlight the extent of the errors introduced by the higher-order coupling. The discrepancies vary in severity depending on design parameters, butwithout compensation, they can have a notable effect on the antenna performance in terms ofsidelobe levels, impedance matching and gain [60]. To date, no solution which compensatesfor these effects during the design phase is available.

TE20 mode coupling was shown to be the most significant of the higher-order modes[83, 50]. As shown in Figure 1.6, TE20 mode is the first higher-order mode and willexperience the least amount of decay over a fixed distance for a half-height waveguide.The second higher order mode is the TE01 mode. At the coupling junction, the dominantmode and all higher order modes are excited. It is evident that TE20 is the only significanthigher order mode that contributes to internal coupling between the coupling slot and theneighbouring radiating slots in half-height waveguide. The cut off frequencies for the otherhigher order modes are substantially higher than that of the TE20 mode, and consequentlythey will experience much higher attenuation.

On the contrary, in standard-height waveguide, the cut-off frequency of the TE01 modeis close to that of the TE20 mode, as seen in Figure 1.7. It has been confirmed that theTE20 mode account for almost all the higher-order mode coupling effects at the junctions inreduced-height guides, while both the TE20 and TE01 modes are the major contributors instandard-height waveguide [83, 85].

1.5.2 Phase discrepancy in coupling junctions

The design equations for the calculation of coupling slot dimensions comprise expressionsfor the series impedance and the coupling coefficient. The equations are based on analysisresults of coupling slot junctions [24, 25].

Expressions for the impedance and the coupling coeffiecients were developed in [89].However, all scattering parameters of junctions are assumed to be either in-phase or out-

12 Introduction

0

100

200

300

400

500

600

700

5 10 15 20 25 30

am

n(N

p/m

), b

mn

(Ra

d/m

)

Frequency (GHz)

a 10

a 20

a 01b 01

fc 10 fc 20 fc 01

b 10

b 20

Figure 1.6 Dispersion curve for reduced-height waveguide with a = 22.86 mm and b = 5.08mm.

of-phase at resonance. This assumption is known to fail for slots with small inclinationangles [56] and leads to errors in slot exciation and thus reduced array performance. Thedesign also fails when implementing shaped beam arrays that require non-resonant couplingslots. Mazzarella and Montisci [56] developed a wideband equivalent circuit consisting of anon-idea transformer with a reactance in parallel with the primary, or an ideal transformerwith a parallel reactance connected to the secondary.

It has been shown that internal higher-order mode coupling between centered-inclinedslots in feed networks can be significant [83]. Higher-order coupling can largely be attributedto the TE20 and TE01 modes. Compensation for higher-order coupling can be done byapplying full-wave analysis to an existing design, and performing simplified optimization byperturbing the slot dimensions to improve the array performance [51, 52]. This approach iscomputationally intensive [57], and access to suitable analysis code or commercial softwareis essential.

1.5 Shortcomings in existing design procedures 13

0

50

100

150

200

250

300

350

400

5 6 7 8 9 10 11 12 13 14 15 16 17 18

am

n(N

p/m

), b

mn

(ra

d/m

)

Frequency (GHz)

a 10 a 20

a 01

b 01

b 11

fc 10 fc 20fc 01

a 11

b 10

b 20

fc 11

Figure 1.7 Dispersion curve for standard-height waveguide with a= 22.86 mm and b= 10.16mm.

1.5.3 Waveguide material-losses in the design of slot arrays

For high-frequency telecommunication applications, the small waveguide and slot dimen-sions dictate the use of substrate-integrated waveguide (SIW) with etched slots instead ofconventional metallic waveguide with machined slots [76, 77, 78].

In these situations, waveguides exhibit inherent losses due to either the reduced con-ductivity of the waveguide walls (in non-metallic waveguides), or leakage and imperfectdielectrics (in SIW) [79]. Existing design procedures for slot arrays do not account for theselosses. This will cause slots to be under-illuminated; especially those located further awayfrom feed points, and may ultimately degrade the overall performance of the array.

1.5.4 Asymmetrical fields in the design of waveguide slot arrays

Isolated longitudinal slots in the broad wall of a waveguide can ideally be represented by anequivalent T-network of impedances. Apart from design techniques based on optimizationof full-wave analysis results [51, 52], all existing design methods assume a symmetric fielddistribution in the slot apertures [90, 82]. In addition, the series elements in the T-networkare discarded and a simple shunt-element equivalent network is used to simplify the design.

14 Introduction

It is known that the aperture field distribution becomes asymmetrical for large slot offsets,especially in reduced-height or dielectric-filled waveguide [41, 42]. In these cases, the serieselements in the equivalent T-network become significant. In SIW, this effect becomes evenmore pronounced. By neglecting the contributions of the series elements, proper control overthe total admittance of individual branches is not possible. As a result, the feed network doesnot function as intended, resulting in element excitation errors and poor impedance matchingat the array input.

1.6 Aims and objectives

This thesis presents an investigation into the design of waveguide slot arrays and proposenew design techniques for accurate designs in metallic and non-metallic waveguide. Slotarrays are currently designed using conventional techniques developed in the 1980s by Elliott.These antennas are extensively used for radar and telecommunication applications and thereis a growing demand for high-precision implementation of them.

The aim of the research reported in this thesis is to provide a new methods to accuratelydesign resonant waveguide slot arrays implemented in metallic, non-metallic and substrate-integrated waveguide. Closed-form design equations that further extend the pioneeringwork by Elliott will be developed, contributing originally to five components : (a) higherorder mutual coupling between coupling slots and the two straddling radiating slots, (b)accurate charcaterisation of coupling junctions and compensation for higher-order mutualcoupling between the coupling slots, (c) compensation for waveguide losses, (d) accuraterepresentation of radiating slots with a T-network equivalent circuit and asymmetric slotfield, and (e) design of single layer feed structures in lossy waveguide. The objectives can bestated as follows:

1. Design equations will be developed in order to extend Elliott’s design to compensate formutual coupling due to higher order TE20 and TE01 modes between the coupling slotsand the two straddling radiating slots. Compensation is achieved through determinationof the electric field in coupling slots, and the addition of two closed-form terms in theexpression for the active admittance of radiating slots straddling the coupling slots.This work is discussd in Chapter 2.

2. A model to accurately represent coupling junctions will be developed. Closed-formexpressions for the series impedance and the coupling coefficients will be developed,taking into account the phase variations between individual scattering parameters inthe junctions. Further amendments will be made to the expressions for the equivalent

1.7 Publications 15

impedance for coupling slots in order to compensate for mutual coupling due to higherorder TE20 and TE01 modes between the coupling slots in the main line of the array.The proposed technique is discussed in Chapter 3.

3. Design expressions that compensate for arbitrary levels of waveguide loss for arrayswill be developed. A design procedure for the calculation of accurate slot dimensionsand positions will be presented in Chapter 4.

4. A design technique based on the full T-network equivalent circuit representation ofslots will be developed in Chapter 5. This will account for asymmetrical slot fields andphase variation along the length of each slot. This technique is primarily aimed at therefinement of shaped-beam and narrow-beam arrays in SIW, but can also be employedfor arrays in dielectric-filled or reduced-height waveguide.

5. The final contribution in this thesis aims at developing a design to implement a singlelayer feed structure that can provide arbitrary excitation for side-fed branches of SIWand conventional metallic waveguide. This is also discussed in Chapter 5.

1.7 Publications

Journal articles

1. J.C. Coetzee and S. Sheel, “Compensation for waveguide losses in the design ofslot arrays,” IEEE Transaction on Antennas and Propagation, vol. 66, no. 3, pp.1271-1279,2018.

2. J.C. Coetzee and S. Sheel,” Waveguide slot array design with compensation for higherorder mode coupling between inclined coupling slots and neighboring radiating slots,“IEEE Transaction on Antennas and Propagation, vol. 67, no. 1, pp. 378-379, 2019.

3. J. C. Coetzee and S. Sheel, "Improved model for inclined coupling slots in the feed ofa planar slot array" IEEE Transactions on Antennas and Propagation (under review)

Conference papers

1. J. C. Coetzee and S. Sheel, “Compensation for waveguide losses in the design of planarslot arrays,” in Proc. 15th Australian Symposium on Antennas, Sydney, Australia,2017.

16 Introduction

2. S. Sheel and J.C. Coetzee,” Waveguide loss compensation in resonant slot arraydesign", in Proc. 2017 IEEE International Symposium on Antennas and Propagationand USNC-URSI Radio Science Meeting, San Diego, California, USA, July 2017.

3. S. Sheel and J.C. Coetzee, “Compact feeding structure for standard waveguide andsubstrate integrated waveguide arrays,” in Proc. 3rd Australian Microwave Symposium,Brisbane, Australia, February 2018.

4. J.C. Coetzee and S. Sheel, “Design of planar slotted waveguide arrays for satellitecommunications and airborne radar applications,” X Headquarters, Mountain View,CA, September 2016.

5. S. Sheel and J.C. Coetzee, "Accurate equivalent circuit model for centred inclinedcoupling slots in planar slotted waveguide Array Feeds," IEEE International Conferenceon Microwaves, Antennas, Communications and Electronic Systems (COMCAS), TelAviv, Israel, 2019 (under review)

6. S. Sheel and J.C. Coetzee, "Compensation for asymmetrical slot fields in the de-sign of SIW slot arrays," IEEE International Conference on Microwaves, Antennas,Communications and Electronic Systems (COMCAS), Tel Aviv, Israel, 2019 (underreview)

In preparation

1. J.C. Coetzee and S. Sheel, “Precision design of substrate integrated waveguide slotarrays,” IEEE Transaction on Antennas and Propagation.

2. J. C. Coetzee and S. Sheel, "Flexible Feed network for Slot Arrays in StandardWaveguide or Substrate Integrated Waveguide" IEEE Transactions on MicrowaveTheory and Techniques.

Other publications not directly related to this thesis

1. S. Sheel, M.M. Albannay and J.C. Coetzee, “Integrated decoupling and matchingnetwork incorporated in the ground plane of a compact monopole array,” Microwaveand Optical Technology Letters, vol. 57, no. 6, pp. 1315–1319, June 2015

2. S. Sheel and J. C. Coetzee, "Switchable-feed reconfigurable ultra-wide band planarantenna," in Proc. 2015 International Symposium on Antennas and Propagation,Hobart, Australia, Nov. 2015.

1.7 Publications 17

3. S. Sheel and J. C. Coetzee, "Electronically-reconfigurable horizontally polarized wide-band planar antenna," in Proc. 2015 International Symposium on Antennas andPropagation, Hobart, Australia, Nov. 2015.

4. S. Sheel and J.C. Coetzee, “A compact dielectric-filled slotted cavity MIMO antenna,”Progress in Electromagnetic Research Letters, vol. 72, pp. 17-22, 2018.

5. S. Sheel and J.C. Coetzee, “Electronically steerable circularly polarized planar an-tenna”, in Proc. 12th European Conference on Antennas and Propagation, London,United Kingdom, April 2018.

Chapter 2

Higher order mode compensationbetween the coupling slots and theradiating slots straddling them

2.1 Introduction

Design procedures for planar slot arrays are largely based on Elliott’s design equations[44, 91, 92]. However, his design equations do not include higher-order coupling effectsbetween coupling slots and neighbouring radiating slots, which are spaced only a quarterguide-wavelength apart. This can cause deviations in achieved slot excitations, especiallyfor the two radiating slots neighbouring the coupling slots [83, 93, 94, 95, 87, 88]. Severalresearchers have analysed examples to highlight the extent of the errors introduced [60]. Todate no direct synthesis method that compensates for these effects during the design phasehas been proposed.

In practice, the deviations can be compensated by a priori adjustment of the slot positionsand lengths, or by adding an additional guide wavelength space between the coupling andthe radiating slots [96].

Commercial software like CST Microwave Studio and Ansoft HFSS can also be employedas an aid to determine the slot positions and lengths, but it is pre-dominantly a trial and errorprocedure. Commercial tools have built-in optimization capabilities but are generally notsuitable for multi-objective optimisation. Changes to any single slot affect the excitation ofother slots through mutual coupling and thus individual adjustments to slots is not feasibleand causes additional errors in the excitation of other elements.

20Higher order mode compensation between the coupling slots and the radiating slots

straddling them

Methodical approaches to compensation for higher-order coupling in the design of slot-arrays involve the application of a full-wave finite element or method of moment analysis toan existing design, and performing optimisation by perturbing the slot dimensions to improvethe array performance [51, 52]. This process requires access to custom analysis code orcommercial software and has been described as being cumbersome and time consuming [57].

In this chapter, a design procedure which compensates for higher-order mode couplingin the waveguide junctions is presented. Compensation is achieved by determining theelectric field in coupling slots, and the addition of two closed-form terms in the expressionfor the active admittance of radiating slots straddling the coupling slots. This change hasthe greatest impact on the dimensions of those slots, but it simultaneously also introducesminor adjustments to the dimensions of other radiating slots. For the calculation of thecoupling slots dimensions, amendments to design relations to avoid assumptions on thephase relationship between the respective scattering parameters of coupling junctions are alsoproposed. The improved performance of the new procedure is demonstrated by analysingdesign examples with commercial full-wave software and comparing results to the casewhere no compensation has been applied.

2.2 Analysis

Consider an array with a total of T branch lines, with branch t having a total of Nt radiatingslots for a total of M slots in the entire array. All waveguides are assumed to have uniformdimensions of width a and height b. The slots are spaced at intervals of λ10/2 , with λ10

being the wavelength of the TE10 mode in the waveguide at the design frequency f0. Thecoupling slot for branch t is located between radiating slots (t,kt) and (t,kt +1), as shown inFigure 2.1 .

The equivalent circuit of the branch consists of shunt elements yat,n, representing the

normalized active admittances of each slot. The voltage across yat,n is denoted as Vt,n. The

main line equivalent circuit consists of normalized series impedances zt , which representthe active impedances of the coupling slots. The current flowing through zt is defined asIt . Without any loss in functionality, we use a normalized characteristic impedance z0 = 1in all calculations involving the equivalent circuits for the branch lines and main line. Therelations provided in [90, 82, 89] were derived for an arbitrary characteristic impedance Z0.Our formulation therefore compares to the special case where Z0 = 1Ω in [90, 82, 89]. Thegeometry for a radiating slot is depicted in Figure 2.2.It has a length L, width w and an offset of xoff from the waveguide centreline. We assume thefield in the slot to be [90]

2.2 Analysis 21

(xi,0,z

i)

Figure 2.1 A planar slotted waveguide array (centre) with equivalent circuits for a branch ofradiating slots (above) and the main line with inclined coupling slots (below).

Figure 2.2 Geometry of a radiating slot.

Eslot(x′,z′) =V slot

tw

cos

(πz′

L

)x′ (2.1)

22Higher order mode compensation between the coupling slots and the radiating slots

straddling them

where V slot is the slot voltage and (x′,y′,z′) is the slot coordinate system. Note that duringthe design process, the slot voltages are chosen to produce the desired radiation pattern, andtherefore the value of V slot is generally known.

2.2.1 Fundamental mode analysis of the coupling junction

Refer to the coupling junction shown in Figure 2.3.

t k,t

t k, +1t

t

qt

lt

d0

ft

z’

x’

z

a

a

q

l

z

x

z'

x'

Branch line

z z=off

Main line

A10

Emn

1

3 4

Fmn

Bmn

Cmn

Figure 2.3 Geometry of a coupling junction.

The coupling slot has a width w, length l, and is rotated by an angle θ with respect to theaxis of the main line. The inclination angle θ is assumed to be positive for clockwise rotation.The coupling slot is centered at (x,y,z) = (a/2,b,zoff). The main line and slot coordinatesystems are related via

(x,y,z) = (x′ cosθ − z′ sinθ +a/2,y′+b,x′ sinθ + z′ cosθ + zoff) (2.2)

x = cosθ x′− sinθ z′, y = y′, z = sinθ x′+ cosθ z′,

Consider an incident field in the form of a TE10 mode wave of amplitude A10 in the mainline, which induces the field Ecpl in the coupling slot. Analogous to the radiating slot, weassume the field in the coupling slot to be

Ecpl(x′,z′) =V cpl

t

wcos

(πz′

l

)x′ (2.3)

2.2 Analysis 23

where V cpl is the coupling slot voltage. This slot voltage is unspecified. In order to relate it toother quantities that are known or can readily be computed, we perform a fundamental modeanalysis of the coupling junction. The reciprocity theorem was used to determine the TE10

mode scattering from a radiating slot in [89]. We apply a similar approach to calculate thescattering from the coupling slot. In response to the incident TE10 mode wave of amplitudeA10, waves with amplitudes B10 and C10 are scattered in the backward and forward directions,while the amplitude of the waves scattered to the left and right in the branch line are D10

and E10. The normalized series impedance in the equivalent circuit shown in Figure 2.4 iszself = 2S11/(1−S11), where S11 is the slot reflection coefficient.

Figure 2.4 Equivalent circuit for coupling slot t.

A voltage wave of amplitude A=KpA10 is incident on zself, where Kp =(π/a)√

ωµ0β10ab/2to ensure uniformity in the power carried by the TE10 wave and the voltage wave. It inducesthe current I = 2A/(zself++ 2) to flow through the series impedance. Voltage waves ofamplitude B = KpB10 and C = KpC10 are scattered in the backward and forward directions.

With the phase reference at z = zoff, the TE10 mode propagating in the ±z direction in awaveguide is defined by E10(x,z) = E0

10(x)e∓ jβ10(z−zoff) and H10(x,z) = H0

10(x)e∓ jβ10(z−zoff).

The cross-sectional parts of the respective field components are

E010y(x) =−ωµ0a

πsin(

πxa

)(2.4)

H010x(x) =∓β10a

πsin(

πxa

)H0

10z(x) = j cos(

πxa

)where β10 =

√k2 − (π/a)2 is the phase constant in the waveguide at the design frequency,

and k = ω√

µ0ε0εr is the wavenumber in the dielectric medium with relative permittivity

24Higher order mode compensation between the coupling slots and the radiating slots

straddling them

εr that fills the waveguide. By using the reciprocity theorem, the forward and backwardscattered amplitudes can be calculated from

B10 =−C10 =

∫slot

(Ecpl ×H10) ·ds

2∫S1

(E010 ×H0

10) · zds(2.5)

where S1 is the waveguide cross section. Using the narrow slot approximation, the numeratoris given by

∫slot

(Ecpl ×H10) ·ds =w/2∫

−w/2

l/2∫−l/2

[Ecpl(z′)×H10(x′,z′)

]· ydz′dx′ (2.6)

≈−w

l/2∫−l/2

Ecpl(z′)H10z′(z′)dz′

where H10z′(z′) is the z′ component of the magnetic field of a TE10 mode travelling in the +zdirection and observed on the centreline of the coupling slot (i.e. x′ = y′ = 0). It is given by

H10z′(z′) =−H10x(−z′ sinθ +a/2,z′ cosθ + zoff)sinθ

+H10z(−z′ sinθ +a/2,z′ cosθ + zoff)cosθ

= e− jβ10 cosθz′[

β10asinθ

πcos(

π sinθz′

a

)+ j cosθ sin

(π sinθz′

a

)]. (2.7)

The denominator is given by

2∫S1

(E010 ×H0

10) · zds =ωµ0β10ab(π/a)2 . (2.8)

Equation (2.5) can be evaluated in closed form as

2.2 Analysis 25

B10 =−C10 =− (2π/a)2

ωµ0β10kabp10(θ , l)V cpl (2.9)

where

p10(θ , l) =k2

l/2∫−l/2

cos

(πz′

l

)[β10 sinθ

(π/a)cos

(π sinθz′

a

)e− jβ10 cosθz′ (2.10)

+ j cosθ sin

(π sinθz′

a

)e− jβ10 cosθz′

]dz′.

By using the trigonometric identities of

cosAcosB = cos(A+B)+ cos(A−B)and

cosAsinB = cos(A+B)+ cos(A−B)

together with the relations

∫e− jax cosbxdx =

e− jax[bsinbx− jacosbx]b2 −a2∫

e− jax sinbxdx =−e− jax[bcosbx+ jasinbx]b2 −a2 ,

26Higher order mode compensation between the coupling slots and the radiating slots

straddling them

we can express (2.10) in closed form as

p10(θ , l) =β10 sinθ

2(π/a)

( π

kl +π sinθ

ka

)cos(

π sinθ l2a

)cos(

β10 cosθ l2

)(

π

kl +π sinθ

ka

)2−(

β10 cosθ

k

)2

+

(β10 cosθ

k

)sin(

π sinθ l2a

)sin(

β10 cosθ l2

)(

π

kl +π sinθ

ka

)2−(

β10 cosθ

k

)2

−

(β10 cosθ

k

)sin(

π sinθ l2a

)sin(

β10 cosθ l2

)(

π

kl −π sinθ

ka

)2−(

β10 cosθ

k

)2

+

(π

kl −π sinθ

ka

)cos(

π sinθ l2a

)cos(

β10 cosθ l2

)(

π

kl −π sinθ

ka

)2−(

β10 cosθ

k

)2

+cosθ

2

( π

kl +π sinθ

ka

)sin(

π sinθ l2a

)sin(

β10 cosθ l2

)(

π

kl +π sinθ

ka

)2−(

β10 cosθ

k

)2

+

(β10 cosθ

k

)cos(

π sinθ l2a

)cos(

β10 cosθ l2

)(

π

kl +π sinθ

ka

)2−(

β10 cosθ

k

)2

+

(π

kl −π sinθ

ka

)sin(

π sinθ l2a

)sin(

β10 cosθ l2

)(

π

kl −π sinθ

ka

)2−(

β10 cosθ

k

)2

−

(β10 cosθ

k

)cos(

π sinθ l2a

)cos(

β10 cosθ l2

)(

π

kl −π sinθ

ka

)2−(

β10 cosθ

k

)2

.

Assuming an identical field in the upper and lower apertures of the slot, the scattered waveamplitudes in the branch line are

E10 =−F10 =

∫slot

(Ecpl ×H10) ·ds

2∫S1

(E010 ×H0

10) · zds. (2.11)

The numerator becomes

2.2 Analysis 27

∫slot

(Ecpl ×H10) ·ds =w/2∫

−w/2

l/2∫−l/2

[Ecpl(z′)×H10(x′,z′)

]· (−y)dz′dx′ (2.12)

≈ w

l/2∫−l/2

Ecpl(z′)H10z′(z′)dz′.

Note that in this case, the normal vector is in the opposite direction, since the couplingslot is located in the bottom wall of the branch line as opposed to the top wall of the main line.With reference to the branch line coordinates, the magnetic field of a TE10 mode travelling inthe +z direction in the branch line observed on the centreline of the coupling slot is given by

H10z′(z′) =−H10x(−z′ sinφ +a/2,z′ cosφ + zoff)sinφ

+H10z(−z′ sinφ +a/2,z′ cosφ + zoff)cosφ

= e− jβ10 cosφz′[

β10asinφ

πcos(

π sinφz′

a

)+ j cosφ sin

(π sinφz′

a

)](2.13)

where φ = θ −90. The range of coupling slot inclination angles is −45 ≤ θ ≤ 45, andtherefore −135 ≤ φ ≤−45. Equation (2.11) can again be evaluated in closed form, giving

E10 =−F10 =− (2π/a)2

ωµ0β10kabp10(φ , l)V cpl. (2.14)

For the equivalent circuit in Figure 2.4, the backscattered and forward scattered voltagewave amplitudes are given by

B =−C = 1/2Izself. (2.15)

Equations (2.6) and (2.8) yield the following expression for the coupling slot voltage and thecurrent flowing through the series impedance:

28Higher order mode compensation between the coupling slots and the radiating slots

straddling them

V cpl =−(a/λ )√

ωµ0β10ab/2zselfI

p10(θ , l)(2.16)

where λ = 2π/k. In order to verify the dominant mode analysis, we used CST MicrowaveStudio to compute the scattering parameters of a crossed-guide junction for a variety ofinclination angles and slot lengths. Scattering parameters of a coupling junction werecalculated for a half-height X-band waveguide slot coupler (a = 22.86 mm, b = 5.08 mm),with θ ranging from 10 to 30 at 10 intervals with a thin wall thickness (τ=0.1 mm) and aslot width of w = 1 mm. With the coupling slot voltage set to the value given in (2.16), theresults for the ratio B10/A10 and S11 are identical. Figure 2.5(a) and 2.5(b) show the resultsfor the magnitude and phase respectively.

