Synthesis of acoustic wave filters : ladder and transversal topologies : towards a practical implementation

Synthesis of acoustic wave filters. Ladder and transversal topologies: towards a practical complementation Alberto Hueltes Escobar ADVERTIMENT La consulta d’aquesta tesi queda condicionada a l’acceptació de les següents condicions d'ús: La difusió d’aquesta tesi per mitjà del repositori institucional UPCommons (http://upcommons.upc.edu/tesis) i el repositori cooperatiu TDX ( h t t p : / / w w w . t d x . c a t / ) ha estat autoritzada pels titulars dels drets de propietat intel·lectual únicament per a usos privats emmarcats en activitats d’investigació i docència. No s’autoritza la seva reproducció amb finalitats de lucre ni la seva difusió i posada a disposició des d’un lloc aliè al servei UPCommons o TDX. No s’autoritza la presentació del seu contingut en una finestra o marc aliè a UPCommons (framing). Aquesta reserva de drets afecta tant al resum de presentació de la tesi com als seus continguts. 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LADDER AND TRANSVERSAL TOPOLOGIES. Towards a practical implementation _________________________________________ PH.D DISSERTATION Author: Alberto Hueltes Escobar Thesis Advisor(s): Prof. Jordi Mateu Mateu Prof. Carlos Collado Gomez SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Acta de calificación de tesis doctoral Curso académico: Nombre y apellidos Programa de doctorado Unidad estructural responsable del programa Resolución del Tribunal Reunido el Tribunal designado a tal efecto, el doctorando / la doctoranda expone el tema de la su tesis doctoral titulada ____________________________________________________________________________________ __________________________________________________________________________________________. Acabada la lectura y después de dar respuesta a las cuestiones formuladas por los miembros titulares del tribunal, éste otorga la calificación: NO APTO APROBADO NOTABLE SOBRESALIENTE (Nombre, apellidos y firma) Presidente/a (Nombre, apellidos y firma) Secretario/a (Nombre, apellidos y firma) Vocal (Nombre, apellidos y firma) Vocal (Nombre, apellidos y firma) Vocal ______________________, _______ de __________________ de _______________ El resultado del escrutinio de los votos emitidos por los miembros titulares del tribunal, efectuado por la Escuela de Doctorado, a instancia de la Comisión de Doctorado de la UPC, otorga la MENCIÓN CUM LAUDE: SÍ NO (Nombre, apellidos y firma) Presidente de la Comisión Permanente de la Escuela de Doctorado (Nombre, apellidos y firma) Secretario de la Comisión Permanente de la Escuela de Doctorado Barcelona a _______ de ____________________ de __________ SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. ABSTRACT The meteoric growth of the mobile communication market for the last three decades has been strongly related with the evolution of the electroacoustic (EA) filter technology. With way over 5 billion cell phone users worldwide in 2017 and 5G standard in the horizon, the radiofrequency spectrum is becoming increasingly crowded whereas demand for mobile data has no expected limits in the short to medium term. In this scenario, where a unique filter needs to be designed for each band of operation, requirements for advanced filtering solutions continue to grow as well as the average value of the RF solutions and the RF content per mobile device. RF and microwave devices based on EA resonators such as Bulk Acoustic Wave (BAW) and Surface Acoustic Wave (SAW) filters overcame the limitations of the existent technologies back in the 1980’s thanks to its compatibility with the manufacturing process of standard Silicon Integrated Circuits (Si-IC). Nowadays sophisticated RF Front-End (RFFE) Modules based on System-in-Package (SiP) are the key solution to integrate the increasing number of electronic parts - Power Amplifiers (PAs), Low Noise Amplifiers (LNAs) and switches - that accompanies acoustic filtering devices like filters, duplexers and multiplexers. Even though the level of complexity and accuracy present on the current EA devices is extraordinary, the design procedures for acoustic filters are still based on the optimization of built-in performance parameters from behavior-based compact models of resonators and the success on this duty relies mostly on the expertise of the designers. However due to the stringent technological constrains and the increasing complexity of the RFFE module architectures devoted to satisfy demanding specifications of the current and forthcoming communications standards, the challenging work of the designers is never getting easier. The aim of this work is focused on providing valuable insights that might help to overcome some of the existent limitations in the design of EA filters. Wider bandwidths, prescribed inclusion of external elements or multiplexing features are taken into consideration. On one hand we show a synthesis formulation and an automated procedure to carry out the synthesis SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. for current well-known topologies, i.e. ladder, taking into consideration realistic values and specifications. On the other hand, a synthesis formulation for novel topologies, named here transversal, is provided. Success of this novel topology will certainly help to guarantee the prevalence of EA filters in the future communication standards, along with the extension to other applications with more stringent requirements and different operating frequencies. The first part of the work is devoted to explain the basic theory related to filter synthesis and its applicability to filters based on EA resonators. Subsequent chapters elaborate the theory to explain the procedures followed to obtain both synthesis and their limitations. Finally, a Chapter devoted to study cases shows the results of applying the filter synthesis to real case scenarios. | x SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. LIST OF FIGURES Figure 1. Nowadays smartphones integrate different front-ends modules to cover (a) low bands (b) mid bands and (c) high bands. Image Source: [5] ......................................................... 17 Figure 2. Filter response with referred parameters .................................................................... 25 Figure 3. Equivalent circuit of a two-port network .................................................................... 25 Figure 4. Ladder topology of a 3rd order Chebyshev lowpass prototype filter. .......................... 29 Figure 5. Coupled resonators topology of a 3rd order Chebyshev lowpass prototype filter. ...... 30 Figure 6. Nodal notation of an in-line three order filter. S stands for source, Ri stands for resonant node, L stands for load and Mij is the notation used for the couplings between nodes. .................................................................................................................................. 32 Figure 7. a) Equivalent network of a transversal filter of order N. The empty circles S and L represent the source and the load, the black filled-in circles (from R1 to RN) represent the resonators and the lines connecting the circles represent the mainline couplings from each resonator to the source and load. b) Equivalent circuit of the k-th resonator and its couplings to the source and load. ........................................................................................ 35 Figure 8. Transfer and reflection coefficients of a polynomial bandpass filtering response as a function of its Q factor. ...................................................................................................... 37 Figure 9. Outline of an acoustic wave resonator on its BAW (Bulk Acoustic wave) configuration. ...................................................................................................................... 38 Figure 10. Outline of an acoustic wave resonator on its SAW (Surface Acoustic wave) configuration. ...................................................................................................................... 38 Figure 11. BVD equivalent circuit model ................................................................................... 39 Figure 12. Magnitude and phase response of an acoustic resonator, throughout evaluation of the BVD model. .................................................................................................................. 41 Figure 13. BVD equivalent circuit model. .................................................................................. 42 Figure 14. mBVD circuit model including acoustic and electrode losses. ................................... 42 Figure 15. a) lowpass prototype corresponding to the BVD reciprocal circuit model. b) lowpass prototype corresponding to the BVD model. ...................................................................... 