Contribution to the design of waveguide-fed compound-slot arrays by means of equivalent circuit modelling

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(1)UNIVERSIDAD POLITÉCNICA DE MADRID ESCUELA TÉCNICA SUPERIOR DE INGENIEROS DE TELECOMUNICACIÓN. CONTRIBUTION TO THE DESIGN OF WAVEGUIDE-FED COMPOUND-SLOT ARRAYS BY MEANS OF EQUIVALENT CIRCUIT MODELLING. TESIS DOCTORAL Autor: IGNACIO MONTESINOS ORTEGO Ingeniero de Telecomunicación. Director: MANUEL SIERRA PÉREZ Doctor Ingeniero de Telecomunicación Catedrático de Universidad 2012.

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(3) TESIS DOCTORAL: CONTRIBUTION TO THE DESIGN OF WAVEGUIDE-FED. COMPOUND-ARRAYS BY MEANS OF EQUIVALENT CIRCUIT MODELLING. AUTOR: Ignacio Montesinos Ortego Ingeniero de Telecomunicación. DIRECTOR: Manuel Sierra Pérez Doctor Ingeniero de Telecomunicación Catedrático de Universidad (SSR-UPM). Departamento de SEÑALES, SISTEMAS Y RADIOCOMUNICACIONES El Tribunal de Calificación, compuesto por: PRESIDENTE: VOCALES:. VOCAL SECRETARIO: VOCALES SUPLENTES:. Acuerda otorgar la CALIFICACIÓN de:. Madrid,. de. de 2012..

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(5) Contents I. Analysis. 2. 1 Introduction. 4. 1.1. State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4. 1.1.1. Slot antennas . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4. 1.1.2. Array Design Methods . . . . . . . . . . . . . . . . . . . . . .. 7. 1.1.3. Mutual coupling effects . . . . . . . . . . . . . . . . . . . . .. 9. 1.2. Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9. 1.3. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11. 1.4. Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13. 2 Equivalent Circuit Modelling. 20. 2.1. Slots in rectangular waveguide walls . . . . . . . . . . . . . . . . . .. 20. 2.2. Structural characteristics of the Slot . . . . . . . . . . . . . . . . . .. 21. 2.3. Scattering theory of waveguide-fed radiators . . . . . . . . . . . . . .. 22. 2.3.1. Scattering from compound slots . . . . . . . . . . . . . . . . .. 26. Circuit Theory in Microwaves Systems . . . . . . . . . . . . . . . . .. 29. 2.4. I.

(6) CONTENTS. 2.4.1. Single port circuits . . . . . . . . . . . . . . . . . . . . . . . .. 32. 2.4.2. Multiple ports circuits . . . . . . . . . . . . . . . . . . . . . .. 34. 2.4.2.1. Properties of the impedance matrix Z . . . . . . . .. 35. Proposed Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 36. 2.5.1. T-Network. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 38. 2.5.2. Π-Network . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 40. Networks’ performance . . . . . . . . . . . . . . . . . . . . . . . . . .. 42. 2.6.1. Complex Power . . . . . . . . . . . . . . . . . . . . . . . . . .. 42. 2.6.2. Power Handling . . . . . . . . . . . . . . . . . . . . . . . . . .. 43. 2.6.3. Radiation Phase . . . . . . . . . . . . . . . . . . . . . . . . .. 44. 2.6.4. Resonance Condition . . . . . . . . . . . . . . . . . . . . . . .. 45. 2.7. Radiation pattern of a linear array of compound slots . . . . . . . . .. 46. 2.8. Previous works discussion . . . . . . . . . . . . . . . . . . . . . . . .. 51. 2.8.1. Main differences with existing bibliography . . . . . . . . . .. 51. 2.8.2. Design Alternative . . . . . . . . . . . . . . . . . . . . . . . .. 52. Chapter conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .. 53. 2.5. 2.6. 2.9. II. CONTENTS. Synthesis. 60. 3 Forward Matching Procedure. 62. 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 62. 3.2. Forward Matching Procedure . . . . . . . . . . . . . . . . . . . . . .. 63. II.

(7) CONTENTS. 3.2.1. 3.2.2. CONTENTS. Setting up Stage . . . . . . . . . . . . . . . . . . . . . . . . .. 64. 3.2.1.1. Definition of the Database. . . . . . . . . . . . . . .. 64. 3.2.1.2. Constraints Definition . . . . . . . . . . . . . . . . .. 65. 1) Power distribution constraint definition . . . . . . .. 65. 2) Radiation phase constraint definition . . . . . . . .. 66. 3) Physical constraint definition . . . . . . . . . . . . .. 66. 4) Short length constraint definition . . . . . . . . . .. 67. Cascading Stage: Steps Definition. . . . . . . . . . . . . . . .. 68. 3.2.2.1. Step I: first slot design. . . . . . . . . . . . . . . . .. 68. 3.2.2.2. Step II: slot connection . . . . . . . . . . . . . . . .. 70. Radiation phase difference . . . . . . . . . . . . . . . .. 70. Step III: linear array termination . . . . . . . . . . .. 72. 3.3. Four-element array design example . . . . . . . . . . . . . . . . . . .. 72. 3.4. Limitations of the method . . . . . . . . . . . . . . . . . . . . . . . .. 75. 3.5. Chapter Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .. 76. 3.2.2.3. 4 Mutual Coupling. 80. 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 80. 4.2. Effects of the mutual coupling . . . . . . . . . . . . . . . . . . . . . .. 82. 4.2.1. 83. Six-element illustrative array . . . . . . . . . . . . . . . . . .. 4.3. Complete Design Process. . . . . . . . . . . . . . . . . . . . . . . . .. 85. 4.4. New Array Circuit Model . . . . . . . . . . . . . . . . . . . . . . . .. 86. III.

(8) CONTENTS. CONTENTS. 4.4.1. New Slot Model . . . . . . . . . . . . . . . . . . . . . . . . . .. 86. 4.4.2. Global Circuit and Coupling Model . . . . . . . . . . . . . . .. 87. Coupling factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 88. 4.5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 88. 4.5.2. Coupling factor calculation . . . . . . . . . . . . . . . . . . .. 89. 4.5.2.1. Coupling factor verification . . . . . . . . . . . . . .. 91. Feeding and Coupling Matrices Connection . . . . . . . . . . . . . .. 94. 4.6.1. Process basics . . . . . . . . . . . . . . . . . . . . . . . . . . .. 94. 4.6.2. Ports numbering process . . . . . . . . . . . . . . . . . . . . .. 96. 4.7. 4-element array example . . . . . . . . . . . . . . . . . . . . . . . . .. 97. 4.8. Chapter conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .. 99. 4.5. 4.6. 5 Experimental and theoretical validation of the design technique. 104. 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104. 5.2. Applicability of the method . . . . . . . . . . . . . . . . . . . . . . . 105. 5.3. 5.2.1. Longitudinal slots array . . . . . . . . . . . . . . . . . . . . . 105. 5.2.2. Transverse slots array . . . . . . . . . . . . . . . . . . . . . . 107. 5.2.3. Compound slot arrays of a large number of elements . . . . . 108. 5.2.4. Higher frequency designs . . . . . . . . . . . . . . . . . . . . . 110. 5.2.5. Arbitrary Tilting Angle . . . . . . . . . . . . . . . . . . . . . 112. 5.2.6. Arrays in reduced height waveguides . . . . . . . . . . . . . . 114. Built Prototypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116. IV.

(9) CONTENTS. CONTENTS. 5.3.1. 6-element prototype . . . . . . . . . . . . . . . . . . . . . . . 116 5.3.1.1. 5.3.2. AWR Microwave Office simulation . . . . . . . . . . 117. 6-Element array’s measurements . . . . . . . . . . . . . . . . 118 5.3.2.1. Hints about MoM-FMP results, CST simulations and measurements . . . . . . . . . . . . . . . . . . . . . 120. 5.3.3. 10-element prototype . . . . . . . . . . . . . . . . . . . . . . . 120 5.3.3.1. 5.3.4. AWR Microwave Office simulation . . . . . . . . . . 122. 10-Element array’s measurements . . . . . . . . . . . . . . . . 124. 5.4. MoM - FMP Computation Time . . . . . . . . . . . . . . . . . . . . 126. 5.5. Chapter conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 128. 6 Conclusions and future tasks. 130. 6.1. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130. 6.2. Future Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132. 7 Annexes. 136. 7.1. Annex A: Method of Moments - MoM Analysis . . . . . . . . . . . . 136. 7.2. Annex B: Slot’s radiated field . . . . . . . . . . . . . . . . . . . . . . 141. 7.3. Annex C: Relative Distance Calculation . . . . . . . . . . . . . . . . 145. 7.4. Annex D: Graphic User Interface . . . . . . . . . . . . . . . . . . . . 149. V.

