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Investigation of a frequency multiplexer design for band splitting in a wideband feed antenna Moazen, Nima 2014

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 Investigation of a Frequency Multiplexer Design for Band Splitting in a Wideband Feed Antenna  by Nima Moazen  B.Sc., Malek-Ashtar University of Technology, 2011  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES  THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan)  January 2014  ? Nima Moazen, 2014    ii  Abstract This thesis describes the design of a frequency multiplexer for band splitting in a wideband feed antenna. The work is motivated by the design of a 1-8 GHz coaxial waveguide feed antenna intended for radio astronomy applications in the square kilometer array (SKA) project. The feed antenna consists of three nested coaxial waveguides each designed to operate over an octave bandwidth. The bands are 1-2 GHz, 2-4 GHz and 4-8 GHz, and a frequency multiplexer is required to connect a common input terminal and split the signal into three contiguous frequency bands. The common input terminal must be matched over the entire frequency range while the output ports are connected to coaxial waveguides.   The requirements for the multiplexer are challenging and a design is proposed based on t he theory of complementary diplexers. Complementary diplexers are three port filter networks which are matched at the input port and whose output ports are split into two complementary and contiguous bands. By cascading two complementary diplexers, a design can be realized that meets the SKA requirements. Different design concepts were evaluated using electromagnetic simulation tools and an important conclusion from the work is that compact capacitor structures are required to obtain broadband stop band responses in the diplexers. Two compact capacitor designs were fabricated and tested in diplexer configurations. One design uses interdigital capacitors and in another design thin dielectric metal insulator metal capacitors are used. Experimental prototypes were fabricated to compare with simulation results and the results are in good agreement. The research shows that complementary diplexer filter concepts can be used to realize broadband multiplexers for the SKA antenna feed.    iii  Table of Contents Abstract .......................................................................................................................................... ii List of Tables ................................................................................................................................ vi List of Figures .............................................................................................................................. vii Acknowledgements ..................................................................................................................... xii Dedication ................................................................................................................................... xiii Chapter 1:  Introduction and Background................................................................................. 1 1.1. Overview Of The Square Kilometer Array Project ............................................................. 1 1.2. Background .......................................................................................................................... 2 1.3. Frequency Multiplexer Requirements .................................................................................. 3 1.4. Research Objectives ............................................................................................................. 6 1.5. Scope .................................................................................................................................... 7 1.6. Summary of Chapters .......................................................................................................... 7 Chapter 2: Lumped Element Complementary Multiplexer Design ......................................... 8 2.1. Complementary Diplexer Theory ........................................................................................ 8 2.2. Singly Terminated Filters .................................................................................................. 13 2.3. Design Methodology .......................................................................................................... 16 2.3.1. First Order Multiplexer ............................................................................................... 19 2.3.2. Third Order Multiplexer ............................................................................................. 21 2.3.3. Fifth Order Multiplexer............................................................................................... 23  iv  2.4. Waveguide Loads at Termination Ports ............................................................................. 24 Chapter 3: Implementation of Complementary Diplexers ..................................................... 28 3.1. Classical Lumped Element to Transmission Line Transformations .................................. 29 3.1.1. Richards? Transformation ........................................................................................... 29 3.1.2. Kuroda?s Identities ...................................................................................................... 35 3.2. Transmission Lines and Bandwidth ................................................................................... 39 3.3. Interdigital Capacitors ........................................................................................................ 40 3.3.1. Closed Form Interdigital Capacitor Model ................................................................. 42 3.3.2. EM simulation of an Interdigital Capacitor ................................................................ 43 3.3.3. Experimental Measurements of an Interdigital Capacitor .......................................... 45 Chapter 4: Simulation and Measurement Results for Diplexers and Multiplexers ............. 53 4.1. Two GHz Diplexer with Interdigital Capacitor ................................................................. 53 4.1.1. Lumped Element Model ............................................................................................. 53 4.1.2. Distributed Model ....................................................................................................... 55 4.1.3. Measurement Results .................................................................................................. 60 4.2. Four GHz Diplexer with Interdigital Capacitor ................................................................. 64 4.2.1. Lumped Element Model ............................................................................................. 64 4.2.2. Distributed Model ....................................................................................................... 65 4.2.3. Measurement Results .................................................................................................. 69 4.3. Multiplexer ......................................................................................................................... 73  v  4.4. MIM Diplexer .................................................................................................................... 75 4.5. Waveguide Load Measurement with MIM Diplexer ......................................................... 78 Chapter 5: Conclusions .............................................................................................................. 82 References .................................................................................................................................... 84                vi  List of Tables Table 1. Singly terminated lowpass prototype values for a 0.5 dB Chebyshev filter. .................. 16 Table 2. First order multiplexer component values. ..................................................................... 19 Table 3.Third order multiplexer component values. ..................................................................... 21 Table 4. Fifth order multiplexer component values. ..................................................................... 23 Table 5. Standard rectangular waveguides. .................................................................................. 26 Table 6. Rogers 5880LZ material properties. ............................................................................... 57              vii  List of Figures Figure 1. Band splitting requirements for the SKA coaxial feed antenna. ..................................... 4 Figure 2. First proposed block diagram of multiplexer. ................................................................. 5 Figure 3. Second proposed block diagram of multiplexer. ............................................................. 6 Figure4. Lumped element circuit model for a Complementarydiplexer.                                                                                             This picture is adapted from [15]. ..................................................................................... 9 Figure5. Normalized input admittance characteristics                                                                              for each filter branch in a 2 GHz complementary diplexer. ........................................... 10 Figure 6. An example of the normalized input admittance characteristics of a 2 GHz complementary diplexer. ................................................................................................ 11 Figure 7. Y-parameter two port model for a filter branch. ........................................................... 13 Figure 8. Doubly terminated filter network. ................................................................................. 14 Figure 9. Singly terminated filter network. ................................................................................... 15 Figure 10. Complementaryfirst order multiplexer. ....................................................................... 19 Figure 11. Complementaryfirst order multiplexer S parameters. The S11  parameter (input      port match) is not shown in the figure due to fact it is below -50 dB. ......................... 20 Figure 12. Complementarythird order multiplexer. ...................................................................... 21 Figure 13. Complementarythird order multiplexer S parameters. ................................................ 22 Figure 14. Complementary lumped element multiplexer. ............................................................ 23 Figure 15. Complementaryfifth order multiplexer S parameters. ................................................. 24 Figure 16. Complementary multiplexer with waveguide terminations. ....................................... 26 Figure 17. S parameters for multiplexer with waveguide terminations at output ports. ............... 27  viii  Figure 18. Transmission line with characteristic impedance ?0 terminated with a load impedance  ??. .............................................................................................................. 30 Figure 19. A 2 GHz first order diplexer after using Richards? transformation. ............................ 32 Figure 20. S21 simulation comparison of lumped element and                                              Richards? transformed first order 2 GHz diplexer. ....................................................... 33 Figure 21. S31 simulation comparison of lumped element and                                                  Richards? transformed first order 2 GHz diplexer. ....................................................... 34 Figure 22. A series transmission line with a series stub. This photo is adapted from [20]. ......... 36 Figure 23. Application of Kuroda's identity to transform the series stub in Figure 19 into a     shunt stub. This photo is adapted from [6]. .................................................................. 36 Figure 24. A 2 GHz first order diplexer with the lowpass filter transformed using Kuroda's identity. ......................................................................................................................... 