The results for the magnitude and phase for the ratio of E10/A10 and S31 are shown inFigure 2.6(a) and 2.6(b) respectively. The results for the magnitude agree well with a slightdiscrepancy for shorter lengths. The phase shows good agreement except for small angleswhere a phase offset can be observed. This phase discrepancy will be further discussed inChapter 3. Equations (2.7) and (2.9) provide a simple relation between the coupling andreflection coefficients of a coupling junction:

S31

S11≈− p10(φ , l)

p10(θ , l). (2.17)

2.2.2 Higher-order mode scattering in a coupling junction

The waveguide dimensions of slot arrays are chosen to only support dominant TE10 modepropagation. If any higher order mode is excited in the waveguide, it will be in the form of anevanescent wave with an intensity that exhibits exponential decay. In response to an incidentTE10 wave, the dominant mode and higher order modes are scattered at the coupling junction.Due to the increasing level of attenuation, only the first few higher-order modes will interactwith the neighbouring radiating slots. In conventional applications where a > 2b, TE20 andTE10 are the first two higher-order modes. They will experience the least amount of decayover a fixed distance, and it is assumed that they are largely responsible for the higher-ordercoupling effects. In this section, we analyse the scattering of these modes from the couplingslot. The results are utilized to derive new design relations in Section 3.3.

2.2 Analysis 29

(a)

(b)

Figure 2.5 (a) Magnitude and (b) phase of S11 for a slot coupler calculated using CSTMicrowave Studio and the approximate dominant mode approach.

Again refer to coupling junction of Figure 2.3. The TE20 mode fields in the main line trav-elling in the ±z direction are E20(x,z)=E0

20(x)e∓α20(z−zoff) and H20(x,z)=E0

20(x)e∓α20(z−zoff)

where

30Higher order mode compensation between the coupling slots and the radiating slots

straddling them

(a)

(b)

Figure 2.6 (a) Magnitude and (b) phase of S31 for a slot coupler calculated using the CSTMicrowave Studio and the approximate dominant mode approach.

E020y(x) =

ωµ0

2π/asin

(2πx

a

)(2.18)

H020x(x) =± jα20

2π/asin

(2πx

a

)

H020z(x) = j cos

(2πx

a

),

with α20 =√

(2π/a)2 − k2 being the attenuation constant. For TE01 mode, the fields in themain line travelling in the ±z direction are E01(x,z) = E0

01(x)e∓α01(z−zoff) and H01(x,z) =

E001(x)e

∓α01(z−zoff) where

2.2 Analysis 31

E001x(y) =

ωµ0

π/bsin

(πyb

)(2.19)

H001y(y) =± jα01

π/bsin

(πyb

)

H001z(y) = j cos

(πyb

),

and α01 =√

(π/b)2 − k2.Assuming that the coupling slot voltage V cpl is known and with the slot field Ecpl as

defined in (2.3), we can determine the backscattered field intensity for the TE20 mode from[89]

B20 =

∫slot

(Ecpl ×H20) ·ds

2∫S1

(E020 ×H0

20) · zds. (2.20)

The numerator in (2.20) is

∫slot

(Ecpl ×H20) ·ds =

w/2∫−w/2

l/2∫−l/2

[Ecpl(z′)×H20(x′,z′)

]· ydz′dx′ (2.21)

≈−w

l/2∫−l/2

Ecpl(z′)H20z′(z′)dz′

where H20z′(z′) is the z′ component of the magnetic field of a TE20 mode travelling in the +zdirection and observed on the centerline of the coupling slot. It is given by

32Higher order mode compensation between the coupling slots and the radiating slots

straddling them

H20z′(z′) =−H20x(−z′ sinθ +a/2,z′ cosθ + zoff)sinθ

+H20z(−z′ sinθ +a/2,z′ cosθ + zoff)cosθ

=− jeα20 cosθz′[

α20asinθ

2πsin(2π sinθz′

a

)+ j cosθ cos

(2π sinθz′

a

)](2.22)

The denominator in (2.20) is given by

2∫S1

(E020 ×H0

20) · zds =ωµ0α20ab(2π/a)2 (2.23)

Equation (2.12) can be evaluated in closed form to give

B20 =− 2(π/a)2

ωµ0α20kabp20(θ , l)V cpl (2.24)

where

p20(θ , l) = k

l/2∫−l/2

cos

(πz′

l

)[± α20 sinθtz′

asin

(2π sinθz′

a

)e∓α20 cosθz′

+ cosθ cos

(2π sinθz′

a

)e∓α20 cosθz′

]dz′ (2.25)

By again using the trigonometric identities of

cosAcosB = cos(A+B)+ cos(A−B)and

cosAsinB = cos(A+B)+ cos(A−B)

together with the relations

2.2 Analysis 33

∫eax cosbxdx =

eax[acosbx+bsinbx]a2 +b2∫

eax sinbxdx =−eax[asinbx−bcosbx]a2 +b2 ,

we can express (2.25) in closed form as

p20(θ , l) =−α20 sinθ

(2π/a)

( π

kl +2π sinθ

ka

)sin(

π sinθ la

)sinh

(α20 cosθ l

2

)(

π

kl +2π sinθ

ka

)2+(

α20 cosθ

k

)2

+

(α20 cosθ

k

)cos(

π sinθ la

)cosh

(α20 cosθ l

2

)(

π

kl +2π sinθ

ka

)2+(

α20 cosθ

k

)2

+

(π

kl −2π sinθ

ka

)sin(

π sinθ la

)sinh

(α20 cosθ l

2

)(

π

kl −2π sinθ

ka

)2+(

α20 cosθ

k

)2

−

(α20 cosθ

k

)cos(

π sinθ la

)cosh

(α20 cosθ l

2

)(

π

kl −2π sinθ

ka

)2+(

α20 cosθ

k

)2

+ cosθ

( π

kl +2π sinθ

ka

)cos(

π sinθ la

)cosh

(α20 cosθ l

2

)(

π

kl +2π sinθ

ka

)2+(

α20 cosθ

k

)2

−

(α20 cosθ

k

)sin(

π sinθ la

)sinh

(α20 cosθ l

2

)(

π

kl +2π sinθ

ka

)2+(

α20 cosθ

k

)2

+

(π

kl −2π sinθ

ka

)cos(

π sinθ la

)cosh

(α20 cosθ l

2

)(

π

kl −2π sinθ

ka

)2+(

α20 cosθ

k

)2

+

(α20 cosθ

k

)sin(

π sinθ la

)sinh

(α20 cosθ l

2

)(

π

kl −2π sinθ

ka

)2+(

α20 cosθ

k

)2

.

The forward-scattered TE20 amplitude is

34Higher order mode compensation between the coupling slots and the radiating slots

straddling them

C20 =

∫slot

(Ecpl ×H20) ·ds

2∫S1

(E020 ×H0

20) · zds. (2.26)

In this case, the numerator in (2.26) becomes

∫slot

(Ecpl ×H20) ·ds ≈−w

l/2∫−l/2

Ecpl(z′)H20(z′)dz′ (2.27)

.

Here H20z′(z′) is the z′-component of the magnetic field of a TE20 mode travelling in the −zdirection and observed on the centreline of the coupling slot, given by

H20z′(z′) =−H20x(−z′ sinθ +a/2,z′ cosθ + zoff)sinθ

+H20z(−z′ sinθ +a/2,z′ cosθ + zoff)cosθ

=− jeα20 cosθz′[

α20asinθ

2πsin(2π sinθz′

a

)+ cosθ cos

(2π sinθz′

a

)]. (2.28)

Therefore

C20 = B20. (2.29)

For the TE01 mode, the backscattered field intensity is given by

B01 =

∫slot

(Ecpl ×H01) ·ds

2∫S1

(E001 ×H0

01) · zds. (2.30)

The numerator in (2.30) is

2.2 Analysis 35

∫slot

(Ecpl ×H01) ·ds =w/2∫

−w/2

l/2∫−l/2

[Ecpl(z′)×H01(x′,z′)

]· ydz′dx′ (2.31)

≈ w

l/2∫−l/2

[Ecpl(z′)×H01(z′)

]· ydz′

=−w

l/2∫−l/2

Ecpl(z′)H01z′(z′)dz′ (2.32)

where H01z′(z′) is the z′ component of the magnetic field of a TE01 mode travelling in the zdirection and observed on the centreline of the coupling slot. It is given by

H01z′(z′) =−H01z(b,z′ cosθ + zoff)cosθ

=− j cosθeα01 cosθz′. (2.33)

The denominator in (2.30) is given by

2∫S1

(E001 ×H0

01) · zds =ωµ0α01ab(π/b)2 . (2.34)

Therefore, it can be evaluated in closed form as

B01 =− (π/b)2

ωµ0α01kabp01(θ , l)V cpl (2.35)

where

36Higher order mode compensation between the coupling slots and the radiating slots

straddling them

p01(φ , l) = k cosφ

l/2∫−l/2

cos

(πz′

l

)e∓α01 cosφtz′dz′ (2.36)

= 2cosφ

(π

kl

)cosh

(α01 cosφ l

2

)(

α01 cosφ

k

)2+(

π

kl

)2 .

The forward-scattered TE01 amplitude is

C01 =

∫slot

(Ecpl ×H01) ·ds

2∫S1

(E001 ×H0

01) · zds. (2.37)

In this case, the numerator of (2.37) is

∫slot

(Ecpl ×H01) ·ds ≈−w

l/2∫−l/2

Ecpl(z′)H01z′(z′)dz′ (2.38)

where H01z′(z′) is the z′ component of the magnnetic field of a TE01 mode travelling in the zdirection and observed on the centreline of the coupling slot, given by

H01z′(z′) = H01z(b,z′ cosθ + zoff)cosθ

=− j cosθeα01 cosθz′. (2.39)

Therefore

C01 = B01. (2.40)

The scattered wave amplitudes of the higher-order modes in the branch line can becalculated in a similar way, and are given by

2.3 Compensation for higher-order coupling 37

E20 = F20 =(2π/a)2

ωµ0α20kabp20(φ , l)V cpl (2.41)

E01 = F01 =− (π/b)2

ωµ0α01kabp01(φ , l)V cpl.

2.3 Compensation for higher-order coupling

The nth slot of branch t has an offset xofft,n and length Lt,n. The slot voltage can be expressed

as the sum of four components [82], i.e.

• V slott,n(1) is due to a TE10 mode of amplitude A10

t,n incident from z′ < Lt,n/2.

• V slott,n(2) is due to a TE10 mode of amplitude D10

t,n incident from z′ > Lt,n/2.

• V slott,n(3) is due to external mutual coupling from other slots.

• V slott,n(4) is due to internal higher-order mode coupling from neighbouring radiating slots.

The individual slot voltage terms are given by

V slotn,1 =

1K ft,n

yselft,n

2+ yselft,n

A10t,n

V slotn,2 =

1K ft,n

yselft,n

2+ yselft,n

D10t,n

V slotn,3 =− 2

K2 f 2t,n

yselft,n

2+ yselft,n

M

∑j=1j =i

V slotj g ji

V slotn,4 =− 2

K3 f 2t,n

yselft,n

2+ yselft,n

[V slot

t,n−1h20t,nh20

t,n−1 +V slott,n+1h20

t,nh20t,n+1

](2.42)

where yselft,n = 1

2 [(1− St,n11 )

2 − (St,n11 )

2]/St,n21 is the normalized self-admittance and St,n is the

scattering parameter matrix of the isolated slot. The constants in (2.42) are given by

38Higher order mode compensation between the coupling slots and the radiating slots

straddling them

K =2(π/a)2

jωµ0kab

K2 =2

j(β10/k)(k0b)(a/λ )3

K3 =− j2(α20/β10)eα20d. (2.43)

The mutual coupling terms are defined by [82, 64]

ft,n =( π

KLt,n)sin(

πxofft,n

a )cos β10Lt,n2

( π

kLt,n)2 − (β10

k )2

h20t,n =

2( π

KLt,n)cos(

πxofft,n

a )cosh(α20Lt,n2 )

( π

kLt,n)2 +(α20

k )2

g ji =

k0L j/2∫−k0L j/2

cosξ jλ0

2L j

λ0

2Li

[e− jk0R1

k0R1+

e− jk0R2

k0R2

]+[1− (

λ0

2Li)2]∫ k0Li/2

−k0Li/2cos(

ξiλ0

2Li)e− jk0R

k0R

dξ j

(2.44)

where k0 is the free space wavenumber, λ0 the free space wavelength,R =

√(x j − xi)2 +[(z j − zi)+(ξ j/k0 −ξ/k0)]2 and

R1,2 =√

(x j − xi)2 +[(z j − zi)+(ξ j/k0 ∓Li/2)]2, with (xi,0,zi) being the coordinatesof the centre of radiating slot i = n+Nt +N2 + ...+Nt−1 (see Figure 2.1).

For the radiating slots straddling the coupling slot, we include two additional slot voltagecomponents, i.e.

V slott,n =

6

∑q=1

V slott,n(q),n ∈ [kt ,kt +1]. (2.45)

• V slott,n(5) is due to TE20 mode coupling between the coupling slot and the radiating slot.

• V slott,n(6) is due to TE01 mode coupling between the coupling slot and the radiating slot.

Due to the distances between the coupling slot and radiating slots other than its neigh-bors, higher-order mode coupling will be negligible and can be ignored. Each slot voltage

2.3 Compensation for higher-order coupling 39

component is associated with a backward and forward scattered TE10 wave, with the modeamplitudes and the corresponding slot voltage component related via [82]

B10t,n(q) =C10

t,n(q) =−K ft,nV slott,n(q). (2.46)

Elliott and O’Laughlin [82] derived the expression for V slott,n(4) to account for TE20 mode

coupling between neighboring radiating slots. I use the same approach to determine V slott,n(5)

and V slott,n(6).

2.3.1 TE20 mode compensation

Refer to the radiating slot in branch t to the left of the coupling slot (n = kt) in Figure 2.7.

t k,t

t k, +1t

t

qt

lt

d0

ft

z’

x’

slot , t kt

slot , +1 t kt

lt

z

x

z'

x'

d0 = /4l10

Branch t

Main line

ft

At,n(a)

10C

t,n(a)

20

Bt,n(5)

10

Figure 2.7 Geometry for calculation of TE20 mode coupling between radiating slot (t,kt) andcoupling slot t

As in [82], consider two scenarios a and b. In situation a, let this slot be illuminated by aTE10 mode wave of amplitude A10(a)

t,n . This sets up an electric field Eslot(a) in the slot with a

slot voltage of V slot(a)t,n . The amplitude of the forward scattered TE20 wave from this slot is

given by [82]

C20(a)t,n =− (2π/a)2

ωµ0α20kabV slot(a)

t,n h20t,n. (2.47)

This decaying TE20 wave sets up a magnetic field in the region of coupling slot t, givenby

40Higher order mode compensation between the coupling slots and the radiating slots

straddling them

H(a)20 (x,z) =C20(a)

t,n H020(x)e

−α20z. (2.48)

In situation b, all other radiating and coupling slots are replaced by equivalent magneticcurrent sheets. The magnetic current sheet in coupling slot t is defined as

M(b)t (z′) = Ecpl

t (z′)× y = z′V cpl

t

wcos

(πz′

lt

). (2.49)

The amplitude of the additional TE10 wave scattered by the radiating slot due to TE20

mode coupling with the coupling slot can then be calculated using the reciprocity theorem. Itis given by

B10t,n(5) =

(π/a)2

ωµ0β10ab

∫St

H(a)20z′

A10(a)t,n·M(b)

t ds (2.50)

where St is the surface of coupling slot t.The coupling slot is centred at (x,y,z) = (a/2,0,d0), and the branch line and coupling

slot coordinate systems are related via

(x,y,z) = (x′ cosφt − z′ sinφt +a/2,y′,x′ sinφt + z′ cosφt +d0) (2.51)

x = cosφt x′− sinφt z′, y = y′, z = sinφt x′+ cosφt z′,

Using the narrow slot approximation, (2.50) reduces to

B10t,n(5) ≈

(π/a)2

ωµ0β10abw

lt/2∫−lt/2

H(a)20z′(z

′)

A10(a)t,n

M(b)t (z′)dz′. (2.52)

H(a)20z′(z

′) is the z′ component of the magnetic field observed on the centreline of the couplingslot (i.e. x′ = y′ = 0 ), given by

2.3 Compensation for higher-order coupling 41

H(a)20z′(z

′) =−H(a)20x(−z′ sinφt +a/2,z′ cosφt +d0)sinφt

+H(a)20z(−z′ sinφt +a/2,z′ cosφt +d0)cosφt

=− jC20(a)t,n e−α20d0e−α20 cosφtz′ ×

[α20 sinφt

2π/asin(

2π sinφtz′

a)

+ cosφt cos(2π sinφtz′

a)

]. (2.53)

By using (2.18) and (2.43), (2.52) can be evaluated in closed form as

B10t,n(5) =

jK2β10eα20d0

α20

V slot,at,n

A10(a)t,n

V cplt p20

t h20t,n (2.54)

where p20t = p20(φt , lt). From the first expression in (2.42), it follows that

V slot(a)t,n

A10(a)t,n

=1

K ft,n

yselft,n

2+ yselft,n

. (2.55)

By substituting (2.46) and (2.55) into (2.54), we obtain

V slott,n(5) =− 2

K4 f 2t,n

yselft,n

2+ yselft,n

h20t,n p20

t,nV cplt (2.56)

where

K4 = j2(α20/β10)e(α20d0). (2.57)

In order to determine V slott,n(5) for radiating slot n = kt +1, we again assume TE10 mode

illumination of amplitude A10(a)t,n and calculate the amplitude of the TE20 wave scattered

backward from this slot as [82]

42Higher order mode compensation between the coupling slots and the radiating slots

straddling them

B20(a)t,n =−

(2π/a)2h20t,n

ωα20kabV slot(a)

t,n . (2.58)

The TE20 mode magnetic field travelling in the z direction in the region of the coupling slotis in this case

H(a)20 (x,z) = B20(a)

t,n H020(x)e

α20(z−2d0). (2.59)

On the centreline of the coupling slot, the z′ component of this magnetic field is given by

H(a)20z′(z

′) =−H(a)20x(−z′ sinφt +a/2,z′ cosφt +d0)sinφt

+H(a)20z(−z′ sinφt +a/2,z′ cosφt +d0)cosφt

=− jB20(a)t,n e−α20d0eα20 cosφtz′ ×

[− α20 sinφt

2π/asin(

2π sinφtz′

a)

+ cosφt cos(2π sinφtz′

a)

]. (2.60)

Substitution of (2.60) into (2.52) and subsequent evaluation yields the same results as in(2.54). The expression for V slot

t,n(5) in (2.56) is thus also valid for slot n = kt +1.

2.3.2 TE01 mode compensation

Following the procedure described in [82], we can calculate the TE01 mode scattering fromradiating slot n = kt under illumination by a TE10 mode wave of amplitude A10(a)

t,n, . Theamplitude of the forward-scattered TE01 wave is given by

C01(a)t,n =

(π/b)2

jωµ0α01ab

∫Sn

(Eslot(a)t,n ×H01) ·ds (2.61)

where St,n is the surface of the radiating slot. Using the narrow slot approximation, we findthat

2.3 Compensation for higher-order coupling 43

C01(a)t,n ≈ (π/b)2

jωµ0α01abw

Lt,n/2∫−Lt,n/2

Eslot(a)t,n (z′)H01z′(z

′)dz′ (2.62)

where H01z′(z′) is the z′ component of a TE01 mode travelling in the −z direction, evaluatedon the centreline of the radiating slot to the left of the coupling slot (n = kt). It is given by

H01z′(z′) = H01z(b,z′+ zoff) =− jeα01z′. (2.63)

Equation (2.61) can then be evaluated in closed form as

C01(a)t,n =

(π/b)2

jωµ0α01kabV slot(a)

t,n h01t,n (2.64)

where

h01t,n = k

Lt,n/2∫−Lt,n/2

cos

(πz′

Lt,n

)e∓α01z′dz′ (2.65)

=2(

π

kLt,n

)cosh

(α01Lt,n

2

)(

α01k

)2+(

π

kLt,n

)2 .

This TE01 wave sets up a magnetic field in the region of coupling slot t of

H(a)01 (y,z) =C01(a)

t,n H001e−α01z. (2.66)

The amplitude of the additional TE10 wave scattered by the radiating slot due to TE01 modecoupling with the coupling slot can then be calculated from

B10t,n(6) =

(π/a)2

ωµ0β10ab

∫St

H(a)01

A10(a)t,n

·Mbt ds. (2.67)

44Higher order mode compensation between the coupling slots and the radiating slots

straddling them

The z′ component of the magnetic field observed on the centerline of the coupling slot is

H(a)01z′(z

′) = H(a)01z(0,z

′ cosφt +d0)cosφt (2.68)

= jC01(a)t,n e−α01d0 cosφte−α01 cosφtz′.

By using the narrow slot approximation together with (2.19) and (2.43), (2.67) can beevaluated in closed form as

B10t,n(6) =

jK2a2β10e−α01d0

4b2α01

V slot(a)t,n

A10(a)t,n

V cplt p01

t (2.69)

with p01t = p01(φt , lt) . As before, the corresponding slot voltage component can be calculated

as

V slott,n(6) =− 2

K5 f 2t,n

yselfn

2+ yselft,n

h01t,n p01

t V cplt (2.70)

where

K5 =− j8(b/a)2(α01/β01)eα01d0. (2.71)

The amplitude of the TE01 wave scattered backward from radiating slot n = kt + 1 inresponse to TE10 mode illumination of amplitude A10(a)

t,n is

B01(a)t,n =− (π/b)2

jωµ0α01ab

∫St,n

(Eslot(a)t,n ×H01) ·ds

≈ (π/b)2

jωµ0α01abw

Lt,n/2∫−Lt,n/2

Eslot(a)t,n (z′)H01z′(z

′)dz′. (2.72)

Here H01z′(z′) is the z′ component of a TE01 mode travelling in the z direction, evaluated onthe centreline of the radiating slot to the right of the coupling slot (n = kt +1). It is given by

2.4 Design equations 45

H01z′(z′) = H01z(b,z′+ zoff) =− jeα01z′. (2.73)

Equation (2.72) can then be evaluated in close form as

B01(a)t,n =− (π/b)2

ωµ0α01kabV slot(a)

t,n h01t,n. (2.74)

The TE01 mode magnetic field travelling in the z direction in the region of the coupling slotis thus

H(a)01 (y,z) = B01(a)

t,n H001(y)e

α01(z−2d0). (2.75)

On the centreline of the coupling slot, the z′ component of this magnetic field is given by

H(a)01z′(z

′) = H(a)01z(0,z

′ cosφt +d0)cosφt

= jB01(a)t,n e−α01d0 cosφteα01 cosφtz′. (2.76)

Substitution of (2.76) into (2.64) and subsequent evaluation yields a result identical to (2.66).The expression for V slot

t,n(6) in (2.70) is therefore also valid for slot n = kt +1.