(10) List of Figures 1.1. slot configurations and their equivalent circuit . . . . . . . . . . . . .. 5. 1.2. Two dimensional arrays and their main planes, E and H. . . . . . . .. 12. 2.1. Basic parameters of a compound slot.. . . . . . . . . . . . . . . . . .. 21. 2.2. Boundaries definition . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22. 2.3. Local coordinate axes definition . . . . . . . . . . . . . . . . . . . . .. 27. 2.4. absolute value of the scattered field ratio, as a function of tilting angle. 30. 2.5. Single port circuit general schematic . . . . . . . . . . . . . . . . . .. 33. 2.6. Multiple port circuit general schematic . . . . . . . . . . . . . . . . .. 35. 2.7. Circuit Simplification . . . . . . . . . . . . . . . . . . . . . . . . . . .. 36. 2.8. Compound slot and equivalent networks . . . . . . . . . . . . . . . .. 37. 2.9. Behavior of the elements of the T - Network . . . . . . . . . . . . . .. 39. 2.10 Behavior of the elements of the Π - Network . . . . . . . . . . . . . .. 41. 2.11 Power handling of Π - Network . . . . . . . . . . . . . . . . . . . . .. 44. 2.12 Available radiating power limits of Π - Network depending on the inclination angle at resonance.. . . . . . . . . . . . . . . . . . . . . .. VI. 46.

(11) LIST OF FIGURES. LIST OF FIGURES. 2.13 Array distribution schemes . . . . . . . . . . . . . . . . . . . . . . . .. 47. 2.14 6-element longitudinal slot array . . . . . . . . . . . . . . . . . . . .. 49. 2.15 6-element 15o -tilted compound slot array . . . . . . . . . . . . . . . .. 49. 2.16 6-element 30o -tilted compound slot array . . . . . . . . . . . . . . . .. 50. 2.17 6-element 45o -tilted compound slot array . . . . . . . . . . . . . . . .. 50. 2.18 Yamaguchi’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 52. 2.19 Return losses of the MoM-FMP designed 3-elements array . . . . . .. 53. 2.20 Small arrays comparison in terms of reflection and radiation . . . . .. 54. 3.1. Stages of the FMP . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 63. 3.2. CSP-Resolution backtracking algorithm . . . . . . . . . . . . . . . .. 64. 3.3. Physical restriction of a compound slot . . . . . . . . . . . . . . . . .. 67. 3.4. Final element constraint definition. . . . . . . . . . . . . . . . . . . .. 68. 3.5. First step of the FMP technique: the requested load impedance ZL (1) to get the input matched is calculated. . . . . . . . . . . . . . . . . .. 69. 3.6. First slot design procedure . . . . . . . . . . . . . . . . . . . . . . . .. 70. 3.7. Second step of the FMP technique. . . . . . . . . . . . . . . . . . . .. 71. 3.8. Slot connection flowchart. This is repeated till the last element of the. 3.9. arrayy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 73. Global equivalent circuit . . . . . . . . . . . . . . . . . . . . . . . . .. 73. 3.10 Four-element array design example. Process and model. . . . . . . .. 74. . . . . . . . . . . . . . . . . . . . . . . . . .. 75. Effects of mutual coupling . . . . . . . . . . . . . . . . . . . . . . . .. 84. 3.11 4-elements array design 4.1. VII.

(12) LIST OF FIGURES. LIST OF FIGURES. 4.2. CST Design Studio analysis . . . . . . . . . . . . . . . . . . . . . . .. 84. 4.3. New equivalent network with coupling port . . . . . . . . . . . . . .. 87. 4.4. Sub-networks interconnection . . . . . . . . . . . . . . . . . . . . . .. 88. 4.5. Coupling characterization block diagram . . . . . . . . . . . . . . . .. 89. 4.6. Scheme of the phenomenon . . . . . . . . . . . . . . . . . . . . . . .. 90. 4.7. Coupling phenomena equivalent circuit . . . . . . . . . . . . . . . . .. 92. 4.8. mutual impedance calculation . . . . . . . . . . . . . . . . . . . . . .. 93. 4.9. Mutual impedance: parallel elements. . . . . . . . . . . . . . . . . .. 93. 4.10 Mutual impedance: parallel-in-echelon elements . . . . . . . . . . . .. 93. 4.11 Port Numbering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 97. 4.12 New 4-element array equivalent network with coupling ports . . . . .. 98. 4.13 4-element linear array . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.1. Longitudinal slot array model . . . . . . . . . . . . . . . . . . . . . . 107. 5.2. Return Loss and Radiation Pattern of the array of longitudinal slots. 5.3. Transverse slot array model . . . . . . . . . . . . . . . . . . . . . . . 108. 5.4. Return Loss and Radiation Pattern of the array of transverse slots . 108. 5.5. Return Loss and Radiation Pattern of the 15-element array of 45º compound slots. 5.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109. Return Loss and radiation pattern of 45º tilted compound-slot array working at 40GHz.. 5.7. 107. . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Return Loss and radiation pattern of 45º tilted compound-slot array working at 100GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . 112. VIII.

(13) LIST OF FIGURES. LIST OF FIGURES. 5.8. 30º tilted compound slot array model . . . . . . . . . . . . . . . . . . 113. 5.9. Return loss and radiation pattern of the 30º compound-slot array. . 113. 5.10 Return loss and radiation pattern of the 15º compound-slot array. . 114. 5.11 45º tilted compound slot reduced-height array model . . . . . . . . . 115 5.12 Return loss and radiation pattern of the 45º tilted compound slot reduced-height array . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.13 6-element real prototype . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.14 AWR Microwave Office global equivalent circuit. . . . . . . . . . . . 118. 5.15 AWR Input Loss Estimation . . . . . . . . . . . . . . . . . . . . . . . 118 5.16 Measurements of the 6-element prototype . . . . . . . . . . . . . . . 119 5.17 Near field reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.18 10-element real prototype . . . . . . . . . . . . . . . . . . . . . . . . 122 5.19 AWR Input Loss Estimation . . . . . . . . . . . . . . . . . . . . . . . 124 5.20 Measurements of the 10-element prototype. . . . . . . . . . . . . . . 125. 5.21 Near field reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . 127 7.1. Method of Moments analysis scheme . . . . . . . . . . . . . . . . . . 137. 7.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141. 7.3. relative distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145. 7.4. Case 1, relative distances as a function of angles and offsets. . . . . . 147. 7.5. Case 2, relative distances as a function of angles and offsets. . . . . . 147. 7.6. Case 3, relative distances as a function of angles and offsets. . . . . . 147. 7.7. Case 4, relative distances as a function of angles and offsets. . . . . . 148. 7.8. Graphic User Interface . . . . . . . . . . . . . . . . . . . . . . . . . . 150. IX.

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(15) Resumen Los arrays de ranuras son sistemas de antennas conocidos desde los años 40, prinicipalmente destinados a formar parte de sistemas rádar de navíos de combate y grandes estaciones terrenas donde el tamaño y el peso no eran altamente restrictivos. Con el paso de los años y debido sobre todo a importantes avances en materiales y métodos de fabricación, el rango de aplicaciones de este tipo de sistemas radiantes creció en gran medida. Desde nuevas tecnologías biomédicas, sistemas anticolisión en automóviles y navegación en aviones, enlaces de comunicaciones de alta tasa binaria y corta distancia e incluso sistemas embarcados en satélites para la transmisión de señal de televisión. Dentro de esta familia de antennas, existen dos grupos que destacan por ser los más utilizados: las antennas de placas paralelas con las ranuras distribuidas de forma circular o espiral y las agrupaciones de arrays lineales construidos sobre guia de onda. Continuando con las tareas de investigación desarrolladas durante los últimos años en el Instituto de Tecnología de Tokyo y en el Grupo de Radiación de la Universidad Politécnica de Madrid, la totalidad de esta tesis se centra en este último grupo, aunque como se verá se separa en gran medida de las técnicas de diseño y metodologías convencionales. Los arrays de ranuras rectas y paralelas al eje de la guía rectangular que las alimenta son, sin ninguna duda, los modelos más empleados debido a la fiabilidad que presentan a altas frecuencias, su capacidad para gestionar grandes cantidades de potencia y la sencillez de su diseño y fabricación. Sin embargo, también presentan desventajas como estrecho ancho de banda en pérdidas de retorno y rápida degradación del diagrama de radiación con la frecuencia. Éstas son debidas a la naturaleza resonante de sus elementos radiantes: al perder la resonancia, el sistema global se desajusta y sus. XI.

