Diplexers and Multiplexers Design by Using Coupling Matrix Optimisation Wenlin Xia A thesis submitted to the University of Birmingham for the degree of Doctor of Philosophy School of Electronic, Electrical and Systems Engineering The University of Birmingham April, 2015 University of Birmingham Research Archive e-theses repository This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder. Abstract Microwave filters and multiplexers are used in many application areas and have been studied for decades. However, with increasing demands on communications and radar systems more complex filters are required which not only have superior performance but also are required to be small and lightweight. This thesis looks at new techniques in microwave filter design to achieve these aims. Coupled resonator circuits are of importance for design of RF/microwave narrow-band filters with any type of resonator regardless its physical structure. The coupling matrix is used to represent the coupled resonator circuit. Each matrix entry value refers to a physical dimension of the circuit. The response of the circuit can also be calculated by using the coupling matrix. Different methods are developed to generate the coupling matrix. This thesis presents designs of the coupled resonator based diplexers and multiplexers by using the coupling matrix local optimisation technique. By altering the values of the matrix, the optimisation program search for a particular matrix with the desired circuit response. The initial values of the matrix, which is used as the input of the optimiser, have a great effect on the convergence of the final result of the optimisation. The principles of generating the good initial values of the matrix are included in this thesis. The design procedures and measurement performance of 3 X-band (8.2-12.4 GHz) rectangular waveguide circuits, including a 10th order diplexer, a 4th order diplexer with cross-couplings and a 4-channel multiplexer, are presented. A novel computer-aided physical structure tuning technique, called Step Tune method, is also presented in this thesis. Instead of conventionally tuning the whole structure, we simulate and tune just part of the circuit by using the new method. As only limited number of physical dimensions is tuned each time, the final result is more reliable. i Acknowledgement I would like to express my deepest gratitude to my supervisor Professor Michael Lancaster for his support and guidance throughout my research. I would also like to express my appreciation to my progress report assessor Dr. Fred Huang for his useful advices. I would like to extend my sincerest thanks to my colleagues in the Emerging Device Technology Research Group at the University of Birmingham for their support over these years. In particular, I would like to thank Dr. Xiaobang Shang for many inspiring discussions over the project, Mofei Guo for both academic and personal help, Ekasit Nugoolcharoenlap for the discussion on the n+2 coupling matrix theory, Rashad Hassan Mahmud for the discussion on my thesis writing, Tianhao He for his talented photography technique. I’m also thankful Mr. Warren Hay for fabricating the waveguide devices presented in the thesis. Thanks to all of my friends at the University of Birmingham, especially, Mao Li, Yang Yue, Hao Li and Bing Hua, for their support after hours; and my Big Chefs, Leyang Xu and Yang Yang, for their guidance on my cooking. My appreciation goes to all of my friends in China, especially, Lei Wang and Hongbing Wang, for their encouragement over my life aboard. Special thanks to Mr Dongqing Cao and Mr Xigen Zhu, my high school teachers, for their words and experiences which raised me up. Finally, my sincere gratitude goes to my family: my father Bangren Xia, my mother Suying Lin and my uncle Zhigang Dai. Their love and support made my success possible. My life is meaningful because of my family. ii Table of Contents CHAPTER 1 INTRODUCTION ........................................................................................ 1 1.1 Overview of Diplexers and Multiplexers, and Their Applications ............................ 1 1.2 Thesis Motivation ....................................................................................................... 2 1.3 Thesis Overview ......................................................................................................... 5 CHAPTER 2 LITERATURE REVIEW ............................................................................ 8 2.1 General Filter Theory ................................................................................................. 8 2.2 Microwave Filter Theory ............................................................................................ 9 2.2.1 Lumped and Distributed Elements for Microwave Filters ..................................... 9 2.2.2 Microwave Filter Theory for Narrowband and Wideband Filters ....................... 10 2.2.3 Immittance Inverters ............................................................................................. 11 2.2.4 Low-Pass to Band-Pass Transformation .............................................................. 13 2.2.5 Coupling Matrix Theory ....................................................................................... 15 2.2.5.1 General n n Coupling Matrix of n-Coupled Resonator Circuit with 2- Port .................................................................................................................. 16 2.2.5.2 n n Coupling Matrix of n-Coupled Resonator Circuit with Multi-Port . 19 2.2.5.3 n+2 and n+X Coupling Matrix .................................................................. 21 2.2.6 Synthesis of the Coupling Matrix.......................................................................... 22 2.2.6.1 Synthesis Method Using Matrix Rotation ................................................ 23 2.2.6.2 Synthesis Method Using Optimisation ..................................................... 24 2.2.6.3 Comparison of Two Synthesis Methods ................................................... 25 2.2.7 Microwave Filter Design ...................................................................................... 26 2.3 Diplexers and Multiplexers ...................................................................................... 28 iii 2.4 Cross Couplings and Transmission Zeros ................................................................ 30 CHAPTER 3 REPRESENTATION OF N+X COUPLING MATRIX ......................... 35 3.1 Node Equation Formulation for Electrically Coupled Circuit.................................. 35 3.2 Loop Equation Formulation for Magnetically Coupled Circuit ............................... 47 3.3 General n+X Coupling Matrix .................................................................................. 56 3.4 Scale the n+X Coupling Matrix during the Frequency Transformation .................. 57 CHAPTER 4 COUPLING MATRIX SYNTHESIS BY OPTIMISATION ................. 61 4.1 Frequency Transformation of the Diplexer .............................................................. 61 4.2 Topologies of the Resonator Based Diplexers ......................................................... 63 4.3 Principles of the Starting Point ................................................................................. 65 4.3.1 Starting Values of the Branch Part....................................................................... 66 4.3.2 Starting Values of the Branches Having Chebyshev Responses........................... 68 4.3.3 Adjustment of the Branch Starting Point .............................................................. 70 4.3.4 Starting Values of the Stem Part .......................................................................... 72 4.3.5 External Quality Factor qe1 of the Stem ............................................................... 72 4.3.6 Coupling Coefficient mi,j and Self Coupling mi,i of the Stem ................................ 73 4.3.7 Initialise the Reflection Zeros ............................................................................... 74 4.4 Cost Function for the Optimisation .......................................................................... 75 4.5 Example A: a Diplexer Matrix Synthesised by Optimisation .................................. 77 4.5.1 The Specifications and Topology of the Diplexer ................................................. 77 4.5.2 Reducing the Total Number of Variables ............................................................. 78 4.5.3 Initialisation of the Starting Point ........................................................................ 80 4.5.3.1 Branch Couplings ..................................................................................... 80 iv 4.5.3.2 Stem Couplings ........................................................................................ 81 4.5.3.3 Reflection Zeros ....................................................................................... 81 4.5.4 Starting Point of the Diplexer ..............................................................................