45 Figure 16. Lowpass prototype (mapped at the operating frequency) and bandpass prototype performance of a single acoustic wave resonator. ................................................................ 46 Figure 17. Typical representation of an acoustic resonator. ....................................................... 47 Figure 18. Outline of a ladder network. ..................................................................................... 50 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Figure 19. S-parameters of a 7th order (4 series, 3parallel -4s3p-) AW ladder filter (Figure 18), and input impedance of the isolated resonators. Note that due to the symmetry of the response only two of the four series resonators are different and two of the three shunt resonators are equal. ........................................................................................................... 50 Figure 20. Working principle of a ladder network based on acoustic resonators a) TZ created by the shunt resonators b) behavior in the in-band region and c) TZ created by the series resonators............................................................................................................................ 51 Figure 21. Lowpass filter response (transmission in solid black and reflection in dashed black) typical logarithmic representation of the impedance magnitude of the resonators (series resonators in blue and shunt resonators in red). Filter network is outlined below. ............ 57 Figure 22. Lowpass prototype topology of an order 5 filter with 3 Transmission Zeros (TZ) at the lower side band and 2 at the upper side band .............................................................. 58 Figure 23. Network topology of an N even order filter, a) starting with a shunt resonator, b) starting with a series resonator. .......................................................................................... 59 Figure 24. Synthesized network of a 7th order filter with 4 transmission zeros at the upper side band and 3 transmission zeros at the lower side band. ....................................................... 59 Figure 25. Impedance of the synthesized resonators. Blue lines correspond to series resonators, and red lines correspond to shunt resonators (above). Synthesized frequency response of the 7th order filter (below). ................................................................................................ 61 Figure 26. Filter response of a 5th order filter for a) 50 MHz bandwidth, b) 200 MHz bandwidth, c) 400 MHz bandwidth. ................................................................................... 62 Figure 27. BVD model with the inclusion of external elements in order to modify the coupling coefficient. a) inclusion of a series capacitor, b) inclusion of a shunt capacitor, c) inclusion of a shunt inductor, and d) inclusion of a series inductor. .................................................. 64 Figure 28. Impedance response of BVD resonator with the inclusion of external capacitors. Legend details on each of the lines in the figure. ................................................................ 65 Figure 29. Impedance response of BVD resonator with the inclusion of external inductors. Legend details on each of the lines in the figure. Effects of the inclusion of external elements in the broadband response can be seen in the inset at the lower right side. ........ 66 Figure 30. First stage on the extraction element process. a) Shunt susceptance and series BVD lowpass prototype. b) Series susceptance and series BVD lowpass prototype. a) Shunt inductance and series BVD resonator. b) Series inductance and series BVD resonator. ..... 69 Figure 31. Screen shot of the GUI Synthesis Software outlining the input filter parameters. .... 71 Figure 32. Frequency responses corresponding to the topologies of Table 2. Details on the inband and band-edges of the filter (above). Details on the wideband response (below). ..... 72 Figure 33. Ladder filter topology with inclusion of series ground inductors in the shunt resonators............................................................................................................................ 74 Figure 34. Typical circuit scheme of the path to ground consisting on a shunt resonator followed by a ground inductor. ........................................................................................... 74 Figure 35. a) BVD model, b) equivalent BVD model, c) equivalent BVD model with the inclusion of the ground inductance, d) resulting BVD model with the inclusion of the parasitic inductance. ........................................................................................................... 76 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Figure 36. In-band transmission and reflection coefficient of a 7th order filter with the effects of a parasitic inductance. ........................................................................................................ 78 Figure 37: Filter response for two different location of the transmission zeros prescribed in the synthesis ............................................................................................................................. 79 Figure 38. Transmission and reflection coefficient of a 7th order filter with the effects of including a ground inductor in the first shunt resonator. The additional transmission zero is fixed to 2.5 GHz. ............................................................................................................. 80 Figure 39. Ladder filter network where each node is connected to an external parasitic network. ............................................................................................................................................ 81 Figure 40. Ladder network of a 5-th order filter with parasitic cross couplings between nodes 1 and 4, and between nodes 2 and 4. Connections of the shunt resonators with the laminate have not been considered here. ........................................................................................... 82 Figure 41. a) and b) Filter responses for different position of transmission zeros and the effects of a given parasitic network. ............................................................................................... 83 Figure 42. a) and b) Filter responses for different position of the parasitic cross-coupling. ....... 84 Figure 43. Filter performance for a) -80 dB cross-coupling, b) Filter performance for -50 dB cross-coupling. ..................................................................................................................... 85 Figure 44. Filter network of the synthesized topology ............................................................... 86 Figure 45. Measured and synthesized response of the B40 receiver band. ................................. 87 Figure 46. Overall filter network including the laminate representation with the ground inductors. ............................................................................................................................ 88 Figure 47. Simulated (blue) and measured (black) S-parameters. Top left figure: input reflection coefficient. Top right figure: output reflection coefficient. Bottom left figure: narrow band transmission coefficient. Bottom right figure: broadband transmission coefficient ............................................................................................................................ 89 Figure 48. Outline of a transversal network, a) without direct source-load coupling, b) with direct source-load coupling. ................................................................................................ 93 Figure 49. Single path of a conventional transversal network. a) non-normalized values, b) scaling transformation has been applied to obtain unitary coupling coefficients. ............... 94 Figure 50. Outlined of the two paths existing in a 2-port filter with symmetric response and without transmission zeros. ................................................................................................. 95 Figure 51. Outlined of the two paths existing in a two-port filter with symmetric response and without transmission zeros, with additional inverters included. ......................................... 95 Figure 52. Outlined of the two paths, with additional inverters included, after initial transformation .................................................................................................................... 96 Figure 53. Outlined of the two paths, with additional inverters included, after initial transformation .................................................................................................................... 