(16) Resumen. prestaciones degeneran. En arrays bidimensionales de slots rectos, el campo eléctrico queda polarizado sobre el plano transversal a las ranuras, correspondiéndose con el plano de altos lóbulos secundarios. Esta tesis tiene como objetivo el desarrollo de un método sistemático de diseño de arrays de ranuras inclinadas y desplazadas del centro (en lo sucesivo “ranuras compuestas”), definido en 1971 como uno de los desafíos a superar dentro del mundo del diseño de antennas. La técnica empleada se basa en el Método de los Momentos, la Teoría de Circuitos y la Teoría de Conexión Aleatoria de Matrices de Dispersión. Al tratarse de un método circuital, la primera parte de la tesis se corresponde con el estudio de la aplicabilidad de las redes equivalentes fundamentales, su capacidad para recrear fenómenos físicos de la ranura, las limitaciones y ventajas que presentan para caracterizar las diferentes configuraciones de slot compuesto. Se profundiza en las diferencias entre las redes en T y en Π y se condiciona la selección de una u otra dependiendo del tipo de elemento radiante. Una vez seleccionado el tipo de red a emplear en el diseño del sistema, se ha desarrollado un algoritmo de cascadeo progresivo desde el puerto alimentador hacia el cortocircuito que termina el modelo. Este algoritmo es independiente del número de elementos, la frecuencia central de funcionamiento, del ángulo de inclinación de las ranuras y de la red equivalente seleccionada (en T o en Π). Se basa en definir el diseño del array como un Problema de Satisfacción de Condiciones (en inglés, Constraint Satisfaction Problem) que se resuelve por un método de Búsqueda en Retroceso (Backtracking algorithm). Como resultado devuelve un circuito equivalente del array completo adaptado a su entrada y cuyos elementos consumen una potencia acorde a una distribución de amplitud dada para el array. En toda agrupación de antennas, el acoplo mutuo entre elementos a través del campo radiado representa uno de los principales problemas para el ingeniero y sus efectos perjudican a las prestaciones globales del sistema, tanto en adaptación como en capacidad de radiación. El empleo de circuito equivalente se descartó por la dificultad que suponía la caracterización de estos efectos y su inclusión en la etapa de diseño. En esta tesis doctoral el acoplo también se ha modelado como una red equivalente cuyos elementos son transformadores ideales y admitancias, conectada al conjunto de redes equivalentes que representa el array.. XII.

(17) Resumen. Al comparar los resultados estimados en términos de pérdidas de retorno y radiación con aquellos obtenidos a partir de programas comerciales populares como CST Microwave Studio se confirma la validez del método aquí propuesto, el primer método de diseño sistemático de arrays de ranuras compuestos alimentados por guía de onda rectangular. Al tratarse de ranuras no resonantes, el ancho de banda en pérdidas de retorno es mucho mas amplio que el que presentan arrays de slots rectos. Para arrays bidimensionales, el ángulo de inclinación puede ajustarse de manera que el campo quede polarizado en los planos de bajos lóbulos secundarios. Además de simulaciones se han diseñado, construido y medido dos prototipos centrados en la frecuencia de 12GHz, de seis y diez elementos. Las medidas de pérdidas de retorno y diagrama de radiación revelan excelentes resultados, certificando la bondad del método genuino Method of Moments - Forward Matching Procedure desarrollado a lo largo de esta tésis.. XIII.

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(19) Abstract The slot antenna arrays are well known systems from the decade of 40s, mainly intended to be part of radar systems of large warships and terrestrial stations where size and weight were not highly restrictive. Over the years, mainly due to significant advances in materials and manufacturing methods, the range of applications of this type of radiating systems grew significantly. From new biomedical technologies, collision avoidance systems in cars and aircraft navigation, short communication links with high bit transfer rate and even embedded systems in satellites for television broadcast. Within this family of antennas, two groups stand out as being the most frequent in the literature: parallel plate antennas with slots placed in a circular or spiral distribution and clusters of waveguide linear arrays. To continue the vast research work carried out during the last decades in the Tokyo Institute of Technology and in the Radiation Group at the Universidad Politécnica de Madrid, this thesis focuses on the latter group, although it represents a technique that drastically breaks with traditional design methodologies. The arrays of slots straight and parallel to the axis of the feeding rectangular waveguide are without a doubt the most used models because of the reliability that they present at high frequencies, its ability to handle large amounts of power and their simplicity of design and manufacturing. However, there also exist disadvantages as narrow bandwidth in return loss and rapid degradation of the radiation pattern with frequency. These are due to the resonant nature of radiating elements: away from the resonance status, the overall system performance and radiation pattern diminish. For two-dimensional arrays of straight slots, the electric field is polarized transverse to the radiators, corresponding to the plane of high side-lobe level.. XV.

(20) Abstract. This thesis aims to develop a systematic method of designing arrays of angled and displaced slots (hereinafter "compound slots"), defined in 1971 as one of the challenges to overcome in the world of antenna design. The used technique is based on the Method of Moments, Circuit Theory and the Theory of Scattering Matrices Connection. Being a circuitry-based method, the first part of this dissertation corresponds to the study of the applicability of the basic equivalent networks, their ability to recreate the slot physical phenomena, their limitations and advantages presented to characterize different compound slot configurations. It delves into the differences of T and Π and determines the selection of the most suitable one depending on the type of radiating element. Once the type of network to be used in the system design is selected, a progressive algorithm called Forward Matching Procedure has been developed to connect the proper equivalent networks from the feeder port to shorted ending. This algorithm is independent of the number of elements, the central operating frequency, the angle of inclination of the slots and selected equivalent network (T or Π networks). It is based on the definition of the array design as a Constraint Satisfaction Problem, solved by means of a Backtracking Algorithm. As a result, the method returns an equivalent circuit of the whole array which is matched at its input port and whose elements consume a power according to a given amplitude distribution for the array. In any group of antennas, the mutual coupling between elements through the radiated field represents one of the biggest problems that the engineer faces and its effects are detrimental to the overall performance of the system, both in radiation capabilities and return loss. The employment of an equivalent circuit for the array design was discarded by some authors because of the difficulty involved in the characterization of the coupling effects and their inclusion in the design stage. In this thesis the coupling has also been modeled as an equivalent network whose elements are ideal transformers and admittances connected to the set of equivalent networks that represent the antennas of the array. By comparing the estimated results in terms of return loss and radiation with those obtained from popular commercial software as CST Microwave Studio, the validity of the proposed method is fully confirmed, representing the first method of systematic design of compound-slot arrays fed by rectangular waveguide. Since these slots do. XVI.

(21) Abstract. not work under the resonant status, the bandwidth in return loss is much wider than the longitudinal-slot arrays. For the case of two-dimensional arrays, the angle of inclination can be adjusted so that the field is polarized at the low side-lobe level plane. Besides the performed full-wave simulations two prototypes of six and ten elements for the X-band have been designed, built and measured, revealing excellent results and agreement with the expected results. These facts certify that the genuine technique Method of Moments - Matching Forward Procedure developed along this thesis is valid and trustable.. XVII.

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(23) Part I. Analysis. 2.

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(25) Chapter 1. Introduction 1.1 1.1.1. State of the Art Slot antennas. Nowadays more and more applications demand efficient and reliable communication systems and techniques at high frequency, such as millimeter and sub-millimeter frequency bands. Among this applications, it is easy to point out some for military purposes, like radar and missile guiding systems [1], and those for biomedical and civil purposes, like automatic anti-collision systems in cars, disease diagnosis, weather analysis or new body-scanning devices [2]. Ultra-fast data links are being installed to avoid the deployment of high-cost optic-fiber links [3]. To obtain the high-speed data transfer cordless links, it is necessary to achieve high frequency bands, although the distance range is limited by the inherent propagation attenuation of the waves. To compensate the attenuation phenomena, high-gain antennas or arrays of a big number of antennas are popular solutions. The most famous high gain antenna could be the reflector antenna and it is common to find them in satellite television broadcast-reception, high-frequency data links, radio telescopes and deep space communications. However their mechanical properties make them inappropriate for low-profile systems or portable devices. Another solutions are arrays of light weight, low profile and surface adaptable planar antennas. Arrays of patch antennas have become really popular during the last years. 4.