37 Figure 25. S21 comparison of three different first order diplexer designs: 1) lumped element,     2) design after using Richards? transformation and 3) design after using Kuroda's identity. ......................................................................................................................... 38 Figure 26. S31 comparison of three different first order diplexer designs: 1) lumped element,      2) design after using Richards? transformation and 3) design after using Kuroda's identity. ......................................................................................................................... 38 Figure 27. Interdigital Capacitor. This photo is adapted from [24]. ............................................. 41 Figure 28. Interdigital microstrip capacitor model in AWR. ........................................................ 42 Figure 29. Attenuation characteristics for an interdigital capacitor. ............................................. 43 Figure 30. Interdigital capacitor with two 4.4mm port transmission lines. .................................. 44 Figure 31: Interdigital capacitor with tapered ports to improve matching to SMA connectors. .. 44  ix  Figure 32. EM simulation results for two different interdigital capacitor designs. ...................... 45 Figure 33. 2 GHz interdigital capacitor with normal ports. .......................................................... 46 Figure 34.  2 GHz interdigital capacitor with tapered ports. ........................................................ 47 Figure 35. Measured results for an interdigital capacitor. The measurement is S21. ................... 48 Figure 36. Measured results for an interdigital capacitor. The measurement is S11. ................... 49 Figure 37. Comparison of measured and simulation results for an interdigital capacitor. ........... 49 Figure 38. A MIM capacitor fabricated with copper tape and a thin dielectric ............................ 51 Figure 39. Measured S-parameters for the MIM capacitor. .......................................................... 52 Figure 40. Comparison of measured responses for 1.6 pF MIM and 1.6 pF                     interdigital capacitors. .................................................................................................. 52 Figure 41.  Lumped element model for a first order 2 GHz diplexer. .......................................... 54 Figure 42.  Simulation results for a lumped element 2 GHz diplexer. ......................................... 55 Figure 43. Distributed model for a 2 GHz first order diplexer. .................................................... 56 Figure 44. 3-D view of first order 2 GHz diplexer. ...................................................................... 57 Figure 45. S parameter simulation results for a distributed 2 GHz diplexer using an            interdigital capacitor. .................................................................................................... 58 Figure 46. Comparison of simulation results for the lowpass filter branch in                                  2 GHz diplexers. ........................................................................................................... 59 Figure 47. Comparison of simulation results for the highpass filter branch in                                 2 GHz diplexers. ........................................................................................................... 59 Figure 48. Photograph of the 2 GHz diplexer with an intedigital capacitor. ................................ 60 Figure 49. Dimensions (in mm) for the fabricated 2 GHz interdigital diplexer. .......................... 61  x  Figure 50. Simulation and measurement results for the highpass filter branch in the 2 GHz interdigital diplexer. ..................................................................................................... 63 Figure 51. Simulation and measurement results for the lowpass filter branch in the 2 GHz interdigital diplexer. ..................................................................................................... 63 Figure 52. 4 GHz lumped element model. .................................................................................... 64 Figure 53. The lowpass and highpass filter responses for the 4 GHz diplexer. ............................ 65 Figure 54. Distributed 4 GHz diplexer schematic. ....................................................................... 66 Figure 55. 3-D view of the first order 4 GHz diplexer. ................................................................ 67 Figure 56. S Parameter simulation results for the 4 GHz diplexer  with an interdigital        capacitor. ...................................................................................................................... 68 Figure 57. Comparison of lumped element and electromagnetic simulation results for the highpass filter branch in the 4 GHz diplexer. ............................................................... 68 Figure 58. Comparison of lumped element and electromagnetic simulation results for the lowpass filter branch in the 4 GHz diplexer. ................................................................ 69 Figure 59. Photograph of the 4 GHz diplexer with an intedigital capacitor. ................................ 70 Figure 60 . Dimensions (in mm) for the 4 GHz interdigital diplexer. .......................................... 71 Figure 61. Simulation and measurement results for the highpass filter branch in the 4 GHz interdigital diplexer. ..................................................................................................... 72 Figure 62. Simulation and measurement results for the lowpass filter branch in the 4 GHz interdigital diplexer. ..................................................................................................... 72 Figure 63. First order multiplexer using interdigital capacitors. .................................................. 73 Figure 64. S parameters simulation for the first order interdigital multiplexer. ........................... 74  xi  Figure 65. The AWR TXLINE tool to estimate the characteristic impedance for an inductor design. ........................................................................................................................... 76 Figure 66. 2 GHz MIM Diplexer .................................................................................................. 77 Figure 67. Cross-section of the first order highpass filter using a MIM capacitor. ...................... 77 Figure 68. The measured frequency response (S21) of the 2 GHz MIM Diplexer. ..................... 78 Figure 69. MIM Diplexer with a waveguide load on the highpass port. ...................................... 79 Figure 70. Lowpass comparison with waveguide load. ................................................................ 81 Figure 71. MIM diplexer with a highpass port termination to model a waveguide in cut-off. ..... 81             xii  Acknowledgements First and foremost, I would like to express my sincere gratitude to my supervisor, Dr. Thomas Johnson, for his research direction and excellent vision, great inspiration, patient guidance, caring, and generous support throughout my studies. I also would like to express my great appreciation to Dr. Thomas Landecker for his exceptional expertise and guidance. I would like to thank all the professors at the school of engineering whom I attended their classes and lectures. I would like to also thank my colleagues for their support throughout the work. I also thank Vipul Vishnoi for his help on laser milling machine. The last but not the least, I would like to thank my family for their continuous support and encouragement and unconditional love.          xiii  Dedication                                                                  To My Parents, Morteza and Tahereh   1  1.Chapter 1:  Introduction and Background 1.1. Overview Of The Square Kilometer Array Project This research project is motivated by a much larger project called the Square Kilometre Array (SKA). The SKA is sponsored by a consortium of countries including Canada and the goal of the project is to implement the world?s largest radio telescope to support scientists in their quest to answer fundamental questions about the universe. The SKA project was initiated by t he International Union of Radio Science (URSI) in September 1993. It?s taken many years to gain the sponsorship of participating countries, but the project is now gaining momentum and prototype equipment is currently being developed for it. The SKA will eventually consist of an array of 2500 parabolic receiving antennas that are distributed over thousands of kilometers. Two locations have been selected for the antenna: South Africa and Australia. The locations are in the middle of desert regions where interference from terrestrial radio frequency sources is minimized. Since the signals received from the farthest reaches of the universe are very small, impairments from terrestrial signals must be minimized.  Scientists have interest in receiving signals over a very broad frequency range from 20 MHz to 800 GHz. However, building antennas, low noise amplifiers, and signal processing hardware that can support this frequency range is extremely challenging, and the frequency range is subdivided into broadband frequency segments which are much more likely to be implementable with current technology. One of the frequency bands spans a frequency range from 1-8 GHz and this is the focus of this work [1].  2  1.2. Background  In a former research project, a collaboration between the University of British Columbia (Okanagan) and the Dominion Radio Astrophysical Observatory (DRAO), a wideband feed antenna was implemented. The design was based on a nested coaxial waveguide structure where each waveguide supported an octave frequency range. The research included the fabrication of a prototype feed antenna and the test results established the feasibility of the design concept. The research also laid groundwork for new research projects to continue to improve and optimize the design.  One of the limitations in the prototype feed was the excitation structure which coupled a single input to the nested waveguides. The design consisted of three nested coaxial waveguides. The outer guide spanned a frequency range of 1-2 GHz, the middle guide spanned a frequency range of 2-4 GHz, and the inner guide spanned a frequency range of 4-8 GHz. An excitation structure was designed to couple the guides to a common output port which connects to the low noise amplifier. The design worked well for some frequencies and not so well for other frequencies. Problems included higher order modes in waveguides and poor matches; a return loss of 10 dB over the entire frequency range is required. The most difficult frequency ranges tended to be in the neighborhood of frequency transitions from one guide to another. The frequency transition from one guide to another requires a transition through the cut-off frequency range of the higher frequency guide to frequencies beyond cut-off, and over this transition the impedance of the guide changes significantly [2, 3]. A conclusion from prior research based on the first prototype of the nested coaxial waveguide feed design is that improvements are required in the implementation of an excitation structure for  3  the waveguides. The hypothesis that is central to the work in this thesis is that a better excitation structure could be designed using the theory of complementary filters. The theory of complementary filters is explained in more detail in the next chapter but the principal concept is to synthesize two filter branches which are shunted at a common input node such that the shunted impedance is matched termination for all frequencies.  The theory of complementary filter design dates back to the 1960?s and was studied extensively by Wenzel [4-7]. However, most of these designs had narrow passbands [8-13] and the implementation methods using distributed microwave transmission line structures are not easily extended to multioctave bandwidths as required in the SKA project. The challenges with coupled line filter structures which were used in early filter design work are described in more detail in Chapter 3.  