96 Figure 54. Lowpass prototype of the BVD model ...................................................................... 97 Figure 55. Value of k as a function of 𝑘𝑒2 ............................................................................... 101 Figure 56. Transversal filter topology based on acoustic wave resonators. .............................. 102 Figure 57. a) Filter response, b) impedance of each individual resonator. ............................... 104 CONTENTS | xiii SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. CONTENTS ABSTRACT i LIST OF FIGURES x LIST OF TABLES xi CONTENTS xiii INTRODUCTION 16 0.1 Objectives of the thesis .................................................................................................. 19 0.2 Contents ......................................................................................................................... 21 THEORETICAL BACKGROUND 23 1.1 Filter transfer function ................................................................................................... 24 1.2 Principles of Synthesis Techniques: Mathematical formulation ..................................... 25 1.3 Element extraction synthesis approach .......................................................................... 26 1.4 Coupling matrix representation ..................................................................................... 30 1.5 Coupling Matrix Synthesis approach ............................................................................. 32 1.6 Dissipation effects .......................................................................................................... 35 1.7 Electro-acoustic resonators and technology .................................................................... 37 1.8 Summary ........................................................................................................................ 47 DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS 49 2.1 Acoustic wave ladder filter topology and concept .......................................................... 50 2.2 Characteristic polynomials for ladder acoustic filters ..................................................... 51 2.3 Synthesis procedure. Element extraction ....................................................................... 52 2.4 Circuit transformation ................................................................................................... 63 2.5 Input series inductance .................................................................................................. 69 2.6 Ground inductors effects ................................................................................................ 73 2.7 General modeling of parasitic effects .............................................................................. 80 CONTENTS | xiv SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. 2.8 Synthesis of a filter for the Band 40 .............................................................................. 85 2.9 Conclusions .................................................................................................................... 89 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY 91 3.1 Transversal Coupling Matrix. Conventional network .................................................... 92 3.2 Transformation to Transversal Topology Based on acoustic wave resonators ............... 93 3.3 Procedure to extract the required cross coupling for a given coupling coefficient 𝒌𝒆𝟐. . 97 3.4 From lowpass prototype to bandpass prototype ............................................................ 99 3.5 Synthesis and Design procedure: Approach I. Prescribed 𝒌𝒆𝟐. .................................... 103 3.6 Flexible Approaches ..................................................................................................... 104 3.7 Synthesis Procedure: Approach II. Prescribed Impedance. .......................................... 109 3.8 Synthesis Procedure Approach III. Prescribed Resonant Frequency. ........................... 112 3.9 Synthesis Procedure: Approach IV. Trade-off. ............................................................. 116 3.10 Advanced filter performances ....................................................................................... 118 3.11 Evaluation of the losses in a transversal filter configuration ........................................ 119 3.12 Sensitivity analysis on transversal topologies ............................................................... 123 3.13 Effects of a non-ideal BALUN stage on the overall filter ............................................. 129 3.14 Practical considerations ............................................................................................... 135 3.15 Practical considerations - External capacitors and shunt inductors ............................. 138 3.16 Conclusions .................................................................................................................. 141 CASE STUDIES 143 4.1 Software for the synthesis of ladder filters ................................................................... 143 4.2 Case studies of ladder filters ........................................................................................ 148 4.3 Case studies on transversal filters ................................................................................ 171 4.4 Conclusions .................................................................................................................. 184 CONCLUSIONS AND FUTURE RESEARCH LINES 186 5.1 Conclusions .................................................................................................................. 186 5.2 Future research lines .................................................................................................... 188 PUBLICATIONS AND OUTCOMES 194 APPENDIX 1: Demonstration of Equivalent Circuits 198 References 202 CONTENTS | xv SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. INTRODUCTION | 16 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. INTRODUCTION Current technological challenges The requirements in mobile communication technologies nowadays are tremendous due to the worldwide steady growth in mobile devices [1] [2] [3] [4]. Every new mobile communications standard is aiming for enabling more demanding applications thus increasing the needed data transmission rates. Requirements for reduced power consumption, low latency, spectrum efficiency and shrinking device footprint are other concerns typycally reflected in communication standards. Just a glimpse of what is discussed here: 4G-LTE defined up to more than forty frequency bands together with Carrier Aggregation (CA) and MIMO technology as some of its most notable features. This increased the complexity of the RFFE Module up to unbelievable limits. Hundreds of possible CA combinations are available inside the RFFE modules existing in the market, meaning standalone filters are being replaced by all sorts of complex multiplexers - including BAW and SAW filter technology – that have to cover low, middle and high frequency bands along with switches for numerous different purposes, diplexers, power amplifiers (PA) or Low Noise Amplifiers (LNA) and antennas. Besides new mobile phone standards must often coexist with both, older mobile phone standards i.e. GSM/3G and different purpose mobile communications standards i.e. Bluetooth, Wi-Fi, GNSS, WiMAX and so on, all of them in a single, light and slim portable device. Therefore integration issues are a nightmare for design engineers and in turn set a lot of pressure on the scientific community and the ICT wireless industry in order to find new solutions that help facing the challenge. Among the RF components, the filtering stage is one of the more critical components, which usually has to be individual for each communication band. Adding to that, many of these bands are very close together and therefore require highly selective filters. When this happens it is crucial to operate without detrimental of mutual interference. Currently modern 4G smart phones may include more than 40 RF filters with very demanding specifications [6]. In fact, more than 100 filters in a phone could be a reality in a near future. The increasing complexity is expected to stay on the forthcoming 5G devices [7] [8] [9], which will operate in an environment that will include many new bands especially at higher portions of the spectrum. This goes along the requirement of enhanced performance demands, smaller size and lower cost as the driving forces for RF enabled products. Natural questions arise then: How does one fit 100+ filters and all the additional RF components into a cell phone