(26) CHAPTER 1. Section 1.1.  .  . . . . . .  . . . . . .  . . . . . . . . . . . . . .  .  . . . . . .  . .  . . . Figure 1.1: slot configurations and their equivalent circuit because of their low cost, easy design and flexibility, although the amount of power that the substrate-based systems can handle is quite limited. Furthermore, their use is limited to frequencies below the X-band [4]. Slot arrays antennas are the best candidate to fulfill high power, low profile and highgain radiation pattern. Their properties of low losses, robust structure and reliability at high frequencies have made slots antennas to be the most common radiator in long distance communication systems or scenarios of adverse conditions. Mainly for radar, spacecraft and missile applications, these antennas consist of waveguides in which slots have been cut setting up a special distribution to achieve matching and radiation requirements. Slots’ length, position, inclination and width control the electrical behavior in terms of radiation and internal impedance [5]. It is then necessary to adjust those structural parameters in order to fulfill a set of specifications like gain, side-lobe level and return loss of the array. In this thesis, four main categories have been established: longitudinal slot, parallel to the waveguide axis, fig. 1.1(a); transverse slot, perpendicular to the propagation direction of the feeding mode, fig. 1.1(b); centered-inclined slot, which center is placed over the waveguide axis and it is tilted an specific angle θ, fig. 1.1(c); compound slots, inclined and displaced from the waveguide axis, fig. 1.1(d). Slots can be also found interrupting the currents sheets that flow on the narrow faces of the guide, but those will not be treated here.. 5.

(27) CHAPTER 1. Section 1.1. It was during the forties when slot arrays were firstly investigated by Stevenson [6], Watson [7] and Oliner [8, 9]. Stevenson developed the basic theory of slots in rectangular waveguides and firstly attempted to find an analogy with transmission lines. Mathematical formulation for the transmission and reflection coefficients and the voltage amplitude generated in the aperture by a given incident wave were calculated. Only the terms of resistance or conductance related with transverse or longitudinal slots respectively could be mathematically obtained, always at resonance and for infinity thin conducting wall. Also Stevenson in [6] proposed integral equations to estimate the resonance length of shunt slots. Oliner studied the impedance properties of the slot cut in the wide face of a rectangular waveguide at resonance and away from it [8, 9]. Again, only the slots subject to be modeled as a single element were studied and the most general case (compound slot) was ignored. Oliner proposed the equivalent circuits for the transverse, longitudinal and centered-inclined and the mathematical expressions to calculate the components (admittances/impedances) of these equivalent networks taking into account the thickness of the wall. Gülick and Elliott demonstrated that this technique is limited, claiming that circuit models based on a single shunt or series element are only valid for standard-height waveguide and/or small offsets [11, 12]. The reason is the slot can be represented by a single element if and only if the forward and backward scattered waves from the slot have same amplitude when it is fed by the T E10 mode. The transverse slot is not popular as a radiating element since the distance between elements is too large and grating lobes are inevitably present. However, it is possible to develop an equivalent network and use it to generate a radiating slot array as it will be shown in the proper section. The employment of the compound slot as a radiating element is extremely unusual and consequently, the related bibliography is not so vast. Watson was the pioneer in the characterization of the compound slot at resonance [7]. He developed a complete mathematical formulation for the analysis of the impedance properties depending on the inclination angle and under the condition of zero thickness. Maxum continues the work carried out by Watson finding out the dependance of the slot voltage with the inclination angle for the case of small excitation, small inclination angles and negligible thickness. It was Maxum the first to define the resonance status for a compound slot in a more general way than previous authors [13].. 6.

(28) CHAPTER 1. Section 1.1. “...resonance of a slot is not that one sees a pure resistance or pure conductance at the plane of the slot, but rather whether or not the transformation of the wave past the slot exhibits a change in phase.”. This definition has been followed to perform a study of the slot radiation characteristics for different slot lengths, offsets and tilting angles. Under this condition, the slot exhibits maximum radiated power for all the distributions shown in fig. 1.1. Reasons why, against the rest of cases, the compound slot arrays cannot be designed under resonance are explained in chapter 2. Rengarajan in [14, 15] contributed to the employment of centered-inclined and compound slot as couplers in multilayered structures of finite conductor thickness [16]. In [17] the characteristics of compound slot for radiation purposes are presented, where Method of Moments (MoM hereafter) with dyadic Green functions are employed to calculate the fields on the apertures of the slot. The advantage of compound slot array compared to the center-inclined slot array is that spacing between elements is reduced to approximately half guided-wavelength and corresponding grating lobes are prevented. The definition of the equivalent circuit of the compound slot is not clearly established and the selection between the simplest networks (Π and T configurations) is subject to the definition of the system although their use was not recommended for array design in [17]. Depending on the waveguide dimensions, tilting angle and offset of the slot, one is preferable to the other and the reasons are discussed and presented along the next chapter.. 1.1.2. Array Design Methods. Since their discovery, longitudinal slots are the most common radiators. Their electromagnetic properties and equivalent circuit models have been deeply studied and as a result, there is an extensive amount of excellent contributions in the field. Among all the scientists of the antenna community, Elliott’s contributions represent the cornerstone of the literature about longitudinal slot antennas. To synthesize a specific aperture illumination employing the equivalent circuit, Elliott in [18] proposed a design procedure for the standing-wave array design that accounts for the external. 7.

(29) CHAPTER 1. Section 1.1. coupling between elements. Lately, it was extended to the case of traveling-wavefed longitudinal slot arrays [19] and the effect of the internal coupling was finally included in [20]. Then, the method of moment (MoM) analysis was employed to enhance the accuracy of the calculations with less computational cost, taking into account the internal high-order mode coupling and finite slot-wall thickness [21, 22]. Stern and Elliott found the mathematical relation between slot offset and resonant length for a given frequency and waveguide dimensions [23]. Methods for designing linear or planar arrays given by Elliott, Sangster and McCormick need an iterative process to determine slots size and offset to realize a required excitation. Such a process requires a knowledge of scattering of fields inside the waveguide of a single slot, the relation between the incident T E10 , the aperture electric field and forward and backward scattered fields. In addition, the electric field induced in each slot due to all other slots via external coupling needs to be calculated to increase the accuracy of the procedures. Besides radiation properties, the input matching is another requirement to be fulfilled being the bandwidth dependent on the specific characteristics of the selected slots and the spacing between the radiators. These methods account only for the case of longitudinal slot regularly spaced since the scattered forward and backward fields within the waveguide seem to be in phase and the slot represents a symmetric circuit. This means that it can be represented by a single shunt element and the amount of unknowns compared to the general case is significantly reduced. For all these cases, several authors continued and improved the equivalent circuit representation and its use was incorporated in array design methods. The most stunning and important contribution came from Elliott, which for decades was considered as the benchmark of slot array design. Elliott conceived a methodology based on the equivalent circuit representation and the scattered waves from a slot. More recent, method of moments was employed and excellent works can be found in the literature, applying equivalent networks for the longitudinal slot case [24] and for the parallel plate slot antenna [25]. After an initial circuit-based design stage, an iterative full-wave method analyze and adjust the system properties to achieve the requested radiation properties. There is not systematic procedure for compound slot array designing and the necessity of investigate and develop a design method of linear arrays for this kind of radiator was pointed out by Joseffsson, Rengarajan and Larson in [26]. This. 8.