2.4 Design equations

2.4.1 Radiating slots

Elliott’s first design equation [90] establishes a relationship between yat,n (the normalized

active admittance for radiating slot t,n), V slott,n (its slot voltage), and Vt,n (the voltage that

appears across yat,n in the equivalent circuit for branch t). This equation still holds, and is

given by

yat,n = K1 ft,n

V slott,n

Vt,n. (2.77)

46Higher order mode compensation between the coupling slots and the radiating slots

straddling them

Since a normalized impedances and admittances are used in the equivalent networks, theconstant K1 can be reduced to

K1 =1

j(a/λ )

√2

ωµ0β10ab. (2.78)

The second design equation provides an expression for the normalized active admittancein terms of its self-admittance and other terms that account for the various forms of mutualcoupling. The active admittance is related to the incident and scattered wave amplitudes via[90]

1ya

t,n=

A10t,n +D10

t,n +B10t,n

2B10t,n

. (2.79)

Manipulation of this expression gives

1ya

t,n=

1yself

t,n+

2+ yselft,n

yselft,n

[V slott,n − (V slot

t,n(1)+V slott,n(2))]

2V slott,n

. (2.80)

Substitution of (2.42), (2.45), (2.56) and (2.70) into (2.80) yields the new form of the seconddesign equation for slots n = kt and n = kt +1 as

1ya

t,n=

1yself

t,n+

1K2 f 2

t,n

M

∑j=1j =i

V slotj

V sloti

g ji +1

K3 f 2t,n

[V slot

t,n−1

V slott,n

h20t,nh20

t,n−1 +V slot

t,n+1

V slott,n

h20t,nh20

t,n+1

]

+1

K4 f 2t,n

V cplt

V slott,n

h20t,n p20

t +1

K5 f 2t,n

V cplt

V slott,n

h01t,n p01

t . (2.81)

For all other slots, the second design equation remains unchanged, i.e. [82]

1ya

t,n=

1yself

t,n+

1K2 f 2

t,n

M

∑j=1j =i

V slotj

V sloti

g ji +1

K3 f 2t,n

[V slot

t,n−1

V slott,n

h20t,nh20

t,n−1 +V slot

t,n+1

V slott,n

h20t,nh20

t,n+1

]. (2.82)

With a slot spacing of d = λ0/2, the voltages across the shunt elements have uniformmagnitude and alternating phase, so that

2.4 Design equations 47

Vt,n = (−1)n−ktVt,kt . (2.83)

2.4.2 Coupling slots

With reference to the geometry shown in Figure 2.3 and the phase reference at the center ofthe slot, the scattering parameter matrix of crossed guide junction t is defined by [56]

St =

St

11 1−St11 St

31 −St31

1−St11 St

11 −St31 St

31

St31 −St

31 St33 1−St

33

−St31 St

31 St33 1−St

33

. (2.84)

The conventional formulation uses resonant coupling slots (i.e. Im[St11] = 0 ), and relies

on the assumption that all scattering parameters can be expressed in terms of the reflectioncoefficient [89, 25], viz.

St31 = sgn θt

√St

11(1−St11), St

33 = 1−St11. (2.85)

This implies that all scattering parameters should be in-phase or out-of-phase at resonance.This assumption is valid for very small waveguide wall thickness or large inclination angleθt , but for realistic values of wall thickness and smaller inclination angles, substantial phasedifferences between St

31 and St11 have been observed [56]. The use of the conventional

approach for the calculation of the coupling slot dimensions may therefore contribute addi-tional errors in the array element excitations, especially in branches with small inclinationangles. We propose a more general approach that takes phase differences between elementsof the scattering parameter matrix into consideration. The coupling slot is represented byan active impedance, which takes mismatches at ports 3 and 4 of the coupling junction intoconsideration. The active impedance of coupling slot t is defined as [97]

zt =2(St

11 +Rt)

1− (St11 +Rt)

(2.86)

with

48Higher order mode compensation between the coupling slots and the radiating slots

straddling them

Rt =St2

31(ytott −2)

ytott −S33(ytot

t −2)(2.87)

and ytott =

Nt∑

n=1ya

t,n being the total active admittance of branch t. In addition, Vt,kt and It are

related via

Vt,kt =− jκtIt (2.88)

with the coupling coefficient, κt given by

κt =2St

31(1+ zt/2)ytot

t −St33(y

tott −2)

. (2.89)

By substituting (2.86) and (2.87) into (2.89), this can also be expressed as

κt =2St

31[(1−St

11)(1−St33)− (St

31)2]ytot

t +2(St33 −St

11St33 +(St

31)2). (2.90)

For cases where the assumptions stated in (2.85) hold, the expressions in (2.86) and (2.88)reduce to the conventional relations in [56] (eq. 28) and [56] (eq. 31), respectively.

Refer to the equivalent circuit for the main line in Figure 2.1. Analogous to the firstdesign equation, we can define a relationship between za

t , V cplt and It . From (2.10) and (2.78),

it follows that

zat =− jK1 p10

tV cpl

t

It(2.91)

where p10t = p10(θt , lt). Using (2.11), p10

t can alternatively be defined as

p10t =−

St11

St31

p10(φt , lt). (2.92)

2.5 Design procedure 49

With half-wavelength spacing between coupling slots, the currents through the seriesimpedances have uniform magnitude and alternating phase, so that

It = (−1)t−1I1. (2.93)

From (2.83), (2.89) and (2.93), the voltages in the equivalent circuits for different branchesare related via

V1,m

Vt,n= (−1)t−1+kt−k1+n−m κ1

κt. (2.94)

2.5 Design procedure

The following section provides a step-by-step design procedure to calculate the slot positionsand slot dimensions.

1. Specify the input resistance for the main line, rin. The most common choice is todesign for an impedance match, i.e. rin = 1.

2. Specify the slot voltages V slott,n for a desired radiation pattern.

3. Select the realistic values for the total conductance of each branch line, gt . Valuestypically range between 1.0 and 4.0 depending on the number of slots in the branch.Increasing the value of gt generally results in larger radiating slot offsets.

4. Select reliable starting values for (|xofft,n|,Lt,n), and (|θt |, lt).

5. Select a reference slot n = mt for each branch.

6. Select an even-valued integer C as the number of iterations to be performed.

7. Set the iteration counter C = 1.

8. Calculate the mutual coupling terms, g ji,h20t,n = h20(xoff

t,n,Lt,n),h01t,n = h01(xoff

t,n,Lt,n), p20t =

p20(φt , lt) and p01t = p01(φt , lt).

9. For the first half of the iterations (c ≤C/2), weight the mutual coupling terms by afactor of 2(c−1)/C. For the second half of the iterations, no scaling is applied. Thisimplies that no mutual coupling is considered for the first iteration (i.e. ya

t,n = yselft,n ) and

the full effect of mutual coupling is only included in the second half of the iterations.

50Higher order mode compensation between the coupling slots and the radiating slots

straddling them

10. For each branch solve the following set of 2Nt nonlinear equations to compute theradiating slot offsets and lengths (xoff

t,n,Lt,n), with yat,n given by (2.81) or (2.82)

Re

[ya

t,n

yat,mt

]=−(1)n−mt

ft,nV slott,n

ft,mtVslot

t,mt

= 0 n = 1,2,3, ...Nt ,n = mt

Im

[ya

t,n

]= 0 n = 1,2,3, ...Nt

Re

[Nt

∑n=1

yat,n

]= gin.

11. Solve the following set of 2T nonlinear equations for the inclination angles and cou-pling slot lengths, (|θt |, lt), with za

t κt given by (2.86) and (2.89)

Re

[ya

t,mt

ya1,m1

]=−(1)t−1+kt−k1+mt−m1

ft,mtVslot

t,mt

f1,m1Vslot1,m1

Re

[κ1

κt

]

Im

[κ1

κt

]t = 2,3, ...T

Re

[T

∑t=1

zt

]= rin

Im

[T

∑t=1

zt

]= 0.

12. Calculate V1,k1 , z1 and I1 using (2.77), (2.86), (2.93) and (2.88).

13. Calculate It , zt and V cplt , using (2.93), (2.86), (2.87), (2.91) and (2.92).

14. Increment c and repeat steps 8-13 until c =C.

2.6 Validation

To confirm the validity of the proposed design procedure, half height and standard heightarrays are designed and analyzed for a design frequency of f0 = 9 GHz. Designs areimplemented in different slot widths , w, and wall thickness in order to identify the effects of

2.6 Validation 51

higher-order mutual coupling when designing arrays with varying specifications, and assessthe significance of the proposed design technique in improving the performance of thesearrays. The arrays are fed using inclined coupling slots at the center of each branch. Thedesigns were carried out using the technique proposed in [92] and the proposed procedure.

The radiating and coupling slot data was generated using the frequency domain solver ofCST Microwave Studio.

2.6.1 Standard-height array

Standard height slot arrays have been known to suffer in performance due to the effects ofhigher-order mutual coupling between the coupling slots and the radiating slots straddlingthem [83]. In this section, a 5×4 element array is implemented and analysed.

The 5×4 element array is implemented using WR90 waveguide with dimensions a =

22.86 mm and b = 10.16 mm, with slot width, w = 1 mm and wall thickness of 0.508 mm.The array was designed to achieve sidelobe levels of −20 dB and −30 dB in the H-plane andthe E-plane respectively.

The slot dimensions are shown in Table 2.3 and Table 2.4. The effect of gradual inclusionof mutual coupling is illustrated in Figure 2.8, where the variation of the offset for the slotwith index (1,3,3) as computed during the iterative steps is shown.

Figure 2.8 Offset of slot (1,3,3) as computed during each iterative step.

52Higher order mode compensation between the coupling slots and the radiating slots

straddling them

All slot offsets and slot lengths are initially set to |xofft,n|= 2 mm and Lt,n = 16 mm. The initial

offset calculated for the case with zero mutual coupling is 5.74 mm, while the final offset isonly 3.75 mm. The gradual introduction of the mutual coupling contributions over the next12 iterative steps avoids abrupt changes in the slot dimensions, and convergence is easilyachieved during the second half of the iterations. A change in initial guess does not impactthe final dimensions if the initial guess is within the database generated for the radiating andcoupling slots.

The radiating slot and the coupling slot dimensions of the conventional and the compen-sated arrays are compared in Table 2.3 and Table 2.4 respectively.

There are small differences in dimensions due to the compensation for higher-ordercoupling, but the effect on array performance is significant. In order to verify results, thecomplete arrays were modelled and analysed using CST Microwave Studio. The complexslot voltages were calculated by integrating the transverse component of the electric field atthe centre and across the width of each slot. For both designs the phase of all slot voltageswas normalized to yield a zero average in order to assess the spread of the phase betweenarray elements. The slot voltage magnitudes were normalized to have the same average as thetarget excitations. The amplitude error for individual slots was calculated as the differencebetween the magnitude and the simulated slot voltage and the target excitation, divided bythe target excitation.

Figure 2.9 and Figure 2.10 show the phase and amplitude errors of the radiating slotvoltages respectively. The amended design procedure achieves a reduction in the phaseerrors, resulting a 13 phase spread in comparison to 17 when using the conventional designapproach. The proposed design also results in reduced amplitude errors with a maximumslot amplitude error of 4% compared to the conventional design technique, which results in amaximum error of 8%.

The slot voltages in the coupling slots were also analysed using CST simulation results,and are compared to values calculated using (2.91). Using the same normalization as for theradiating slot voltages, the phase and amplitude errors in the coupling slot voltages for thetwo designs are compared in Figure 2.11 and Figure 2.12 respectively. The proposed designachieves a slight improvement in the phase spread, while the amplitude errors are significantlyreduced with a standard deviation of 4.5% as compared to 9.5% for the conventional design.

Figure 2.13 and Figure 2.14 compare the principal plane radiation patterns of the twosimulated designs. In spite of the excitation errors, the H-plane pattern of the conventionaldesign is consistent with that of the proposed design. In contrast, the E-plane pattern ismore sensitive to excitation errors, as shown in Figure 2.14. For the conventional design,the sidelobe specification in the E-plane is not met with the maximum sidelobe level at −26

2.6 Validation 53

Slot index Slot offset (mm) Slot length (mm)

(t,n) Uncompensated Compensated Uncompensated Compensated

(1,1) 1.77 1.80 16.24 16.25

(1,2) -3.90 -3.94 16.37 16.38

(1,3) 3.63 3.64 16.59 16.60

(1,4) -2.16 -2.12 16.07 16.05

(2,1) 2.07 2.07 16.02 16.01

(2,2) -3.78 -3.75 16.16 16.09

(2,3) 3.61 3.57 16.30 16.22

(2,4) -2.12 -2.15 15.87 15.87

(3,1) 2.14 2.13 15.90 15.89

(3,2) -3.72 -3.75 16.19 16.25

(3,3) 3.72 3.75 16.19 16.25

(3,4) -2.14 -2.13 15.90 15.89

(4,1) 2.12 2.15 15.87 15.87

(4,2) -3.61 -3.57 16.30 16.22

(4,3) 3.78 3.75 16.16 16.09

(4,4) -2.07 -2.07 16.02 16.01

(5,1) 2.16 2.12 16.07 16.05

(5,2) -3.63 -3.64 16.59 16.60

(5,3) 3.90 3.94 16.37 16.38

(5,4) -1.77 -1.80 16.24 16.25

Table 2.1 Dimensions of the radiating slots of the 5×4 planar array.

54Higher order mode compensation between the coupling slots and the radiating slots

straddling them

Slot index Inclination angle (degrees) Slot length (mm)

(t) Uncompensated Compensated Uncompensated Compensated

1 -6.73 -6.72 15.88 15.90

2 16.24 15.99 15.90 15.92

3 -21.47 -21.41 15.92 15.93

4 16.24 15.99 15.90 15.92

5 -6.73 -6.72 15.88 15.90

Table 2.2 Dimensions of the coupling slots of the 5×4 planar array.

Figure 2.9 Phase errors in the excitations of the radiating slots in a 5×4 element array.

dB. the sidelobes are not well defined and have been flattened. Conversely, the amendeddesign yields a radiation pattern that closely mirrors a theoretical Chebyshev pattern. Ithas a well-defined main beam and sidelobes, and improved sidelobe performance achievinga maximum sidelobe level of −29.5 dB within an angular sector of ±80 off boresight.Both the conventional and the proposed designs have lobes of −25 dB at ±90. This isinevitable, since the chosen design specifications and waveguide dimensions yield a nominalinter-element spacing of 0.73λ0, where the formation of grating lobes can strat to becomenoticeable. A remedy would have been to use non-standard waveguide dimensions toachieve an element spacing closer to 0.7λ0, for which the far-out sidelobes can be suppressedsufficiently.

2.6 Validation 55

Figure 2.10 Amplitude errors in the excitations of the radiating slots in a 5×4 element array.

Figure 2.11 Phase errors in the excitations of the coupling slots in a 5×4 element array.

The return loss performance is shown in Figure 2.15. The amended design results ina reflection of less than −21 dB at f0, with the minimum reflection shifted 0.4% off thedesign frequency. This shift in frequency is potentially due to higher order mutual couplingbetween the adjacent coupling slots, which results in an input mismatch along with a slightdegradation in sidelobe level [52].

56Higher order mode compensation between the coupling slots and the radiating slots

straddling them

Figure 2.12 Amplitude errors in the excitations of the coupling slots in a 5×4 element array.

Figure 2.13 H-plane radiation pattern of the 5×4 element array.

2.6 Validation 57

Figure 2.14 E-plane radiation pattern of the 5×4 element array.

Figure 2.15 Return loss of the 5×4 element array.

58Higher order mode compensation between the coupling slots and the radiating slots

straddling them

2.6.2 Half-height array

The effect of higher order mutual coupling is less significant in half-height waveguide arrays[86], especially in larger arrays where the number of radiating slots affected by coupling isrelatively small in comparison to the total number of slots. However, higher order couplingcan still degrade the performance in low-sidelobe applications. To illustrate this, this sectionwill compare the performance of a half height arrays designed using the conventional and theproposed design technique.

The 8×8 element array is implemented in half-height waveguide (a = 22.86 mm andb = 5.08 mm), with −45 dB sidelobes in the E-plane and −30 dB in the H-plane. The slotwidth, w = 1 mm and with a wall thickness of 0.508 mm. The dimensions for the arraysdesigned using the conventional and the proposed design technique are compared in Table2.3 and Table 2.4.

Figure 2.16 and Figure 2.17 compare the principal plane radiation patterns of the twosimulated designs. Similar to the standard-height designs, the H-plane pattern of the conven-tional design is consistent with that of the proposed design. However, the sidelobes of theE-plane pattern of the conventional design exceed the specification by 6 dB with a maximumsidelobe level of −39 dB. In contrast, the E-plane pattern achieved using the proposed designtechnique achieves sidelobe levels with a maximum of −44 dB.

Table 2.3 Dimensions of the radiating slots of the 8×8 planar array in half-height waveguide.

Slot index Slot offset (mm) Slot length (mm)(t,n) Uncompensated Compensated Uncompensated Compensated(1,1) 0.32 0.32 15.93 15.93(1,2) -0.92 -0.91 16.38 16.38(1,3) 1.54 1.53 16.71 16.71(1,4) -2.20 -2.24 16.71 16.72(1,5) 1.94 1.96 16.85 16.86(1,6) -1.70 -1.69 16.56 16.55(1,7) 0.83 0.83 16.49 16.49(1,8) -0.33 -0.33 15.93 15.93(2,1) 0.47 0.46 16.04 16.04(2,2) -1.18 -1.19 16.20 16.20(2,3) 1.79 1.79 16.51 16.51(2,4) -2.45 -2.44 16.62 16.61

Continued on next page

2.6 Validation 59

Table 2.3 – continued from previous page

Slot index Slot offset (mm) Slot length (mm)(t,n) Uncompensated Compensated Uncompensated Compensated(2,5) 2.27 2.26 16.69 16.68(2,6) -1.93 -1.94 16.43 16.43(2,7) 1.09 1.09 16.28 16.28(2,8) -0.55 -0.56 16.04 16.04(3,1) 0.62 0.62 16.05 16.05(3,2) -1.29 -1.28 16.18 16.18(3,3) 2.04 2.04 16.51 16.51(3,4) -2.63 -2.64 16.65 16.67(3,5) 2.59 2.60 16.73 16.75(3,6) -2.08 -2.07 16.43 16.43(3,7) 1.25 1.25 16.22 16.22(3,8) -0.64 -0.64 16.03 16.03(4,1) 0.63 0.64 16.03 16.03(4,2) -1.26 -1.26 16.16 16.16(4,3) 2.06 2.06 16.47 16.47(4,4) -2.62 -2.61 16.66 16.63(4,5) 2.61 2.60 16.69 16.66(4,6) -2.05 -2.06 16.43 16.43(4,7) 1.27 1.27 16.19 16.19(4,8) -0.63 -0.64 16.01 16.01(5,1) 0.63 0.63 16.01 16.00(5,2) -1.27 -1.27 16.19 16.19(5,3) 2.05 2.05 16.43 16.43(5,4) -2.61 -2.62 16.69 16.72(5,5) 2.62 2.63 16.66 16.68(5,6) -2.06 -2.05 16.47 16.46(5,7) 1.26 1.26 16.16 16.16(5,8) -0.63 -0.63 16.03 16.03(6,1) 0.64 0.64 16.03 16.03(6,2) -1.25 -1.25 16.22 16.22(6,3) 2.08 2.08 16.43 16.43

Continued on next page

60Higher order mode compensation between the coupling slots and the radiating slots

straddling them

Table 2.3 – continued from previous page

Slot index Slot offset (mm) Slot length (mm)(t,n) Uncompensated Compensated Uncompensated Compensated(6,4) -2.59 -2.58 16.73 16.71(6,5) 2.63 2.62 16.65 16.63(6,6) -2.04 -2.04 16.51 16.50(6,7) 1.29 1.29 16.18 16.18(6,8) -0.62 -0.62 16.05 16.05(7,1) 0.55 0.55 16.04 16.04(7,2) -1.09 -1.09 16.28 16.28(7,3) 1.93 1.93 16.43 16.42(7,4) -2.27 -2.27 16.69 16.70(7,5) 2.45 2.46 16.62 16.63(7,6) -1.79 -1.79 16.51 16.51(7,7) 1.18 1.18 16.20 16.20(7,8) -0.47 -0.47 16.04 16.05(8,1) 0.33 0.33 15.93 15.93(8,2) -0.83 -0.83 16.49 16.48(8,3) 1.70 1.71 16.56 16.56(8,4) -1.94 -1.91 16.85 16.83(8,5) 2.20 2.16 16.71 16.70(8,6) -1.54 -1.56 16.71 16.72(8,7) 0.92 0.94 16.38 16.39(8,8) -0.32 -0.32 15.93 15.92

Table 2.4 Dimensions of the coupling slots of the 8×8 planar array.

Slot index Inclination angle (degrees) Slot length (mm)(t) Uncompensated Compensated Uncompensated Compensated

1 6.56 5.21 17.78 16.842 -7.71 -6.77 17.39 16.853 9.32 8.70 17.15 16.86

Continued on next page

2.6 Validation 61

Table 2.4 – continued from previous page

Slot index Inclination angle (degrees) Slot length (mm)(t) Uncompensated Compensated Uncompensated Compensated4 -13.66 -13.12 16.95 16.905 13.66 13.32 16.95 16.906 -9.32 -9.10 17.16 16.877 7.71 7.30 17.40 16.858 -6.56 -5.78 17.78 16.85

Figure 2.16 H-plane radiation pattern ofthe 8×8 element array.

62Higher order mode compensation between the coupling slots and the radiating slots

straddling them

Figure 2.17 E-plane radiation pattern ofthe 8×8 element array.

Chapter 3

Design refinements for the feed of aplanar slot array

3.1 Introduction

Planar slot arrays are conventionally implemented with multiple rectangular waveguidesarranged side-by-side, and fed via centred-inclined coupling slots in the upper wall of awaveguide located behind and at right agles to the branch lines. The radiating slots of aresonant array are spaced half a guide wavelength, and the same applies to the spacingbetween the coupling slots in the main line.

The conventional design procedure [90, 82, 89] utilises resonant coupling slots in themain line of the planar arrays. The slot is modelled as a series impedance in the equivalentcircuit for the main line. The design equations for the calculation of coupling slot dimensionscomprise expressions for the series impedance and the coupling coefficient which relates thecurrent through the series impedance to the voltages across the shunt elements in the branchline equivalent circuit. The equations are based on analysis results of coupling slot junctions[24, 25] and simple expressions for the impedance and the coupling coefficient are obtained[89].

The design assumes that all scattering parameters of junctions are either in-phase orout-of-phase at resonance. Mazzarella and Montisci highlighted some shortcomings in thisconventional approach [56]. They showed that the equivalent circuit model is only valid formoderate to large inclination angles, small wall thickness between the main line and thebranch line and over a frequency range close to the resonant frequency. The conventionalmodel was also deemed to be unsuitable for the design of shaped-beam arrays that requirenon-resonant coupling slots.

64 Design refinements for the feed of a planar slot array

This chapter provides a detailed anlysis of coupling slot junctions, illustrating the severityof the phase variation between the elements of the scattering matrix. It is shown that thecoupling slot can still be modelled as a series impedance, but with modified expressions forthe impedance and the coupling coefficient that are derived in this chapter. This approachtakes phase differences between individual scattering parameters into consideration andaccommodates non-resonant slots.

It has also been discussed that internal higher-order mode coupling between centered-inclined slots in feed networks can be significant [83]. It was found that higher-order couplingcan largely be attributed to the TE20 and TE01 modes.

Compensation for higher-order coupling can be done by applying full-wave analysis toan existing design, and performing simplified optimization by perturbing the slot dimensionsto improve the array performance [51, 52]. This approach is computationally intensive [57],and access to suitable analysis code or commercial software is essential.

Thus, TE20 and TE01 mode coupling through further amendments to the expression forthe equivalent impedance of a coupling slot is also derived in this chapter. The proposedclosed-form expression for the active impedance remains valid for non-resonant slots and forarbitrary slot inclination angles.