considering the ever-tighter size restrictions? And, how can they be integrated for an efficient solution without losing performance? INTRODUCTION | 17 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Figure 1. Nowadays smartphones integrate different front-ends modules to cover (a) low bands (b) mid bands and (c) high bands. Image Source: [5] In the last two decades, electro-acoustic wave filters have been, without any doubt, the key technology for the development of highly miniature high-performance RF/Microwave filters [3]. Such a technological and engineering effort has been one of the driving forces for expansion of handset wireless devices, such as smartphones, tablets and other portable gadgets. The main features of current AW filters are low insertion loss, sharp frequency response and highly miniaturized, in comparison with other technologies. To cope with the worldwide global transition to 4G/5G networks, a demand for even more exceptional high-performance filters is significantly increasing and may become the technological bottleneck in the definition of new advanced services in future standards. This challenge demands for continuous research in the RF acoustic technology to drastically improve the performance of the current filtering structures, reduce further their area, and lower their cost and time-to-market as much as possible. The most critical aspects that limit the performance of Acoustic Wave (AW) filters go from intrinsic limitations of the technology itself, to the filter design and manufacturing process. Despite these huge efforts, achievements and expansion on the development of AW filters, there are still some important drawbacks that limit their transition to even more enhanced responses, their complete expansion into other domains, and the certainty that those will give response to the near future demands. The limitations go from:  Lack of mathematical formulation of the filter responses based on AW filter, even for well-established topologies. INTRODUCTION | 18 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES.  Systematic procedure to go from the mathematical formulation to the network configuration.  Poor prediction of degrading undesired parasitic effects.  Achievable response: o Limited achievable filter bandwidth. o Non-arbitrary location and number of transmission zeros. o Non (or hardly) achievable advance filter performances such as multiband response, and the always desired self-equalized filter response, required for space applications.  Strong dependence on technological constrains. The achievable response mainly depends on: o The electro-acoustic coupling coefficients, o The impedance of the resonators, o The amount of different resonant frequencies of the resonators and their position. Those limitations come mostly from the filter synthesis and the resulting topology used in conventional AW filters. Note that due to the nature of the AW configuration, which gives rise to a pole/zero performance in a single resonator, makes the synthesis and circuit transformation used on conventional technology might not always be applicable. State of the art Even though the contributions of two pioneers like K. S. Van Dyke and W. P. Mason [10] on the characterization of piezoelectric resonators and their integration in electric networks such as filters were published between the late ‘20s and the ‘40s of the past century it wasn’t until the late ’70s and early ’80s when the work of K.M. Lakin, J.S. Wang and G.R. Kline - among others - on thin film SAW and BAW started to become more significant for the manufacturers of filtering devices. Their work was the precursor of the nowadays existent technology and made remarkable advancements in the miniaturization of filters [11] [12] [13], involving fabrication techniques, suitability of materials, piezoelectric film growth and characterization of electromechanical resonators for microwave frequencies. Later in the early ‘90s R. Ruby started out a program for the development of FBAR technology at Hewlett Packard Labs and officially laid the first stone of what currently is a multi-billion dollar industry [6] [9] [14] [15]. It does not come as a surprise that some authors have employed expressions like the “Golden Age” [16] [17] to refer to the acoustic filter design industry nowadays. Major contributors to the technology, like R. Aigner [18] [19] [20] [21] [22], showed their concern as well on the issues related to the performance optimization and practical manufacturing for high-volume production having a key role on the success of the technology and enabling it for today’s advanced performance wireless applications. INTRODUCTION | 19 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Thus, this technology has been under development for decades in one form or another, and it has required very significant advances in integrated circuit processing and high-volume manufacturing procedures to become what it is currently. In addition to [17], [6], there are in the literature some readings that are highly recommended for those interested in this topic. They comprehend a good overview and provide the reader with bright insights regarding key aspects of the thin film acoustic filter technology, helping to follow the path from the filtering circuits used within the first existing wireless cellphones (GSM and before) to the complex multiplexer architectures within the modern RF front-end modules (4G-LTE) as well as foreseen trends (5G and tunable filters) [23]- [22]. In parallel, the evolution of filter synthesis techniques has reached outstanding results and many researchers have worked on it for many decades now. The filter transfer function synthesis date back to the late ‘40s and early ‘50s and some of the most significant contributions, also considering the early published works, may be attributable to G. L. Matthaei [24] . More recent authors like R. J. Cameron, C. M. Kudsia and R. R. Mansour were very prolific generalizing the coupling matrix synthesis method [25], first introduced by A.E. Atia et al. [26] and therefore providing with a very flexible and powerful tool to obtain all sort of filter topologies and filtering responses. Other distinguished authors like J-S. Hong, M. J. Lancaster [27] or I. Hunter [28], also made significant contributions in order to apply the filter synthesis concept to all sort of applications and technologies. Filter synthesis not only helps making easier and more flexible the filter design procedure but also covering all sorts of features regarding filtering requirements, i.e. multiplexing architectures, group delay equalization, predistortion [29], multiband, lossy [30] [31] [32] [33] or tunable. In turn, being such a generalized procedure, it is suitable for any kind of technology where two fundamental pieces coexist, resonators (cavity, waveguide, lumped element, planar, etc.) and couplings mechanisms between them. Oddly enough the paths of filter synthesis and acoustic wave filters had not cross until recent years. Actually, the first works related to the topic didn’t use closed-form expression synthesis techniques but optimization methods of lumped element models based on preliminary given data related to the acoustic technology [34], [35] and [36]. The coupling matrix used to synthesize acoustic filters appears in [37], however the related results are only valid for Coupled Resonators Filters (CRF), which have a very particular working principle and are still a residual percentage of all the EA filters in the market. Finally, in [38] a method to synthesize BAW and SAW filters using the coupling matrix is discussed for the first time. Some more recent works [39], [40] and [41] give a natural continuation to the prior work. 0.1 Objectives of the thesis The goal of this work is to contribute to overcome the above limitations and significantly help to the achievement of higher performance filters to give response to future requirements and further expansion of the application of AW filters. INTRODUCTION | 20 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Current methodologies on the filters and multiplexer design rely on an initial design of the basic resonator and a given ladder pre-existing filter topology from a previous design. Iterations over the initial design are performed until the final filter design reaches the electrical and implementation requirements. This limits the achievable filter performance since does not consider all possible solutions, and makes the procedure very time consuming and with lack of flexibility. To this end the general objective of the thesis is to establish a synthesis procedure for designing BAW filters under the technological constraints given by the technology. The synthesis procedure will allow design novel filter topologies, beyond the conventional ladder topology, which better fulfills given requirements. The specific objectives are:  Develop a mathematical description of the AW filter response based on an equivalent low pass prototype model of the AW filters.  