(30) CHAPTER 1. Section 1.2. thesis tries to fill this gap in the antenna theory and it represents a step forward in the antennas’ field.. 1.1.3. Mutual coupling effects. The effect of internal higher order mode coupling between adjacent radiating slots in a waveguide was reported in [27], where it was shown that the T E20 mode is the only responsible for reduced height waveguides and almost within the standard height ones. For the coupling slots, both for the centered-inclined coupling slot and the longitudinal-transverse coupling slot, a similar effect was reported in [28]. In this thesis, the contribution of the mode T E20 to the inter-slot internal coupling has been neglected since the attenuation constant of this mode within the standard waveguides is significant and the separation between slots is large enough to get the mode highly attenuated. Elliott included the effects of the external mutual coupling between longitudinal slots resorting to an equivalent array of cable-fed slots and a complementary array of dipoles. Rengarajan completed his own study accounting for the external coupling between radiating slots, for transverse and longitudinal broad wall slots and other configurations [29, 30, 31, 32], although its applicability to the design of compound slot arrays could not be found in the literature.. 1.2. Objectives. As it was previously commented, the purpose of this doctoral work is to develop a systematic methodology of compound-slot array designing. This is based on the equivalent circuit modelling and MoM analysis and it represents a complete engineering work that must be turned into a bunch of smaller hints, listed below. 1. Study and comprehension of the physical phenomena within the single compound slot scenario. The inherent properties of the scattered fields must be analyzed to understand how the feeding wave is altered because of the presence of the slot and its dependance on mechanical properties of the aperture. The understanding of the fields behavior allows the determination. 9.

(31) CHAPTER 1. Section 1.2. of the minimum requirements that the network must fulfill. To perform this study, a general case is set out and the Reciprocity Theorem will be applied, always under the situation of monomode propagation inside the waveguide. 2. Study and applicability of the possible equivalent networks to be employed. Once the nature of the required network is known, it is mandatory to select the most suitable equivalent circuit and to study the validity of its physical interpretation under the circumstances of design. Structural properties as inclination angle, length, waveguide dimensions and offset are the critical parameters that control the values of the lumped elements that compose the equivalent network. Sometimes, these values fulfill mathematical expressions but they don’t provide a real behavior in terms of physics (like, for example, a negative resistance). 3. To develop an algorithm for cascade connection of slots in order to keep the input matched at the frequency of interest and to obtain an specific aperture distribution. The elements of the circuit have to recreate the properties of radiation and internal matching of the slot. Taking into account the impedance properties of the equivalent circuits, those circuits that once connected to the rest of elements radiate with the power and phase that the specifications demand will be selected to be part of the array. This algorithm gives an array with the expected radiation specifications and matched at the input. 4. To create a mutual coupling model to be included in the design process. In all kind of arrays the mutual coupling between elements has a vital importance and must be taken into account to properly predict the array behavior. The mutual coupling effects characterization has been carried out using Circuit Theory and Scattering Matrix Connection Theory. The array given in the previous point is regarded as the initial model where to include these phenomena and their consequences in the array performance. 5. Design, construction and measurement of prototypes in the 12GHz band. From the beginning, there was no method of checking if the procedure was being developed in a correct way. Furthermore, since this is the first attempt for compound slot array systematic design procedure existing in the bibliography, it was impossible to compare to results from other authors. The only way to validate the design methodology here proposed is to simulate the. 10.

(32) CHAPTER 1. Section 1.3. calculated arrays in full-wave software environments like CST and HFSS. After this, next step consists of building prototypes and compare expected results and measurements. It is also necessary to take into account limitations and tolerances in the fabrication stage. The employed circuit are the more general ones so it is believed that the range of applicability of the here proposed methodology is wide enough to cover all the possible slot configurations shown in fig. 1.1. All these different cases will be analyzed. Once it is confirmed that the method is fully applicable to longitudinal arrays, then it is possible to think about extending its use to two dimensional case. High gain (>35dBi) and low side lobe level (<-30dB) antennas at any frequency could be designed choosing the proper number of elements and inclination angle on the basis of the two following facts: 1. Polarization of the field: the field is polarized in the low side-lobe-level plane of the rectangular array. Choosing for example a squared array, it would be interesting to select 45º slots in order to make the E-plane coincide to the low side-lobe-level plane, as depicted in the right side of fig. 1.2. 2. It is possible then to achieve a rhombic aperture distribution concentrating more elements in the center of the array, so that the performance of the grouping in terms of gain and side-lobe-level are improved.. 1.3. Motivation. During the last 20 years the Ando-Hirokawa laboratory, formerly Goto-Ando laboratory, in Tokyo Institute of Technology has been leading the developing of analysis techniques and construction methodologies for slot antennas. The Method of Moments is the main tool employed for the study of many different kinds of structures, from feeding networks to radiating arrays, cavities or scattering scenarios. The relations between Tokyo Institute of Technology and Radiation Group of Technical University of Madrid began thirteen years ago, when Prof. Sierra stayed as a research student under the advice of Prof. Hirokawa. Some years later, Mr. Sudo. 11.

(33) CHAPTER 1. Section 1.3. .  . . . . .  . Figure 1.2: Two dimensional arrays and their main planes, E and H. became research student of Prof. Sierra at Radiation Group in Madrid. The core of this thesis was developed at Tokyo Institute of Technology during two different stays of two years of duration in total, under the supervision of Prof. Ando and Prof. Hirokawa. The topic of the thesis came up as a suggestion of both supervisors in order to continue and complete their dilated experience in electromagnetics associated with slot radiators and feeding networks. Their proposal included a simple requirement that had to be adopted to design compound slot arrays: the use of a verified equivalent circuit. Professors knew about the necessity of developing such a systematic methodology for the complex slot, as it was pointed out by some famous professors, Prof. Rengarajan, Prof. Josefsson and Prof. Peterson in [26]: “There is a need to develop and investigate a design procedure for compound slot arrays.” The proposed topic was new for the author so that a lot of papers and existing bibliography had to be consulted and studied. During this period, a contribution from Professor Rengarajan increased the uncertainty of it it was possible or not to carry out the request from Professors Ando and Hirokawa.. 12.

(34) CHAPTER 1. Section 1.4. “The equivalent-circuit representation becomes substantially complicated away from the resonant frequency. Therefore, it is desirable to work with a scattering wave representation of a compound slot for array applications.” Disregarding Prof. Rengarajan’s advice this thesis consists on a complete and genuine methodology that fills a gap in the antenna theory. It represents a powerful tool for the design of any kind of slot array in rectangular waveguide, specially the compound one, which use for radiation purposes was really limited. Inside the Radiation Group of the Technical University of Madrid, this PhD dissertation could be regarded as a continuity of Prof. Sierra thesis, replacing the parallel plate waveguide by a rectangular waveguide. As it has been commented in the previous section, the slot antennas field is an old and well known field, where the development of new ideas and concepts is quite complicated. For the sake of completion, the intention of the author is to develop a model of equivalent circuit, a trustful methodology of array design and to validate of the whole process by means of the measurements of real prototypes.. 1.4. Contents. This thesis is divided in two main parts. The first one is titled as “Analysis” and it covers the basic study that has been developed in order to fully characterize and understand the slot as a single device, the study of the scattered fields within the waveguide and their dependence with the slot properties. After this, the employed networks and their performance in terms of power and radiation phase. This fits with points 1 and 2 of section 1.2. Second part, “Synthesis”, addresses points 3, 4 and 5 of the same section and explains the secrets of circuit cascading and the algorithm in charge of global circuit composition. The effects of the externally inter-exchanged power between elements and its characterization is also explained in this part. Chapter 5 exhibits the versatility of the proposed technique, showing excellent results when comparing genuine predicted results to CST simulations for several array configurations and cases. Also in this. 13.

(35) CHAPTER 1. Section 1.4. chapter, two real prototypes and their measurements in terms of radiation and return loss are shown. Near field reconstruction plots have been included and they reveal, together with regular measurements, the good performance of both systems and the validity of the developed MoM-FMP. As usual for this kind of document, this thesis ends with a chapter dedicated to future tasks, achieved hints and general conclusions.. 14.