3.2 Analysis

Consider an array with a total of T branch lines, with branch t having a total of Nt radiatingslots. All waveguides are assumed to have uniform dimensions with width a and height b.The slots are spaced at intervals of d = λ10/2, with λ10 being the wavelength of the TE10

mode in the waveguide at the design frequency, f0. The coupling slot for branch t is locatedbetween the radiating slots (t,kt) and (t,kt +1), as shown in Figure 3.1.

The equivalent circuit of the branch consists of the shunt elements yat,n, representing

the normalized active admittances of each slot. The voltage across yat,n is denoted as Vt,n.

The main line equivalent circuit comprises normalized series impedances separated by half-wavelength sections of transmission line. The current flowing through zt is denoted as It . Weuse a normalized characteristic impedance z0 = 1 in all calculations involving the equivalentcircuits.

3.2.1 Conventional formulation

Refer to coupling junction t in Figure 3.2. The coupling slot has a width w, length lt , and isrotated by an angle θt with respect to the axis of the main line. The inclination angle θt is

3.2 Analysis 65

(xi,0,z

i)

Figure 3.1 A planar slotted waveguide array (center) with equivalent circuits for a branch ofradiating slots (above) and the main line with inclined coupling slots (below).

assumed to be positive for clockwise rotation. With the phase reference at the centre of theslot, the scattering parameter matrix of the junction t is given by [56]:

St =

St

11 1−St11 St

31 −St31

1−St11 St

11 −St31 St

31

St31 −St

31 St33 1−St

33

−St31 St

31 1−St33 St

33

. (3.1)

The conventional formulation uses resonant slots (i.e Im[S11] = 0) in the design of the feed.It also relies on the assumption that all scattering parameters can be expressed in terms of thereflection coefficient [91, 25], viz.

St31 = sgn(θt)

√St

11(1−St11), St

33 = 1−St11. (3.2)

66 Design refinements for the feed of a planar slot array

Figure 3.2 Geometry of coupling junctions.

Under these conditions, the series impedance in the main line becomes:

zt = χ2t ytot

t , (3.3)

with χt =√

St11/(1−St

11) being the coupling coefficient, and ytott be the total admittances

seen from radiating slots (t,kt) and (t,kt +1) looking toward the shorted ends of the branch,i.e.

ytott,kt

=kt

∑n=1

yat,n, ytot

t,kt+1 =Nt

∑n=kt+1

yat,n. (3.4)

With at1 − at

2 = It(1 + zt/2), the wave intensities eminating from ports 3 and 4 can becalculated as

3.2 Analysis 67

bt3 = sgn(θt)χt

yt,kt +12

It (3.5)

bt4 =−sgn(θt)χt

yt,kt+1 +12

It .

The voltages across the neighboring shunt admittances in the equivalent circuit for the branchline are

Vt,kt =− j2bt

3ytot

t,kt+1

, Vt,kt+1 =− j2bt

4ytot

t,kt+1 +1. (3.6)

Substitution of (3.5) into (3.6) thus yields

Vt,kt =−Vt,kt+1 =− j sgn(θt)χtIt (3.7)

3.2.2 Coupling junction

Restrictions in the validity of the assumptions stated in (3.2) were first pointed out in [56]. Theauthors analysed slots with different values of slot width w, wall thickness τ and inclinationangle θ . They identified cases with phase discrepancies between the coupled and reflectedsignals of coupling junctions, especially for wide slots and thick waveguide walls. Theirinvestigations were limited to geometries with θ ≥ 15. The theoretical range for a slotcoupler is 0 ≤ |θ | ≤ 45, but in practice the angles of a typical slot array feed vary between5 and 25.

In order to further assess the validity of the assumptions stated in (3.2), we analysedcoupling junctions in a standard WR90 waveguide (width a = 22.86 mm, height b = 10.16mm) at 9 GHz. We considered slots of varying width and wall thickness, and includedsmaller tilt angles. The calculations were performed using the frequency domain solver ofCST Microwave Studio. With a fixed value for the inclination angle, simulations were carriedout for different lengths to determine the scattering parameters. The simulation results wereinterpolated to identify the resonant slot length, defined as the length where Im[S11] = 0. Fig.3.3 compares the phase difference between S31 and S11 at resonance, with ports 1 and 3 asshown in Fig. 2.3. This was done for slot widths of 1 mm, 2 mm and 3 mm, waveguidewall thickness 0.508 mm and 1.27 mm and inclination angles ranging from 5 to 20. The

68 Design refinements for the feed of a planar slot array

results shown in Fig. 3.3 suggest that phase discrepancies in resonant junctions are notrestricted to wide slots and thick waveguide walls, but are also prevalent for small tilt angleswhere marked phase differences between S31 and S11 are observed. The influence of the wallthickness is less pronounced, but phase discrepancies become more severe for wider slots.Without compensation, the discrepancies can give rise to element excitation errors in theE-plane of planar slot arrays fed by inclined slots. A similar trend can be seen for half-heightwavguides as seen in Figure 3.4.

w=3 mm

w=2 mm

w=1 mm

Figure 3.3 Phase variation between the reflected and scattared wave in a coupling junction instandard-height waveguide.

3.2.3 Amended formulation

The relations given in (3.2) imply that all scattering parameters should be in-phase or out-of-phase at resonance. This assumption is valid for small waveguide wall thickness or largeinclination angle (θ ), but for realistic values of wall thickness and smaller inclination angles,substantial phase differences between S31 and S11 have been observed [56]. In this section,expressions for the equivalent impedance and the coupling coefficient are derived withoutrelying on the assumptions in (3.2).

In an array environment, ports 3 and 4 of coupling junction t are generally not matched,and reflection coefficients and are observed at ports 3 and 4. They are given by [97]

Γt3 =

ytott,kt

−1

ytott,kt

+1, Γ

t4 =

ytott,kt+1 −1

ytott,kt+1 +1

. (3.8)

3.2 Analysis 69

w=3 mm

w=2 mm

w=1 mm

Figure 3.4 Phase variation between the reflected and scattared wave in a coupling junction inhalf-height waveguide.

With bt3 = at

3/Γt3 and bt

4 = at4/Γt

4, the relations between the wave intensities at the portsbecome

bt

1

bt2

bt3 = at

3/Γt3

bt4 = at

4/Γt4

=

St

11 1−St11 St

31 −St31

1−St11 St

11 −St31 St

31

St31 −St

31 St33 1−St

33

−St31 St

31 1−St33 St

33

at1

at2

at3

at4

(3.9)

From the last two equations in (3.9), we find that

at4 =−

Γt3Γt

4 −Γt4

Γt3Γt

4 −Γt3

at3 (3.10)

at3 −at

4at

1 −at2=

St31(Γ

t3 +Γt

4 −2Γt3Γt

4)

(1−Γt3Γt

4)−St33(Γ

t3 +Γt

4 −2Γt3Γt

4).

Using the first two equations of (3.9), the terminated 4-port network can be reduced to alossy 2-port network with S-parameters given by

[bt

1

bt2

]=

[St

11 +Rt 1− (St11 +Rt)

1− (St11 +Rt) St

11 +Rt

][at

1

at2

](3.11)

70 Design refinements for the feed of a planar slot array

where

Rt = St31

at3 −at

4at

1 −at2=

St2

31(Γt3 +Γt

4 −2Γt3Γt

4)

(1−Γt3Γt

4)−S33(Γt3 +Γt

4 −2Γt3Γt

4). (3.12)

Substituting these relations given in [97] (eq. 25) into (3.12) reduces the expression forRt at the design frequency f0 to

Rt =St2

31(ytott −2)

ytott −S33(ytot

t −2)(3.13)

where ytott = ytot

t,kt+ ytot

t,kt+1. The equivalent circuit for the 2-port network with scatteringparameters given by (3.11) is a series impedance, given by

zt =2(St

11 +Rt)

1− (St11 +Rt)

. (3.14)

The wave intensities emanating from ports 3 and 4 are

bt3 =

St31(1−Γt

4)(at1 −at

2)

(1−Γt3Γt

4)−St33(Γ

t3 +Γt

4 −2Γt3Γt

4)

=St

31(ytot +1)(1+ zt/2)

ytott −St

33(ytott −2)

It

bt4 =

−St31(1−Γt

3)(at1 −at

2)

(1−Γt3Γt

4)−St33(Γ

t3 +Γt

4 −2Γt3Γt

4)

=−St

31(ytot+1 +1)(1+ zt/2)

ytott −St

33(ytott −2)

It . (3.15)

Substitution of (3.15) into (3.6) yields

Vt,kt =−Vt,kt =− jκtIt (3.16)

with the coupling coefficient, κt given by

3.2 Analysis 71

κt =2St

31(1+ zt/2)ytot

t −St33(y

tott −2)

. (3.17)

By substituting (3.13) and (3.14) into (3.17), the coupling coefficient can also be expressedas

κt =2St

31[(1−SSt

11)(1−St

33)−St2

31

]ytot

t +2(St33 −St

11 +St2

31). (3.18)

For cases where the assumptions stated in (3.2) hold, the expressions in (3.14), (3.15)and (3.16) reduce to the conventional relations in (3.3), (3.5) and (3.7), respectively.

Compensation for higher order mode coupling

Refer to he geometry of coupling slot t in Figure 3.2. The slot is centered at (x,y,z) =(a/2,b,zoff

t ). The main line and slot coordinate systems are related via

(x,y,z) = (x′ cosθt − zsinθt +a/2,y′+b,x′ sinθt + z′ cosθt + zofft )) (3.19)

x = cosθt x′− sinθt z′, y = y′, z = sinθt x+ cosθt z.

With the phase reference at z = zoff, the TE10 mode fields propagating in the ±z directionin the main line are E10(x,z) = E0

10(x)e∓ jβ10(z−zofft ) and H10(x,z) = H0

10(x)e∓ jβ10(z−zofft )

where β10 =√

k2 − (π/a)2 is the phase constant in the waveguide at the design frequency,k = ω

√ωµ0εrε0 is the wavenumber in the dielectric medium with relative permittivity εr

that fills the waveguide and the cross-sectional parts of the respective field components asdefined in [97].

When TE10 mode waves of amplitude A10t and D10

t are incident from z < zofft and z > zoff

t ,they induce a field Ecpl

t in the coupling slot. We assume the field in the coupling slot to be

Ecplt (x′,z′) =

V cplt

wcos(

πz′

lt

)x’ (3.20)

72 Design refinements for the feed of a planar slot array

where V cplt is the coupling slot voltage. Waves with amplitudes B10

t and C10t are scattered in

the backward and forward directions, while the amplitude of the waves scattered to the leftand right in branch line t are E10

t and F10t .

If reflection coefficients Γt4 and Γt

4 are observed at ports 3 and 4 of coupling junction t,the amplitude of the backscattered TE10 wave from slot t can be expressed as the sum of twocomponents, i.e.

B10t = B10

t(1)+B10t(2). (3.21)

The components are defined as follows:

• B10t(1)is due to a TE10 mode wave of amplitude A10

t incident from z < zofft

• B10t(2)is due to a TE10 mode wave of amplitude D10

t incident from z > zofft .

Using (3.11), they can be expressed as

B10t(1) = (St

11 +Rt)A10t (3.22)

B10t(2) =−(St

11 +Rt)D10t .

Each of the backscattered wave components is associated with a coupling slot voltagecomponent, and they are related via

B10t(q) =−C10

t(q) =− jK p10(θt , lt)Vcpl

t(q) (3.23)

with the function p10(θ , l) and constant K as defined in (2.11).From [97, eq(8)] and [97, eq(11)], the corresponding wave components scattered into thebranch line can be written as

E10t(q) =−F10

t(q) = jK p10(θt −90, lt)Vcpl

t(q)

≈− jKSt

31St

11p10(θt , lt)V

cplt(q). (3.24)

From (3.22) and (3.23), the coupling slot voltage components can be calculated as

3.2 Analysis 73

V cplt(1) =−

St11 +Rt

jK p10(θt , lt)A10

t (3.25)

V cplt(2) =

St11 +Rt

jK p10(θt , lt)D10

t .

To compensate for higher-order mode coupling, we include two additional scatteringcomponents, i.e.

B10t = B10

t(1)+B10t(2)+B10

t(3)+B10t(4). (3.26)

• B10t(3) is due to internal TE20 mode coupling from neighbouring coupling slots.

• B10t(4) is due to internal TE01 mode coupling from neighbouring slots.

Due to the distances between coupling slots other than its neighbours, higher-order modecoupling will be negligible and can be ignored. Elliott and O’Laughlin [91] derived theexpressions to account for TE20 mode coupling between neighbouring radiating slots of aslot array shown in Chapter 2. We use a similar approach to determine B10

t(3) and B10t(4).

With the phase reference at z = zofft , the higher-order TEmn fields in the main line

travelling in the ±z direction are Emn(x,y,z) = E0mn(x,y)e

∓αmn(z−zofft ) and Hmn(x,y,z) =

H0mn(x,y)e

∓αmn(z−zofft ) with the cross-sectional parts of the field components for the TE20 and

TE01 modes as defined in [97] and α01 =√

(mπ/a)2 +(nπ/b)2 − k2 being the attenuationconstant.

As in [91], consider two scenarios a and b. In situation a, only coupling slot t is presentand all other slots in the main line are covered. Slot t is illuminated by a TE10 mode wave ofamplitude A10(a)

t . It is assumed that port 2 of coupling junction t is terminated in a matchedload, but reflection coefficients Γt

3 and Γt4 are observed at ports 3 and 4. The incident wave

sets up an electric field Ecpl(a)t in the slot with a slot voltage of V cpl(a)

t . Using relationsprovided in [97] together with (3.26), the amplitude of the backward and forward scatteredTE20 and TE01 waves can be calculated as

B20(a)t =C20(a)

t =− (2π/a)2

ωµ0α20kabp20(θt , lt)V

cpl(a)t (3.27)

=2β10(St

11 +Rt)

α20

p20(θt , lt)p10(θt , lt)

A10(a)t

74 Design refinements for the feed of a planar slot array

and

B01(a)t =C01(a)

t =− (π/b)2

ωµ0α01kabp01(θt , lt)V

cpl(a)t (3.28)

=β10a2(St

11 +Rt)

2α01b2p01(θt , lt)p10(θt , lt)

A10(a)t

with p20(θ , l) defined in (2.26) and

p01(θ , l) = k cosθ

l/2∫−l/2

cos

(πz′

l

)e∓α01 cosθtz′dz′ (3.29)

= 2cosθ

(π

kl

)cosh

(α01 cosθ l

2

)(

α01 cosθ

k

)2+(

π

kl

)2

The decaying TEmn waves set up magnetic fields of

H(a)20 (x,z) =

B20(a)t H0

20(x)eα20(z−zoff

t ) z < zofft

C20(a)t H0

20(x)e−α20(z−zoff

t ) z > zofft

(3.30)

and

H(a)01 (y,z) =

B01(a)t H0

01(y)eα01(z−zoff

t ) z < zofft

C01(a)t H0

01(y)e−α01(z−zoff

t ) z > zofft .

(3.31)

In situation b, only coupling slot t is again present, but all other slots in the main line arereplaced by equivalent magnetic current sheets. The magnetic currents in coupling slot t −1and t +1 are defined as

M(b)t−1(z

′) = Ecplt−1(z

′)× (−y) =−V cpl

t−1

wcos(

πz′

lt−1)z′ (3.32)

M(b)t+1(z

′) = Ecplt+1(z

′)× (−y) =−V cpl

t+1

wcos(

πz′

lt+1

)z′

3.2 Analysis 75

Following the argument presented in [91], the amplitude of the additional TE10 wavescattered by the coupling slot t due to TE20 mode coupling with the neighbouring couplingslots can be calculated using the reciprocity theorem. It is given by

B10t(3) =

(π/a)2

ωµ0β10ab

[ ∫St−1

H(a)20

A10(a)t

·M(b)t−1ds+

∫St+1

H(a)20 (z

′)

A10(a)t

·M(b)t+1ds

](3.33)

where St−1 and St+1 are the surfaces of coupling slots t −1 and t +1 respectively. Using thenarrow slot approximation, (3.33) reduces to

B10t(3) =− jKkw

2

[ lt−1/2∫−lt−1/2

H(a)20z′

A10(a)t

M(b)t−1(z

′)dz′+

lt+1/2∫−lt+1/2

H(a)20z′(z

′)

A10(a)t

M(b)t+1(z

′)dz′]. (3.34)

H(a)20z′(z

′) is the z′ component of the magnetic field observed in the centreline of the couplingslots t −1 and t +1, where x′ = y′ = 0. In slot t −1, it is given by

H(a)20z′(z

′) =−H(a)20x(−z′ sinθt−1 +a/2,z′ cosθt−1 + zoff

t −d)sinθt−1

+H(a)20z(−z′ sinθt−1 +a/2,z′ cosθt−1 + zoff

t −d)cosθt−1

=− jB20(a)t e−α20d eα20 cosθt−1z′

×[− α20 sinθt−1

2π/asin(2π sinθt−1z′

a

)+ cosθt−1 cos

(2π sinθt−1z′

a

)(3.35)

and in slot t +1 it is

H(a)20z′(z

′) =−H(a)20x(−z′ sinθt+1 +a/2,z′ cosθt+1 + zoff

t −d)sinθt+1

+H(a)20z(−z′ sinθt+1 +a/2,z′ cosθt+1 + zoff

t −d)cosθt+1

=− jC20(a)t e−α20de−α20 cosθt−1z′

×[

α20 sinθt+1

2π/asin(2π sinθt+1z′

a

)+ cosθt+1 cos

(2π sinθt+1z′

a

). (3.36)

Equation (3.34) can be evaluated in closed form as

76 Design refinements for the feed of a planar slot array

B10t(3) =−

Kβ10e−α20d(St11 +Rt)

α20

p20(θt , lt)p10(θt , lt)

×[V cpl

t−1 p20(θt−1, lt−1)+V cplt+1 p20(θt+1, lt+1)

](3.37)

The amplitude of the additional TE10 scattered by the coupling slot t due to TE01 modecoupling with the neighbouring coupling slot is given by

B10t(4) =

(π/a)2

ωµ0β10ab

[ ∫St−1

H(a)01

A10(a)t

·M(b)t−1ds+

∫St+1

H(a)01

A10(a)t

·M(b)t+1ds

](3.38)

The z′ component of the TE01 mode magnetic field on the centreline of the coupling slott −1 given by

H(a)01z′(z

′) = H(a)01z(b,z

′ cosθt−1 + zoff −d)cosθt−1

=− jB01(a)t e−α01d cosθt−1eα01 cosθt−1z′ (3.39)

and on the centreline of slot t +1, it is

H(a)01z′(z

′) = H(a)01z(b,z

′ cosθt+1 + zoff −d)cosθt+1

=− jC01(a)t e−α01d cosθt+1e−α01 cosθt+1z′. (3.40)

By using the narrow slot approximation, (3.38) can be evaluated in closed form as

B10t(4) =−

Kβ10a2e−α01d(St11 +Rt)

4α01b2p01(θt , lt)p10(θt , lt)

×[V cpl

t−1 p01(θt−1, lt−1)+V cplt+1 p01(θt+1, lt+1)

].

(3.41)

The normalized impedance of coupling slot t can be expressed in terms of the incidentand scattered wave amplitudes as

zt =2B10

t

A10t −B10

t −D10t. (3.42)

3.2 Analysis 77

Alternatively, we can write (3.42) with the aid of (3.22) and (3.25) as

1zt=

A10t −D10

t

2B10t

− 12

(3.43)

=B10

t − (B10t(3)+B10

t(4))

2B10t (St

11 +Rt)− 1

2

=1− (St

11 +Rt)

2(St11 +Rt)

−B10

t(3)+B10t(4)

2B10t (St

11 +Rt).

Substitution of (3.23), (3.37) and (3.41) into (3.43) then gives

1zt=

1− (St11 +Rt)

2(St11 +Rt)

(3.44)

+jβ10e−α20d

2α20(p10(θt , lt))2

[V cpl

t−1

V cplt

p20(θt , lt)p20(θt−1, lt−1)+V cpl

t+1

V cplt

p20(θt , lt)p20(θt+1, lt+1)

+jβ10a2e−α01d

8α01b2(p10(θt , lt))2

[V cpl

t−1

V cplt

p01(θt , lt)p01(θt−1, lt−1)+V cpl

t+1

V cplt

p01(θt , lt)p01(θt+1, lt+1)

].

At the design frequency with half-wavelength slot spacing, Rt can conviviently be cal-culated using (3.13). However, by using (3.12) instead and utlizing the relations providedin [64] to compute Γt

3 and Γt4, the expression for the active impedance in (3.44) can also be

used at other frequencies.The current flowing through z1 is related to the incident wave intensity via

I1 =2a1

1zin +1

(3.45)

where zt is the total impedance of the main line. At the design frequency, it is given by

zin =T

∑t=1

zt . (3.46)

With half wavelength spacing between coupling slots, the currents flowing through the seriesimpedances have uniform magnitude and alternating phase, so that

78 Design refinements for the feed of a planar slot array

It = (−1)t−1I1. (3.47)

The coupling slot voltages can also be calculated using

V cplt =

ztIt− jK1 p10(θt , lt)

(3.48)

with the constant K1 as defined in (2.80). For a given set of inclination angles and slot lengths(θt , lt), t = 1,2, ..,T , it follows from (3.48) and (3.44) that the vectors Vcpl = [V cpl

1 V cpl2 ...V cpl

T ]

and z = [z1 z2..zT ] are interdependent. This issue can be overcome by assuming initial valuesz. In the absence of reliable data, (3.14) can be utilized for this purpose. With an arbitraryfixed value a1

1, we can employ (3.44) and (3.48) to repeatedly calculate Vcpl and z until bothsets of values converge. At frequencies other than the design frequency, relations provided in[64] can be used to determine zin and It in the iterative calculation of Vcpl and z.

The inclusion of the higher-order effects in the expression for the equivalent impedanceadjusts the phase and the amplitude of the coupling into the branch lines. The expressionsin (3.15) and (3.17) thus remain good approximations for bt

3,bt4 and κt , provided that zt is

calculated using (3.44) instead of (3.14).

3.3 Design procedure

The objective is to design a feed with T branch lines and an impedance match at the input ofthe main line, so that zin = 1. The signals emanating from port 3 of each coupling junctionshould have a chosen distribution, i.e. b1

3 : b23 : b3

3 : ... : bT3 = χ1

3 : χ23 : χ3

3 : .. : χT3 where χt

can in general be complex for applications where phase variations between branches arecalled for (eg: shaped beam arrays). The design requires data on the scattering parameters ofan isolated coupling junction in terms of slot inclination angle and slot length. A bivariatespline interpolation scheme can then be used for the calculation of the terms St

11 = S11(θt , lt),St

31 = S31(θt , lt), and S33 = S33(θt , lt) for dimensions where explicit data is not available.The design steps are as follows:

1. Select a reference slot with index t = r.

2. Select reliable starting values for (θt , lt), t = 1,2, ..,T .

3. By setting a11, it follows from (3.46) and (3.48) that It = (−1)t−1.

3.4 Validation 79

4. Solve the following set of 2T non linear equations to obtain appropriate values forinclination angles and coupling slot lengths (θt , lt): Re[zin] = 1Im[zin] = 0Re/Im[bt

3/br3] = Re/Im[χt/χr] t = 1,2,3, ...,T, t = r

3.4 Validation

As an example, a feed with T = 5 branches was designed, giving a single input port atthe main line and 10 output ports: 5 at the ends of the branches to the left of the couplingjunctions and 5 to the right. Ports are numbered as port 1 for the input, port t + 1 for theoutput on the left and port t +6 for the output on the right of branch t as shown if Figure 3.5.

2

4

5

3

6

11

10

9

8

7

1

Figure 3.5 Geometry of the prototype feed structure.