Develop a synthesis procedure for the design of the most commonly used filter and multiplexer topologies. Those are based on ladder configurations.  Develop a new synthesis procedure to obtain novel topologies and generate a solutionspace of filters (different topologies that are complaint with the required specifications)  Propose and demonstrate the synthesis of an innovative topology for the design of AW filters, in both SAW and BAW configurations, which allows to obtain high performance filter with arbitrary response, not achievable with the current state of the art electro-acoustic topologies.  Account for the technological constrains of the filter synthesis from the very beginning. This thesis will perform activities on:  Achieving a good knowledge of the current design procedures of BAW filters, the manufacturing process and the technological constraints.  Mathematical formulation to: o Obtain the characteristic polynomials of the lowpass prototype of AW filters. o Define the lowpass prototype of an AW resonator  Synthesis of ladder filters. o Define a procedure based on the element extraction approach to automatically synthesize conventional ladder filters with prescribed topology. o Use in the synthesis of multiplexers. Since those activities will focus on existing and widely used topologies a software tool has been developed along the mathematical formulation and synthesis procedure. This software tools not only provides the synthesized network but also it allows for the evaluation of INTRODUCTION | 21 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. undesired effects, like parasitic effects, nonlinearities [42] [43] and effects of lateral modes [44] and border ring modes [45]. The major contribution of this work is the novel filter topologies based on a transversal arrangement of the AW resonators forming the filter [47] [48]. The objectives on this last topic are: o Define a new filter topology along with a tailored synthesis procedure. o Evaluate the electrical filter performance of the new topology and its limitations due to its particular nature. o Evaluate the technological requirements of the topology according to the available materials and manufacturing processes. o Provide a design process based on useful designing curves The above objectives and activities define the content of this work. 0.2 Contents This thesis is divided into five chapter in addition of this introductory initial chapter. Among those five chapter we also include the chapter gathering the conclusions of this work and the outline of future research lines. We consider this last chapter very significant since the novel proposed topology opens up a new way of designing acoustic filters and therefore the existance of many different topologies, applications and the possibility of using other materials. Chapter 1 is referred as a background chapter that presents the previous required knowledge for contributing on the synthesis of AW filters. In that sense, this chapter essentially outlines the initial concepts and formulation for the definition of microwave filters along with some of the existing synthesis methodologies that will be used along the thesis. This chapter also includes the description of a BVD (Butterworth-Van-Dike) model which is the basic building block describing the behaviour of an AW resonator around the resonant frequency. The equivalent lowpass prototype circuit model of the BVD is also presented and evaluated. Note that the lowpass prototype is the initially required circuit in a synthesis procedure [24]. Chapter 2 presents all the mathematical formulation and detailed development of the synthesis procedure used for obtaining the ladder networks based on AW resonators. Practical considerations on the filter implementation such as the existance of shunt inductances to ground, or the way the filters need to be ended for a proper connection to the antena even when the filter will be part of a multiplexer, will be also considered on the synthesis process. Examples of designed filters are shown in this chapter. All the formulation and synthesis procedure detailed in this chapter are then used to create a comprehensive software tool for the synthesis of ladder filters. The tool showed to be very INTRODUCTION | 22 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. usefull to provide a quick design of an AW filter. The code also includes the features to provide the design of multiplexing structures. Some details and examples of real case studies designed with this tool, will be shown in Chapter 4. Chapter 3 proposes a novel filter topology, referred along the thesis as transversal network, which allows to overcome many of the limitations observed in the ladder topologies of Chapter 2. This new topology essentially can be used to implement any transfer function without restriction on the electro-acoustic coupling coefficient of each individual resonator. The mathematical details and synthesis procedure to go from the filter characteristic polynomials to the novel transversal network are fully detailed. This method shows to be very flexible and several examples of enhanced performance filters are illustrated at the end of the chapter. As done for ladder filters developed in Chapter 2, this chapter is supplemented with the case studies resported in Chapter 4. Chapter 4 applies the synthesis procedures developed in previous chapters, 2 and 3, respectively for ladder and transversal topologies to evaluate some practical cases, this is using real electrical requirements and implementation constrains. These cases correspond to scenarios of existing sytems and/or bands specificifactions but also to some cases which are under definition for future systems. This chapter clearly illustrates the validity of the methodology proposed in this thesis and breifly details on the implemented software that includes all the mathematical formulation and features detailed in Chapter 2. Finally, Chapter 5 summarizes the conclusions and outlines future research activies that are consecuence of the findings of this work. At this point, it is worth to mention that the novel topologies presented in Chapter 3 opens up the possibility of developing other novel topology based on the transversal concept. Also the advantages of this topology in terms of implementation constrains give rise to the possibility of using those topologies for novel applications or for being suitable to be used with other material or resonator configuration where the coupling coefficient it is not an issue. 1 THEORETICAL BACKGROUND | 29 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. By doing so the remaining admittance is Y1(s) = 1/Z1(s). Analogously, and since the remaining input admittance has a numerator defined by polynomial of one order higher than the denominator, we can extract a shunt capacitor as: 𝑌 1(𝑠) 𝑠∣𝑠→∞=1.154  C2=1.154 F Y2(s)=Y1(s)-C2s= 1.383 1.548s+1.383 Z 2 (s) L 1 C 2 Z 0 =1 Now, from the remaining impedance Z2(s) we can again extract a series inductance as: 𝑍2(𝑠) 𝑠∣𝑠→∞=1.12  𝐿3=1.12 𝐻 Then, we can write down Z3(s) as: Z3(s)=Z2(s) - L3s=1 Which results in a non-dependent frequency component, a resistor to a normalized unitary value, that is the output impedance. The element extraction technique results in a ladder network as in figure below. L 1 =1.12H C 2 =1.154F Z 0 =1 Z 0 =1 L 3 =1.12H Figure 4. Ladder topology of a 3rd order Chebyshev lowpass prototype filter. This example, although a very simple one, clearly illustrates the element extraction process. It is worth to emphasize that the extraction of each component at each step of the procedure is not unique and different filter networks could give rise to the same filter performance [28]. Note that this corresponds to the lowpass prototype, which would become the bandpass filter based on LC series and LC shunt resonators by applying circuit transformations [24]. 1 THEORETICAL BACKGROUND | 30 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. 