(36) Bibliography [1] W. Swelam, A. A. Mitkees, ; M. M. Ibrahim, “Wideband planar phased array antenna at Ku frequency-band for synthetic aperture radars and radar-guided missiles tracking and detection”, 2006 IEEE Conference on Radar, 24-27 April 2006. [2] Hasan, M.M. ; Jayawardene, R. ; Hirano, T. ; Hirokawa, J. and M. Ando, “Localized behaviors of rain measured in Tokyo Tech millimeter-wave wireless network,” Proceedings of the 5th European Conference on Antennas and Propagation (EUCAP), 11-15 April 2011. [3] S. Holzwarth, Baggen, L. “Planar antenna design at 60 GHz for high date rate point-to-point connections,” Antennas and Propagation Society International Symposium, 2005. [4] A. Garcia-Aguilar, J. Inclan-Alonso, J. ; L. Vigil-Herrero, ; J. M. FernandezGonzalez and M. Sierra-Perez, “Low-Profile Dual Circularly Polarized Antenna Array for Satellite Communications in the X Band,” IEEE Transactions on Antennas and Propagation, 60 , Issue: 5, May 2012. [5] R. S. Elliott, “Antenna Theory and Design,” Englewood Cliffs: Prentice-Hall, 1981. [6] A.F. Stevenson. Theory of Slots in Rectangular Wave-Guides. Journal of Applied Physics, 19, pp. 24-38, Jan. 1948. [7] W. H. Watson, “The coupling of a resonant slot in a rectangular waveguide,” in The Physical Principles of Wave Guide Transmission and Antenna Systems, Oxford, Great Britain, Oxford University Press., 1947, ch. VI.. 15.

(37) CHAPTER 1. Section 1.4. [8] A. A. Oliner, “The impedance Properties of Narrow Radiating Slots in the Broad Face of Rectangular Waveguide. Part I Theory,” IRE Transactions on Antennas and Propagation, January 1957. [9] A. A. Oliner, “The impedance Properties of Narrow Radiating Slots in the Broad Face of Rectangular Waveguide. Part II Comparison with Measurements,” IRE Transactions on Antennas and Propagation, January 1957. [10] T. B Khac, C. T. Carson, “Impedance properties of a longitudinal slot antenna in the broad face of a rectangular waveguide,” IEEE Trans. Antennas Propag., vol. AP-21, pp. 708-710, Sep. 1973. [11] J. J. Gulick, G J Stern, and R. S. Elliott. “The equivalent circuit of a rectangularwaveguide-fed longitudinal slot.” In Proceedings of the Antennas And Propagation Society Symposium, pages 685–688, 1986. [12] J. J. Gulick and R. S. Elliott. “A general design procedure for small arrays of waveguide-fed slots”. In Proceedings of the Antennas And Propagation Society Symposium, pages 302–305, 1987. [13] B. J. Maxum, “Resonant Slots with Independent Control of Amplitude and Phase,” IRE Transactions on Antennas and Propagation, July 1960. [14] S. R. Rengarajan, “Analysis of a Centered-Inclined Waveguide Slot Coupler,” IEEE Trans. Microwave Theory Tech., vol.37, no. 5, pp. 884-889, May 1989. [15] [S. R. Rengarajan, “Compound Coupling Slots for Arbitrary Excitation of Waveguide-Fed Planar Slot Arrays,” IEEE Trans. Antennas Propag., vol. 38, no. 2, pp. 276- 280, Feb. 1990. [16] J. Hirokawa, M. Zhang, Y. Miura M. Ando, “Double-Layer Slotted Waveguide Array Antennas with Corporate-Feed by Diffusion Bonding of Laminated Thin Metal Plates”, Proc. of the 4th European Conf. on Antennas and Propagation, Barcelona, Spain, April 12-16, 2010. [17] S. R. Rengarajan, “Compound Broad-Wall slots for Array Applications”, IEEE Antennas Propag Mag., vol 32, no 6, pp 20-26, Dec. 1990. [18] R. S. Elliott, L. A. Kurtz, “The Design of Small Slot Arrays,” IEEE Trans. Antennas Propag., vol. AP-26, no. 2, pp. 214-219, Mar. 1978.. 16.

(38) CHAPTER 1. Section 1.4. [19] R. S. Elliott. “On the design of traveling-wave-fed longitudinal shunt slot arrays,” IEEE Transactions on Antennas and Propagation, 27(5):717–720, September 1979. [20] R. S. Elliott. “An Improved Design Procedure for Small Arrays of Shunt Slots,” IEEE Transactions on Antennas and Propagation„ (1):48–53, 1983. [21] R.W. Lyon and A. J. Sangster. Efficient moment method analysis of radiating slots in a thick-walled rectangular waveguide. IEE Proceedings H Microwaves, Optics and Antennas, 128(4):197, 1981. [22] A.J. Sangster and A.H.I. McCormick, “Moment method applied to round-ended slots,” IEE Proc., vol.3, Pt. H, no.3, pp.310–314, June 1987. [23] G. Stern and R. Elliott, “Resonant length of longitudinal slots and validity of circuit representation: Theory and experiment,” IEEE Transactions on Antennas and Propagation, vol. 33, no. 11, pp. 1264-1271, Nov. 1985. [24] M. Zhang, J. Hirokawa and M. Ando, “Full-Wave Design Considering Slot Admittance in 2-D Waveguide Slot Arrays with Perfect Input Matching,” IEICE Trans. Commun., Vol.E94-B, No.03, pp. 725-734, Mar. 2011. [25] M. Sierra-Castañer, M. Sierra-Pérez, M. Vera-Isasa, and J. L. FernándezJambrina, “Fast analysis model for radial-line slot antennas,” Microwave and Optical Tech. Letters, vol. 44, No.1, pp. 17-21, Jan 2005. [26] S. R. Rengarajan, L. Josefsson and R. Petersson, “Recent Developments in broad wall slots in rectangular waveguides for array applications”, 7th International Conf. on Antennas and Propagation, vol. 2, pp. 729-731, April 1991. [27] S. R. Rengarajan and D. D. Nardi, "On internal higher-order mode coupling in slot arrays," IEEE Trans. Antennas Propag., vol. 39, no. 5, pp. 694-698, May 1991. [28] S. R. Rengarajan, " Higher order mode coupling effects in feeding waveguide of a planar slot array," IEEE Trans. Microwave Theory Tech., vol. 39, pp. 12191223, July 1991. [29] S. R. Rengarajan, and E. Gabrelian, "Efficient and accurate evaluation of external mutual coupling between compound broad wall slots," IEEE Trans. Antennas Propag., vol. 40, no. 6, pp. 733-737, June 1992 .. 17.

(39) [30] S. R. Rengarajan, "Mutual coupling between slots cut in rectangular cylindrical structures: Spectral domain analysis," Radio Science, vol. 31, pp. 1651-1661, Nov. - Dec., 1996. [31] S. R. Rengarajan, "Mutual coupling between waveguide-fed longitudinal broad wall slots radiating between baffles," Electromagnetics, vol. 16, no. 6, pp. 671683, Nov. - Dec., 1996. [32] M. Dich, and S. R. Rengarajan, "Mutual coupling between waveguide-fed transverse broad wall slots radiating between baffles," Electromagnetics, vol. 17, no. 5, pp. 421-435, Sep. - Oct., 1997..

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(41) Chapter 2. Equivalent Circuit Modelling 2.1. Slots in rectangular waveguide walls. It is well known that the electromagnetic field within a waveguide is related to an inseparable current distribution over the metallic boundaries of the guide. Those current sheets can be regarded as those required to prevent the field to escape to the external region. If a narrow slot is cut in a waveguide wall such that its axis runs along a current line or over a region without current density, the coupling of the field into the external region is negligible. These non-radiating slots are not useless and they allow the measurement of the standing wave ratio for impedance purposes. On the other hand, a slot cut in a wall transverse to the direction of the current lines causes an important alteration of the current sheets, causing the guided field to be coupled to the external region. The degree of radiation depends on the current density perturbed by the slot and the component of the length of the slot transverse to the current lines. This means that position within the waveguide face, length and orientation control the contribution of the slot to the radiated field, [1-12]. As it will be later explained, the waveguide can be seen as a bunch of transmission lines, each of them for each of the modes that are propagating inside the pipe. The structure of the equivalent circuit that the radiating element presents to the modal transmission line is then fully dependent of the position and the orientation of the slot. For longitudinal slots, the offset from the waveguide axis controls the amount. 20.