1.3. Frequency Multiplexer Requirements The current design of the nested coaxial waveguide feed for the SKA project spans a frequency range of 1-8 GHz. The design consists of three concentric coaxial waveguides where each waveguide spans one octave. The frequency bands for the three guides are: i) 1-2 GHz, ii) 2-4 GHz, and iii) 4-8 GHz. The entire frequency range must be covered contiguously including frequency transitions between the different waveguides. Figure 1 shows the three frequency bands.  4   Figure 1. Band splitting requirements for the SKA coaxial feed antenna. The three band waveguide feed antenna needs to be fed from a single microwave port which connects to a low noise amplifier (LNA). The single input port then needs to be frequency divided between the three frequency bands and coupled to the excitation probes in each waveguide. Therefore, a three band frequency division multiplexer is required. The common input port needs to be matched over the entire frequency range from 1 to 8 GHz. An input match with a 10 dB return loss over the 1-8 GHz frequency range is proposed. One of the most challenging requirements for the multiplexer is to maintain the input match with output ports that are mismatched. For example, if the signal falls within the 1-2 GHz band, then the impedance presented by the coaxial waveguides for the other two bands is highly reflective because both guides are below the cut-off frequency. Conversely, in the high frequency band of 4-8 GHz, all the guides can propagate the frequency band and isolation is required to ensure only the high band guide is propagating.  As a way to meet these requirements, a multiplexer design using the concepts of complementary diplexers is proposed. A diplexer is a three terminal filter structure which consists of two filter branches which are connected in shunt at a common input terminal.  The two filter branches are complementary if they can be shunted at a common input node in such a way that the input  5  impedance is matched for all frequencies including the passband and stopband responses of the filter branches. More background on complementary diplexer theory is given in chapter 2, but accepting this approach for now, the multiplexer can then be implemented by a  cascade of complementary diplexers as shown in Figure 2 and Figure 3. In Figure 2, a high band split is preformed first, while in Figure 3 a low band split is performed first. Given that implementation losses usually increase as frequency increases, the high band split is recommended. Also, although the required frequency bands could be separated with three bandpass filters, the low band and the high band do not necessarily need to have bandpass responses. The low band signal separation could be achieved with a lowpass filter and the high band could be separated with a high pass filter. The dashed lines in Figure 1 show the corresponding response for a lowpass and high pass filter response in the multiplexer.  Common InputBand 3Band 21-4 GHzBand 12-4 GHz1-2 GHz4-8 GHz Figure 2. First proposed block diagram of multiplexer.    6  Common InputBand 1Band 22-8 GHzBand 32-4 GHz4-8 GHz1-2 GHz Figure 3. Second proposed block diagram of multiplexer.  1.4.  Research Objectives The objective of this research project was to investigate methods to implement wideband complementary filters with passband and stopband characteristics that would meet the requirements of the multioctave coaxial feed antenna. In pursuing this objective, many design approaches turned out to be unsuitable and could not meet the bandwidth requirements.  Eventually the research led to the conclusion that electrically small components (i.e. physical designs for capacitors and inductors that are significantly less than a wavelength long) are required to limit spurious responses in either the stopband or passband of the filters. In fact, although the work began with attempts to use classical microwave distributed structures, the conclusion after exploring these designs was that direct implementations of filter structures which mimic lumped element filter models worked better. Two physical designs show promise:    7  1) designs which use interdigital capacitor structures, and  2) small area parallel plate capacitors implemented with thin dielectrics.  1.5. Scope Within the time frame of this project, it was possible to investigate the design and implementation of first order diplexing networks. Simulation results of higher order filters are also shown. However, there remains further work to investigate and implement complete three band multiplexers for the SKA feed project. This work will be continued in future research projects.  1.6. Summary of Chapters In the next chapter, the theory of complementary diplexers is reviewed and lumped element models for diplexer and multiplexer designs are shown. Chapter 3 t hen builds on t he lumped element filter models and transforms the designs into distributed element designs that are physically realizable at high frequencies. The chapter includes the design of compact capacitors implemented using interdigital structures and thin dielectric parallel plate structures. In Chapter 4, measurement results are shown for first order diplexer designs. Chapter 4 also shows measurement results for a diplexer with a waveguide load. Finally, conclusions from the research project are given in Chapter 5.    8  2.Chapter 2: Lumped Element Complementary Multiplexer Design As described in Chapter 1, a multioctave frequency multiplexer is required for the SKA nested coaxial antenna feed. The multiplexer is implemented using a cascade of complementary diplexers which are configured to create a single-input three-output filter structure which could also be called a triplexer. In this chapter, the synthesis of complementary filter structures is reviewed. The complementary diplexer filter block is then used to synthesize a lumped element model for the three band multiplexer. In this chapter the design procedure to achieve ideal lumped element complementary multiplexer is explained and before starting the distributed design of the project in the next chapter. The idea of using complementary multiplexer design with waveguide loads is tested.   2.1. Complementary Diplexer Theory By definition, a complementary diplexer is a three port device that splits the input frequency spectrum into two complementary frequency bands. An example of a complementary diplexer is shown in Figure 4 and consists of two complementary filter branches that are connected in shunt at an input node. The term ?complementary? imposes constraints on the synthesis of the two filter branches such that the input impedance at the common port is independent of frequency. For example, an input impedance of 50 Ohms is required in this design. In this case, an input impedance is required that is purely real for all frequencies and the reactive part is ideally zero. The frequency response of the filter branches are also complementary and include lowpass and highpass filters as well as bandpass and bandstop filters [4, 7, 14].  9   Figure 4. Lumped element circuit model for a complementary diplexer. This picture is adapted from [15]. Given the shunt connection of two filter branches at the input node, it is much more convenient to work in terms of admittance. Further, relations are simplified by normalizing the admittance such that the input admittance is unity. With reference to Figure 4, the total input admittance is Yin =  YinLP + YinHP = 1  (2.1) In general the input admittance of each branch is complex. Therefore, in order for (2.1) to be satisfied, we require Re[YinLP] + Re[YinHP] = 1 (2.2 ) Im[YinLP] + Im[YinHP] = 0 (2.3 )   Equations (2.2) and (2.3) define the complementary admittance conditions that are required to ensure the input admittance of the common input node is constant and independent of frequency. Equation (2.3) clearly requires the susceptance characteristics of each filter branch to be conjugates over the entire frequency range of the design. The real part of the admittance of each  10  filter branch must also be complementary so the net sum is always unity. An example of admittance characteristics which satisfy the complementary condition is shown in Figure 5 and Figure 6. The results are for a filter design which is described later in Chapter 2. In Figure 5, the input admittance characteristics for each filter branch (????? and ?????) are shown and the conjugate characteristics of susceptance are clearly seen. In Figure 6, the overall admittance at the input port (?????) is shown. As shown, the normalized input admittance is purely real and equal to unity [15, 16].  Figure 5. Normalized input admittance characteristics for each filter branch in a 2 GHz complementary diplexer. 0 2 4 6 8 10Frequency (GHz)2 GHz Complementary Diplexer-2-1012Normalized AdmittanceReal_Normalized Input Admittance_LPReal_Normalized Input Admittance_HPImaginary_ Normalized Input Admittance_HPImaginary_Normalized Input Admittance_LP 11   Figure 6. An example of the normalized input admittance characteristics of a 2 GHz complementary diplexer. Another expression for the complementary condition is found by modeling each filter branch as an equivalent two port network as shown in Figure 7. If Y-parameters are used to describe a single filter branch, and if the filter branch is lossless, then the power delivered to the input of the network must equal the power delivered to the output of the network. From these constraints it can be shown that Re[Yin] = |Y12|2 (2.4) where Y12 is the corresponding Y-parameter for the equivalent two port network representation of the filter branch. Equation (2.4) is derived as follows.  0 2 4 6 8 10Frequency (GHz)Normalised Input Admittance-2-1012Normalised AdmittanceReal AdmittanceImaginary Admittance  12  With reference to Figure 7 the Y-parameters and terminal currents and voltages are related by the following equation ?I1I2? = ?Y11 Y12Y21 Y22? ?V1V2? (2.5)  Assume the filter branch is terminated in a resistive load ? as shown in Figure 7. If the filter network is lossless, then all the input power at port 1 must be dissipated in the load ? at the output. Conservation of power therefore requires  ??? = ???? (2.6) which can be expressed as  I1. I1?Re(Yin)=V2. V2?R (2.7) From matrix equation (2.5), ?12 is defined as ?1 ?2?  when  ?1 =  0 . Rearranging (2.7) then gives |I1|2|V2|2= |Y12|2 =Re(Yin)R (2.8) For a normalized load of 1 Ohm, equation (2.8) then reduces to (2.4).  Using the relation in (2.4), equation (2.2) can then be expressed as |Y12|LP2 + |Y12|HP2 = 1 (2.9)   13   Figure 7. Y-parameter two port model for a filter branch.  Although equations (2.1) through (2.9) explain conditions required to meet the complementary conditions required for a diplexer, they do not give insight into how to synthesize the required filter characteristics. As will be shown in the next section, the synthesis of each filter branch is possible using the concept of singly terminated filters.  2.2.  Singly Terminated Filters Most introductory filter theory begins with the synthesis of doubly terminated filters. In a doubly terminated filter, as shown in Figure 8, the filter matches a source with a resistance of ?1 to a load of resistance ?2 over a specified frequency range. Filter tables and frequency transformations can be used to synthesize LC ladder networks with the required passband and stopband attenuation characteristics. Unfortunately doubly terminated filter theory cannot be applied to the complementary diplexer problem because it treats the design of a filter branch in isolation without considering the change in source impedance that occurs when the input is shunted by another filter branch.  14   Figure 8. Doubly terminated filter network. Complementary filter structures must be synthesized under a loaded condition taking into account the impedance (admittance) characteristics of the shunting filter branch. A key observation to make is that when the two filter inputs are shunted, the net input impedance is constant and independent of frequency. Under this condition, the voltage across the input terminal is independent of frequency, in contrast to the doubly terminated filter which has an input terminal voltage that depends on f requency because the filter input impedance changes with frequency. Observing that the complementary condition corresponds to a constant input voltage, this is equivalent to applying a voltage source directly across the input terminals of the diplexer. If we now return to the doubly terminated filter and let ?1 go to zero, the filter input is then driven directly by a voltage source. When a filter is driven by an ideal voltage source or the dual, an ideal current source, the filter is called a singly terminated filter. The only resistance in the circuit is ?2, the load.  