1.4 Coupling matrix representation As any network the synthesized filter of Figure 5 allows for a matrix representation. In order to introduce the Coupling matrix synthesis approach, detailed section 1.5 and used for the development of novel filter topologies based on acoustic filters in chapter 3, this subsection uses the network of Figure 5 above to find the coupling matrix representation. Coupling matrix representation requires of the existence of couplings between resonators, thus a circuit transformation needs to be applied to the circuit of Figure 5 for the inclusion of those couplings. This transformation consists on the introduction of impedance/admittance inverters in order to transform the ladder topology in a coupled resonators topology [25]. This results in the filter topology shown in Fig. 5. C 2 J 01 C 1 C 3 J 12 J 23 J 34 Node 1 Node 2 Node 3Node S Node L G S G L Figure 5. Coupled resonators topology of a 3rd order Chebyshev lowpass prototype filter. The general admittance matrix of a 3x3 network relates the current getting into the node Ii with the voltage dropped at the corresponding node Vi. ⎣ ⎢ ⎡ 𝐼1 𝐼2 𝐼3 ⎦ ⎥ ⎤ = ⎣ ⎢ ⎡ 𝑌 11 𝑌 12 𝑌 13 𝑌 21 𝑌 22 𝑌 23 𝑌 31 𝑌 32 𝑌 33 ⎦ ⎥ ⎤ ⎣ ⎢ ⎡ 𝑉1 𝑉2 𝑉3 ⎦ ⎥ ⎤ The resulting admittance matrix of the red-squared circuit in Fig. 5, without including the source and load impedances neither the input-output couplings (admittance inverters), is: ⎣ ⎢ ⎡ 𝐼1 𝐼2 𝐼3 ⎦ ⎥ ⎤ = ⎣ ⎢ ⎡ 𝑠 𝐶 1− 𝑗𝐽 12 0 − 𝑗𝐽 12 𝑠 𝐶 2− 𝑗𝐽 23 0− 𝑗𝐽 23 𝑠 𝐶 3 ⎦ ⎥ ⎤ ⎣ ⎢ ⎡ 𝑉1 𝑉2 𝑉3 ⎦ ⎥ ⎤ Which can be extended to include the source and the load as: 1 THEORETICAL BACKGROUND | 31 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. ⎣ ⎢ ⎢ ⎢ ⎡ 𝐼𝑠 𝐼1 𝐼2 𝐼3 𝐼𝐿 ⎦ ⎥ ⎥ ⎥ ⎤ = ⎣ ⎢ ⎢ ⎢ ⎡ 𝐺𝑠 − 𝑗𝐽 01 000 − 𝑗𝐽 01 𝑠𝐶1 −𝑗𝐽12 00 0 − 𝑗𝐽 12 𝑠𝐶2 − 𝑗𝐽 23 0 00 −𝑗𝐽23 𝑠 𝐶 3 − 𝑗𝐽 34 000 − 𝑗𝐽 34 𝐺𝐿 ⎦ ⎥ ⎥ ⎥ ⎤ ⎣ ⎢ ⎢ ⎢ ⎡ 𝑉𝑠 𝑉1 𝑉2 𝑉3 𝑉𝐿 ⎦ ⎥ ⎥ ⎥ ⎤ This admittance matrix is also referred as the (N+2)×(N+2) admittance matrix, where N is the order of the filter, N=3 in this case. The (N+2)×(N+2) admittance matrix can then be written down as the summation of three matrices, ⎣ ⎢ ⎢ ⎢ ⎡ 𝐺𝑠 0000 00000 00000 00000 0000 𝐺𝐿 ⎦ ⎥ ⎥ ⎥ ⎤ −𝑗 ⎣ ⎢ ⎢ ⎢ ⎡ 0 𝐽 01 000 𝐽 01 0 𝐽12 00 0 𝐽 12 0 𝐽 23 0 00 𝐽23 0 𝐽 34 000 𝐽 34 0 ⎦ ⎥ ⎥ ⎥ ⎤ +𝑠 ⎣ ⎢ ⎢ ⎢ ⎡ 00000 0 𝐶 1 000 00 𝐶2 00 000 𝐶 3 0 00000 ⎦ ⎥ ⎥ ⎥ ⎤ Where the first matrix includes the source and the load impedances, the second matrix include the coupling between resonators and the third matrix accounts for the frequency dependence. The values of the matrices above can be scaled without affecting the relation between Ii and Vi [25] therefore without affecting the filter performance and obtaining the following matrices: ⎣ ⎢ ⎢ ⎡ 10000 00000 00000 00000 00001 ⎦ ⎥ ⎥ ⎤ −𝑗 ⎣ ⎢ ⎢ ⎢ ⎡ 0 𝑀 01 000 𝑀 01 0 𝑀12 00 0 𝑀 12 0 𝑀 23 0 00 𝑀23 0 𝑀 34 000 𝑀 34 0 ⎦ ⎥ ⎥ ⎥ ⎤ +𝑠 ⎣ ⎢ ⎢ ⎡ 00000 01000 00100 00010 00000 ⎦ ⎥ ⎥ ⎤ It is now clear, from this notation, that all the information on the filter can be gathered into the coupling matrix, whose values Mij are the normalized coupling values. The resulting coupling matrix for the characteristic polynomials of Section 1.3.1 would be [𝑀]= ⎣ ⎢ ⎢ ⎡ 0 0.9453 000 0.9453 0 0.8779 00 0 0.8779 0 0.8779 0 00 0.8779 0 0.9453 000 0.9453 0 ⎦ ⎥ ⎥ ⎤ Although more details will be given in the following section, during this thesis the filter topologies will be represented with the conventional scheme that appears in Figure 6, where the 1 THEORETICAL BACKGROUND | 32 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. filled-in black circles corresponds to the resonators (Ri) forming the filter, the empty circles represent the source (S) and load (L), and the connecting lines indicate the coupling mechanisms (Mij). Note that in this topology there are no cross-couplings, this is coupling between non-adjacent nodes, and therefore results in an in-line topology [25]. Figure 6. Nodal notation of an in-line three order filter. S stands for source, Ri stands for resonant node, L stands for load and Mij is the notation used for the couplings between nodes. 1.5 Coupling Matrix Synthesis approach In contrast with the element extraction approach where the characteristic polynomials of (1) are used to obtain the input impedance and its circuit network afterwards, the matrix approach uses these polynomials to obtain the coupling matrix of the filter. As in the example above (section 1.3.1) we will use the coupling matrix in its (N+2)×(N+2) format, instead of the N×N format. Note that the N×N matrix does not consider possible couplings between the source and the load, and as will be seen in Chapter 3, the existence of the source and load nodes in the coupling matrix might be convenient on the synthesis of novel topologies based on a transversal configuration. The synthesis procedure starts by recalling the scattering parameters of the network as a function of the characteristic polynomials (1). The 2-port scattering matrix can then be transformed to the 2-port Y matrix, by following matrix conversion from S to Y [50]. Doing so, the admittance matrix of a 2-port network can be written as: [ 𝑌 11(𝑠) 𝑌 12(𝑠) 𝑌 21(𝑠) 𝑌 22(𝑠)]= 1 𝑌 𝑑(𝑠)[ 𝑌 11𝑛(𝑠) 𝑌 12𝑛(𝑠) 𝑌 21𝑛(𝑠) 𝑌 22𝑛(𝑠)](5) Where the numerators and denominator of the admittance matrix may be written as a function of the characteristic polynomials as detailed in the following set of equation: MS1 M23 M3L S L R1 R 2 R3 M1 2 1 THEORETICAL BACKGROUND | 33 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. 𝑌𝑑(𝑠)=(𝐸(𝑠)+ 𝐹 (𝑠) 𝜀𝑟)(𝐸(𝑠)+(−1)𝑁 𝐹 ∗(𝑠) 𝜀𝑟)−(𝑃(𝑠) 𝜀)2 2𝐸(𝑠) 𝑌21𝑛(𝑠)=𝑌12𝑛(𝑠)=𝑃(𝑠)/𝜀 𝑌𝑑11𝑛(𝑠)=(𝐸(𝑠)−𝐹(𝑠)/𝜀𝑟)(𝐸(𝑠)+(−1)𝑁𝐹∗(𝑠)/𝜀𝑟)+(𝑃(𝑠)/𝜀)2 2𝐸(𝑠) 𝑌 𝑑22𝑛(𝑠)=(𝐸(𝑠)+ 𝐹 (𝑠)/𝜀𝑟)(𝐸(𝑠)−(−1)𝑁 𝐹 ∗(𝑠)/𝜀𝑟)+(𝑃(𝑠)/𝜀)2 2𝐸(𝑠) (6) * Note that we used a normalized source and load impedance to 1. As outlined in (6), it results in a set of rational fractions where both the numerator and the denominator are polynomials. In such case, we may perform partial fraction expansion (also known as partial fraction decomposition) [51]. This operation consists in expressing the fraction as a sum of polynomials with a simpler denominator. From the nature of the characteristic polynomials, the resulting rational fraction above can be decomposed in a sum of polynomials where the numerators are residuals and the denominators are first order polynomials. This is illustrated in the expression below, where ri are the residuals and pi are the poles of the rational function. 𝐵(𝑠) 𝐴(𝑠)=𝑟1 𝑠− 𝑝 1+𝑟2 𝑠− 𝑝 2+𝑟3 𝑠− 𝑝 3+⋯+ 𝑟4 𝑠− 𝑝 4(7) By applying this to the rational functions of the 2-port admittance matrix the decomposition results on: [ 𝑌 11(𝑠) 𝑌 12(𝑠) 𝑌 21(𝑠) 𝑌 22(𝑠)]= 1 𝑌 𝑑(𝑠)[ 𝑌 11𝑛(𝑠) 𝑌 12𝑛(𝑠) 𝑌 21𝑛(𝑠) 𝑌 22𝑛(𝑠)]=∑ 1 (𝑠− 𝑗 𝜆𝑘) 𝑁 𝑘=1 [𝑟11𝑘 𝑟12𝑘 𝑟21𝑘 𝑟22𝑘](8), where rijk are the residues and  k are the eigenvalues. The eigenvalues are obtained from the roots of Yd(s) and, in case of characteristic polynomials corresponding to an ideal lossless response, they result in purely imaginary roots, therefore purely real eigenvalues. The residues can be obtained from: 1 THEORETICAL BACKGROUND | 34 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. 𝑟𝑖𝑗𝑘= 𝑌 𝑖𝑗𝑛(𝑠) 𝑌 ′𝑑(𝑠)∣𝑠= 𝑗 𝜆𝑘(9), where Y'd(s) is the first derivative of Yd(s). The conclusions of that procedure is very significant and states that any response following the performance of the characteristic polynomials in (1) can be synthesized by a network consisting on transversal connections of first order filters, i.e., resonators. This type of topology is known as transversal topology (see Figure 7a, below). This well-known transversal topology is the basis for the new developed AW transversal topology. As an example (10) shows the coupling matrix of a 4th order canonical transversal filter. [𝑀]= ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ 0 𝑀 𝑆1 𝑀1𝑆 𝑀11 𝑀 𝑆2 𝑀 𝑆3 00 𝑀 𝑆4 𝑀 𝑆𝐿 0𝑀 1𝐿 𝑀2𝑆 0 𝑀3𝑆 0 𝑀22 0 0𝑀 33 0 𝑀2𝐿 0 𝑀3𝐿 𝑀 4𝑆 0 𝑀 𝐿𝑆 𝑀 𝐿1 00 𝑀 𝐿2 𝑀 𝐿3 𝑀 44 𝑀 4𝐿 𝑀 𝐿4 0 ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ (10) where each resonator is only coupled to the source (MSi or MiS) and load (MLi or MiL), and there are no couplings between resonators. The terms Mii on the diagonal indicate the mutual coupling, which defines the resonant frequency of each individual resonator . Considering the equivalent circuit model of each resonator coupled to the source and load of Figure 7.b, we can obtain the 2-port admittance matrix of the network as: [ 𝑌 ]=[ 𝑌 11(𝑠) 𝑌 (𝑠) 𝑌 21(𝑠) 𝑌 22(𝑠)]= 𝑗 [0 𝑀 𝐿𝑆 𝑀 𝑆𝐿 0]+∑ 1 𝑠𝐶𝑘+ 𝑗 𝐵𝑘[ 𝑀 𝑆𝑘 2 𝑀 𝑆𝑘 𝑀 𝐿𝑘 𝑀 𝑆𝑘 𝑀 𝐿𝑘 𝑀 𝐿𝑘 2] 𝑁 𝑘=1 (11)  Note that the elements of the admittance matrix Yij (s) can be obtained as a function of the characteristic polynomials. 1 THEORETICAL BACKGROUND | 35 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. a) SL Conventional Resonator Convention a Coupling R 1 R 2 R N-1 R N b) M sk M Lk C k jB k Conventional Resonator Conventional Coupling Figure 7. a) Equivalent network of a transversal filter of order N. The empty circles S and L represent the source and the load, the black filled-in circles (from R1 to RN) represent the resonators and the lines connecting the circles represent the mainline couplings from each resonator to the source and load. b) Equivalent circuit of the k-th resonator and its couplings to the source and load. Now the outer terms of the coupling matrix MkS, MSk, MkL and MLk can be found from the residues of the admittance matrix terms, using (9). The other elements in matrix M, the diagonal elements Mkk, can be found by equating the real and imaginary part of the elements of (8) and (11), respectively. Doing so results in: 𝐶 𝑘=1, 𝐵𝑘(≡ 𝑀 𝑘𝑘)=−𝜆𝑘(12) Therefore, the diagonal of the coupling matrix is formed by the eigenvalues of the coupling matrix. As previously mentioned the conclusions resulting from this section are very significant and will be also valid when applied to acoustic wave filters. This means than any filter response could be synthesized as a transversal network. This concept will imply a significant advantage over the conventional responses achievable with AW ladder filters. 