All output ports are matched, which implies that ytott,kt

= ytott,kt+1 = 1 and ytot

t = 2 for eachbranch. The feed was designed to be quasi-phase and we selected output amplitude ratios of0.281:0.753:1.000:0.753:0.281 as in [51]. It was implemented at a frequency of 9GHz instandard WR90 waveguide (a = 22.86 mm, b = 10.16 mm) with a waveguide wall thicknessof 1.2 mm and coupling slot width of w = 1.5 mm. Slot data was generated using thefrequency domain solver of CST Microwave Studio. In order to accurately model the slotfield, a vacuum block was inserted into the volume occupied by a slot and a limit of 0.021λ0

mesh step size was enforced. Adaptive mesh refinement was used to achieve a 1% accuracyin the scattering parameters and 0.5% accuracy in port mode impedance calculations. With a

80 Design refinements for the feed of a planar slot array

phase reference at the slot centre, scattering parameters for coupling junctions with differentinclination angles and lengths were obtained.

Three designs were performed:

• Conventional with zt and bt3 calculated using (3.3) and (3.5).

• Amended with zt and bt3 calculated using (3.14) and (3.15).

• Compensated with zt and bt3 calculated using (3.44) and (3.15).

The slot dimensions for the respective designs are shown in Table 3.1. The slot dimensionsfor the amended and compensated designs are similar, with only minor differences in slotlengths and angles. Compared to the conventional design, the most significant differencesare in the dimensions of slots with small inclination angles, i.e. slots 1 and 5.

The designs were analyzed using CST Microwave Studio, and the results for the scatteringparameters are compared in Figure 3.6 and Figure 3.7.

At the design frequency, the target scattering parameters are |S21| = |S61| = −17.64dB, |S31|= |S51|=−9.08 dB and |S41|=−6.61 dB. All of the designs perform well withrespect to the amplitude of the coupled signals at the output ports. The maximum amplitudeerrors for the conventional, compensated and amended designs at the design frequencyare in theroy below 0.3 dB. Conversely, the amended and compensated designs producesignificant improvements with regard to the phase consistency of the output signals. Due tothe small inclination angles for the first and last branches, and associated inaccuracy in theconventional formulation, phase errors of 20.2 and 19.3 were observed at ports 2 and 6 ofthe conventional design. The amended and compensated designs produced similar results andlimited phase spread between the branches to 6.5 at 9 GHz. Figure 3.8 shows the return lossresponse of the respective designs as a function of frequency. The inclusion of higher ordercoupling effects in the compensated design results in the improved impedance matching atthe input port. A reduction of the frequency shift of the reflection minimum is achieved. Atthe design frequency, the compensated design realizes a reflection coefficient of |S11|=−33dB, as opposed to −23 dB for the conventional and amended designs.

3.4.1 Manufactured prototype

In order to assess the validity of the designed and simulated model, a prototype was manufac-tured in Aluminium. In order to make all output ports accessible for measurement purposes,90 bends were incorporated in alternate branches. Differences in electrical length betweenrespective ports and coupling slot centres were taken into consideration in the de-embedding

3.4 Validation 81

Table 3.1 Slot dimensions of the feed network.

Slot Index Inclination angle (degrees)

(t,n) Uncompensated Amended Compensated

1 6.17 5.97 5.96

2 -16.16 -16.16 -16.07

3 21.57 21.59 21.47

4 -16.16 -16.16 -16.07

5 6.17 5.97 5.96

Slot length (mm)

Uncompensated Amended Compensated

1 16.16 16.02 15.97

2 16.03 16.04 15.99

3 16.04 16.05 16.00

4 16.03 16.04 15.99

5 16.16 16.02 15.97

82 Design refinements for the feed of a planar slot array

Figure 3.6 Amplitude distribution for the 5 branches.

Figure 3.7 Phase spread for the 5 branches.

of scattering parameters. The feed was implemented at a frequency of 9 GHz in standardWR90 waveguides with a waveguide wall thickness of 1.2 mm and coupling slot width ofw = 1.5 mm.

The model was constructed in three parts, the main line, the middle plate with couplingslots, and the branches were cut into the top metal block incorporating the 90 bends asshown in Figure 3.9. The parts were put together in three stages as shown in Figure 3.10 with

3.4 Validation 83

Figure 3.8 Reflection coefficient for the conventional and proposed designs.

multiple screws holding the structure to avoid any air gaps and to ensure electrical connectionbetween the parts. The fully assembled prototype is shown in Figure 3.12.

Figure 3.9 Manufactured sections of the model.

Table 3.2 shows the measured magnitude and phase of the scattering parameters, andFigure 3.13 shows the measured input reflection coefficient of the manufactured prototype.

84 Design refinements for the feed of a planar slot array

(a) Feed waveguide

(b) Coupling slots cut in a 1.2 mm thick sheet.

(c) Branch lines with the 90 bends.

Figure 3.10 Prototype assembly

3.4 Validation 85

Figure 3.11 Assembled prototype.

Figure 3.12 Measurement set-up with two N-type transitions and matched terminations atother ports.

The measured results closely agree with the theoretical results. The phase spread with theproposed design technique is restricted to 6.6 in comparison to the 20.5 for the conventional

86 Design refinements for the feed of a planar slot array

Table 3.2 Comparison of the simulated and measured scattering parameters of the prototype.

Scattering parameter Magnitude (dB) Phase (degrees)

Simulated Measured Simulated Measured

S21 -17.41 -17.53 -3.80 -3.37

S31 -9.35 -9.58 2.38 3.21

S41 -6.52 -6.79 2.30 -1.76

S51 -9.36 -9.80 2.77 2.75

S61 -17.43 -17.68 -2.80 -2.42

Figure 3.13 Measured reflection of the prototype.

design. The amplitude distribution agrees with the theoretical values. The frequency shift inreflection is eliminated and provides a reflection of less than −35 dB at the design frequency.

Chapter 4

Compensation of waveguide losses in thedesign of slot arrays

4.1 Introduction

For satellite applications, there is a tendency to implement these antennas using lightweight,non-metallic materials such as carbon fiber reinforced plastic (CFRP). Metal plating withinthe waveguides is sometimes used to achieve the desired levels of efficiency [66, 67]. Directimplementation in CFRP without metal plating has also been reported [71, 70].

For high-frequency telecommunication applications, the small waveguide and slot dimen-sions dictate the use of substrate-integrated waveguide (SIW) with etched slots instead ofconventional metallic waveguide with machined slots [76, 77, 78].

In these situations, waveguides exhibit inherent losses due to either the reduced con-ductivity of the waveguide walls (in non-metallic waveguides), or leakage and imperfectdielectrics (in SIW) [79]. Existing design procedures for slot arrays do not account for theselosses. This will cause slots to be under-illuminated; especially those located further awayfrom feed points, and may ultimately degrade the overall performance of the array.

In this chapter an amended design procedure which compensates for both main line andbranch line losses in order to maintain uniform phase and achieve the desired excitationamplitude for each slot is presented. The proposed design is an extension to the designspresented in the previous chapters.

Modified non-linear equations are employed to compute the slot dimensions iteratively,resulting in modest adjustments to the final dimensions of the slots. The method accounts forarbitrary levels of waveguide loss, and the only additional input required is accurate data forthe phase and attenuation constants of the lossy waveguide.

88 Compensation of waveguide losses in the design of slot arrays

4.2 Slot characterization

Array design requires data on the self-admittance of isolated radiating slots as a functionof the slot offset and length, as well as scattering parameters for isolated coupling slots interms of slot inclination angle and slot length. Resonant lengths and the slot conductance forwaveguide implemented in metal and lossy materials, although at the same frequency candiffer significanctly. In the following section, slot characterisitcs of standard and half-heightwaveguide implemnented in CFRP are discussed. Additionally, the scattering parameters ofcoupling junctions implemennted in CFRP is also analysed.

4.2.1 Carbon fibre reinforced plastic slot arrays

It is important to note that the slot characteristics can differ significantly in lossy and losslesswaveguide. To illustrate this, isolated radiating slots are characterized in perfect electricconductor (PEC) and CFRP waveguide. CFRP is anisotropic and its electrical properties canvary widely depending on the specific composition of the material. Values of the conductivityranging from 1.8×104S/m to 2.8×104S/m have been reported [98, 99].

Experiments were conducted on CFRP waveguide in the X-band frequency range, andit was concluded that it can be modelled as having walls made of a lossy conductor withan effective conductivity of σ = 2.45×104S/m [68]. For the purpose of this study, we usethe same approximation to simulate the waveguide losses. Simulations were carried out forhalf-height and standard-height waveguide with dimensions a = 22.86mm and b = 5.08 mmand a = 22.86 mm and b = 10.16 mm respectively at a frequency of 9 GHz. All slots have awidth of w = 1.5875 mm, with a waveguide wall thickness of t = 1.27mm. The calculationswere performed using the frequency domain solver of CST Microwave Studio.

In order to accurately model the slot field, a vacuum block was inserted into the volumeoccupied by a slot and a limit of 0.021λ0 mesh step size was enforced. Accurate representa-tion of the losses due to finite conductivity was achieved by using a maximum mesh stepsize of 0.036λ0 for the CFRP waveguide walls. The mesh was refined using adaptive meshrefinement to achieve a 1% accuracy in the scattering parameters and 0.5% accuracy in portmode impedance calculations. Simulations were carried out for different offsets and lengthsto calculate the self-admittance yself = gself + jbself. For a given offset, the simulation resultswere interpolated to identify the resonant slot length where bself = 0. The resonant lengthand the corresponding self-conductance as a function of slot offset for half-height waveguideare shown in Figure (4.1) and Figure (4.2), respectively.

4.2 Slot characterization 89

A bivariate interpolation scheme can be used for the calculation of the self-admittanceyself

t,n = yself(xofft,n,Lt,n), and the scattering parameters, St

11 = S11(θt , lt), St31 = S31(θt , lt) and

St33 = S33(θt , lt) for dimensions where explicit data is not available.

Figure 4.1 Resonant slot length as a function of slot offset in PEC and CFRP half-heightwaveguide.

Figure 4.2 Normalized resonant slot conductance as a function of slot offset in PEC andCFRP half-height waveguide.

As discussed in the previous chapter, the characteristics of a coupling junction, in thedesign of slot arrays plays an important role in achieving the desired slot excitations. Figure4.3 shows the self-conductance as a function of slot inclination angles implemented inhalf-height CFRP waveguide. Figure 4.4, 4.5 and 4.6 compare the slot characteristics for astandard height CFRP waveguide.

It is evident that slot characteristics of designs implemented in CFRP waveguide differssignificantly to those implemented in lossless waveguide. Assuming similar slot characteris-tics for waveguide implemented in different materials can thus lead to significant errors andresult in performance degradation.

90 Compensation of waveguide losses in the design of slot arrays

Figure 4.3 Normalized resonant slot conductance as a function of slot inclination angle inPEC and CFRP half-height waveguide.

Figure 4.4 Resonant slot length as a function of slot offset in PEC and CFRP standard-heightwaveguide.

Figure 4.5 Normalized resonant slot conductance as a function of slot offset in PEC andCFRP standard-height waveguide.

It can also be noted that the variation in slot characteristics is more pronounced instandard-height waveguide. The resonant length for larger slot offsets in CFRP standard-height waveguide are much smaller than those in PEC.

4.3 Design 91

Figure 4.6 Normalized resonant slot conductance as a function of slot inclination angle inPEC and CFRP standard-height waveguide.

4.3 Design

A lossy waveguide is characterized by the fact that at frequencies above cut-off, waves areable to propagate while experiencing some degree of attenuation. The axial wave number forthe dominant TE10 mode thus becomes a complex quantity, defined as

k10 = β10 − jα10, (4.1)

where β10 (rad/m) and α10 (Np/m) are the phase constant and attenuation constant, respec-tively. The attenuation constant can be determined through measurement, full-wave numericalmodelling or using analytical expression provided in the literature [29]. It is important tonote that the value of the phase constant may differ somewhat in lossy waveguide comparedwith lossless waveguide. The proper value of β10 therefore also needs to be defined, since itwill ultimately determine the slot spacing.

4.3.1 Design equations for radiating slots

Consider an array with a total of T branch lines, with branch t having a total of Nt radiatingslots for a total of M slots in the entire array. All waveguides are assumed to have uniformdimensions of width a and height b. The slots are spaced at intervals of λ10/2, withλ10 = 2π/β10 being the wavelength of the TE10 mode in the waveguide at the designfrequency f0. The coupling slot of branch t is located between the radiating slots (t,kt) and(t,kt +1), as shown in Figure 4.7.

The equivalent circuit of the branches consists of shunt elements yat,n is denoted as Vt,n.

For the main line, the equivalent circuit comprises normalized series impedances zt , and lossyhalf-wavelength line sections. The current flowing through zt is defined as It . Without any

92 Compensation of waveguide losses in the design of slot arrays

(xi,0,z

i)

Figure 4.7 A planar slotted waveguide array (centre) with equivalent circuits for a branch ofradiating slots (above) and the main line with inclined coupling slots (below).

loss in functionality, we use a normalized characteristic impedance z0 = 1 in all calculationsinvolving the equivalent circuits for the branch lines and main line. The relations providedin [44, 91, 92] were derived for an arbitrary characteristic impedance Z0. Our formulationtherefore compares to the special case where Z0 = 1 Ω in [44, 91, 92].

The geometry for the nth slot of branch t is depicted in Figure 4.8. It has a length Lt,n,width w and an offset xoff

t,n from the waveguide centreline. If the loss in the waveguide isrelatively low, the fields of the fundamental TE10 mode can be approximated as having a fielddistribution similar to that of a lossless waveguide, but with the addition of an axial attenuationfactor. With the phase reference at the centre of the slot (z = zoff

t,n), the field components ofthe TE10 mode propagating in the ±z direction inside a rectangular waveguide are given by

4.3 Design 93

Figure 4.8 Geometry for the nth slot in branch t.

E010y(x,z) =

ωµ0aπ

sin

(πxa

)e∓ jβ10(z−zoff

t,n)e∓α10(z−zofft,n)

H010x(x,z) =∓β10a

πsin

(πxa

)e∓ jβ10(z−zoff

t,n)e∓α10(z−zofft,n) (4.2)

H010z(x,z) = j cos

(πxa

)e∓ jβ10(z−zoff

t,n)e∓α10(z−zofft,n),

We assume the field in the slot to be

Eslott,n (x′,z′) =

V slott,n

wcos

(πz′

Lt,n

)x, (4.3)

where V slott,n is the slot voltage, and (x′,y′,z′) is the slot coordinate system. Under the assump-

tion of a narrow slot and following the procedure described in [44], the first design equationis found as

yat,n = K1 ft,n

V slott,n

Vt,n, (4.4)

where

K1 =1

j(a/λ )

√2(k/k0)

η0(β10/k)(ka)(kb)(4.5)

94 Compensation of waveguide losses in the design of slot arrays

and ft,n = f (xofft,n,Lt,n) with

f (x,L) = sinπxa

k2

L/2∫−L/2

cos

(πz′

L

)e− jβ10z′e−α1−z′dz′ (4.6)

=

(π

kL

)sin

(πxa

)×

cos(β102 )cosh(α10L

2 )+ j sin(β10L2 )+ j sin(β10

2 )sinh(α10L2 )

( π

kL)2 − (β10

k )2 +(α10k )2 + j2(β10

k )(α10k )

.

The terms k0 and k are the wave numbers in free space and in the dielectric mediumfilling the waveguide, λ = 2π/k is the wavelength in the dielectric, and η0 is the intrinsicimpedance of free space.

The second design equation provides an expression for the active admittance in termsof the admittance of the isolated slot, yself

t,n , and two other terms which account for externalhigher-order internal mutual coupling between the radiating elements. The external mutualcoupling is not affected by losses inside the waveguide. The effect of the marginal increasein the attenuation constant is also neglected. The second design equation is thus given by

1ya

t,n=

1yself

t,n+

1K2 f 2

t,n

M

∑j=1j =i

V slotj

V sloti

g ji +1

K3 f 2t,n

[V slot

t,n−1

V slott,n

ht,nh20t,n−1 +

V slott,n+1

V slott,n

ht,nht,n+1

], (4.7)

with the terms K2, L3, g ji and ht,n as defined in [64], and where the summation in thesecond term is carried out over all slots in all branches. Note that both design equations areessentially identical to those for the lossless case [91], apart from the modified expressionfor ft,n in (4.6).

4.3.2 Equivalent network analysis

In the conventional design approach for lossless waveguides [100], the voltages across shuntelements in the equivalent circuit for a specific branch all have the same magnitude andalternating phase, so that Vt,n = −Vt,n+1. In order to achieve slot voltages with uniformphase, consecutive slots are placed on either side of the waveguide centreline to accountfor the 180 phase difference of the half-wavelength inter-element spacing. Since the termft,n in (4.4) is purely real, the active admittances in a branch should have uniform phase as

4.3 Design 95

well. The convention is to adjust the slot dimensions to produce resonant slots with purelyreal admittances. The coupling slot will then also be resonant with real-valued equivalentimpedances.

In the lossy case, the complex term ft,n will contribute a phase shift in (4.4) that dependson the dimensions of individual slots. Consequently, the radiating slots cannot be designedto be resonant, since they will need to cancel this phase offset. With non-resonant activeadmittances, the total branch admittance is not purely real, and therefore we also need toaccommodate non-resonant coupling slots. The voltage across the shunt elements in theequivalent circuit for a branch will still display alternating phase as a result of the halfwavelength spacing, but due to attenuation, they will no longer have the same magnitude.The Vt,n term may be obtained by analysing the equivalent circuits. We follow the proceduredescribed in [64], but in this case, the frequency is fixed at f0 and transmission line lossesare accounted for.

In the equivalent network for branch t in Figure 4.7, the total admittance seen from thenth element looking toward the shorted end of the branch is given by

ytott,1 = ya

t,1 + tanh(α10d0) (4.8)

ytott,1 = ya

t,n +ytot

t,n−1 cosh(α10d)+ sinh(α10d)

cos(α10d)+ ytott,n−1 sinh(α10d)

n = 2,3, ...kt

and

ytott,Nt

= yat,Nt

+ tanh(α10d0) (4.9)

ytott,n = ya

t,nytot

t,n−1 cosh(α10d)+ sinh(α10d)

cos(α10d)+ ytott,n+1 sinh(α10d)

n = Nt −1,nt −2, ...kt +1.

Refer to the t th coupling junction depicted in Figure 4.9. With the centre of coupling slott as reference for all ports, the normalized load admittances observed at ports 3 and 4 aregiven by

96 Compensation of waveguide losses in the design of slot arrays

a1

t

a2

t

qt

1

3 4G3

t

y3t

Branch t

slot ( , )t kt

slot ( , +1)t kt

y4t

d0 = /4l

10

G4

t

Figure 4.9 Geometry of t th coupling junction. The coupling slot of length lt is rotated throughan angle θt with respect to the axis of the main line, with θt positive for clockwise rotation.The ports are numbered as indicated.

yt3 =

ytott,kt

sinh(α10d0)+ cos(α10d0)

ytott,kt

sinh(α10d0)+ cos(α10d0)(4.10)

yt4 =

ytott,kt+1 sinh(α10d0)+ cos(α10d0)

ytott,kt+1 sinh(α10d0)+ cos(α10d0)

.

The voltage reflection coefficients at ports 3 and 4 are defined in terms of the normalizedadmittances via

Γt3 =

1− yt3

1+ yt3, Γ

t4 =

1− yt4

1+ yt4. (4.11)

The voltages at ports 4 and 5 due to excitation at ports 1 and 2 can be calculated from

V t3 =

St31(1+Γt

3)(1−Γt4)(a

t1 −at

2)

(1−Γt3Γt

4)−S33(Γt3 +Γt

4 −2Γt3Γt

4)(4.12)

V t4 =−

St31(1−Γt

3)(1+Γt4)(a

t1 −at

2)

(1−Γt3Γt

4)−S33(Γt3 +Γt

4 −2Γt3Γt

4),

where the difference in incident wave intensities at ports 1 and 2 is

4.3 Design 97

(at1 −at

2) = It(1+ zt/2). (4.13)

The equivalent impedance of a non-resonant coupling slot is given by

z =2(St

11 +Rt)

1− (St11 +Rt)

, (4.14)

where

Rt =St2

31(Γt3 +Γt

4 −2Γt3Γt

4)

(1−Γt3Γt

4)−St33(Γ

t3 +Γt

4 −2Γt3Γt

4). (4.15)

The terms St11, St

31 and St33 are the scattering parameters of the junction, de-embedded for

phase and magnitude to the slot centre.The voltages across the shunt admittances can be calculated from

Vt,kt =− jV t

3sinh(α10d0)+ ytot

t,ktcosh(α10d0)

(4.16)

Vt,n =−Vt,n+1

cosh(α10d)+ ytott,n sinh(α10d)

n = kt −1,kt −2, ...,1 ,

and

Vt,kt+1 =− jV t

4sinh(α10d0)+ ytot

t,ktcosh(α10d0)

(4.17)

Vt,n =−Vt,n−1

cosh(α10d)+ ytott,n sinh(α10d)

n = kt +2,kt +3, ...,Nt .

For the equivalent network of the main line shown in Figure 4.7, the total normalizedimpedance seen from the coupling slot t looking toward the shorted end is given by

98 Compensation of waveguide losses in the design of slot arrays

ztotT = zT + tanh(α10d) (4.18)

ztott = zt +

ztott+1 cosh(α10d)+ sinh(α10d)

cosh(α10d)+ ztott+1 sinh(α10d)

t = T −1,T −2, ...,1

With an incident wave intensity of a11 at the first coupling slot, it holds that

I1 =2a1

1ztot

1 +1(4.19)

The current flowing through the successive series impedances can be calculated from

It =−It−1

cosh(α10d)+ ztott sinh(α10d)

(4.20)

t = 2,3, ...,T.

4.3.3 Design procedure for arrays fed via coupling slots

The procedure consists of the following steps:

1. Specify the input resistance for the main line, rin. The most common choice is todesign for an impedance match, i.e. rin = 1.

2. Select a reference slot, n = mt for each branch.

3. Specify the slot voltages, V slott,n , for a desired radiation pattern.

4. Select realistic values for the total conductance of each branch line, gt . Values typicallyrange between 1.0 and 4.0 depending on the number of slots in each branch. Anincrease in the value of gt generally results in larger radiating slot offsets.

5. Select reliable values for (xofft,n,Lt,n) and (θt , lt) A suitable approach would be to design

using the conventional procedure for lossless waveguide and to use the result for thispurpose.

6. Calculate g ji and ht,n.

7. Compute yat,n, using (4.7).

4.3 Design 99

8. Calculate ytott,n, using (4.8) and (4.9).

9. Calculate yt3 and yt

4 for each branch, using (4.10).

10. Calculate Γt3 and Γt

4 for each branch, using (4.11).

11. Calculate the equivalent impedance for each coupling slot zt , using (4.14) and (4.15).

12. Calculate ztott , using (4.18).

13. Set a11 = 1 and calculate It , using (4.19) and (4.20).

14. Compute (at1 −at

2) from (4.13).

15. Calculate V t3 and V t

4 from (4.12).

16. Compute the voltages Vt,n, using (4.16) and (4.17).

17. For each branch, solve the following set of 2Nt nonlinear equations to obtain newvalues for the radiating slot offsets and length (xoff

t,n,Lt,n):

Re

[ya

t,n ft,mtVt,n

yat,mt ft,nVt,mt

]=

V slott,n

V slott,mt

Im

[ya

t,n ft,mtVt,n

yat,mt ft,nVt,mt

]= 0 n = 1,2, ...,Nt ,n = mt

Re

[Nt

∑n=1

yat,n

]= gt

Im

[Nt

∑n=1

yat,n

]= 0.