(42) CHAPTER 2. Section 2.2. ! l. Port 2. D w. t x. Port 1. z. y b. a Figure 2.1: Basic parameters of a compound slot. of radiated power, so that the larger the offset the more power has the equivalent circuit to dissipate [13]. The length is intimately related to the phase of the radiated field. For the case of compound slot such a distinction offset-power and length-phase cannot be applied. The inclusion of a tilting angle causes the slot to cut parallel and transverse current sheets simultaneously, so that inclination angle, slot length and offset participate in radiation phase and radiated power matters. The main objective of this chapter is to qualitatively find out the dependence of the equivalent circuit lumped elements on the structural properties of the compound slot.. 2.2. Structural characteristics of the Slot. The structural characteristics of a generic slot placed in the wide face of a rectangular waveguide are represented in fig. 2.1. In this geometry, a and b are the width and height of the waveguide, respectively. These determine the propagation constant γ of the feeding mode and the cutoff frequency, as well as the attenuation constant of the higher order modes. The offset (D) is the distance between centre of the slot and the centre of the waveguide; l and w are the length and width of the slot, respectively; the tilting angle θ is defined as the angle between the longitudinal axis of the waveguide and the longitudinal axis of the slot. The thickness of the conductor is denoted by t. Nowadays, the main limitation on the array implementation comes at the fabrication stage, since the thickness must be solid enough to keep the robustness of the structure and electrically short in order not to become a resonator of high Q. As the frequency. 21.

(43) CHAPTER 2. Section 2.3. x. S3. S2. S1. z. z = z2 z=0 z = z1. Figure 2.2: Boundaries definition goes up, conductor loss must be taken into account and construction details like round edges have effects that are not negligible anymore.. 2.3. Scattering theory of waveguide-fed radiators. Lets consider a rectangular waveguide which internal dimensions a and b are set up in order to cut off all modes except T E10 , it is infinitely long and the generator is in z = −∞. The waveguide has a slot cut in the broad-wall at z = 0, y = b; there is no other media than vacuum inside the waveguide and the conductor is considered as. perfect. Under these conditions the loss in power is exclusively due to the radiation phenomenon. Supposing the slot is contained in a rectangular loss-free parallelepiped S, bounded by S1 at z = z1 and S2 at z = z2 , being S3 the rest of the surface of S as shown in fig. 2.2. Since evanescent modes are created at the discontinuity, the planes must be chosen far enough so that these decaying waves do not contribute to the power at the reference planes. The dominant T E10 mode then propagates in the positive Z-direction, feeds the slot and causes different phenomena: forward and backward scattered modes inside the pipe and power radiated to the free space region. Taking into account the described system depicted in fig. 2.2, two pairs of fields satisfy the Maxwell’s equations in the region V : 1.. !. " " 1, H " 1 , outgoing from the slot scattered fields in both directions due to to E. interaction between the feeding mode and the slot. The transverse component. 22.

(44) CHAPTER 2. Section 2.3. of the electric and magnetic fields of the scattered waves can be written as " 1t = #Ca E " at e−γa z z > z2 E a. " 1t = #Ba E " at eγa z z < z1 E a. " 1t = H. # a. (2.1). " at e−γa z z > z2 Ca H. " 1t = −#Ba H " at eγa z z < z1 H a. where the amplitude of the backward and forward-scattered mode, Ba and Ca respectively, are not necessarily equal and have to be calculated. Despite the waveguide is normally fed by the fundamental T E10 , the higher order modes are scattered from the slot. The subscripts a accounts for their indexes, a = m, n. In order to calculate the amplitudes Ca and Ba , a relation between feeding and scattered modes must be included. In the region z1 < z < z2 the scattered fields are not represented by eq. 2.1 because of the presence of the slot and the non-propagating evanescent modes.. 2.. !. " 2, H "2 E. ". is the incident feeding mode that propagates along the Z-direction.. Both electric and magnetic fields have two main components: the longitudinal component parallel to the direction of propagation of energy, denoted by the subscript z and the transverse component, addressed by the subscript t. " 2 = E2zp ẑ + E"2tp E (2.2) " 2 = H2zp ẑ + H"2tp H It is believed that the reader is familiar with the distribution of electromagnetic fields in the rectangular waveguide so that the complete mathematical derivation of modes will be omitted [14]. Nevertheless, the generic expressions for the TE and TM modes’ fields, divided in longitudinal (z-axis direction). 23.

(45) CHAPTER 2. Section 2.3. and transverse components are stated below:. H2zp. T Emn modes $ % $ nπ % ∓γ z p = j· cos mπ x · cos a b y ·e. $ mπ % $ nπ % $ % ω·µ0 $ −x̂· nπ b · cos a x · sin b y γp2 +k2 $ % $ mπ % $ nπ %% ∓γ z +ŷ· mπ · sin x · cos ·e p a a b y. " 2tp = E. (2.3). $ mπ % $ nπ % $ % −j·γp $ −x̂· nπ b · cos a x · sin b y + γp2 +k2 $ % $ % $ nπ %% ∓γ z +ŷ· mπ · sin mπ ·e p a a x · cos b y. " 2tp = H. E2zp. T Mmn modes $ % $ nπ % ∓γ z p = ±j· sin mπ a x · sin b y ·e. $ % $ nπ % −j·γp $ $ mπ % x̂· a · cos mπ a x · sin b y γp2 +k2 $ % $ mπ % $ nπ %% ∓γ z +ŷ· nπ ·e p b · sin a x · cos b y. " 2tp = E. $. " 2tp = H. $. %. $. %. $. −ω·ε0 mπ nπ x̂· nπ b · sin a x · cos b y γp2 +k2 $ % $ % $ nπ %% ∓γ z −ŷ· mπ · cos mπ ·e p a a x · sin b y. %. (2.4). −. where the subscript p is a shorthand for the double index mn, γp is the propagation constant of the p mode. The upper signs in the previous equations need to be taken for the propagation in the positive Z-direction, being the lower one the required for propagation in the negative Z direction. In this thesis, it has been assumed that the fundamental T E10 mode is the only-feeding mode, so that its particular expression will be obtained for the sake of understanding during the following steps. " z,10 = j· cos H " 2t = E " y,10 = E 10. $ π % ∓γ z 10 a x ẑ·e. $ π % % ∓γ z j·ω·µ0 $$ π % 10 2 +k 2 · a · sin a x ŷ ·e γ10. " 2t = H " x,10 = ∓ 2γ10 2 · H 10 γ +k 10. $$ π % a. · sin. (2.5). $ π % % ∓γ z 10 a x x̂ ·e. Since the volume V represents a region free of sources, the application of the Reciprocity Theorem [15] inside the volume V for both families of fields previously des-. 24.

(46) CHAPTER 2. Section 2.3. cribed could be written as: ˆ ! S. " "1 × H "2 − E "2 × H " 1 ·dS "=0 E. (2.6). " is normal to the surface S and directed outwards from V. Applying the where dS " 2 on the surface S3 and the nullity of the tangential nullity of the tangential field of E " 1 in all the surface S3 except the part of the slot, one obtains: component of E ´. slot. !. " " ´ ! "1 × H " 2 · dS "+ "1 × H "2 − E "2 × H " 1 · dS+ " E E S1 +. ´. S2. ˆ I1 =. I2 =. ´. ´. S1. S2. &. &. !. (2.7). " "1 × H "2 − E "2 × H " 1 · dS "=0 E. slot. !. " "1 × H " 2 · dS " = I1 + I2 E. " 1at · eγa z × H " 2pt · e−γb z " 2pt · e−γb z × #B1a H " 1at · eγa z + #B1a E E a. a. (2.8) '. " 1at · e−γa z × H " 2pt · e−γb z " 1at · e−γa z − #C1a E " 2pt · e−γb z × #C1a H E a. a. " · dS. '. " · dS (2.9). where p accounts again for the index of the feeding modes p = m# n# . Taking into account the orthogonal properties of the formulated modes (eq. 2.1), I1 is different from zero if a = mn is equal to p = m# n# and both modes belong to the same family, TE or TM. Contrary to I1 , I2 becomes zero when a = mn = p = m# n# . I1 = 2Ba ·. ´. S1. !. " " 2pt × H " 1pt · dS " E. I2 = 0. (a = p). (2.10). " " 2pt × H " 1pt · dS " E. (2.11). Taking eq. 2.8 and substituting, one gets ˆ. slot. !. ˆ " "1 × H " 2 · n̂·dS = 2Ba · E. S1. !. where n is the unitary vector directed outwards from the waveguide. The back-. 25.