Since a voltage source has zero internal impedance and can drive any impedance, it means the common input node of the diplexer can be decoupled.      15   Figure 9. Singly terminated filter network. Singly terminated filter design tables are available in the literature with classical attenuation characteristics such as Butterworth and Chebyshev responses. An example of a singly terminated design table for a normalized lowpass filter is given in Table 1. Complementary filters are designed by transforming the lowpass prototype from the filter table into complementary lowpass and highpass filters with identical cut-off frequencies. A lowpass prototype in the filter table can also be transformed into complementary bandpass and bandstop filters. In this project, complementary lowpass and highpass filters are used. One difference between singly terminated filters and doubly terminated filters is that singly terminated filters are always asymmetric in that every component value in an LC ladder network is different, while in a doubly terminated filter, with equal source and load resistances, there is symmetry in the LC ladder values about the midpoint of the network.  The singly terminated filter structure is used in next section to design complementary diplexers for the SKA multiplexer.      16  2.3. Design Methodology Singly terminated filter prototypes are used in this work to design lumped element diplexers and multiplexers. All filter designs are based on a  Chebyshev 0.5 dB in-band ripple lowpass filter prototype with a 1 rad/s cut-off frequency and a 1 Ohm termination impedance. The lowpass prototype component values are summarized in Table 1. Frequency, impedance, and lowpass to highpass filter transformations [17, 18] are used to map the lowpass prototype values into filters which meet the requirements of the SKA project.   Table 1. Singly terminated lowpass prototype values for a 0.5 dB Chebyshev filter. Order ?1 ?2 ?3 ?4 ?5 ?6 ?7 1 1.00000       2 1.36144 1.01565      3 1.57200 1.51790 0.93182     4 1.45345 1.91162 1.53945 0.92395    5 1.62994 1.73996 1.92168 1.51377 0.90343   6 1.46994 1.99089 1.79019 1.93593 1.51606 0.90305  7 1.64643 1.77716 2.03065 1.78918 1.92388 1.50337 0.89478      17  The SKA antenna feed requires a three band split which can be implemented using two diplexers. Therefore, there are four filter designs required for a multiplexer. These are summarized below: ? Lowpass filter with a 2 GHz cut-off frequency ? Highpass filter with a 2 GHz cut-off frequency ? Lowpass filter with a 4 GHz cut-off frequency ? Highpass filter with a 4 GHz cut-off frequency Other important specifications are that the common input port should have an input return loss of at least 10 dB over the frequency range from 1 GHz to 8 GHz. The design procedure for the multiplexer design is summarized below:  1) Select the filter order and the normalized component values from Table 1. These values correspond to a lowpass filter with a cut-off frequency of 1 r ad/s and a termination impedance of 1 Ohm. 2) Frequency scale to shift the cut-off frequency to 2 GHz. Inductor and capacitor values for the prototype in step 1 are divided ??. 3) Scale the filter impedance in step 2 to increase the termination resistance from 1 Ohm to 50 Ohms. Inductor values are multiplied by ? and capacitor values are divided by ? where ? is 50 Ohms.  4) Steps 2 and 3 complete the design of the 2 GHz lowpass filter. 5) Create a co mplementary highpass filter with a 2 GHz cut-off frequency by f irst creating a highpass filter prototype. The LC ladder consists of series capacitors and shunt inductors with values given by Table 1.  18  6) Apply the same frequency and impedance scaling as in steps 2 and 3. This completes the design of the 2 GHz high pass filter. 7) The 4 GHz lowpass filter and highpass filter design repeat steps 2 through 6 except the cut-off frequency is changed from 2 GHz to 4 GHz. The above procedure was used to design three different lumped element multiplexers. Each multiplexer has the same cut-off frequencies and the designs differ only in the order of the filters used in the filter branches. Filters with first, third and fifth order networks were designed. The circuit diagrams for the multiplexers are shown in Figure 10, Figure 12, and Figure 14. The associated component values are summarized in Table 2, Table 3, and Table 4, respectively. The multiplexer designs were simulated using a so ftware tool called AWR [19] and the S-parameters for the filter designs are shown in Figure 11, Figure 13, and Figure 15. In Figure 11, it can be seen that the first order filter response affects the passband attenuation characteristics especially for the middle band from 2-4 GHz. The in-band attenuation is impacted by the slow roll-off (20 dB/decade) of the stopband from the first order response. Although the attenuation characteristics are not acceptable for a f irst order design, it does demonstrate complementary filter characteristics and the input terminal has a return loss in excess of 50 dB and falls outside the graph axes. The results for the third order design in Figure 13 and the fifth order design in Figure 15 shows that passband attenuation characteristics are improved significantly by increasing filter order. The fifth order design is excellent and ultimately experimental designs of this order are required for the SKA project. However, within the scope of this project, experimental results were measured for the first order design only. These results are shown later in Chapter 4.  19  2.3.1. First Order Multiplexer  Table 2. First order multiplexer component values. ?1,??(nH) ?2??(pF) ?3,??(nH) ?1,??(pF)  3.979 1.592 1.989 0.7958   Figure 10. Complementary first order multiplexer. INPUT PORT:11-2GHz OUTPUT PORT:22-4GHz OUTPUT PORT:34-8GHz OUTPUT PORT:4PORTP=4Z=50 OhmPORTP=3Z=50 OhmPORTP=2Z=50 OhmPORTP=1Z=50 OhmINDID=L2L=L2lp nHCAPID=C2C=C2hp pFINDID=L1L=L1lp nHCAPID=C1C=C1hp pF 20   Figure 11. Complementary first order multiplexer S parameters. The S11 parameter (input port match) is not shown in the figure due to fact it is below -50 dB.        0 2 4 6 8 10Frequency (GHz)First Order Multiplexer S Parameter Simulation  -50-40-30-20-1005Magnitude (dB)3.99 GHz-3.994 dB2.308 GHz-3.676 dBDB(|S(1,1)|)First_Order_Complementary_MultiplexerDB(|S(2,1)|)First_Order_Complementary_MultiplexerDB(|S(3,1)|)First_Order_Complementary_MultiplexerDB(|S(4,1)|)First_Order_Complementary_Multiplexer 21  2.3.2. Third Order Multiplexer  Table 3.Third order multiplexer component values.       Figure 12. Complementary third order multiplexer. 1-2GHz OUTPUT PORT:22-4GHz OUTPUT PORT:34-8GHz OUTPUT PORT:4PORTP=4Z=50 OhmPORTP=3Z=50 OhmPORTP=2Z=50 OhmPORTP=1Z=50 OhmINDID=L6L=L3lp2 nHINDID=L3L=L2hp2 nHINDID=L2L=L1lp2 nHCAPID=C6C=C3hp2 pFCAPID=C3C=C2lp2 pFCAPID=C2C=C1hp2 pFINDID=L5L=L3lp nHINDID=L4L=L2hp nHINDID=L1L=L1lp nHCAPID=C5C=C3hp pFCAPID=C4C=C2lp pFCAPID=C1C=C1hp pF?1,?? (nH) ?2,?? (pF) ?3,?? (nH) ?1,??(pF) ?2,?? (nH) ?3.?? (pF) 6.255 6.255 3.708 1.012 2.621 1.708 ?1,??2 (nH) ?2,??2 (pF) ?3,??2 (nH) ?1,??2 (pF) ?2,??2 (nH) ?3.??2 (pF) 3.127 1.208 1.845 0.5062 1.311 1.708  22   Figure 13. Complementary third order multiplexer S parameters.          0 2 4 6 8 10Frequency (GHz)Third Order Multiplexer S Parameter Simulation  -50-40-30-20-1005Magnitude (dB)2.008 GHz-3.095 dB4.133 GHz-3.09 dBDB(|S(3,1)|)Third_Order_Complementary_MultiplexerDB(|S(2,1)|)Third_Order_Complementary_MultiplexerDB(|S(1,1)|)Third_Order_Complementary_MultiplexerDB(|S(4,1)|)Third_Order_Complementary_Multiplexer 23  2.3.3. Fifth Order Multiplexer  Table 4. Fifth order multiplexer component values.   Figure 14. Complementary lumped element multiplexer. PORTP=4Z=50 OhmPORTP=3Z=50 OhmPORTP=2Z=50 OhmPORTP=1Z=50 OhmINDID=L15L=L4hp2 nHINDID=L14L=L2hp2 nHINDID=L13L=L5lp2 nHINDID=L12L=L3lp2 nHINDID=L11L=L1lp2 nHINDID=L12L=L4hp nHINDID=L4L=L1lp nHINDID=L3L=L5lp nHINDID=L2L=L3lp nHINDID=L1L=L1lp nH 1-2GHz OUTPUT PORT:22-4GHz OUTPUT PORT:34-8GHz OUTPUT PORT:4INPUT PORT:1CAPID=C1C=C2lp pFCAPID=C4C=C3hp pFCAPID=C3C=C1hp pFCAPID=C15C=0.881 pFCAPID=C14C=C3hp2 pFCAPID=C13C=C1hp2 pFCAPID=C12C=C4lp2 pFCAPID=C11C=C2lp2 pFCAPID=C17C=C5hp pFCAPID=C2C=C4lp pF?1,?? (nH) ?2,?? (pF) ?3,?? (nH) ?4,?? (pF) ?5,?? (nH) 6.488 2.77 7.65 2.41 3.596 ?1,??(pF) ?2,?? (nH) ?3.?? (pF) ?2,?? (nH) ?3.?? (pF) 0.976 2.28 0.828 2.62 1.761 ?1,??2 (nH) ?2,??2 (pF) ?3,??2 (nH) ?2,??2 (pF) ?3,??2 (nH) 3.243 1.385 3.824 1.204 1.797 ?1,??2 (pF) ?2,??2 (nH) ?3.??2 (pF) ?2,??2 (nH) ?3.??2 (pF) 0.4883 1.143 0.4142 1.315 0.881  24   Figure 15. Complementary fifth order multiplexer S parameters.  2.4. Waveguide Loads at Termination Ports In the previous section, results are shown for broadband terminations at the output ports of the multiplexer. However, for the SKA project, the port loads are coaxial waveguides designed to be matched for the dominant ??11 mode. Although the waveguide is matched over the specific output band, the termination impedance presented by the waveguide is frequency dependent. An important port termination condition is to evaluate the multiplexer design when a coaxial waveguide is operating in the cut-off frequency range. For example, over the 1-2 GHz band, the port impedance presented by the coaxial waveguides at the other two output ports corresponds to the cut-off regime for the waveguides. In the cut-off region, the input impedance to the guide becomes highly reflective and at very low frequencies it approaches an open circuit. The 0 2 4 6 8 10Frequency (GHz)Fifth Order Multiplexer S Parameter Simulation  -50-40-30-20-1005Magnitude (dB)4 GHz-3.007 dB2 GHz-2.937 dBDB(|S(4,1)|)Fifth_Order_Complementary_MultiplexerDB(|S(3,1)|)Fifth_Order_Complementary_MultiplexerDB(|S(2,1)|)Fifth_Order_Complementary_MultiplexerDB(|S(1,1)|)Fifth_Order_Complementary_Multiplexer 25  excitation structure in the guide consists of probes which couple to the electric field and the probes are electrically open at low frequencies. Therefore, the performance of the multiplexer with waveguide loads needs to be evaluated. As a way to investigate the impact of waveguide loads in the cut-off regime, a simulation model in AWR (Microwave office Simulation software) is evaluated. AWR has models for rectangular waveguide loads and the circuit model is shown in Figure 16. Working with rectangular waveguide, there are standard waveguide bands, and the cut-off frequencies of the filter branches are adjusted to work with the standard waveguides. A summary of the waveguide loads is given in Table 5.  Before proceeding to a discussion of simulation results with the waveguide load, further comments on the complementary filter design are made. The complementary filter is matched at the input over all frequencies but output ports are matched only over the in-band frequency range of the filter and become reflective in the stop band. The output port is therefore similar to a classical filter design which is matched in-band and reflective out-of-band. Therefore, if a waveguide is in cut-off in the stop band of the filter, we do not expect significant changes in performance. The simulation results for waveguide terminations are shown in Figure 17. As shown, the return loss is still very good and S11 is less than -20 dB which is a very acceptable match. Therefore, we conclude that the multiplexer design should be acceptable for waveguide loads.    26   Figure 16. Complementary multiplexer with waveguide terminations.  Table 5. Standard rectangular waveguides.  