18. Solve the following set of 2T nonlinear equations to obtain new values for the inclina-tion angles and coupling slot lengths, (θt , lt):

Re

[ya

t,mtf1,m1Vt,mt

ya1,m1

ft,mtV1,m1

]=

V slott,mt

V slot1,m1

Im

[ya

t,mtf1,m1Vt,mt

ya1,m1

ft,mtV1,m1

]= 0 t = 2,3, ...,T

Re[ztot

1

]= rin

Im[ztot

1

]= 0.

100 Compensation of waveguide losses in the design of slot arrays

19. Repeat steps 6 to 18 until the solution converges.

4.3.4 Design procedure for a single-layer array

The design approach may be adapted for a single layer array, with branches fed from one endand the other end shorted. The slots in each branch are numbered with slot 1 being the oneclosest to the feed, and slot Nt the one closest to the short. The design procedure comprisesthe following steps:

1. Perform steps 3 and 4 of the design procedure for arrays fed by inclined coupling slots.

2. Select reliable starting values for (xofft,n,Lt,n).

3. Calculate g ji and ht,n.

4. Compute yat,n, using (4.7).

5. Set kt = 0 in (4.9) to calculate ytott,n.

6. Calculate the voltage ratios for each branch as

Vt,n

Vt,1=

−(Vt,n−1/Vt,1)

cosh(α10d)+ ytott,n sinh(α10d)

n = 2,3, ...,Nt

7. For each branch, solve the following set of 2Nt nonlinear equations to obtain newvalues for slot offsets and length (xoff

t,n,Lt,n):

Re

[ya

t,n ft,1ya

t,1 ft,n

Vt,n

Vt,1

]=

V slott,n

V slott,1

Im

[ya

t,n ft,1ya

t,1 ft,n

Vt,n

Vt,1

]= 0 n = 2,3, ...,Nt

Re

[Nt

∑n=1

yat,n

]= gt

Im

[Nt

∑n=1

yat,n

]= 0.

8. Repeat steps 3 to 7 until the solution converges.

4.3 Design 101

9. Calculate the voltage ratios for the first element in the different branches from

Vt,1

V1,1=

V slott,1 ya

1,1 ft,1V slot

11 yat,1 f1,1

t = 1,2, ...,T.

10. The ratios for the required wave intensities to be delivered to the branches by anexternal power splitter can then be calculate as per [101]

at

a1=

Vt,1

V1,1

1+gt

1+g1t = 1,2, ...,T.

4.3.5 Validation

To confirm the validity of the proposed design procedure, planar arrays fed via couplingslots, and fed from the side were designed. Designs were performed using the conventionalapproach which neglects losses [92], and the amended procedure which compensates forthe losses. Waveguides made from CFRP with dimension, a = 22.86 mm and b = 5.08mm were used. The dimensions are consistent with half-height standard X-band waveguide.All slots have a width of w = 1.5875 mm, and a waveguide wall thickness of t = 1.27 mmwas selected. The values of β10 and α10 were obtained by modelling a section of emptywaveguide approximately one wavelength using CST Microwave Studio and observing thephase shift and attenuation over the length of the waveguides. At the design frquency off0 = 9 GHz, values of β10 = 130.33 rad/m and α10 = 1.12 Np/m were obtained. Note thatthe corresponding values for PEC waveguide are β10 = 129.2 rad/m and α10 = 0.

In order to verify results, the arrays were analysed using CST Microwave Studio. Theindividual slot voltages were calculated by integrating the transverse component of theelectric field at the centre and across the width of the slot, and the results obtained using bothdesign techniques are compared to the desired excitations to assess the accuracy.

Planar 8×8 element array fed via inclined coupling slot

The 8× 8 planar array has been designed to achieve a radiation pattern that resembles aChebyshev pattern with −25 dB sidelobes in both the E-plane and the H-plane. Table 4.1and Table 4.2 compare the dimensions of the conventional and the proposed design. Whilethere is a slight change in the dimension of the radiating slots and the inclination angles of

102 Compensation of waveguide losses in the design of slot arrays

the coupling slots, the lengths of the coupling slots can be seen to be drastically changed asthe slots are not necessarily resonant when using the proposed design technique.

Table 4.1 Dimensions of the radiating slots of the 8th branch of the 8×8 planar array.

Slot index Slot offset (mm) Slot length (mm)(t,n) Uncompensated Compensated Uncompensated Compensated(1,1) 1.44 1.48 16.29 16.29(1,2) -2.57 -2.62 16.61 16.63(1,3) 3.05 3.09 16.8 16.81(1,4) -4.58 -4.45 17.70 17.62(1,5) 3.56 3.48 17.00 16.92(1,6) -3.90 -3.92 17.30 17.30(1,7) 2.06 2.11 16.48 16.49(1,8) -1.59 -1.64 16.23 16.25(2,1) 1.26 1.29 16.08 16.10(2,2) -2.53 -1.56 16.10 16.12(2,3) 3.51 3.53 17.26 17.26(2,4) -2.56 -2.52 16.55 16.52(2,5) 4.68 4.50 18.06 17.91(2,6) -2.18 -2.22 16.41 16.41(2,7) 2.07 2.13 16.48 16.50(2,8) -1.19 -1.22 15.98 15.99(3,1) 1.32 1.35 16.03 16.05(3,2) -1.74 -1.77 16.22 16.23(3,3) 2.37 2.40 16.57 16.57(3,4) -3.16 -3.08 16.94 16.89(3,5) 2.70 2.65 16.78 16.74(3,6) -2.65 -2.67 16.66 16.66(3,7) 1.63 1.66 16.18 16.19(3,8) -1.32 -1.34 16.08 16.09(4,1) 1.26 1.28 16.09 16.10(4,2) -2.07 -2.11 16.32 16.34(4,3) 2.54 2.56 16.56 16.56(4,4) -3.75 -3.65 17.16 17.09

Continued on next page

4.3 Design 103

Table 4.1 – continued from previous page

Slot index Slot offset (mm) Slot length (mm)(t,n) Uncompensated Compensated Uncompensated Compensated(4,5) 2.99 2.92 16.75 16.71(4,6) -3.16 -3.17 16.85 16.85(4,7) 1.73 1.76 16.24 16.25(4,8) -1.37 -1.41 16.04 16.06(5,1) 1.37 1.41 16.04 16.06(5,2) -1.73 -1.76 16.24 16.25(5,3) 3.16 3.17 16.85 16.85(5,4) -3.00 -2.92 16.75 16.71(5,5) 3.75 3.66 16.16 17.10(5,6) -2.54 -2.56 16.56 16.56(5,7) 2.07 2.11 16.32 16.33(5,8) -1.26 -1.28 16.09 16.10(6,1) 1.32 1.35 16.08 16.10(6,2) -1.63 -1.67 16.18 16.19(6,3) 2.65 2.67 16.66 16.66(6,4) -2.70 -2.65 16.78 16.74(6,5) 3.16 3.08 16.94 16.89(6,6) 1.19 -2.40 16.57 16.58(6,7) 1.74 1.77 16.22 16.24(6,8) -1.32 -1.35 16.03 16.05(7,1) 1.19 1.22 15.98 15.99(7,2) -2.07 -2.13 16.48 16.50(7,3) 2.18 2.22 16.40 16.41(7,4) -4.68 -4.50 18.06 17.92(7,5) 2.57 2.52 16.55 16.53(7,6) -3.51 -3.53 17.26 17.26(7,7) 1.53 1.56 16.11 16.13(7,8) -1.26 -1.30 16.09 16.10(8,1) 1.59 1.64 16.23 16.41(8,2) -2.06 -2.11 16.48 16.49(8,3) 3.91 3.92 17.30 17.30

Continued on next page

104 Compensation of waveguide losses in the design of slot arrays

Table 4.1 – continued from previous page

Slot index Slot offset (mm) Slot length (mm)(t,n) Uncompensated Compensated Uncompensated Compensated(8,4) -3.56 -3.48 16.95 16.92(8,5) 4.58 4.45 17.70 17.62(8,6) -3.05 -3.09 16.80 16.81(8,7) 2.58 2.62 16.61 16.63(8,8) -1.44 -1.48 16.29 16.30

Table 4.2 Dimensions of the coupling slots of the 8×8 planar CFRP array.

Slot index Inclination angle (degrees) Slot length (mm)(t) Uncompensated Compensated Uncompensated Compensated

1 6.56 5.21 17.78 16.842 -7.71 -6.77 17.39 16.853 9.32 8.70 17.15 16.864 -13.66 -13.12 16.95 16.905 13.66 13.32 16.95 16.906 -9.32 -9.10 17.16 16.877 7.71 7.30 17.40 16.858 -6.56 -5.78 17.78 16.85

The slot voltage amplitude errors in the branch line of the array are compared to thedesired excitations in Figure 4.10. The slot voltages were normalized to the average amplitudeof the two centre slots of the fourth branch. Without compensation, the slots further awayfrom the centre are under-illuminated. These errors are more pronounced in the last branchdue to the additional main line losses. The proposed design compensates for losses in boththe main line and the branches of the array, significantly reducing the slot excitation errors inall branches.

Without compensation, the phase of the slot voltages is also inconsistent. The phaseof all slot voltages is also normalized to yield a zero average. In this case, the proposed

4.3 Design 105

design procedure achieves a considerable reduction in phase errors, resulting in a 4 standarddeviation over the entire array in comparison to 13 for the uncompensated design.

Figure 4.10 Slot voltage amplitude errors for the 8×8 element slot array.

Figure 4.11 Slot voltage phase errors for the 8×8 element slot array.

Figure 4.12 compares the H-plane radiation pattern of the conventional and the proposeddesign. In spite of the excitation errors, the H-plane pattern of the uncompensated design islargely consistent with that of the uncompensated design. In contrast, the E-plane pattern ismore sensitive to excitation errors, as shown in Figure 4.13. For the uncompensated design,

106 Compensation of waveguide losses in the design of slot arrays

the sidelobe specification in the E-plane is not met and the first sidelobes have been absorbedinto the main lobe. The amended design yields a radiation pattern that closely mirrors thetheoretical pattern. It shows a well-defined main beam and improved sidelobe performance.The proposed design procedure also improves the return loss performance of the array, asshown in Figure 4.14. At the design frequency, the reflection coefficient is below −22 dB.

Figure 4.12 H-plane pattern for the 8×8 element slot array.

4.3 Design 107

Figure 4.13 E-plane pattern for the 8×8 element slot array.

Figure 4.14 Reflection of the 8×8 element slot array.

Linear side-fed 1×8 element array

For the 8-element linear array, we chose excitation elements to produce a radiation patternthat resembles a chebyshev pattern with −28 dB sidelobes. The linear array is fed from one

108 Compensation of waveguide losses in the design of slot arrays

end, while the other end is shorted at a distance of one-quarter guide wavelength beyond thelast slot.

Table 4.3 compares the slot dimensions of the linear arrays designed using the conven-tional method and the proposed method.

In both cases, the sum of the active admittances were specified as g1 = 3.5. The amendedtechnique results in an asymmetry in slot offsets to compensate for the increasing lossexperienced by slots further from the feed point.

Without compensation, the slots further away from the the feeding end are under-illuminated due to the attenuation of the standing waves in non-metallic waveguide asshown in Figure 4.15.

Table 4.3 Dimensions of the radiating slots of the 1×8 linear array.

Slot index Slot offset (mm) Slot length (mm)(t,n) Uncompensated Compensated Uncompensated Compensated(1,1) 1.64 1.15 16.21 16.41(1,2) -2.56 -2.21 16.60 17.05(1,3) 3.77 3.75 17.17 17.92(1,4) -4.40 -4.96 17.52 18.82(1,5) 4.40 5.35 17.52 19.15(1,6) -3.77 -4.84 17.17 18.86(1,7) 2.56 3.18 16.60 17.79(1,8) -1.64 -1.90 16.22 17.13

4.4 Conclusion 109

Figure 4.15 Slot voltage amplitude errors for the 1×8 element liner slot array.

Figure 4.16 H-plane pattern for the 1×8 element slot array.

4.4 Conclusion

In this chapter, an amended design procedure for waveguide slot arrays that successfullycompensates for losses in non-metallic waveguide, dielectric filled waveguide or substrateintegrated waveguide is presented. The procedure not only corrects the amplitude of the ele-ment excitation, but also improves phase consistency for planar arrays. The results presented

110 Compensation of waveguide losses in the design of slot arrays

for the design examples show that although the amended design procedure compensatesfor losses, errors in element excitation are still evident. Commercial full-wave simulationsoftware like CST Microwave Studio is well suited to analyse a given design and capableof providing reliable results that can be used as benchmark. The accuracy of the full wavesimulation results is only limited by the meshing density and the inherent limitations ofthe underlying numerical method it employs. It is computationally intensive, making itimpractical as a design tool, especially for larger arrays. On the other hand, the procedurepresented in this paper has modest computational capacity requirements and is capable ofrapid design for arrays of arbitrary size.

While the design procedure is primarily aimed at improving the performance of arraysimplemented in lossy waveguide, it also has the potential to be utilized in high-precisiondesign to compensate for low losses due to non-ideal effects like finite conductivity or surfaceroughness in metallic waveguide. The amended design technique is an extension of theconventional procedure, and implementation is relatively straightforward.

Chapter 5

Compensation for asymmetrical fields inthe design of waveguide slot arrays

5.1 Introduction

For high-frequency applications, the small waveguide and slot dimensions dictate the useof substrate integrated waveguide (SIW) with etched slots instead of conventional metallicwaveguide with machined slots [76, 102, 77, 78, 103]. SIW exhibits inherent losses due toleakage and imperfect dielectrics, but losses can be compensated for in order to maintain thedesired excitation for each slot as shown in Chapter 4 [97].

Isolated longitudinal slots in the broad wall of a waveguide can ideally be representedby an equivalent T-network of impedances as discussed in the previous chapters. Apartfrom design techniques based on optimization of full-wave analysis results [51, 52], allexisting design methods assume a symmetric field distribution in the slot apertures [90, 82].In addition, the series elements in the T-network are discarded and a simple shunt-elementequivalent network is used to simplify the design. It has been shown that the aperture fielddistribution becomes asymmetrical for large slot offsets, especially in reduced-height ordielectric-filled waveguide [41, 42]. In these cases, the series elements in the equivalentT-network become significant. In SIW, this effect becomes even more pronounced. Byneglecting the contributions of the series elements, proper control over the total admittanceof individual branches is not possible. As a result, the feed network does not function asintended, resulting in element excitation errors and poor impedance matching at the arrayinput.

The reduced dimensions of SIW also makes the use of inclined coupling slots as feedingmechanism for slot arrays impractical. Single layer feed structures have been proposed

112 Compensation for asymmetrical fields in the design of waveguide slot arrays

as alternative feeds for planar arrays. These structures are simpler to manufacture but aregenerally restricted to providing equal power to branches or assumes an input impedancematch. A design procedure to implement single layer feed structures that can provide arbitrarypower split ratios for varying input impedance is also presented in this chapter. The designalso takes into account the losses due to reduced conductivity in the waveguide.

A coaxial cable-to-SIW transition is also proposed. This is used to connect SIW to amicrowave source using a single standard coaxial cable.

5.2 Characterisation of a single slot

The isolated longitudinal slot in SIW is shown in Figure 5.1(a) has a length L, width w andan offset of xoff from the waveguide centreline. The SIW has a height b and width aSIW

as measured between the centres of the vias forming the virtual side walls. Vias have adiameter of dvia and are spaced at intervals of p. The SIW can be modelled as a conventionalmetal-walled, dielectric-filled waveguide of width a [76, 104]. The equivalent width isdetermined by equating the phase constant in the SIW and the rectangular waveguide.

L xoff

zx

zʹ

xʹ

w

aaSIW

p dvia

(a)

zself

zself

yself

(b)

Figure 5.1 (a) Geometry of longitudinal slot in SIW and (b) its equivalent circuit.

The slot can be represented by the equivalent circuit shown in Figure 5.1(b). Thescattering parameters of this circuit are identical to those of the physical slot, referenced inboth amplitude and phase to the slot centre. The self-admittance and self-impedance aredefined in terms of the slot scattering parameters via

yself =(1−S11)

2 − (S21)2

2S21(5.1)

zself =1−S2

11 −2S21 +S221

(1−S11)2 − (S21)2

5.2 Characterisation of a single slot 113

Conventional design techniques assume the field distribution in the slot aperture to besymmetrical. Irrespective of whether the slot is excited from the left or the right, the slot fieldis approximated as

Eslot(x′,z′) =V slot

wcos(πz′/L)x (5.2)

where V slot is the slot voltage at the center of the slot and (x′,y′,z′) is the slot coordinatesystem. The equivalent circuit then reduces to a simple shunt-element. This field producessymmetrical scattering (i.e. equal forward and backward scattered fields). At the expenseof discrepancies between the scattering parameters of the physical slot and the equivalentcircuit, the equivalent circuit reduces to a simple shunt element, The assumption is validwhen the discrepancies are acceptably small.

5.2.1 Analysis of slot fields in SIW

Individual slots are modelled in SIW using commercial full-wave analysis code. The trans-verse component of the aperture slot fields were recorded for a range of slot offsets andlengths. Figure 5.2(a), Figure 5.2(c) and Figure 5.2(e) depict amplitude of the normalizeddistribution of the transverse electric field where the slot is excited by a wave incidentfrom the left in SIW with dimensions aSIW = 12.44 mm,a = 11.67 mm,b = 3.175 mm,w =

0.5 mm, wall thickness, t = 35µm,εr = 2.22, frequency= 12 GHz and xoff = 1 mm, 2 mm,and 3 mm respectively. Figure 5.2(b), Figure 5.2(d) and Figure 5.2(f) depict the phase distri-bution of the transverse component of the field aperture for 1 mm, 2 mm, and 3 mm slotsrespectively.

The field in the slot in SIW shows a notable asymmetry for all offsets. The phase variationalong the length of the slot is also significant, with up to 50 variation for some offset andlength combinations. To account for the asymmetry in the slot field, a finite element full-waveanalysis of longitudinal slots in SIW is carried out for a range of offsets and lengths withthe slot excited from the left. The transverse electric field distribution on the centre line ofthe slot, Ea = Ea(z′)x is sampled. This field is expanded in terms of a finite number of basisfunctions, i.e.

Ea(z′) =Q

∑q=1

cqeq(z′,L), (5.3)

114 Compensation for asymmetrical fields in the design of waveguide slot arrays

(a) (b)

(c) (d)

(e) (f)

Figure 5.2 (a) Slot field distribution for (a) slot with 1 mm offset illustrating the normalisedamplitude and (b) the phase variation along the length of the slot for different slot lengths, (c)slot with 2 mm offset illustrating the normalised amplitude and (d) the phase variation, and(e) slot with 3 mm offset illustrating the normalised amplitude and (f) the phase variation.

5.3 Design relations 115

where eq(z′,L) = sin [qπ(1/2+ z′/L)]. The complex expansion coefficients can be obtainedas

cq =2L

∫ L/2

−L/2Ea(z′)eq(z′,L)dz′. (5.4)

The coefficients are then normalized as c′q = cq/(c1 − c3 + c5 − c7...) to yield unityamplitude for the expansion at z′ = 0. For a slot excited by a forward or backward travellingwave that induces a voltage V slot at the centre of the slot, a more accurate representation forthe slot field is

Eslot(x′,z′) =V slot

w

Q

∑q=1

c′q(xoff,L)eq(±z′)x. (5.5)

The expression in (5.5) accounts for asymmetry in the amplitude of the slot field andphase variations along the length of the slot. A bivariate spline interpolation scheme canbe used for the calculation of the terms c′q(x

off,L) for slot offsets and lengths where explicitaperture field data are not available.

5.3 Design relations

5.3.1 Calculation of slot voltage components

SIW exhibits inherent losses due to leakage and imperfect dielectrics. At frequencies abovecut-off, waves are able to propagate while experiencing some degree of attenuation. Theaxial wave number for the dominant TE10 mode thus becomes a complex quantity, defined as

k10 = β10 − jα10, (5.6)

where β10 (rad/m) and α10 (Np/m) are the phase constant and attenuation constant respec-tively. If the loss in the waveguide is relatively low, the fields of the fundamental TE10 modecan be approximated as discussed in Chapter 4, i.e. having the same cross-sectional fielddistribution as for a lossless dielectric-filled waveguide, but with the addition of an axialattenuation factor.

Consider an array with a total of T branch lines, with the t th branch having a total ofNt radiating slots for a total of M slots in the entire array. The slots are spaced at intervals

116 Compensation for asymmetrical fields in the design of waveguide slot arrays

of d = λ10/2, with λ10 = 2π/β10 being the wavelength of the TE10 mode in the waveguideat the design frequency f0. In single-layer SIW slot arrays, branches are fed at one endvia a dedicated feed network, while the other end of the branch is shorted at a distance ofd0 = λ10/4 beyond the last slot. To implement an effective shorted quarter-wavelength linesection in SIW, the physical placement of the centres of shorting vias beyond the centreof the last slot is at a distance of λ10/4+ δ with δ = (aSIW − a)/2. Slots in branch t arenumbered as n = 1,2, ...,Nt starting from the feed end of the branch. The nth slot of branch thas a length Lt,n, width w and an offset of xoff

t,n from the waveguide centreline.The nth of branch t has a length Lt,n, width w and an offset of xoff

t,n from the waveguidecentreline. Its slot voltage can be expressed as the sum of four components [102], i.e.

V slott,n =V slot

t,n(1)+V slott,n(2)+V slot

t,n(3)+V slott,n(4). (5.7)

The slot voltage components are defined as:

1. V slott,n(1) is due to a TE10 mode wave of amplitude A10

t,n incident from z′ <−Lt,n/2.

2. V slott,n(2) is due to a TE10 mode wave of amplitude D10

t,n incident from z′ > Lt,n/2.

3. V slott,n(3) is due to external mutual coupling from other slots.

4. V slott,n(4) is due to internal TE20 and TE01 mode coupling with neighbouring slots.

The slot field can similarly be considered as a superposition of four different components.We are able to accurately predict the associated slot field distribution due to the first twocomponents, being

Eslott,n(1)(x

′,z′) =V slot

t,n(1)

w

Q

∑q=1

c′t,n,qeq(z′,Lt,n)x (5.8)

Eslott,n(2)(x

′,z′) =V slot

t,n(2)

w

Q

∑q=1

c′t,n,qeq(z′,Lt,n)x,

where c′t,n,q = c′q(xofft,n,Lt,n). Without resorting to a full-wave analysis, it is not possible to

anticipate the slot field ditribution pertaining to the mutual coupling contributions. For thesake of simplicity, the mutual coupling-induced field is assumed to be essentially symmetrical,as shown in [105]. Thus,

5.3 Design relations 117

Eslott,n(i)(x

′,z′) =V slot

t,n(i)

wcos(πz′/Lt,n)x, i = 3,4. (5.9)

From [105], it is found that

V slott,n(1) =

ξt,n

K f ′t,nA10

t,n, V slott,n(2) =

ζt,n

K f ′′t,nD10

t,n. (5.10)

and

V slott,n(3) =− 2

K2( ft,n)2

yselft,n

2+ yselft,n

N

∑j=1j =i

V slotj g ji (5.11)

V slott,n(4) =− 2

K3( ft,n)2

yselft,n

2+ yselft,n

[V slott,n−1h20

t,nh20t,n−1 +V slot

t,n+1h20t,nh20

t,n+1]

− 2K6( ft,n)2

yselft,n

2+ yselft,n

[V slott,n−1h01

t,nh01t,n−1 +V slot

t,n+1h01t,nh01

t,n+1],

where

ξt,n

ζt,n

=

yselft,n

2+ yselft,n + yself

t,n zselft,n

∓zselft,n

zselft,n +1

f ′t,nf ′′t,n

=

k sin(πxofft,n/a)

2

Q

∑q=1

c′t,n,q(qπ/Lt,n)(Rt,n,q ± jIt,n,q)

(qπ/Lt,n)2 − k210

(5.12)

Rt,n,q = cos(k10Lt,n/2)[1− cos(qπ)]

It,n,q = sin(k10Lt,n/2)[1+ cos(qπ)]

Compensation for higher-order mode internal coupling between slots has conventionallybeen limited to TE20 mode [82], but TE01 mode coupling in SIW arrays can be signif-icant and was therefore included in the expression for V slot

t,n(4) in (5.11). The constantsK,K2 and K3 and mutual coupling terms g ji,h20

t,n and h01t,n are defined in Chapter 2, while

K6 =− j8(b/a)2(α01/β01)eα01d . For lossy waveguide, ft,n is given in Chapter 4.