1.6 Dissipation effects Dissipation effects produce degradation of the filter response, mainly in the in-band and on the roll-off skirts of the band-edges. These effects are due to existence of lossy components on the ultimate designed filter network. In filters based on conventional coupled resonators the major contribution of the losses are due to the limited Q of the resonators, being the couplings between resonators usually negligible on the overall dissipation effect. In that case, and when the limited Q of the resonators is uniform, i.e., all the resonators have identical Q, the 1 THEORETICAL BACKGROUND | 36 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. dissipation effects can be predicted in advance from the characteristic polynomials, without needing to perform the synthesis of the filter [24]. This is not the case of most acoustic implemented filters, where the resonators forming the filter have different Q (non-uniform Q ), and the overall losses of each resonator needs to account for the Q of both resonances (resonance and anti-resonant) and the external losses elements like electrodes and additional inductors and capacitors which are very common on the resulting topologies. Despite of that, the possibility to account for the losses in advance (prior to the synthesis), sets the best filter achievable performance. Note that the accurate modeling of those losses needs to evaluate the resulting network topology and account for the losses of each individual component. Further details of the losses in an AW resonator are presented in section 1.6. The text below outlines how to account for the dissipative effects into the characteristic polynomials. This method will be used along the thesis and compared with the accurate modeling that considers all the lossy sources when needed. To do that one could simply add a positive real factor  to the purely imaginary variable s = j   on the transfer function expression S21(s). This is equivalent to move all the poles of the transfer function to the left, in the s-plane representation. The value of  for a bandpass filter can be expressed as: 𝜎= 1 𝑄·𝐹𝐵𝑊 (13),  where FBW is the fractional bandwidth of the filter. Then the scattering parameters can be evaluated at s=  +j  Figure 8 depicts the effects of losses in the transfer function and reflection coefficient for a 4th order Quasi-Elliptic filter of 2.5% fractional bandwidth, with a single pair of transmission zeros. The black curve corresponds to the ideal response of the filter, whereas the blue, green and red curves represent the filter response for finites Qs of 1000, 500 and 250, respectively. In any case, the effects of dissipation in a bandpass filter are:  The insertion loss increases  A rounding off of the insertion loss curve at the band edges occurs. This diminishes the width of the passband and reduces the selectivity of the filter.  Transmission and reflection zeros are less distinct. These three features are more prominent when the losses increase or when the fractional bandwidth of the filters decreases. 1 THEORETICAL BACKGROUND | 37 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Figure 8. Transfer and reflection coefficients of a polynomial bandpass filtering response as a function of its Q factor. 1.7 Electro-acoustic resonators and technology This work frames on the electro-acoustic technology and in particular in the electro-acoustic filters and its application into the telecommunication industry. Although an extensive and detailed description of the electro-acoustic technology is not necessary for a full comprehension of the work developed during this thesis, outlining the most common AW resonator configuration is a must. In particular, a detailed description of the equivalent circuits describing the AW resonators, is necessary. The synthesis techniques developed in this work are based on a basic building block described by the BVD equivalent circuit model. This model is detailed in this section and it is used to describe the performance of AW resonator in both configuration: Surface Acoustic Wave (SAW) resonators and Bulk Acoustic Wave (BAW) resonators. 1.7.1 Bulk Acoustic Wave Resonator The BAW resonator is widely used on filter development for wireless applications operating at frequency nearly and above 2 GHz. Figure 9 sketches the configuration of a BAW resonator, which consists of a piezoelectric plate sandwiched between to metallic electrodes. Additional layers (or air gaps) are included on at bottom and/or top of the piezoelectric layer to define the boundary conditions that confine the acoustic wave and therefore define the resonant mode. As in a conventional resonator the resonant mode occurs when a standing wave is patterned inside the resonator, which defines the resonant frequency. The fundamental resonant frequency also known as longitudinal frequency in BAW resonators is mainly defined by the thickness and material properties of the piezo-electric layer. This happens approximately when 1 THEORETICAL BACKGROUND | 38 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. the piezo-electric thickness is half-wavelength. Note however, the resonant frequency is strongly affected by any additional layer, including those used as reflector layers. Piezoelectric film V Top electrode Bottom electrode Reflector Layers Substrate Piezoelectric film V Top electrode Bottom electrode Air Gap Suppor t layer Figure 9. Outline of an acoustic wave resonator on its BAW (Bulk Acoustic wave) configuration. Note that the configuration of Figure 9 corresponds to a parallel plate capacitor, where the dielectric constant of the piezo-electric defines the capacitance between electrodes. Being a piezo-electric material, a voltage applied between the top and bottom electrodes gives rise to an acoustic wave. 1.7.2 Surface Acoustic Wave Resonator In contrast with BAW resonators where the acoustic wave travels through the thickness of the piezo-electric layer, the acoustic wave travels along the surface of the piezo-electric. A conventional configuration of a SAW resonator consists of an interdigited capacitor (IDC), as outlined in Figure 10. The resonant frequency in this case is defined by the separation between the fingers of the IDC. The feasibility of manufacturing these structures with very small gap between the fingers, limits the application of this configuration to frequencies below 2 GHz. As occurs in the BAW configuration, an external voltage is applied to create a travelling acoustic wave, note as well that an electric capacitor also exists between the two ports where the voltage is applied. Piezoelectric IDC V Figure 10. Outline of an acoustic wave resonator on its SAW (Surface Acoustic wave) configuration. 1.7.3 BVD circuit model. Lossless Figure 11 shows a Butterworth-Van-Dike (BVD) circuit model that describes the main resonances of an acoustic wave resonator, for both, the BAW and SAW configuration. The 1 THEORETICAL BACKGROUND | 45 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Now, those parameters need to be related with the circuit parameters of the BVD model, 𝐿𝑠, 𝐶𝑠 and 𝐶𝑝. a) C LP Z=jX LP Y=jB LP b) Z=jB’ LP Y=jX’ LP L LP Figure 15. a) lowpass prototype corresponding to the BVD reciprocal circuit model. b) lowpass prototype corresponding to the BVD model. Each of these normalized frequencies can be related with their corresponding bandpass frequencies, 𝑓𝑠 and 𝑓𝑝, by following the frequency transformation equation of (27). Note that, this results in a second order equation as 𝑓𝑠2−𝑓𝑠·𝐹𝐵𝑊·𝑓0·Ωs−𝑓02=0, for the case of 𝑓𝑠, with two solutions. In practice, and as will be shown in chapter 2 in the synthesis of ladder filters, the equivalent circuit of Figure 15.a is extracted as a series resonator, and its de-normalized shunt resonance 𝑓𝑝, should result in a transmission zeros at the upper bandpass. Note that this value is prescribed on the definition of the characteristic polynomials. Then, its counterpart, circuit of Figure 15.b, is extracted as a shunt resonator whose de-normalized series resonance 𝑓𝑠 should match the prescribed transmission zeros at the lower passband. By applying circuit transformation and frequency transformation to the lowpass prototypes above, we can relate the parameters of the pass band prototypes with the circuit parameters defining the lowpass prototype. The two set of equations corresponding to both circuits are detailed in (30) and (31), below: 1 THEORETICAL BACKGROUND | 46 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. 