Port 1 Port 2  Port 3 Waveguide Standard WR-430 WR-284 WR-187 Width (mm) 109.22 72.136  47.5488  Height (mm) 54.61  34.036 22.1488  Frequency Band R (1.70 to 2.60  GHz ) S (2.60 to 3.95 GHz ) G (3.95 to 5.85 GHz ) Cut-off Frequency 1.3724 GHz 2.0780 GHz 3.1525 GHz  COMMON PORT:11-2GHz PORT2-4GHz PORT4-8GHz PORTINDID=L2L=5.88 nHINDID=L1L=4.987 nHCOAXRWG_TE10 COAXRWG_TE10COAXRWG_TE10 COAXRWG_TE10RWG_TEmnRWG_TEmnRWG_TEmnINDID=L15L=1.01 nHINDID=L14L=0.8793 nHINDID=L13L=1.3822 nHINDID=L12L=2.94 nHINDID=L11L=2.49 nHCAPID=C15C=0.6775 pFCAPID=C14C=0.318 pFCAPID=C13C=0.3755 pFCAPID=C12C=0.9269 pFCAPID=C11C=1.065 pFCAPID=C5C=1.3548 pFCAPID=C4C=0.637 pFCAPID=C3C=0.75 pFCAPID=C2C=1.8532 pFCAPID=C1C=2.13 pFINDID=L5L=2.02 nHINDID=L4L=1.75 nHINDID=L3L=2.7645 nHPORTP=4Z=50 OhmPORTP=3Z=50 OhmPORTP=1Z=50 OhmPORTP=2Z=50 OhmCOAXRWG_TE10 COAXRWG_TE10 27   Figure 17. S parameters for multiplexer with waveguide terminations at output ports.        1.4 3.4 5.4 7.4 9.4 10Frequency (GHz)S Parameter Simulation of  Complimentary Multiplexer-30-25-20-15-10-505Magnitude (dB)5.206 GHz-3 dB2.6 GHz-3 dBDB(|S(2,1)|)Complementry_Multiplexer_Waveguide_LoadDB(|S(3,1)|)Complementry_Multiplexer_Waveguide_LoadDB(|S(4,1)|)Complementry_Multiplexer_Waveguide_LoadDB(|S(1,1)|)Complementry_Multiplexer_Waveguide_Load 28  3.Chapter 3: Implementation of Complementary Diplexers  In the previous chapter, lumped element components are used to implement the multiplexer circuits. The lumped components are ideal, have zero length, and infinite Q. Implementing the design at microwave frequencies presents many challenges. These challenges include the physical length of components which start to become a significant fraction of a wavelength. When the physical length of components and transmission lines is significant, the phase shift along the length of the component is significant and the signals which combine and split at circuit nodes will no longer lead to the same response as the lumped element model with zero length component models. Other challenges include losses due to parasitic capacitance and parasitic inductance which limit component Q, dielectric losses in the substrate, conduction losses from the skin effect, and possibly radiation losses. Consequently, it is difficult to implement low-loss broadband filter structures and these limitations are illustrated in this chapter. It will be shown that physical length significantly impacts the bandwidth of stop bands and compact filter structures are required to realize broadband performance. After reviewing different implementation methods, the conclusion is that compact capacitor structures implemented with interdigital or metal insulator metal (MIM) capacitors have better performance than classical transformations of the lumped components into transmission line components.       29  3.1. Classical Lumped Element to Transmission Line Transformations A common approach to microwave filter design is to transform lumped element prototype circuits such as t hose shown in Chapter 2 into transmission line circuits. In this approach, capacitors are transformed into open circuit transmission stubs and inductors are transformed into short circuit stubs. The transformation is called Richards? transformation. However, a direct application of Richards? transformation for series lumped element components does not yield realizable transmission line structures for microstrip. The reason is that the stub requires access to two terminals while microstrip transmission lines have a common ground for all transmission lines. The limitation of series stubs is addressed by transforming the series stubs into equivalent shunt stubs using Kuroda?s identity. These transformations are reviewed in the next two sections. The broadband response of the filter is then shown to have a periodic response which significantly limits the stopband bandwidth of the filter.  3.1.1. Richards? Transformation Consider a transmission line of characteristic impedance ?0, length ?, and terminated in a load ?? as shown in Figure 18. If the transmission line is lossless, then it is well known [17] that the input impedance is given by Zin = Z0ZL + jZ0tan??Z0 + jZLtan?? (3.1) where ? is the phase constant of the transmission line.  30   Figure 18. Transmission line with characteristic impedance ?? terminated with a load impedance ??. If the load impedance ?? is a short, then the input impedance simplifies to  Zins = jZ0tan?? = jZ0tan? (3.2) where ? = ?? . The variable ? is commonly referred to as the electrical length of the line. If the transmission line is less than ?/4, where ? is the wavelength of the wave propagating on t he transmission line, the input impedance is purely a positive reactance that is similar to an inductor (j?L). If on the other hand, the load is an open circuit, the input impedance is Zino = ?jZ0cot?? = ?jZ0cot? (3.3) If the physical length of the open circuit transmission line is less than ?/4, then the input impedance is a n egative reactance and the stub is equivalent to a capacitance. The term stub refers to a transmission line that is terminated with either a short circuit or an open circuit. Although equations (3.2) and (3.3) are expressions for impedance, it is often useful to think  31  about problems in terms of admittance. Let ?0 be the characteristic admittance of the line, then the short circuit stub has an input admittance of Yin =?jZ0cot(??) (3.4) while the open circuit stub has an input admittance of  Yin =jZ0tan(??) (3.5) It is very important to note that the impedance characteristics of transmission line stubs is periodic because the expression includes a tangent function. Consequently, the bandwidth of the equivalence between a stub and a lumped component is limited. The mapping between a lumped element prototype component value and a transmission line equivalent circuit element is called Richards? transformation. In the mapping, all stubs are constrained to a commensurate length and the characteristic impedance of the transmission line is adjusted to provide the required inductance or capacitance. A stub length of ? 8?  provides the most direct mapping. If the stub length is ? 8? , then tan (?l) is 1 because ? =2??? . Consequently, ?? = ? 4?  and tan (?4? ) is unity. Richards? transformation defines the frequency map  ? = tan (??) (3.6) which can then be used to define the input impedance of the stub as ???0 for a short circuit stub. Using equation (3.6) the physical length of the stub ? is chosen to correspond to the cut-off frequency of the filter. Therefore, equation (3.6) implicitly handles the frequency scaling which had to be applied directly in the scaling of an equivalent lumped element design.   32  With Richards? transformation we can map transmission lines directly to lumped element prototype filter values where ? = ?0 for short circuit stubs and ? = 1 ?0? for open circuit stubs. It must be noted that although frequency mapping is handled by R ichards? transformation, impedance scaling is not. Therefore, since the lowpass prototype inductor and capacitor values are associated with a 1  Ohm load, the complete mapping needs to include impedance scaling. Thus, ?0 = ? ? ? for inductors and ?0 = 1 ? ? ??  for capacitors. As an example of Richards? transformation consider ??,?? for the first order diplexer design. For this case, the low pass prototype value for ??,?? is unity (see Table 1). After impedance scaling by 50, the final value for the transmission line characteristic impedance is 50 Ohms. The transmission line is ? ??  long for the cut-off wavelength in the filter and terminated with a short circuit. Similar application of Richards? transformation to the high pass branch leads to the circuit in shown in Figure 19.  Figure 19. A 2 GHz first order diplexer after using Richards? transformation. TLSC2ID=TL4Z0=50 OhmEL=45 DegF0=2 GHzTLOC2ID=TL2Z0=50 OhmEL=45 DegF0=2 GHzPORTP=1Z=50 OhmPORTP=2Z=50 OhmPORTP=3Z=50 Ohm 33  The simulation results for the diplexer obtained by the direct application of Richards? transformation are shown in Figure 20 and Figure 21.   Figure 20. S21 simulation comparison of lumped element and Richards? transformed first order 2 GHz diplexer. 0 2 4 6 8 10Frequency (GHz)S21 Parameter Comparison-50-40-30-20-1005Magnitude (dB)DB(|S(2,1)|)Richards TransformationDB(|S(2,1)|)Lumped Element 34   Figure 21. S31 simulation comparison of lumped element and Richards? transformed first order 2 GHz diplexer. For comparison, the lumped element responses are included in the graphs. As shown, the transmission line filter structure exhibits a periodic passband and stop band response. The periodic response relates to the tangent function. Clearly the periodic response is a limitation and better designs are required for the SKA project. A drawback of the direct application of Richards? transformation is that series stubs are required for series components in the filter. A series transmission line is difficult to implement in microstrip and a more realizable filter design is obtained by applying Kuroda?s identity to replace series stubs with equivalent shunt stubs.   0 2 4 6 8 10Frequency (GHz)S31 Parameter Comparison-50-40-30-20-1005Magnitude (dB)DB(|S(3,1)|)Lumped ElementDB(|S(3,1)|)Richards Transformation 35  3.1.2. Kuroda?s Identities Kuroda?s identities [17] provide tools to transform transmission line filters into equivalent structures that may be more practical to implement. A key concept in Kuroda?s identities is to introduce a fixed length of transmission line called a unit element. For example, the unit element could be a ? /8 transmission line, and unit elements can be added to the input and output ports of the filter without changing the attenuation characteristics of the filter. The additional unit elements only modify the phase response by adding electrical length to the design. There are four Kuroda identities of which only one is used in this work. The identity is illustrated by the equivalence between two different designs shown in Figure 22 and Figure 23. In Figure 22, a unit element is followed by a short circuit series stub. This would be equivalent to adding a unit element to a filter branch followed by a  series inductor. Using Kuroda?s identity, an equivalent network shown in Figure 23 can be implemented which consists of a shunt stub followed by a unit element. Assuming all transmission lines (unit elements and stubs) have the same physical length, it can be shown [6, 9] that the equivalence requires the following conditions to be satisfied: ??,? = ?1??,?+1??,???1  (3.4) ??,? = ?1??,??1??,???1 (3.5) ??,? = ??,? ? ??,? (3.6)  ??,? = ??,? + ??,? (3.7) where ??,?,??,?, ??,?,??,? are characteristic impedances for the transmission lines and stubs in Figure 22 and Figure 23.  36    Figure 22. A series transmission line with a series stub. This photo is adapted from [20].  Figure 23. Application of Kuroda's identity to transform the series stub in Figure 19 into a shunt stub. This photo is adapted from [6]. As an example of applying Kuroda?s identity, consider the series inductor ?1,?? for the first order diplexer design. A unit element with a normalized characteristic impedance of 1 Ohm (??,? = 1) is added in series with port 2 in Figure 19.  The normalized impedance of the stub is found using equation (3.7) and Table 1 giving a value ??,? = 1. Using equations (3.5) and (3.7), the circuit is  37  transformed into a unit element impedance with  ??,? = 2  and an open circuit stub with ??,? = 2. After scaling the impedance to 50 Ohms, ??,? and ??,? are 100 Ohms. The simulated response of the equivalent filter is identical to the Richards? transformation design as shown in Figure 25. Despite improving the physical realization of the filter by removing series stubs, it does not resolve the periodic response and the filter structure is still inadequate for the SKA project.   Figure 24. A 2 GHz first order diplexer with the lowpass filter transformed using Kuroda's identity. TLINID=TL5Z0=100 OhmEL=45 DegF0=2 GHzTLOCID=TL1Z0=100 OhmEL=45 DegF0=2 GHzTLOC2ID=TL2Z0=50 OhmEL=45 DegF0=2 GHzPORTP=1Z=50 OhmPORTP=2Z=50 OhmPORTP=3Z=50 Ohm 38   Figure 25. S21 comparison of three different first order diplexer designs: 1) lumped element, 2) design after using Richards? transformation and 3) design after using Kuroda's identity.  Figure 26. S31 comparison of three different first order diplexer designs: 1) lumped element, 2) design after using Richards? transformation and 3) design after using Kuroda's identity. 0 2 4 6 8 10Frequency (GHz)S21 Parameter Comparison-50-40-30-20-1005Magnitude (dB)DB(|S(2,1)|)Kurodas IdentityDB(|S(2,1)|)Lumped ElementDB(|S(2,1)|)Richards Transformation0 2 4 6 8 10Frequency (GHz)S31 Parameter Comparison-50-40-30-20-1005Magnitude (dB)DB(|S(3,1)|)Kurodas IdentityDB(|S(3,1)|)Lumped ElementDB(|S(3,1)|)Richards Transformation 39  3.2.  