118 Compensation for asymmetrical fields in the design of waveguide slot arrays

5.3.2 Equivalent network analysis

The equivalent circuit of branch t in Figure 5.3(a) consists of shunt admittances yat,n and series

impedances zat,n representing the normalized active admittances and impedances of each slot,

with elements separated by lossy half-wavelength transmission line sections. In contrastto the self-terms, the active admittances and impedances incorporate mutual coupling withother slots in the array. The voltages, currents and admittances associated with the T-networkrepresenting slot n are defined in Figure 5.3(b). A normalized characteristic impedancez0 = 1 is used in all equations involving the equivalent circuit.

d0d

z0=1

t,1 t,Ntt,2

d1

ytin

at

(a)

+

Vt,n

A

-

+

Vt,n

-

+

Vt,n

B

-

It,n

A

It,n

B

It,n

yt,n

a

zt,n

a zt,n

a

yt,n

Lyt,n

tot

At,n Dt,n

(b)

Figure 5.3 (a) Equivalent circuit for the t th branch and (b) voltages, currents and admittancesassociated with the T-network representing the nth slot of branch t.

The total admittance seen from the nth T-network looking toward the shorted end ofbranch t is given by

ytott,n = [(1/yL

t,n + zat,n)

−1 + yat,n]

−1 + zat,n

−1, n = Nt ,Nt −1, ...,1, (5.13)

where

5.3 Design relations 119

yLt,n =

− j cotk10d0 n = Ntytot

t,n+1 cos(k10d)+ j sin(k10d)cos(k10d)+ jytot

t,n+1 sin(k10d) n = Nt −1,Nt −2, ...,1.(5.14)

The admittance seen at the input of branch t at a distance d1 from the centre of the first slot isgiven by

yint =

ytott,n+1 cos(k10d1)+ j sin(k10d1)

cos(k10d1)+ jytott,1 sin(k10d1)

(5.15)

The voltages and currents in the equivalent circuit are given by

V At,n =

V Bt,n−1

cos(k10d)+ jytott,n sin(k10d)

n = 2,3, ...,Nt

IAt,n =V A

t,nytott,n

Vt,n =V At,n − IA

t,nzat,n

It,n =Vt,nyat,n

IBt,n = IA

t,n − It,nV B

t,n =Vt,n − IBt,nza

t,n

n = 1,2, ...,Nt (5.16)

The term V Bt,n can alternatively be expressed as

V Bt,n =

IBt,n

yLt,n. (5.17)

The incident voltage wave amplitudes at each T-network can be calculated from

At,n =12(V A

t,n + IAt,n), Dt,n =

12(V B

t,n − IBt,n). (5.18)

The corresponding TE10 mode amplitudes are related to those in (5.18) via

A10t,n =

At,n

Kp, D10

t,n =Dt,n

Kp, (5.19)

where Kp = (a/π)√

ωµ0β10ab/2. For n = 1, substitution of (5.16)-(5.18) into (5.19) gives

120 Compensation for asymmetrical fields in the design of waveguide slot arrays

A10t,1 =

12

V At,1(1+ ytot

t,1)/Kp

D10t,1 =

12

V At,1(y

tott,1 − ya

t,1 + ytott,1ya

t,1zat,1)(1/yL

t,1 −1)/Kp (5.20)

With assumed approximations for V slott,1(3) and V slot

t,1(4), substitution of (5.10) and (5.20) into(5.7) yields

V At,1 = [V slot

t,1 − (V slott,1(3))+(V slot

t,1(4))] (5.21)

×[ξt,1(1+ ytot

t,1)

K1 f ′t,1+

ζt,1(ytott,1 − ya

t,1 + ytott,1ya

t,1zat,1)(1/yL

t,1 −1)

K1 f ′′t,1

]−1

where K1 = 2KpK, as defined in Chapter 2.Slot field component is associated with a backward and forward-scattered wave amplitude

B10t,n(i) and C10

t,n(i). The total scattered wave amplitudes are given by [105]

B10t,n = B10

t,n(1)+B10t,n(2)+B10

t,n(3)+B10t,n(4)

=−ξt,nA10t,n −ζt,nD10

t,n −K ft,n(V slott,n(3)+V slot

t,n(4)) (5.22)

C10t,n =C10

t,n(1)+C10t,n(2)+C10

t,n(3)+C10t,n(4)

=−ζt,nA10t,n −ξt,nD10

t,n −K ft,n(V slott,n(3)+V slot

t,n(4)). (5.23)

Finally, the active impedance and admittance can be expressed in terms of the incidentand scattered TE10 mode amplitudes as [105]

zat,n =

B10t,n −C10

t,n

2(A10t,n −D10

t,n)− (B10t,n −C10

t,n)(5.24)

yat,n =

−2(B10t,n +C10

t,n)

2(A10t,n +D10

t,n)+(B10t,n +C10

t,n)(1+ zat,n)

. (5.25)

5.4 Design procedure

Array design requires data on the self-admittance and self-impedance of an isolated slot as afunction of the slot offset and length. A bivariate spline interpolation scheme can be used for

5.4 Design procedure 121

the calculation of the terms yselft,n = yself(xoff

t,n,Lt,n) and zselft,n = zself(xoff

t,n,Lt,n) for dimensionswhere explicit data are not available. The design procedure consists of the following steps:

1. Specify the slot voltages V slott,n for a desired radiation pattern.

2. The slot offset for each slot is defined as xofft,n = ϒt,n|xoff

t,n|, where ϒt,n =±1 is the slotpolarity. The slot polarity can be predetermined by selecting ϒt,1 and calculating thepolarity of other slots from

ϒt,n =−ϒt,n−1sgncos [arg(V slott,n /V slot

t,n−1)] n = 2,3, ...Nt

3. Select realistic values for the input conductance of each branch line, gt . Values typicallyrange between 1.0 and 4.0 depending on the number of slots in the branch. Increasingthe value of gt generally results in larger radiating slot offsets.

4. Select reliable initial guesses for the radiating slot offsets and lengths (|xofft,n|,Lt,n).

5. Specify an even-valued integer C as the number of iterative steps to be performed.

6. Set the iteration counter c =1.

7. Calculate the mutual coupling terms, g ji, h20t,n and h01

t,n for the current values for(xoff

t,n,Lt,n).

8. For the first half of the iterations (c ≤ C/2), weigh the mutual coupling terms by afactor of 2(c−1)/C. For the second half of the iterations, no scaling is applied. Thisimplies that no mutual coupling is considered for the first iteration (i.e. ya

t,n = yselft,n ) and

the full effect of mutual coupling is only included in the second half of the iterationssimilar to designs presented in previous chapters.

9. Formulate relations for branch t = 1,2, ..,T to obtain new values for the slot offsetsand lengths (xoff

t,n,Lt,n), n = 1,2, ...,Nt :

9.1. From (5.11), compute the slot voltage due to mutual coupling, V slott,n(3) and V slot

t,n(4).

9.2. Calculate ytott,1 and yL

t,1 using the recursive relations in (5.13) and (5.14).

9.3. Calculate V At,1 using (5.21). This choice adjusts the slot voltage of the first slot to

the required value of V slott,1

9.4. Using the relations in (5.16), determine the other voltages and currents in theequivalent circuit.

122 Compensation for asymmetrical fields in the design of waveguide slot arrays

9.5. Find A10t,n, D10

t,n, B10t,n and C10

t,n using (5.18), (5.19) and (5.22).

9.6. Calculate yat,n and za

t,n from (5.23).

9.7. Repeat steps 9.1 to 9.6 until the values for yat,n and za

t,n converge. Five repetitionsare adequate.

9.8. The goal is to determine updated slot dimensions for branch t to satisfy thefollowing relations with current level of mutual coupling:

Solve the following set of 2Nt non-linear equations with the current values for(xoff

t,n,Lt,n) as initial guesses:

ξt,nA10t,n

K f ′t,n+

ζt,nD10t,n

K f ′′t,n+V slot

t,n(3)+V slott,n(4) =V slot

t,n 2 ≤ n ≤ Nt

ytott,1 = gt (5.26)

This can be done by solving these nonlinear equations for variables (xofft,n,Lt,n), n=

1,2, ...,Nt . Alternatively, we can use an optimization scheme to minimize thefollowing objective function

f =Nt

∑n=2

∣∣∣∣∣ξt,nA10t,n

K f ′t,n+

ζt,nD10t,n

K f ′′t,n+V slot

t,n(3)+V slott,n(4)−V slot

t,n

∣∣∣∣∣2

+

∣∣∣∣∣ytott,1 −gt

∣∣∣∣∣2

10. Increment c and repeat steps 7 to 9 until c =C.

11. The ratios for the required wave intensities to be delivered to the branches by anexternal power divider are at/a1 = A10

t,1/A101,1 with feed point admittances of yin

t asgiven by (5.15). The procedure described in [106] can be used to design a feedstructure capable of accomplishing this.

Note that for shaped-beam arrays, the requirement of purely real values for ytott,1 may not

always be feasible. Phase variations in the element excitations necessitate the use of detunednon-resonant slots. Certain slots will thus need to be substantially shorter or longer than theresonant length. To avoid the potential of overlapping slots, the upper limit of the allowableslot lengths is λ10/2. This restriction may in some cases make it impossible to achieve therequired element excitations with purely real total branch admittances. To resolve this, wesubstitute the objective function in step 9.8 with

5.5 Single-layer feeding structure for slot arrays 123

f =Nt

∑n=2

∣∣∣∣∣ξt,nA10t,n

K f ′t,n+

ζt,nD10t,n

K f ′′t,n+V slot

t,n(3)+V slott,n(4)−V slot

t,n

∣∣∣∣∣2

+

(∣∣∣∣∣1− ytott,1

1+ ytott,1

∣∣∣∣∣−∣∣∣∣∣1−gt

1+gt

∣∣∣∣∣)2

and opt for the optimization approach. Phase differences in the input reflection coefficient ofthe various branches can be accommodated for in the design of the feed network.

5.5 Single-layer feeding structure for slot arrays

Waveguide slot arrays find wide applications in systems that require narrow-beam or shaped-beam patterns. Conventionally, they consist of a number of slotted waveguide placedside-by-side and shorted at both ends. A separate feed wavegeuide with inclined couplingslots is used to excite the radiating branches. However, this feeding technique requirescomplex manufacturing and inhibits mass production. Implementation of slot arrays insubstrate integrated waveguide (SIW) has become popular. SIW arrays have the sameattractive features, but are simpler to manufacture and are easily implemented at higherfrequencies. The reduced dimensions of SIW makes the use of inclined coupling slots asfeeding mechanism for slot arrays practically impossible. The length of the coupling slots(≈ λ10/2) tend to be greater than the width of the radiating branch in SIW, which restrictsthe inclination angles that can be achieved and the designs that can be implemented.

Single-layer feed structures that feed branches both in-phase [107] and with alternatephase [108] have been proposed as an alternative to arrays fed via coupling slots. Althoughthese feed structures simplify the manufacturing of arrays, they are generally restricted toproviding equal power to the branches, or assume matched output ports. This limits thecontrol over slot positions in the design of slot arrays, resulting in excessively large or smallslot offsets from the branch centre-line.

A single layer feed structure with alternating phase and capable of providing arbitrarypower split ratios has been proposed in [109]. A similar design has been used to feed a slottedSIW array [108]. The structure feeds all branches from the same end and with alternatingphase. It has limited bandwidth, especially when the number of branches increases. It is notsuitable for the implementation of shaped-beam arrays with arbitrary slot phase distributions,or differential feeding mechanisms that require branches fed with alternate phase from both

124 Compensation for asymmetrical fields in the design of waveguide slot arrays

ends [110]. A corporate feed consisting of cascaded T-junction power splitters is commonlyused to feed standard waveguide or SIW arrays [111]. Corporate feed structures are physicallylarge and limited to feeding an even number of branches.

This section proposes the design of a feed network capable of feeding an arbitrary numberof branches from either both sides or alternate ends, and with arbitrary complex power splitratios and variable output port impedance terminations. The proposed feeding accommodatesthe design of shaped-beam or wideband arrays in either conventional metallic waveguide orSIW. Figure 5.4(a) and Figure 5.4(b) show the geometry of planar arrays fed via single layerfeed structures for differential arrays and arrays with branches fed from alternate sides.

SIW exhibits inherent losses due to leakage and imperfect dielectrics. At frequenciesabove cut-off, waves are able to propagate while experiencing some degree of attenuation.The axial wave number for the dominant TE10 mode thus becomes a complex quantity,defined as k10 = β10 − jα10, where β10 (rad/m) and α10 (Np/m) are the phase constantand attenuation constant respectively. Conventional metallic waveguide made of imperfectconductors and/or filled with low-loss dielectrics will have a small but non-zero value for theattenuation constant. Our proposed design procedure also compensates for these losses.

5.5.1 Design

The design objective is to realize a chosen power and phase distribution between brancheswith given port terminations. The feed network consists of two arms, each comprising acascaded series of T-junctions as shown in Figure 5.5(a). The cascaded T-junctions arespaced by a distance l. Typically, l = λ10 when alternate neighbouring branches are fed fromopposite ends or l = λ10/2 for differential feeds, with λ10 being the guide wavelength at thedesign frequency. Each arm is terminated by a shorted waveguide section of length l0. Toimplement an effective shorted line section with an electrical length of β10l0 [rad] in SIW,the physical placement of the centers of shorting vias beyond the center of the last junctionis at a distance of l0 +δ where δ = (aSIW −a)/2. A central H-plane or E-plane splitter canbe utilized to feed the arms. The position of the splitter can be adjusted by varying the feedlengths l1 and l2 in order to achieve additional adjustment of phase.

Modelling of a single T-junction

Figure 5.5(b) shows the different geometries of single waveguide T-junctions that can beutilised in this design, where a1 and a2 are excitations from opposite ends, and b3 is thethe wave intensity injected into the output branch terminated in an arbitrary impedancecharacterised by the reflection coefficient Γ3. Parameters t, ∆z and d can be adjusted to

5.5 Single-layer feeding structure for slot arrays 125

a

b

Feed structure

Coupling junctions

Radiating slots

Branch lines

Coupling junctions

(a)

a

b

Feed structure

Coupling junctions

Shorted branch ends

Radiating slots

Branch lines

(b)

Figure 5.4 Geometry of single layer planar arrays with (a) a differential and (b) an alternatingfeed structure.

126 Compensation for asymmetrical fields in the design of waveguide slot arrays

l

l0

1

2

b3

(M)

l

l

a3

( )c

b3

( )c

b3

(2)

b3

(1)

(a)

zD

t

G3

d

2

13 a1

b1

a2

b2

b3

a3

zD

t

G3

d

2

13 a1

b1

a2

b2

b3

a3

zD

t

G3

d

2

13 a

1

b1

a2

b2

b3

a3

(b)

Figure 5.5 (a) Waveguide feed structure with a cascaded series of T-junctions, and (b)geometries of a single T-junction (left: multiple pins, center: 1 pin and right: rectangularplate).

achieve the desired wave intensity at port 3. The junction is characterised by a scatteringparameter matrix S, de-embedded in both amplitude and phase to the line of symmetry (ports1 and 2) and the aperture of the junction (port 3). With port 3 terminated, the scatteringparameters of the equivalent 2-port network are

5.5 Single-layer feeding structure for slot arrays 127

Se =

S11 +

(S13)2Γ3

1−S33Γ3S12 +

S13S23Γ3

1−S33Γ3

S12 +S13S23Γ3

1−S33Γ3S22 +

(S23)2Γ3

1−S33Γ3

(5.27)

With a normalized characteristic impedance z0 = 1, the impedance matrix of the equivalent2-port network can be obtained from Ze = (I+Se)(I−Se)−1, where I is the 2×2 identitymatrix. The two port network can be represented by a T-network of impedances shown inFigure 5.6.

Figure 5.6 Two port equivalent network of the a single junction.

The circuit components are defined by

z1 =Ze11 −Ze

12

z2 =Ze22 −Ze

12 (5.28)

z3 =Ze12

128 Compensation for asymmetrical fields in the design of waveguide slot arrays

If the voltages V1 and V2 and currents I1 and I2 are known, the wave intensities at the ports ofthe junction can be computed from

a1 =12(V1 + I1)

b1 =12(V1 − I1)

a2 =12(V2 − I2)

b2 =12(V2 + I2) (5.29)

b3 =S13a1 +S23a2

1−S33Γ3

a3 = Γ3b3.

Feed arm design

The equivalent network of the entire feed structure is shown in Figure 5.7. The two arms ofthe feed structure are designed independently, and the process is completed by specifyingthe requirements for the central splitter. The scattering parameters for a T-junction areprecomputed for a range of dimensions t, ∆z and d. An interpolation technique can be usedto determine the scattering parameters S(t,∆z,d) when explicit data are not available.

z1(2)

z3(2) z2

(2)

z1( )M

z3( )M z2

( )M

z1(1)

z3(1) z2

(1)

l0 z0=1

z0=1

z0=1

l

l1

z1c z2

c

z3c

l2

Figure 5.7 Equivalent network of the entire feed structure.

5.5 Single-layer feeding structure for slot arrays 129

Considering the left arm of the feed with a total of M T-junctions as an example, wenumber the junctions with m = 1 corresponding to the T-junction closest to the centralsplitter. From the design of the radiating part of the array, the required excitation and inputadmittance for each branch is known. The objective is for the left arm thus to split the powerso that b(1)3 : b(2)3 : ... : b(M)

3 = k(1) : k(2) : ... : k(M) where k(m) represents the complex waveintensity that the branch connected to T-junction m needs to be excited with while presentinga mismatch at its input with a reflection coefficient Γ

(m)3 . At the same time, we also design

the structure for an impedance match at port 1 of the first T-junction.With given dimensions for each T-junction, the equivalent impedances z(m)

1 , z(m)2 and

z(m)3 can be calculated using (5.27) and (5.28). The total impedance seen from the port 1 of

junction m is given by

z(m)tot =

[(z(m)

L + z(m)2 )−1 +1/z(m)

3]−1

+ z(m)1 (5.30)

m = M,M−1...,1,

where

z(m)L =

j tan(k10l0) m = M

z(m+1)tot + j tan(k10l)

1+ jz(m+1)tot tan(k10l)

m = M−1,M−2, ..1.(5.31)

By setting V (1)1 = 1, the other voltages and currents of the equivalent network can be

calculated using the following recursive relations

130 Compensation for asymmetrical fields in the design of waveguide slot arrays

V (m)1 =

V (m−1)2

cos(k10l)+ j sin(k10l)/z(m)tot

m = 1

I(m)1 =

V (m)1

z(m)tot

V (m) =V (m)1 − I(m)

1 z(m)1 (5.32)

I(m) =V (m)

z(m)3

I(m)2 = I(m)

1 − I(m)

V (m)2 =V (m)− I(m)

2 z(m)2

Using the relations in (5.29), we can compute a(1)1 , b(1)1 and b(m)3 for each T-junction.

With l0 and T-junction dimensions t(m),∆z(m),d(m)∀m as variables, we can employ anoptimization scheme to minimize the following cost function

f =

∣∣∣∣∣b(1)1

a(1)1

∣∣∣∣∣2

+M

∑m=2

∣∣∣∣∣b(m)3

b(1)3

− k(m)

k(1)

∣∣∣∣∣2

(5.33)

We can use the same procedure to compute the dimensions of the right arm.

Central splitter design

1. Approximate solution: If the loss is relatively low, the minor difference in transmissionloss over waveguide lengths l1 and l2 can be neglected. That enables us to specify thescattering parameters of the central splitter as

|Sc13|=

√√√√ ∣∣k(1)L/b(1)L3

∣∣2∣∣k(1)L/b(1)L3

∣∣2 + ∣∣k(1)R/b(1)R3

∣∣2|Sc

33|=0 (5.34)

where L and R refer to quantities pertaining to the left and right arms. Since deem-bedded scattering parameters are used, the splitter can be considered as being lossless,and therefore |Sc

23|2 = 1−|Sc13|2. With a suitable fixed value for one of the parameters

5.5 Single-layer feeding structure for slot arrays 131

of the central splitter (∆zc, tc, or dc), we solve the non-linear equations in (5.34) toobtain the two remaining unknowns. In order to provide the appropriate phase to therespective arms, the proper positioning of the central splitter is given by

∆l = l1 − l2 =1

β10arg(

b(1)L3 k(1)RSc13

b(1)R3 k(1)LSc23

). (5.35)

2. Exact Solution: If a more exact solution is required, we can replace the expression for|Sc

13| in (8) with

|Sc13|=

√√√√ ∣∣k(1)L/b(1)L3

∣∣2e2α10∆l∣∣k(1)L/b(1)L3

∣∣2e2α10∆l +∣∣k(1)R/b(1)R3

∣∣2(5.36)

With an initial value of ∆l = 0, we can calculate the dimensions of central splitter bysolving the non-linear equations, compute ∆l from (5.35) and repeat the two steps untilthe values for the dimensions and ∆l converge.

5.5.2 Design example

An SIW feed structure with 8 ports was implemented in Rogers Duroid RT5880 with thicknesst = 3.175 mm, aSIW = 11.77 mm, d = 1 mm and the vias separated by a distance p = 1.5mm, operating at 12 GHz. The geometry of the design is shown in Figure 5.8 which utilisesthe geometry with multiple pins for the junctions.Ports are placed at a distance a apart. Adjacent ports are fed from alternate ends with anamplitude taper ratio of 1:1.5:2:2.5:2.5:2:1.5:1. Ports on the left are labelled 2 to 5 whileports on the right are labelled 6 to 9. All ports are fed-in-phase from one end with ports onthe right having a +100 phase difference relative to the ports fed from the left. All ports areterminated in matched impedances.

The scattering parameters of a feed network can be calculated using commercial softwarelike CST Microwave Studio. For the sake of brevity, results for only four sets of ports areshown in Figure 5.9. The amplitude for four ports fed from the left is shown in Figure 5.9(a)and the phase for ports 2,3,7 and 8 are shown in Figure 5.9(b). The simulated results showgood agreement with the required theoretical results at the design frequency with a slight

132 Compensation for asymmetrical fields in the design of waveguide slot arrays

1

2

3

4

5

6

7

8

9

Figure 5.8 Geometry of the single layer SIW feed structure with 8 branches fed from alternateends.

variation in phase as shown in Table 5.1. The reflection coefficient of the implemented designis shown in Figure 5.10. The design achieves a reflection of less than −20 dB at 12 GHz.

Table 5.1 Validation of the feed structure.

Port number Magnitude Phase (degrees)(m) Target Achieved Target Achieved(2) 1.0 1.0 0 0.0(3) 1.5 1.4 100 98.0(4) 2.0 2.0 0 0.7(5) 2.5 2.4 100 98.1(6) 2.5 2.5 0 0.9(7) 2.0 1.9 100 98.2(8) 1.5 1.5 0 1.4(9) 1.0 0.9 100 98.3

5.5 Single-layer feeding structure for slot arrays 133

(a)

98.37º

(b)

Figure 5.9 (a) The amplitude in ports 2, 3, 4 and 5, and (b) phase distribution in ports 2, 3, 8and 9 of the feed structure versus frequency.

Figure 5.10 Reflection of the feed structure.