(47) CHAPTER 2. Section 2.3. scattered mode amplitude Ba can be then calculated as " " "1 × H " 2 · dS E slot ! " Ba = ´ " " 2pt × H " 1pt · dS 2 E ´. !. (2.12). s1. If the ! process " is repeated changing the direction of propagation of the feeding mode " " in E2 , H2 , the amplitude of the forward-scattered waves can be found in the same way.. " " "1 × H " 2 · dS E slot " Ca = ´ ! " " 2pt × H " 1pt · dS 2 s2 E ´. !. (2.13). Depending on the inclination angle, the amount of field coupled to the slot and consequently the value of the numerator of eq. 2.12 and eq. 2.13 will vary. The mathematical relation between both quantities will be essential to characterize the slot as an equivalent network. Elliott in [16] develops the application of these coefficients to characterize a longitudinal slot. For this inclination angle, θ = 0o , B = C and the slot shows a symmetrical behavior, allowing the slot to be represented as a single shunt element. Next subsection will demonstrate this fact and the impossibility of representing a compound slot with a single element, shunt or series.. 2.3.1. Scattering from compound slots. A compound slot can be seen as the most generic slot configuration in a waveguide and the rest of the shapes presented on fig. 1.1 would be simplifications of this general one. Similarly, different slot configurations bring different equivalent networks that represent simplifications of a less restrictive case. Regardless of slot’s inclination, offset or length, the scattering parameters of the single slot in a waveguide reveals a common reciprocal behavior that allows the treatment of any slot as the same equivalent network. When the slot in the waveguide is modeled as a two-port device with losses, the S matrix is always reciprocal and it can be represented as a circuit of three complex and independent elements [17], as the T or Π circuits. Some specific configurations like longitudinal, transverse or centered-inclined slots show symmetry besides reciprocity, so that they can be modeled by means of a single shunt or series element with two unknowns (real and imaginary parts). It is related to the continuity of the. 26.

(48) Section 2.3. !. !. CHAPTER 2. An. !. l Bn. D. ws. x z. Cn. Figure 2.3: Local coordinate axes definition. transverse component of the electromagnetic field at the center of the slot [16, 15]. To demonstrate this fact, the amplitude of the forward and backward scattered field from a generic compound slot, Ca and Ba respectively, and their ratio, Ba /Ca = Da will be calculated.. For the sake of the orientation and size independence, it is convenient to use a locallydefined coordinate system (fig. 2.3) which axis (ξ, η) are placed along the length and width of the slot, respectively.. Lets start addressing to the equation system written in eq. 2.1, describing the scattered electric and magnetic fields inside the volume depicted in fig. 2.2 when the waveguide is fed by the fundamental T E10 mode that faces a compound slot of tilting angle θ. The amplitude of the scattered fields C10 and B10 will be calculated and their dependency on the tilting angle and other physical properties will be pointed " 2 in eq. 2.12 and eq. 2.13 ) out. The magnetic field on the aperture of the slot (H comes from the longitudinal and transverse components of the magnetic field of the feeding mode, eq. 2.5. The electric field is assumed to be cosinusoidal: " 2 = ξ" H. &. $π % −jβ10 · sin π a x · sin (θ) ( a )· E"1 =. V0 w. ' $π % + j· cos a x · cos (θ) ·e−jβ10 z. cos. $π. 2·l ·ξ. %. (2.14). ξ". To obtain B10 and C10 , both expressions must be re-written according to the local. 27.

(49) CHAPTER 2. Section 2.3. coordinate system. " 2 = ξ" H + j· cos. $π a. &. $π −jβ10 · sin π a ( a )·. % (D + a/2 + ξ· sin (θ) + η· cos (θ)) · sin (θ). % % (D + a/2 + ξ· sin (θ) + η· cos (θ)) · cos (θ) ·e−jβ10 (ξ·cos(θ)−η·sin(θ)) E"1 =. V0 w. cos. $π. 2·l ·ξ. %. ξ". (2.15). Placing eq. 2.15 and eq. 2.5 in eq. 2.12 and eq. 2.13, one obtains the relation between the physical parameters of the slot/waveguide and the amplitude of the scattered waves. B10 =. (π/a)2 ω·µ0 ·β10 ·a·b. (&. ˜. S. $π. + j· cos. a. $ −jβ10 · sin πa ( πa )·. % %) (D + a/2 + ξ· sin (θ) + η· cos (θ)) · cos (θ). · Vw0 cos C10 =. (π/a)2. ω·µ0 ·β10 ·a·b. ˜. + j· cos. S. (&. $π a. % (D + a/2 + ξ· sin (θ) + η· cos (θ)) · sin (θ). $π. jβ10. % −jβ (ξ·cos(θ)−η·sin(θ)) ·ξ ·e 10 dS 2·l. ( πa )·. · sin. $π a. $π. 2·l ·ξ. %. (2.16). (D + a/2 + ξ· sin (θ) + η· cos (θ)) · sin (θ). % %) (D + a/2 + ξ· sin (θ) + η· cos (θ)) · cos (θ). · Vw0 cos. %. ·ejβ10 (ξ·cos(θ)−η·sin(θ)) dS. (2.17). Once C10 and B10 are obtained as a function of the voltage in the slot’s aperture, the relation between them will condition the continuity of the electric and magnetic field at z = 0. Three cases deserve to be pointed out: 1. C10 = B10 , taking into account the expressions of eq. 2.1, at the plane z = 0 the tangential component of the electric field is zero, while the magnetic field in z = 0− and z = 0+ are in opposite phase. This case squares with an slot parallel to the waveguide axis. With respect to the running mode, the slot behaves like a shunt element in a transmission line, fig.1.1(a). 2. C10 = −B10 , forces the tangential component of the electric field to be discon-. tinuous and the magnetic field to show continuity at z = 0 plane. In this case,. 28.

(50) CHAPTER 2. Section 2.4. the slot is oriented with its axis perpendicular to the guide axis and it behaves like a series element in a transmission line, fig.1.1(b)(c). 3. C10 %= ±B10 None of the tangential components of the electric and magnetic. fields are continuos nor zero at z = 0. This fact causes that the slot cannot be characterized using a single shunt or series element. Since the equivalent currents and voltages are not continuos more complicated schemes are needed, fig.1.1(d).. If a new term E10 = from θ =. 0◦. to θ =. B10 C10. is defined and its absolute value is plotted for orientations. 180◦. for five different slot configurations, the three presented. cases can be identified in all of the curves, fig.2.4. Independently of the offset and length values, there are three points where all the graphs converge. For the longitudinal slot (θ = 0◦ and θ = 180◦ ) and for the transverse slot (θ = 90◦ ) addressing the first and the second case, respectively, the absolute value of the ratio E10 becomes one. If one goes deeper and the longitudinal or transverse slot is regarded as a two-port device with internal loss (to account for the radiated power), the corresponding impedance matrix can be completely defined with a single complex value. Away from these specific angles, the ratio between the amplitude of the forward and backward scattered waves depends on the structural parameters like offset and length. Contrary to points 1 and 2, assumptions about the fields in the slot plane can not be done and the corresponding Z matrix is symmetric so it requires three complex values to be univocally defined. This is the case of the compound slot, squaring case 3.. 2.4. Circuit Theory in Microwaves Systems. Working at milimetric and submilimetric bands, voltage and current concepts are not always suitable to be used as in low-frequency projects. However, an extension of those is sometimes useful to explain the physical behavior of the device and to analyze it by means of low frequency techniques.. 29.

(51) CHAPTER 2. Section 2.4. |En| quotient 10 D=!0.2"0, l="0/2 D=!0.12"0, l="0/2. 8. D=0.08"0, l="0/2. |En|. D=0.2"0, l=0.44"0 6. D=0.2"0, l=0.52"0. 4. 2 1 0. 30. 60. 90 120 Tilting angle ! [deg]. 150. 180. Figure 2.4: absolute value of the scattered field ratio, as a function of tilting angle. Generally speaking, inside transmission lines in which TEM modes are propagating, voltage and current waves exist and can be unequivocally ascribed to the transversal electric and magnetic fields, respectively. For transmitting TE and TM modes inside a waveguide, this is not possible. First, since there is only one conductor it is not easy to find the proper points where the voltage should be measured. Second, if the voltage is defined as the circulation of the electric field along a line between two designated points, it is found that for TM modes this integral becomes zero, while for TE this value depends on the integration line that is selected. Because of these reasons, the voltage and current waves inside a waveguide have a formal meaning, in contrast to the physical meaning of transmission lines. It is well known that within a waveguide it is possible to have more than one mode simultaneously propagating. Every single mode can be mathematically expressed as a combination of the transverse and longitudinal components: " + = C + ("et + "ez )·e−j·β·z E (2.18) " − = C − ("et − "ez )·ej·β·z E " + or for the electric field when the mode propagates along the positive z direction E. 30.