Transmission Lines and Bandwidth The problem with transmission line filter implementations is the periodic highpass and lowpass attenuation characteristics which are caused by the periodic impedance characteristics of the transmission line elements in the circuit. One way of improving the bandwidth of the design is to reduce the physical length of transmission lines in the design to be much shorter than the shortest wavelength which is required in the design. Clearly 8 GHz is the highest frequency and therefore the shortest wavelength which the multiplexer must separate. If transmission lines can be kept below a quarter wavelength, then responses should be improved. In the previous section, transmission line stubs and unit elements had a physical length of ?/8. In order to shrink physical lengths even more, we need to consider other types of structures for capacitors and inductors.  For inductors, the primary method for reducing length is to increase the characteristic impedance of the transmission line. In terms of microstrip transmission line implementations, this means that the trace width needs to be as narrow as is possible where the limitation is determined by the fabrication process. High impedance transmission lines are fabricated using a laser mill process or using very fine gauge wire and will be described later in Chapter 4. For capacitors there are other alternatives besides the open circuit stub. Two other capacitor designs which are evaluated in this work are 1) the interdigital capacitor and 2) the thin dielectric metal insulator metal (MIM) capacitor. These designs are described in more detail in the following sections.      40  3.3. Interdigital Capacitors An interdigital capacitor is an extended form of coupled line resonators and can be considered as a distributed equivalent of a lumped element capacitor. An example of an interdigital capacitor is shown in Figure 27. Like other planar capacitor structures, the capacitance value depends on the geometry of the capacitor.  A detailed analysis of the design of interdigital capacitors can be found in [21-23]; however the primary design variables are the length of the fingers, the width of the fingers, and the gap between the fingers. If the gap is very narrow, then the length of the fingers can be short providing enough fingers are connected in parallel. The interdigital capacitor will eventually have resonant characteristics associated with the length of the fingers as well as resonant characteristics associated with the total width of the capacitor which in turn depends on the number of fingers. The frequency of these resonances can be found using lumped element models of the interdigital capacitor or by electromagnetic (EM) field solvers. In this work, the software design tool (AWR) includes both comprehensive lumped element models for interdigital capacitors as well as EM simulation tools called AXIEM used to simulate physical layouts.  These models are used to evaluate the feasibility of using interdigital capacitors in the multiplexer design.   41   Figure 27. Interdigital Capacitor. This photo is adapted from [24]. An important factor that constrains the design of interdigital capacitors is the limitations of the fabrication process. In this research project, laser milling was used to fabricate interdigital capacitors. The laser mill has a 10 ?m beam width which was used to mill a copper clad circuit board with a low loss dielectric. The circuit board material substrate was Rogers 5880LZ with a dielectric thickness of 1.52 mm and a copper laminate thickness of 35 ?m. It took many iterations to optimize the laser mill process and eventually it was found that 100 ?m gaps could be fabricated reliably. However, one problem which was not resolved was that copper residue always remained in the trench and a `clean? gap free of copper residue was not obtained. Despite this limitation relatively good results were obtained. Experimental results are described later in 3.3.3.  42  3.3.1. Closed Form Interdigital Capacitor Model Once the minimum gap constraint was imposed on t he design, the simulator could be used to evaluate different capacitor designs which traded off finger length versus the number of fingers versus the broadband frequency response of the capacitor.  The closed form model for interdigital capacitors in AWR is shown in Figure 28. The model was used to design the first order high pass filter branch in the multiplexer shown in Figure 30. A capacitor value of 1.6 pF is required. After iterating with the model, it was found that a design with nine fingers and a finger length of 4.5 mm would meet the requirements. The model was then used to evaluate the response of a first order high pass filter. The S21 characteristics of the filter are shown in Figure 29. The results show that the passband is free of spurious resonances up 8 GHz which is satisfactory for the SKA project.    Figure 28. Interdigital microstrip capacitor model in AWR.  MICAP1ID=MI2W=0.4 mmS=0.1 mmG=0.1 mmL=l mmN=nW1=4.4 mmW2=4.4 mmMSUB=SUB1MSUBEr=1.96H=1.52 mmT=0.035 mmRho=1Tand=0ErNom=3.38Name=SUB1 PORTP=1Z=50 OhmPORTP=2Z=50 Ohml=4.5n=9 43   Figure 29. Attenuation characteristics for an interdigital capacitor.   3.3.2. EM simulation of an Interdigital Capacitor For a more accurate model of the interdigital capacitor, an EM simulation was made using the layout of the capacitor. An EM simulator called AXIEM is available in AWR which uses a method of moments solver to efficiently model planar circuit designs. Two different capacitor layouts were evaluated. The first layout is shown in Figure 30 and consists of an interdigital capacitor with identical dimensions to the closed form model described in the previous section. The model also includes two 4.4 mm transmission lines to provide connections to the capacitor. The transmission lines are required for experimental verification and are used to interface with 0 5000 10000 15000Frequency (MHz)S Parameter Simulation of Interdigital Capacitor-50-40-30-20-1005Magnitude (dB)12703 MHz-10.92 dBDB(|S(1,1)|)Microstrip Interdigital Capacitor_HPDB(|S(2,1)|)Microstrip Interdigital Capacitor_HP 44  SMA connectors in the test fixture. A second design is shown in shown in Figure 31. In the second design, the transmission lines are tapered to reduce the line width down to a narrow point slightly larger than the width of the SMA connector pin. The tapering reduces the discontinuity between the board and the connector. The EM simulation results for this design are shown in Figure 32. As shown, the tapered port model has better return loss due to the fact that the discontinuity at the ports is reduced.  Figure 30. Interdigital capacitor with two 4.4mm port transmission lines.  Figure 31: Interdigital capacitor with tapered ports to improve matching to SMA connectors.  45    Figure 32. EM simulation results for two different interdigital capacitor designs.  3.3.3. Experimental Measurements of an Interdigital Capacitor Two interdigital capacitor designs, shown in Figure 30 and Figure 31, were fabricated and tested to compare with simulation results. Pictures of the fabricated designs are shown in Figure 33 and Figure 34 . Both designs were fabricated using a laser mill process described earlier. The gap between conductors is 100 ?m and the milling process was not perfect and some copper residue in the gaps is visible under a microscope. SMA connectors (shown in Figure 33) were soldered to the ports on t he board. The test fixtures were then characterized using a vector network 0 2000 4000 6000 8000 10000Frequency (MHz)S-Parameter comparison of two HP Filters-60-50-40-30-20-1005Magnitude (dB)1582.3 MHz-3 dBDB(|S(2,1)|)Highpass_Non-taperDB(|S(2,1)|)Highpass_TaperedDB(|S(1,1)|)Highpass_Non-taperDB(|S(1,1)|)Highpass_Tapered 46  analyzer which measures two port S-parameters. Port extension was used in all measurements to remove the effect of the connectors and shift the measurement reference plane to the edge of the boards.   Figure 33. 2 GHz interdigital capacitor with normal ports.  47   Figure 34.  2 GHz interdigital capacitor with tapered ports. Figure 35 and Figure 36 show experimental results for the tapered and non-tapered interdigital capacitor designs. The tapered design has a better frequency response (S21) than the non-tapered design up to a frequency of about 7 GHz. The input match (S11) is also better for the tapered design over the same frequency range. The measurements confirm that tapering reduces the discontinuity at the connector interfaces. As a final comparison, experimental and simulation results for the interdigital capacitor designs are shown in Figure 37.   48   Figure 35. Measured results for an interdigital capacitor. The measurement is S21. 0 2 4 6 8 10Frequency (GHz)Measurement Comparison of Interdigital Capacitors_S21-50-40-30-20-1005Magnitude (dB)DB(|S(2,1)|)Interdigital_capacitor_taper portsDB(|S(2,1)|)Interdigital_capacitor_non-taper ports 49   Figure 36. Measured results for an interdigital capacitor. The measurement is S11.   Figure 37.  Comparison of measured and simulation results for an interdigital capacitor.  0 2 4 6 8 10Frequency (GHz)Measurement Comparison of Interdigital Capacitors_S11-50-40-30-20-1005Magnitude (dB)DB(|S(1,1)|)Interdigital_capacitor_ taper portsDB(|S(1,1)|)Interdigital_capacitor_ non-taper ports0 2000 4000 6000 8000 10000Frequency (MHz)Interdigital Capacitor Comparison-60-50-40-30-20-1005Magnitude (dB)DB(|S(2,1)|)Interdigital_capacitor_simulationDB(|S(2,1)|)Interdigital_capacitor_measurementDB(|S(2,1)|)Interdigital_capacitor_analytical model_simulation 50  Although the interdigital capacitor has improved bandwidth characteristics relative to a transmission line design based on Richards? transformation, it is  still challenging to meet the requirements for the SKA feed antenna. Recall that the required operating bandwidth of the final multiplexer is from 1-8 GHz. The interdigital capacitors come close to meeting the requirements with an upper frequency limit in the range of 7 G Hz. Given the results for the interdigital capacitor design, it was decided to continue investigating different design concepts and see if further improvements in bandwidth could be realized.  At this point in the project it was decided to directly test an implementation of a thin insulator capacitor. Other researchers have reported broadband filter designs with this kind of capacitor structure [25-29]. A rudimentary metal insulator metal (MIM) capacitor was built using copper tape and Scotch Magic Tape (model 810) as a dielectric( ?? = 3.3, tan ? = 0.1). Measurements with a micrometer were made and the thickness of the tape is 58?5 ?m. The MIM structure is identical to a parallel plate capacitor and the term MIM relates to thin film and semiconductor fabrication processes where thin oxide layers are used to separate the metal layers and form a capacitor structure [30, 31].  A photograph of the MIM capacitor with input and output microstrip lines connected to two ports is shown in Figure 38. The copper tape has a thickness of 60 ?m and the MIM capacitor is mounted on a Rogers 4350B substrate. Interfaces to the capacitor are constructed using 50 Ohm microstrip lines. The area of the capacitor plates is 1.65 mm x 4.0 mm and has a capacitance of approximately 1.6 pF. The test results for a capacitor were remarkable given how crude the fabrication process was. The measured S-parameters are shown in Figure 39. The S21 response is free of resonances up to 10 GHz and the input match is broad and S11 is approximately -10 dB from 2 ? 10 GHz. As a  51  final comparison, the S21 response of the MIM capacitor is compared with the interdigital capacitor described in section 3.3. The responses show that the MIM capacitor is definitely more broadband compared to the interdigital design. The experimental results are very encouraging and demonstrate the potential of fabricating diplexer designs with MIM capacitor structures. In the next chapter, simulation and experimental results are shown for diplexer and multiplexer designs using interdigital and MIM capacitor designs.   Figure 38. A MIM capacitor fabricated with copper tape and a thin dielectric.  