134 Compensation for asymmetrical fields in the design of waveguide slot arrays

A compact, easy to manufacture single layer feed structure capable of feeding any numberof branches with arbitrary power split ratios and terminating impedances is proposed. Thedesign can be implemented in either standard waveguide or substrate integrated waveguide,and is suitable for both narrow-beam or shaped-beam arrays. The design is easy to implementfor different geometries and only requires a database of the chosen geometry for individualT-junctions.

5.6 Coaxial line to SIW transition

In order to feed the SIW using a coaxial cable, Deslandes and Wu [112] proposed a planarmicrostrip to SIW transition integrated on the same substrate shown in Figure 5.11. Thedesign provides a 12% bandwidth with a return loss of 20 dB and an insertion loss of betterthan 0.5 dB.

Figure 5.11 Microstrip to SIW transition.

They implemented the design on a substrate with thickness, b = 0.254 mm for a centrefrequency of 28 GHz. While the design provides excellent response for thin substrates, theproposed geometry becomes impractical for common substrate thickness used for slot arraysimplemented in SIW. Measured results for a prototype manufactured on Rogers DuroidRT5860, with a relative permittivity, εr = 2.33 and substrate thickness, b = 0.254 mm ispresented in [113]. They report an insertion loss of more than 2 dB.

5.6 Coaxial line to SIW transition 135

Similar designs presented in [114] were implemented on substrates with thickness of0.508 mm and 0.8 mm. A return loss of 20 dB and an insertion loss greater than 1.2 dB isreported. Designs implemented in thicker substrates with b = 1 mm, 2 mm and 3 mm arepresented in [115, 116]. They report a reflection coefficient of −10 dB, but the insertion lossis more than 3 dB and even 5 dB for extreme cases. With the use of very thick substrates, thegeometry completely fails resulting in an insertion loss of 15 dB [117]. This is reported to bedue to the leakage in the microstrip structure with increased substrate thickness. Designs thatreduce this leakage by using vias around the microstrip structure have also been reported[114], but an insertion loss of 1 dB is reported.

For designs implemented in thick substrates and at lower frequencies, the design proposedby Deslandes and Wu becomes impractical. Extreme microstrip widths inhibit the use ofcoaxial connectors. The design presented in [118] reports an insertion loss of less than 0.5dB, but it requires an microstrip width of 5.05 mm.

In this section, a new interconnect between coaxial cable and SIW via an SMA connectoris presented.

5.6.1 Transition design

The proposed design utilises an SMA connector to excite a field in the SIW. The centre pinof the SMA runs through the substrate from one copper sheet to the other. The bottom sheethas a circular gap of diameter Gb, resulting in a floating centre pin not physically connectedto the copper. The top copper has a hole with diameter Gt , same as the thickness of the SMAjacket. The outer conductor of the SMA is connected to the top copper layer. A tapered wallwith width, Wt and length L f is implemented using metallic vias to achieve a low reflectionat the input. It is terminated in a circular section of radius Lr. The SMA is fed at an offsetOSMA, from the centre of the rounded wall. Figure 5.12 shows the details of the proposedgeometry.

5.6.2 Design example

A design is implemented at 12 GHz in Rogers Duroid RT5880 with a thickness of 3.175 mm.The dimensions of the design example are shown in Table 5.2. The structure is shown inFigure 5.13.

Figure 5.14 shows the response of the proposed design. The design provides a return lossof greater then 30 dB at the design frequency and an insertion loss of less than 0.3 dB. Thestructure also has in a 15% bandwidth. The proposed design will be used to feed the SIWslot array in the next section.

136 Compensation for asymmetrical fields in the design of waveguide slot arrays

Wt

aSIW

OSMA

Gt

Gb

b

Lf

Lr

Figure 5.12 Geometry of the proposed feed structure.

Table 5.2 Dimensions of the implemented SMA to SIW transition.

Design parameter Length (mm) Design parameter Length (mm)

aSIW 12.44 Wt 26.17

b 3.175 Wt 26.17

Gb 0.735 L f 35.21

Gt 4.09 Lr 13.09

OSMA 8.9

5.7 Validation of proposed design method

To confirm the validity of the proposed design procedure, slot arrays were designed in SIWand PEC to achieve a desired radiation pattern and not restricting slot excitations to uniformphase over the entire array. In order to achieve this, the lengths of the slots in the arrays varysignificantly, that results in varying slot fields in these slots. To illustrate the significance ofaccommodating asymmetrical slot fields in the design of arrays with shaped beams, designs

5.7 Validation of proposed design method 137

Figure 5.13 Implementation of a back-to-back coaxial line to SIW transition.

Figure 5.14 Scattering parameters of a back-to-back coaxial line to SIW transition.

138 Compensation for asymmetrical fields in the design of waveguide slot arrays

were carried out using the technique proposed in Chapter 4 in SIW and PEC. Results arecompared to those of a design implemented using the procedure described in this chapter. Inthe latter case, a total of Q = 10 basis functions were employed to model the asymmetricfield in each slot.

5.8 An 8×8 element SIW array with circular beam radia-tion pattern

In this section, an 8× 8 element SIW array is implemented and analysed. The designwas designed to achieve a circular-shaped beam pattern with slot excitations as definedin [53]. The array is implemented using substrate integrated waveguide with dimensionsaSIW = 12.44 mm and b = 3.175 mm, with slot width, w = 0.5 mm and wall thickness of35 µm. The substrate used is Rogers RT5880 with a dielectric constant, εr = 2.22 and athickness of 3.175 mm. The values of β10 and α10 were obtained by modelling a section ofempty waveguide approximately one wavelength using CST Microwave Studio and observingthe phase shift and attenuation over the length of the waveguide, similar to the proceduredescribed in Chapter 4. At the design frequency of f0 = 12 GHz, values of β10 = 258.37rad/m and α10 = 0.316 Np/m were obtained.

The design is fed using a single layer feed structure [106], with branches fed fromalternate sides. Figure 5.15 shows the model of the 8×8 SIW array.

The dimensions for the arrays designed using the conventional technique [90] and theproposed design technique are compared in Table 5.3.

Table 5.3 Dimensions of the radiating slots of the 8×8 SIW array with a circular main beam.

Slot index Slot offset (mm) Slot length (mm)(t,n) Uncompensated Compensated Uncompensated Compensated

(1 1) 5.17 4.59 8.84 8.40(1 2) -3.20 -3.56 9.18 8.31(1 3) 1.07 2.14 9.65 9.42(1 4) -2.53 -2.00 10.05 9.21(1 5) 1.41 3.49 9.91 10.08(1 6) -5.46 -3.34 8.60 9.33(1 7) 4.16 3.29 9.65 8.49

Continued on next page

5.8 An 8×8 element SIW array with circular beam radiation pattern 139

Table 5.3 – continued from previous page

Slot index Slot offset (mm) Slot length (mm)(t,n) Uncompensated Compensated Uncompensated Compensated(1 8) -4.44 -3.17 8.97 7.92(2 1) -1.08 -1.01 9.51 9.70(2 2) 1.98 3.52 10.06 10.84(2 3) -2.20 -4.14 10.02 9.76(2 4) 2.33 2.07 8.58 7.37(2 5) -2.69 -2.66 10.24 8.55(2 6) 3.64 3.90 10.89 9.63(2 7) -2.95 -2.41 11.02 9.58(2 8) 2.45 4.49 9.80 7.90(3 1) 4.08 4.09 8.28 7.58(3 2) -2.72 -1.45 8.50 8.21(3 3) -1.58 -0.48 8.48 9.94(3 4) 1.89 2.08 10.56 10.48(3 5) -1.49 -1.93 10.19 10.35(3 6) 4.39 3.43 7.99 8.12(3 7) 2.85 3.29 8.16 7.61(3 8) -1.83 -2.40 8.75 8.16(4 1) -4.11 -2.62 6.03 6.04(4 2) 0.41 1.08 7.66 6.04(4 3) 1.02 1.42 9.68 9.89(4 4) -2.29 -4.85 10.64 11.11(4 5) 2.22 3.46 10.59 10.80(4 6) -0.69 -1.15 9.53 9.16(4 7) -0.71 -0.47 7.06 6.15(4 8) 0.55 0.69 6.57 7.31(5 1) 4.11 2.62 6.03 6.04(5 2) -0.41 -1.08 7.66 6.04(5 3) -1.02 -1.42 9.68 9.89(5 4) 2.29 4.85 10.64 11.11(5 5) -2.22 -3.46 10.59 10.80(5 6) 0.69 1.15 9.53 9.16

Continued on next page

140 Compensation for asymmetrical fields in the design of waveguide slot arrays

Table 5.3 – continued from previous page

Slot index Slot offset (mm) Slot length (mm)(t,n) Uncompensated Compensated Uncompensated Compensated(5 7) 0.71 0.47 7.06 6.15(5 8) -0.55 -0.69 6.57 7.31(6 1) -4.08 -4.09 8.28 7.58(6 2) 2.72 1.45 8.50 8.21(6 3) 1.58 0.48 8.48 9.94(6 4) -1.89 -2.08 10.56 10.48(6 5) 1.49 1.93 10.19 10.35(6 6) -4.39 -3.43 7.99 8.12(6 7) -2.85 -3.29 8.16 7.61(6 8) 1.83 2.40 8.75 8.16(7 1) 1.08 1.01 9.51 9.70(7 2) -1.98 -3.52 10.06 10.84(7 3) 2.20 4.14 10.02 9.76(7 4) -2.33 -2.07 8.58 7.37(7 5) 2.69 2.66 10.24 8.55(7 6) -3.64 -3.90 10.89 9.63(7 7) 2.95 2.41 11.02 9.58(7 8) -2.45 -4.49 9.80 7.90(8 1) -5.17 -4.59 8.84 8.40(8 2) 3.20 3.56 9.18 8.31(8 3) -1.07 -2.14 9.65 9.42(8 4) 2.53 2.00 10.05 9.21(8 5) -1.41 -3.49 9.91 10.08(8 6) 5.46 3.34 8.60 9.33(8 7) -4.16 -3.29 9.65 8.49(8 8) 4.44 3.17 8.97 7.92

The 3-D patterns for the two designs are compared in Figure 5.16(a) and Figure 5.16(b).Figures 5.17(a) and 5.18(a) further compares the H-plane and E-plane cuts of the tworadiation patterns. The design when assuming perfectly symmetrical slot fields fails toachieve a radiation pattern with a flat main beam as desired and suffers from slight skewness

5.8 An 8×8 element SIW array with circular beam radiation pattern 141

Figure 5.15 8×8 element SIW slot array fed using a single layer feed, with branches fedfrom alternate sides.

of the main lobe as well. The conventional design results in a 1.7 dB and 4.9 dB ripple inthe H-plane and the E-plane respectively compared to a 0.15 dB and 0.6 dB ripple for theproposed designs.

During the design process, the input admittance of each branch is specified. The feednetwork design relies on branch admittances satisfying these goals. The conventional designmethod neglects the contributions of the series elements in the equivalent circuit, resulting ininaccuracies in the input admittance of individual branches. The feed network consequentlyfails to produce the intended signal at each branch input, giving rise to element excitationerrors in especially the E-plane. The improved performance of the new method is evident.The proposed procedure also improves the impedance matching at the input and eliminatesthe frequency shift with a minimum at 12.008 GHz, as shown in Figure 5.19.

142 Compensation for asymmetrical fields in the design of waveguide slot arrays

(a)

(b)

Figure 5.16 (a) 3-D radiation pattern using conventional design, (b) proposed design.

5.9 An 8×8 element array in quarter-height PEC waveguide with circular beam radiationpattern 143

(a)

(b)

Figure 5.17 (a) H-plane radiation pattern of the 8×8 element array with with a circular mainbeam, and (b) zoomed-in comparison of the main lobe.

5.9 An 8× 8 element array in quarter-height PEC wave-guide with circular beam radiation pattern

Stern and Elliot [41] have shown that the conventional design technique has limitations whenimplementing slot arrays in quarter-height waveguide. They found that the simple-shuntelement equivalent network is inappropriate for slots in quarter-height metallic waveguide.

144 Compensation for asymmetrical fields in the design of waveguide slot arrays

(a)

(b)

Figure 5.18 (a) E-plane radiation pattern of the 8×8 element array with with a circular mainbeam, and (b) zoomed-in comparison of the main lobe.

They suggested that the conventional design procedure should under these circumstances bereplaced by one that directly deals with scattered waves without using equivalent circuits.The equivalent T-network proposed in this chapter accounts for these scattered fields. In thissection, an 8×8 element quarter-height array in PEC is implemented and analysed to furtherassess the validity of the proposed design technique. The design was implemented to achieve

5.9 An 8×8 element array in quarter-height PEC waveguide with circular beam radiationpattern 145

Figure 5.19 Comparison of the input reflection coefficients between the conventional and theproposed design techniques.

a circular-shaped beam pattern with slot excitations as defined in [53]. The dimensions forthe array designed using the proposed technique are given in Table 5.4.

Table 5.4 Dimensions of the radiating slots of the 8×8 quarter-height array with circularbeam radiation pattern.

Slot Index Slot offset (mm) Slot length (mm)(t,n) Uncompensated Compensated Uncompensated Compensated

(1 1) -3.15 -3.17 19.16 19.00(1 2) -3.69 -5.29 22.41 16.11(1 3) 1.54 4.96 16.58 14.13(1 4) -1.82 -2.40 16.70 16.95(1 5) 2.52 2.54 18.11 17.63(1 6) -2.89 -1.73 17.22 16.44(1 7) 4.09 5.12 15.53 15.09(1 8) 3.37 2.85 15.21 18.33(2 1) -1.88 -2.07 17.18 17.26(2 2) 5.14 1.68 22.54 16.86

Continued on next page

146 Compensation for asymmetrical fields in the design of waveguide slot arrays

Table 5.4 – continued from previous page

Slot index Slot offset (mm) Slot length (mm)(t,n) Uncompensated Compensated Uncompensated Compensated(2 3) -2.15 -2.55 16.82 17.09(2 4) 1.52 3.59 23.53 16.02(2 5) -3.46 -2.90 14.21 16.48(2 6) 3.10 4.10 17.19 18.09(2 7) -1.24 -1.81 16.80 16.97(2 8) 3.34 1.58 13.34 18.32(3 1) 1.61 1.70 16.20 16.28(3 2) -4.45 -5.98 16.82 17.45(3 3) -1.02 -5.82 16.16 13.47(3 4) 2.95 2.72 18.92 18.22(3 5) -2.41 -2.76 17.89 17.89(3 6) 3.73 1.03 22.33 16.77(3 7) 5.27 2.79 17.28 15.72(3 8) -1.19 -1.24 16.28 16.09(4 1) -1.00 -1.08 15.96 15.88(4 2) 1.24 4.92 23.86 14.89(4 3) 1.33 1.35 16.77 16.61(4 4) -3.57 -3.02 18.75 17.81(4 5) 2.14 2.70 17.16 17.60(4 6) -1.48 -1.50 16.70 16.49(4 7) -1.03 -3.83 14.82 13.91(4 8) 2.29 1.00 13.65 18.66(5 1) 1.00 1.08 15.96 15.88(5 2) -1.24 -4.92 23.86 14.89(5 3) -1.33 -1.35 16.77 16.61(5 4) 3.57 3.02 18.75 17.81(5 5) -2.14 -2.70 17.16 17.60(5 6) 1.48 1.50 16.70 16.49(5 7) 1.03 3.83 14.82 13.91(5 8) -2.29 -1.00 13.65 18.66(6 1) -1.61 -1.70 16.20 16.28

Continued on next page

5.9 An 8×8 element array in quarter-height PEC waveguide with circular beam radiationpattern 147

Table 5.4 – continued from previous page

Slot index Slot offset (mm) Slot length (mm)(t,n) Uncompensated Compensated Uncompensated Compensated(6 2) 4.45 5.98 16.82 17.45(6 3) 1.02 5.82 16.16 13.47(6 4) -2.95 -2.72 18.92 18.22(6 5) 2.41 2.76 17.89 17.89(6 6) -3.73 -1.03 22.33 16.77(6 7) -5.27 -2.79 17.28 15.72(6 8) 1.19 1.24 16.28 16.09(7 1) 1.88 2.07 17.18 17.26(7 2) -5.14 -1.68 22.54 16.86(7 3) 2.15 2.55 16.82 17.09(7 4) -1.52 -3.59 23.53 16.02(7 5) 3.46 2.90 14.21 16.48(7 6) -3.10 -4.10 17.19 18.09(7 7) 1.24 1.81 16.80 16.97(7 8) -3.34 -1.58 13.34 18.32(8 1) 3.15 3.17 19.16 19.00(8 2) 3.69 5.29 22.41 16.11(8 3) -1.54 -4.96 16.58 14.13(8 4) 1.82 2.40 16.70 16.95(8 5) -2.52 -2.54 18.11 17.63(8 6) 2.89 1.73 17.22 16.44(8 7) -4.09 -5.12 15.53 15.09(8 8) -3.37 -2.85 15.21 18.33

The simulated H-plane and E-plane radiation patterns of the designs using the con-ventional and the proposed design techniques are compared in Figure 5.20(a) and 5.21(a)respectively. It is evident that the conventional design fails to achieve a radiation pattern witha flat main beam in the E-plane with a 2 dB ripple, while the design using the proposed tech-nique results in maximum 0.5 dB ripple in both the E-plane and the H-plane. The achievedreflection coefficient for the two designs is also compared in Figure 5.22. The proposeddesign technique results in a reflection coefficient of -25 dB at the the design frequency

148 Compensation for asymmetrical fields in the design of waveguide slot arrays

compared to the reflection coefficient of -15 dB and an extremely narrow bandwidth whenusing the conventional design technique.

(a)

(b)

Figure 5.20 (a) H-plane radiation pattern of the 8×8 element array with with a circular mainbeam, and (b) zoomed-in comparison of the main lobe.

5.9 An 8×8 element array in quarter-height PEC waveguide with circular beam radiationpattern 149

(a)

(b)

Figure 5.21 (a) E-plane radiation pattern of the 8×8 element array with with a circular mainbeam, and (b) zoomed-in comparison of the main lobe.

150 Compensation for asymmetrical fields in the design of waveguide slot arrays

Figure 5.22 Comparison of the input reflection coefficients between the conventional and theproposed design techniques of the 8×8 element quarter-height array.

Chapter 6

Conclusion

Waveguide slot arrays are extensively used for radar and telecommunication applicationsand there is a growing demand for accurate implementation of these arrays in metallic andnon-metallic materials. This thesis presented refinements to the design of waveguide slotarrays and demonstrated that accurate designs in metallic and non-metallic waveguides canbe implemented. These slot arrays are currently designed using conventional techniquesdeveloped in the 1980s by Elliott [43, 90].

This research presented five refinements to slot array design to address known limitations:

1. Higher-order mode compensation between the coupling slots and the two radiat-ing slots straddling them: An amended design procedure is proposed which compen-sates for higher-order mode coupling between inclined coupling slots and neighbouringradiating slots. Compensation is achieved at minimal cost to the design complexity, asit primarily involves the addition of closed-form terms in the expression for the activeadmittance of radiating slots. It introduces minor adjustments to the dimensions ofslot radiators that effectively improves amplitude accuracy and phase consistency ofelement excitations. Planar slot arrays in standard-height waveguides are successfullydesigned with a desired radiation pattern.

2. Refinement in the design of main line waveguide in the feed of planar arrays: Anaccurate model for inclined coupling slots as employed in the feed of planar slotarrays was presented. The model accommodates phase differences between scatteringparameters of coupling junctions with small slot inclination angles. It introduces minoradjustments to the dimensions of slots that effectively improves phase consistencyof coupled signals into the branches of an array. The phase errors are limited to amaximum of about 5 degrees.

152 Conclusion

To compensate for higher-order mode coupling, we include two additional scatteringcomponents. The inclusion of higher order coupling effects enhances impedancematching and limits the frequency shift in the reflection minimum, thus reducing theneed for tuning.

3. Compensation of waveguide losses: An amended design procedure for waveguideslot arrays that successfully compensates for losses in non-metallic waveguide, dielectric-filled waveguide or substrate integrated waveguide is proposed. The procedure not onlycorrects the amplitude of the element excitation, but also improves phase consistencyfor planar arrays.

While the design procedure is primarily aimed at improving the performance of arraysimplemented in lossy waveguide, it also has the potential to be utilized in high-precisiondesigns to compensate for low losses due to non-ideal effects like finite conductivity orsurface roughness in metallic waveguide. The amended design technique is an exten-sion of the conventional procedure, and implementation is relatively straightforward.In the limit of zero losses, the proposed method reduces to the conventional designapproach

4. Compensation for asymmetrical slot fields: Accurate representation of slots using afull T-network equivalent circuit in order to account for asymmetrical slot fields andthe phase variation along the length of each slot in order to design SIW slot arraysespecially with shaped beam radiation patterns is demonstrated. This not only helpsachieve the desired radiation patterns but also improves the impedance match at theinput of the branch lines.

5. Implementation of a single layer feed structure in lossy waveguide: A compact,easy to manufacture single layer feed structure capable of feeding any number ofbranches with arbitrary power split ratios and terminating impedances was proposed.The design can be implemented in either standard waveguide or substrate integratedwaveguide, and is suitable for both narrow-beam or shaped-beam arrays. The design iseasy to implement for different geometries and only requires a database of the chosengeometry for individual T-junctions.

6.1 Limitations in the proposed design procedures

The design procedures presented in this thesis enable accurate implementation of slot arrays,but still have some limitations.

6.2 Future work 153

The formulation presented in Chapter 2 and Chapter 3 does not account for all the higherorder effects that a detailed full wave analysis is capable of, and discrepancies in the elementexcitation can be expected. Similar variations have been observed for the design of arraysusing the conventional procedure. The compensation technique proposed in Chapter 2 alsodoes not distinguish between “hard” and “soft” coupling orientations, as identified in [85].

Additionally, in cases where overlapping of radiating slots or the projection of couplingslots and neighbouring radiating slots cannot be avoided by careful selection of waveguidedimensions or total branch admittances, accuracy of the proposed methods in this thesis maybe limited. This is because the eigen function expansion of fields in terms of waveguidemodes fails when there is a singularity in the Green’s function for the coupling term.

In Chapter 5, the compensation of asymmetric slot fields only accounts for the funda-mental TE10 mode and the shape of mutual coupling-induced slot fields is assumed to besymmetric. The slots are also modelled as narrow slots with a uniform field across the widthof each slot. The slot field is also assumed to be purely transverse. While the axial componentis minimal, it is not accounted for.

The results presented in this thesis validate the assumptions made. It enables accurateslot array designs using different materials and for arbitrary design specifications.

6.2 Future work

This thesis addresses several limitations in the design of slot arrays. However, there are stillsome areas that future work could address. These include the following:

1. Integrating compensation of asymmetrical slot-fields presented in Chapter 5 for singlelayer side-fed planar arrays, with techniques presented in Chapter 2 and Chapter 3that utilise a feed with coupling slots to design shaped-beam antennas and improvesidelobe level accuracy.

2. Various techniques can be employed to accurately predict ideal element excitationsin an array to achieve specific radiation patterns. In the design of planar slot arrays,slot dimensions dictate the achieved excitation and the overall radiation pattern. Asym-metrical slot fields and inconsistent slot spacing often restricts radiation patterns thatcan be achieved. Dynamic calculation of excitation coefficients by integration ofthe expressions for the radiation pattern [64] into the design techniques proposed inthis thesis will allow accurate implementation of slot arrays with complex radiationpatterns.

154 Conclusion

3. Implementation of travelling-wave arrays is another prospective research topic forthe future. The current design procedures for travelling wave slot arrays suffer frompattern degradation and reduced performance over a wide frequency range. Relationspresented in this thesis can be extended to implement travelling-wave slot arrays,taking into account the losses and asymmetry in slot fields. Arrays with varying slotspacing in individual branches can be implemented to provide greater control overradiation patterns over a wide frequency range with low reflection at the input.

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