52   Figure 39. Measured S-parameters for the MIM capacitor.  Figure 40. Comparison of measured responses for 1.6 pF MIM and 1.6 pF interdigital capacitors. 0 2 4 6 8 10Frequency (GHz)S Parameter Simulation of  MIM Capacitor-40-30-20-10010Magnitude (dB)0.9197 GHz-3.027 dBDB(|S(1,1)|)MIM_CapDB(|S(2,1)|)MIM_Cap10 2010 4010 6010 8010 10000Frequency (MHz)Measurement Comparison of the Highpass Filters-60-50-40-30-20-1005Magnitude(DB)DB(|S(2,1)|)Interdigital capacitor_measurementDB(|S(2,1)|)MIM capacitor_measurement 53  4.Chapter 4: Simulation and Measurement Results for Diplexers and Multiplexers In this chapter, simulation and measurement results are shown for diplexers and multiplexers. Since the multiplexer is a cascade of diplexers, we begin with results for individual multiplexer blocks. The diplexers cover two frequency bands. The first is a 2 GHz lowpass/highpass split and the second is a 4  GHz lowpass/highpass split. The results are constrained to first order filter designs implemented with either interdigital capacitors or MIM capacitors. Finally, simulation results for a complete first order filter design are shown. The simulation results show that interconnect between the diplexer is important and physical line lengths must be very short to maintain broadband properties.   4.1. Two GHz Diplexer with Interdigital Capacitor In this section, the design of a 2 GHz complementary diplexer is described. The results include simulations of an ideal lumped element diplexer and an electromagnetic simulation of the physical diplexer which was fabricated. The simulation results are then compared with measurement results.  4.1.1. Lumped Element Model The 2 GHz lumped element diplexer model is shown in Figure 41 and component values were given earlier in Table 2.  The simulated S-parameter responses for this circuit are shown Figure  54  42. As shown, the lowpass and high pass filter branches are complementary and cross over at 2 GHz. The input match given by S11 is below the -50 dB reference in the graph and therefore shows that the input port is matched over the entire frequency range.  Figure 41.  Lumped element model for a first order 2 GHz diplexer.  CAPID=C1C=1.592 pFINDID=L2L=3.98 nHPORTP=1Z=50 OhmPORTP=2Z=50 OhmPORTP=3Z=50 OhmFirst Order Lumped Element DiplexerTransition Frequency at 2 GHz 55   Figure 42.  Simulation results for a lumped element 2 GHz diplexer.  4.1.2. Distributed Model The lumped element model in Figure 41 is converted into a physical layout as shown in Figure 43. The lumped element capacitor is implemented as an interdigital capacitor and the inductor is implemented as a short length of high impedance transmission line. A 3-D image of the physical layout implemented in the electromagnetic simulator (AXIEM) is shown in Figure 44. The design uses a Rogers 5880LZ substrate with material properties that are summarized in Table 6. 0 2000 4000 6000 8000 10000Frequency (MHz)S-parameter Simulation for Lumped Element-50-40-30-20-1005Magnitude (dB)2000 MHz-3.012 dBDB(|S(2,1)|)2GHz_Diplexer_Lumped_ElementDB(|S(3,1)|)2GHz_Diplexer_Lumped_ElementDB(|S(1,1)|)2GHz_Diplexer_Lumped_Element 56   Figure 43. Distributed model for a 2 GHz first order diplexer.   57   Figure 44. 3-D view of first order 2 GHz diplexer.  Table 6. Rogers 5880LZ material properties. Parameter   Value Dielectric Constant (??) 1.96 ? 0.04 Substrate Thickness  1.52 mm Copper Cladding 35 ?m  The electromagnetic simulation results for the 3-D model of the 2 GHz diplexer are summarized in Figure 45. Unlike the lumped element design which had a perfect input match, the input match (S11) is still very good up to a frequency of 8 GHz. The stopband response of the lowpass filter branch (S21) is also good except for a small resonant dip near 5.5 GHz, and then a more significant resonance at the band edge near 8 GHz. The highpass filter response (S31) is good up to a frequency of about 7.5 G Hz, while the insertion loss is slightly more than the lumped element design. A comparison between the lumped element design and the physical design are  58  shown in Figure 46 and Figure 47. The EM simulation results highlight the importance of considering the physical layout of the circuit and clearly the response is not as good as an ideal lumped element model.  Figure 45. S parameter simulation results for a distributed 2 GHz diplexer using an interdigital capacitor. 0 2000 4000 6000 8000 10000Frequency2GHz Distributed Diplexer S Parameter Simulation-50-40-30-20-1005Magnitude (dB)2000 MHz-3.382 dBDB(|S(1,1)|)2GHz_Diplexer_with_interdigital_capacitorDB(|S(2,1)|)2GHz_Diplexer_with_interdigital_capacitorDB(|S(3,1)|)2GHz_Diplexer_with_interdigital_capacitor 59   Figure 46. Comparison of simulation results for the lowpass filter branch in 2 GHz diplexers.   Figure 47. Comparison of simulation results for the highpass filter branch in 2 GHz diplexers. 0 2000 4000 6000 8000 10000Frequency (MHz)Comparison Between Lumped Element and Distributed Diplexer_LPF-50-40-30-20-1005Magnitude (dB)DB(|S(3,1)|)2GHz_Diplexer_with_interdigital_capacitorDB(|S(3,1)|)2GHz_Diplexer_Lumped_Element0 2000 4000 6000 8000 10000Frequency (MHz)Comparison Between Lumped Element and Distributed Diplexer_HPF-50-40-30-20-1005Magnitude (dB)DB(|S(2,1)|)2GHz_Diplexer_Lumped_ElementDB(|S(2,1)|)2GHz_Diplexer_with_interdigital_capacitor 60  4.1.3. Measurement Results   The 2 GHz diplexer using an interdigital capacitor was fabricated using a laser mill process. A picture of the prototype is shown in Figure 48 and the exact dimensions of the design are shown in Figure 49. Note that in the picture the connectors are removed. SMA connectors are soldered to the board for measuring the S-parameters with a vector network analyzer.     Figure 48. Photograph of the 2 GHz diplexer with an intedigital capacitor.   61   Figure 49. Dimensions (in mm) for the fabricated 2 GHz interdigital diplexer.            62  In Figure 50, the measurement results for the highpass filter branch are shown. The measurement is made by connecting port 1 of the vector network analyzer to the input port of the diplexer, port 2 is connected to the highpass output, and the lowpass output port is terminated in 50 Ohms. Included in the figure is an overlay of the EM simulation result. There is some misalignment between the simulation and experimental results, but the shape of the responses are similar. For example, they both show a minor resonance in the range of 5.5-6.0 GHz and a major resonance just above 8 GHz. There is also about 1-2 dB difference in the frequency response with higher losses in the experimental prototype. One of the main limitations in the fabrication process is copper residue in the milled trenches. This residue is not modelled and the trenches in the simulation are assumed to be perfect. Improved fabrication methods, perhaps using photolithography, are likely to reduce the discrepancy in between measured and simulation results. Measurement and simulation results for the lowpass filter branch of the diplexer are shown in Figure 51. Recall that the inductor is implemented as a short high impedance transmission line at a frequency of 2 GHz. As the frequency increases in the stopband, the electrical length of the line increases and eventually resonant characteristics are observed at frequencies around 5.5 GHz and 8 GHz.   63   Figure 50. Simulation and measurement results for the highpass filter branch in the 2 GHz interdigital diplexer.  Figure 51. Simulation and measurement results for the lowpass filter branch in the 2 GHz interdigital diplexer. 0 2000 4000 6000 8000 10000Frequency (MHz)Simulation vs Measurement of Diplexer Highpass Branch-20-15-10-505Magnitude (dB)DB(|S(2,1)|)Highpass_measurementDB(|S(2,1)|)Highpass_simulation0 2000 4000 6000 8000 10000Frequency (MHz)Simulation vs Measurement of Diplexer Lowpass Branch-20-15-10-505Magnitude (dB)DB(|S(3,1)|)Lowpass_simulationDB(|S(2,1)|)Lowpass_measurement 64  4.2. Four GHz Diplexer with Interdigital Capacitor In this section, simulation and measurement results are shown for a complementary diplexer with a transition frequency of 4 GHz. The design is similar to the 2 GHz diplexer except for the higher transition frequency. Results are shown for the lumped element model, the electromagnetic simulation of the layout, and measured S parameters for the experimental prototype.  4.2.1. Lumped Element Model Figure 52 shows the lumped element model of 4 GHz first order diplexer. S parameter simulation results for the 4 GHz diplexer are shown in Figure 53. As shown, the 3 dB cut-off frequency for both the lowpass and highpass filter branches is 4 GHz. The input match measured by S11 is nearly ideal and less than -50 dB.  Figure 52. 4 GHz lumped element model. CAPID=C1C=0.796 pFINDID=L1L=1.99 nHPORTP=1Z=50 OhmPORTP=2Z=50 OhmPORTP=3Z=50 OhmFirst Order Lumped Element DiplexerTransition Frequency at 4 GHz 65   Figure 53. The lowpass and highpass filter responses for the 4 GHz diplexer.  4.2.2. Distributed Model The series capacitance in the highpass branch of the 2 GHz diplexer is 1.6 pF (see Figure 41) and the 4 GHz diplexer has a ser ies capacitance of half this value, 0.8 pF, as shown in Figure 52. Consequently, the interdigital capacitor is more compact for the 4 GHz design and the number of fingers is reduced compared to the 2 GHz diplexer. In a similar way, the inductor in the lowpass branch is half the value of the 2 GHz design. A reduction in inductance is achieved by shortening the high impedance transmission line which is also beneficial in shifting spurious resonant frequencies to be well above the 8 GHz frequency limit for the SKA feed requirement. The 0 2000 4000 6000 8000 10000Frequency (MHz)4 GHz Lumped Element Diplexer S Parameter-50-40-30-20-1005Magnitude (dB)4000 MHz-3.012 dBDB(|S(1,1)|)4 GHz Lumped Element DiplexerDB(|S(3,1)|)4 GHz Lumped Element DiplexerDB(|S(2,1)|)4 GHz Lumped Element Diplexer 66  layout of the diplexer with the interdigital filter is shown in Figure 54 and a 3-D model of the diplexer is shown in Figure 55.   Figure 54. Distributed 4 GHz diplexer schematic.  67   Figure 55. 3-D view of the first order 4 GHz diplexer.  The S parameter simulation results for the 4 GHz diplexer are shown in Figure 56. The lowpass response, the highpass response, and the input match (< -10 dB) are all acceptable and meet the requirements for the SKA project.  A comparison of the electromagnetic simulation results with the ideal lumped element diplexer model are shown in Figure 57 and Figure 58.  The highpass responses in Figure 57 have slightly more discrepancy than the lowpass responses. However, overall the models compare well.   68   Figure 56. S Parameter simulation results for the 4 GHz diplexer with an interdigital capacitor.   Figure 57. Comparison of lumped element and electromagnetic simulation results for the highpass filter branch in the 4 GHz diplexer.  0 2000 4000 6000 8000 10000 (MHz)4 GHz Distributed Diplexer Simulation-50-40-30-20-1005Magnitude (dB)4000 MHz-3.505 dBDB(|S(1,1)|)4GHz_Diplexer_with_interdigital_capacitorDB(|S(2,1)|)4GHz_Diplexer_with_interdigital_capacitorDB(|S(3,1)|)4GHz_Diplexer_with_interdigital_capacitor0 2000 4000 6000 8000 10000Frequency (MHz)Comparison Between Lumped Element and Distributed Diplexer_HPF-50-40-30-20-1005Magnitude (dB)DB(|S(2,1)|)4GHz_Diplexer_with_interdigital_capacitorDB(|S(2,1)|)4 GHz Lumped Element Diplexer 69   Figure 58. Comparison of lumped element and electromagnetic simulation results for the lowpass filter branch in the 4 GHz diplexer.  4.2.3. Measurement Results  A picture of the 4 GHz diplexer is shown in Figure 59. SMA connectors are used to interface to the board and the dimensions for the physical layout of the diplexer are shown in Figure 60. The measurement results for the highpass and lowpass filter branches are shown in Figure 62 and Figure 63, respectively. The figures also include electromagnetic simulation results for the corresponding filter branches. Similar to the 2 GHz diplexer results, the lowpass filter responses are well matched, while the highpass filter with the interdigital capacitor has more discrepancy.  0 2000 4000 6000 8000 10000Frequency (MHz)Comparison Between Lumped Element and Distributed Diplexer_LPF-50-40-30-20-1005Magnitude (dB)DB(|S(3,1)|)4 GHz Lumped Element DiplexerDB(|S(3,1)|)4GHz_Diplexer_with_interdigital_capacitor 70  As described earlier, fabricating the design using a laser mill tends to leave copper residue in the narrow gaps between the fingers. This could be impacting the measured results leading to a greater discrepancy between simulation and experimental results.    Figure 59. Photograph of the 4 GHz diplexer with an intedigital capacitor.  71   Figure 60 . Dimensions (in mm) for the 4 GHz interdigital diplexer.    72   Figure 61. Simulation and measurement results for the highpass filter branch in the 4 GHz interdigital diplexer.  Figure 62. Simulation and measurement results for the lowpass filter branch in the 4 GHz interdigital diplexer. 0 2000 4000 6000 8000 10000Frequency (MHz)Simulation vs Measurement of Diplexer Highpass Branch-50-40-30-20-1005Magnitude (dB)DB(|S(2,1)|)Highpass_SimulationDB(|S(2,1)|)Highpass_Measurement0 2000 4000 6000 8000 10000Frequency (MHz)Simulation vs Measurement of Diplexer Lowpass Branch-50-40-30-20-1005Magnitude (dB)DB(|S(3,1)|)Lowpass_SimulationDB(|S(2,1)|)Lowpass_Measurement 73  4.3. Multiplexer As a final step within the scope of this project, two first order diplexers are configured to implement a three band multiplexer. The layout for the multiplexer combines the 2 GHz and 4 GHz interdigital capacitor designs and is shown in Figure 63.   Figure 63.  First order multiplexer using interdigital capacitors.   74  S-Parameter simulation results for the multiplexer are shown in Figure 64. The responses in this plot can be compared with Figure 11 shown in Chapter 2. The first lowpass band has a corner frequency close to the required 2 GHz band split. The midband has an insertion loss of about 3 dB and is similar to the lumped element model. As discussed in Chapter 2 the high insertion loss is related to the order of the filter structures and lower attenuation would be expected for higher order structures. The highpass response for the third band is satisfactory although further reductions in insertion loss are desirable and could be obtained with higher order structures. The input match (S11) shows that there are two resonances around 5.5 GHz and 8 GHz associated with the 2 GHz (1.6 pF) interdigital capacitor. Overall, the multiplexer response is similar to the result expected by cascading the 2 GHz and 4 GHz interdigital diplexers.   Figure 64. S parameters simulation for the first order interdigital multiplexer.  0 2000 4000 6000 8000Frequency (MHz)Distributed Multiplexer-50-40-30-20-1005Magnitude (dB)DB(|S(1,1)|)Final_MultiplexerDB(|S(2,1)|)Final_MultiplexerDB(|S(3,1)|)Final_MultiplexerDB(|S(4,1)|)Final_Multiplexer 75  4.4. MIM Diplexer  Given the broadband response of the MIM capacitor a first order 2 GHz diplexer was built to evaluate the design. The MIM capacitor in the high pass filter branch was the same design as described in section 3.4. In order to fabricate a lowpass branch, a very high impedance transmission line is required to implement an inductor. Given the rudimentary construction techniques employed to fabricate the MIM capacitor (copper tape and Magic tape), a technique to fabricate high impedance transmission lines was required. For this prototype, the transmission line was implemented using a very thin wire (32 gauge) taped to the substrate. Since the wire is above a ground plane it can be modelled very crudely as a microstrip line with a very narrow width equal to the diameter of the wire. As an approximation of the characteristic impedance of this configuration, the AWR transmission line design tool was used.  A screen capture of the tool with the values used to estimate the characteristic impedance is shown in Figure 65. The estimated characteristic impedance of the transmission line with the 32 gauge wire is 106 Ohms.    76   Figure 65. The AWR TXLINE tool to estimate the characteristic impedance for an inductor design. Once the characteristic impedance was found, the required physical length of the transmission line was calculated. An inductance of 4.0 nH is required. The reactance of the inductor at 2 GHz is given by  ??? = ??? (4.1)  Using a short circuit stub model, the effective input impedance of the transmission line is   ??? = ???tan (??) (4.2)   Equating (4.1) and (4.2) and solving for the physical length ? gives a length of 7 mm. The phase constant ? is given in Figure 65 and required to calculate the line length. The thin 7 mm wire is shown in Figure 66 and implements the lowpass filter branch of the first order diplexer. A cross-section of the highpass branch which includes a MIM capacitor is shown in Figure 67. The MIM  capacitor has a plate area of 4 mm by 1.65 mm.  77  The measured S-parameters for the MIM diplexer are shown in Figure 68. The results are very good and free of any significant resonances up to 10 GHz. Future research work is recommended to continue investigating the potential of MIM capacitor concepts in broadband multiplexers.    Figure 66. 2 GHz MIM Diplexer    Figure 67. Cross-section of the first order highpass filter using a MIM capacitor.  78     Figure 68. The measured frequency response (S21) of the 2 GHz MIM Diplexer.  4.5. Waveguide Load Measurement with MIM Diplexer  As a final experiment to evaluate the feasibility of implementing a broadband multiplexer for the SKA project using the complementary diplexing structures, a waveguide load was connected to the highpass port of the 2 GHz MIM diplexer. A photograph of the experimental setup is shown in Figure 69.  In this setup, a SMA to N connector adapter is used to interface the highpass port to a WR-284 coax to waveguide adapter. The waveguide adapter is then connected to a 0 2 4 6 8 10Frequency (GHz)S Parameter Simulation of 2 GHz MIM Diplexer-50-40-30-20-1005Magnitude (dB) 2 GHz-3.618 dBDB(|S(2,1)|)BandpassDB(|S(2,1)|)Bandstop 79  waveguide horn antenna and radiates into free space. The free space match was stable and no significant changes in measurements were observed when objects were clear of the near field.   Figure 69. MIM Diplexer with a waveguide load on the highpass port.   With the waveguide load on port 2, a measurement of the lowpass filter branch was made. The results are shown in Figure 70. The figure also includes a trace which corresponds to the ideal response of the lowpass filter branch using the lumped element model. As shown, the lowpass result is impaired by a couple of resonances; one occurs around 1 GHz and the second occurs around 2 GHz. At these frequenciencs, the waveguide connected to port 2 is at or below the cut-off frequency for WR-284 which is 2.08 GHz. Of these resonances, the 1 GHz dip is undesirable  80  and is not acceptable for the SKA project. So further investigation as to why the resonance occurred was required. When the waveguide is operated below cut-off, the impedance is very high and much larger than the matched impedance of 50 Ohms. In this design, the waveguide adapter uses a probe to couple to the electric field in the waveguide which has a dominant mode of ??10. The open circuit impedance presented by the probe at frequencies below cut-off is connected through a transmission line to the lowpass port of the diplexer. The transmission line in this case consists of the physical length of the coaxial connector on the waveguide adapter and the physical length of the N to SMA adapter. The total physical length of the ?transmission line? is at least 70 mm and the dielectric in the coaxial adapters is teflon (??= 2.1). Therefore, the equivalent electrical length of the transmission line is about 122 degrees. Since the line length is between 90 and 180 degrees at 1 GHz, the open circuit line will be inductive. The inductance will then be resonant with the 1.6 pF highpass capacitor and creates a short circuit at the input node of the diplexer. The short circuit then creates a dip in the S21 response.  As a way to model the cut-off behaviour in the experimental setup, a simulation was setup using the lumped element diplexer model with an open circuit transmission line. The model is shown in Figure 71. The model was then simulated to find the S-parameters for the lowpass branch. The S21 response is shown in Figure 70. As shown, the open circuit transmission line creates a dip in the S21 response at a frequency of 1 GHz similar to the measured results. These results highlight a very important design consideration that must be considered in the SKA antenna feed. The physical length between the multiplexer ports and the waveguide probes must  81  be electrically small to ensure spurious responses are not created that will affect the passband response of the filters.   Figure 70. Lowpass comparison with waveguide load.   Figure 71. MIM diplexer with a highpass port termination to model a waveguide in cut-off. 0 2 4 6 8 10Frequency (GHz)S Parameter Lowpass Comparison-50-40-30-20-1005Magnitude (dB)DB(|S(2,1)|)MIM diplexer with waveguide termination measurement at HPFDB(|S(2,1)|)Ideal diplexer with waveguide termination modelINDID=L1L=3.98 nHCAPID=C1C=1.592 pFTLOCID=TL4Z0=50 OhmEL=128 DegF0=1 GHzPORTP=1Z=50 Ohm PORTP=2Z=50 Ohm 82  Chapter 5: Conclusions In this project, the concept of complementary diplexers was investigated as a way to implement a broadband multiplexer for the SKA feed antenna. Beginning with lumped element models, it was shown that a cascade of complementary diplexer stages can realize the required band splitting. After verifying first, third and fifth order lumped element multiplexer designs, the focus shifted to designing physically implementable microwave structures. It was shown that the classical filter design approach using Richards? transformation and Kuroda?s identities would not yield the bandwidth required for the SKA feed because the transmission line structures exhibit periodic passband responses that are too narrow.  As a way to overcome the limitations of classical filter transformations, two compact capacitor structures were evaluated. The capacitors must have a practical geometry that are sub-wavelength to ensure the components operate free from spurious resonances. The first design used interdigital capacitors and the second design used a thin dielectric metal insulator metal structure. The interdigital capacitors were fabricated on low loss substrates using laser milling to create 100 ?m separation between fingers. A 1.6 pF capacitor design showed good performance up to about 7 GHz. The second capacitor design was fabricated by hand using copper tape and Magic tape. The test results for this design were superior to the interdigital design and good performance was obtained up to 10 GHz. After verifying capacitor structures, several first order diplexer designs were built and tested. The designs include a 2 GHz interdigital diplexer, a 4 GHz interdigital diplexer, and a 2 GHz MIM diplexer. The experimental results were also compared with simulation results using an electromagnetic simulator. Overall the diplexers worked as expected and the best results were  83  obtained using the MIM diplexer despite the crude fabrication method. Finally, a simulation of a first order multiplexer using interdigital structures showed that diplexing structures can be integrated to create a three band multiplexer. Another important part of the research project was to evaluate the effect of waveguide load terminations at the output ports. The evaluation was carried out using simulations and an experiment with the 2 GHz MIM diplexer. The waveguide load is highly reflective at frequencies below cut-off and it was shown that consideration must be given to the physical length between the guide and the multiplexer port. Broadband performance will require that this distance be minimized. There are many additional steps that can be investigated in future research projects. The results for thin dielectric capacitors are very encouraging and higher order diplexing structures need to be experimentally verified. Also, the physical layout of the multiplexer must be considered in the context of the nested coaxial waveguide feed antenna. The multiplexer needs to be integrated with the waveguide probes and the physical distance, especially for the low frequency guide, must be minimized. Research work to design and verify the operation of the multiplexer with a coaxial waveguide feed is another significant project. 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