𝐶 2= 𝐶 𝐿𝑃 𝐹𝐵𝑊𝜔0 𝐿2=1 𝐶2(2𝜋𝑓𝑝)2 𝐶 3= 𝐶 2[( 𝑓 𝑝 𝑓 𝑠)2−1] (30) 𝐿𝑠=𝐿𝐿𝑃 𝐹𝐵𝑊𝜔0 𝐶𝑠=1 𝐿𝑠(2𝜋𝑓𝑠)2 𝐶 𝑝 = 𝐶 𝑠( 𝑓 𝑠 𝑓 𝑝 )2[1−( 𝑓 𝑠 𝑓 𝑝 )2]−1 (31) Figure 16, shows the frequency response of a BVD model mapped down and normalized in frequency to overlap the frequency response of the lowpass prototypes defined in the current section. Figure 16. Lowpass prototype (mapped at the operating frequency) and bandpass prototype performance of a single acoustic wave resonator. In contrast with conventional passband circuit transformation to lowpass prototype of conventional LC resonators, - both in its series and shunt configuration -, where the overlap is perfect all over the frequency range, this is not the case for the lowpass prototypes presented in this section. The reason for that is because in the conventional case the circuit transformation is consistent with the frequency transformation, whereas in the present case the developed lowpass prototypes are simply circuit artifacts to emulate a resonator with a resonant and antiresonant frequency, but does not fully follow the same frequency response. In spite of that the proposed circuit model has the advantage of having only one frequency dependent component (CLP or LLP), being therefore an order one resonator. This implies a suitable formulation of filters based on acoustic wave resonators by means of the presented characteristic polynomials, where an N order polynomial requires of N order 1 resonators. It is also worth to mention at this point that both lowpass circuit prototypes of Figure 15, do not follow the exact same frequency dependence. 1 THEORETICAL BACKGROUND | 47 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. On the other hand, the fact that passband and lowpass circuits do not follow exactly the same frequency dependence will result in differences between the lowpass prototype filter and the bandpass filter (based on BVD resonators). Although these effects will be mentioned along the thesis when synthesis and case studies will be shown, its major effects occur in the in-band of very wideband filters and in the out-of-band rejection. In the latter case, the final bandpass filter shows actually a better rejection than the lowpass prototype, due to the frequency dependence of the resonators itself. Despite this latter explanation, the proposed lowpass prototype shows to be very useful for the mathematical formulation, synthesis and to provide good designs at passband frequencies. To finalize this chapter, Figure 17 shows the conventional representation of an acoustic resonator, which will be used along the whole document. This representation corresponds to any of the equivalent circuits presented in previous sections, for both the lowpass prototypes, Figure 15.a and Figure 15.b, and for the bandpass prototypes, Figure 11 and Figure 13, and even if we refer to SAW or BAW configurations. Figure 17. Typical representation of an acoustic resonator. 1.8 Summary This chapter presented most of the theoretical background to track on the mathematical development reported in following Chapters 2 and 3, where the synthesis of ladder filters and transversal acoustic wave filters are developed. To this end, the main topics have been addressed: the mathematical formulation required on the synthesis procedure, which is included in sections 1.2 - 1.6, and the circuit modelling of acoustic resonators, which is described in section 1.7. This chapter is complemented with further details on the mathematical description of the characteristic polynomials in the references provided along the document. Despite the existence of this initial chapter, all the mathematical details and procedures are carefully detailed throughout this thesis. | 48 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. 2 DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS | 49 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS Ladder configuration is probably the most common topology used on the development of acoustic filters, for both SAW and BAW configurations. In spite of that, there is barely none straightforward reported methods for the synthesis of this type of filters. This chapter develops a direct synthesis method based on the very well-known extraction element process, described in section 1.3. Due to the nature of the acoustic resonator, and the fact of performing a resonant and anti-resonant response in a single resonator, some considerations need to be accounted for the application of the element extraction method. The chapter starts by outlining the concept of a ladder topology detailing the way the filter performance is profiled. From this, we identify a kind of characteristic polynomials that can be synthesized through a ladder network based on AW resonators as a main building block. Then the element extraction synthesis procedure is described for these particular polynomials. Previous considerations for the application of this method are described along the test. Additional circuit transformations are latter applied to the resulting synthesized topology in order to address and provided further useful topology from the technological point of view. Some illustrative examples will be shown at the end of this chapter. Additional examples considering real case studies, including multiplexing structures will be detailed in Chapter 4. Chapter 4 will also describe the user-friendly software tool created for the application of the current synthesis technique. CHAPTER 2 2 DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS | 50 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. 2.1 Acoustic wave ladder filter topology and concept Figure 18 outlines the ladder filter topology based on acoustic resonators. As occurs in a conventional ladder filter configuration, this topology consists on cascading series and shunt AW resonators. Series path Shunt Resonators Figure 18. Outline of a ladder network. In contrast with conventional resonators, AW resonators exhibit a pole/zero response. This singular characteristic allows to achieve sharper filter responses due to the existence of transmission zeros without the need of including cross-coupling effects [28]. Nevertheless this also limits the type of responses that can be achieved with this configuration, and therefore, it will set the characteristic polynomials that define the filter responses. Figure 19. S-parameters of a 7th order (4 series, 3parallel -4s3p-) AW ladder filter (Figure 18), and input impedance of the isolated resonators. Note that due to the symmetry of the response only two of the four series resonators are different and two of the three shunt resonators are equal. 1.85 1.9 1.95 22.05 2.1 2.15 Frequency (GHz) -100 -80 -60 -40 -20 0 20 40 60 80 dB Acoustic Filter Frequency Response 𝑓 𝑎 𝑝 𝑓 𝑟 𝑝 𝑓 𝑎𝑠 𝑓 𝑟𝑠 2 DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS | 51 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Figure 19 shows the S-parameters (black lines) corresponding to a 7th order ladder filter based on acoustic resonators and the resonators impedances (blue & red lines). This figure clearly shows how due to the nature of acoustic resonators, that is, each of them creating a transmission zero, the transfer response offers a pronounced selectivity. The transmission zeros at the lower side band are created at the frequency point where the shunt resonator offers a very small impedance (short circuit - known as resonance or series resonance), whereas the transmission zeros at the upper side band are created by the series resonator at the frequency point where the impedance of the resonator is very large (open circuit - known as anti-resonance or shunt resonance). To better illustrate this concept, Figure 20 depicts the circuit behavior for different frequencies. Note that, in general, impedance of series resonators are higher than impedances of shunt resonators at the out-of-band, 𝑍𝑅𝑠𝑒𝑟≫ 𝑍𝑅𝑠ℎ𝑢 and just the other way around, 𝑍𝑅𝑠ℎ𝑢≫𝑍𝑅𝑠𝑒𝑟 for most of the in-band frequency range, especially around 𝑓0 where the difference is especially significant. R ser1 R ser2 R ser3 R ser4 R shu1 R shu2 R shu3 Assuming all shunt resonators have the same 𝑓 𝑟 𝑝 , (which here is an approximation and not always necessarily true), when 𝑓=𝑓𝑟𝑝, the impedances of the shunt resonators are very low (they act as a short-circuit). The signal finds a clearer path to the ground. R ser1 R ser2 R ser3 R ser4 R shu1 R shu2 R shu3 Around the center frequency of the filter we are close to two singular frequency points, 𝑓 ≅ 𝑓 𝑟𝑠, where all 𝑍𝑅𝑠𝑒𝑟 are very low (SC) and 𝑓≅𝑓𝑎𝑝, where the impedances of the shunt resonators are very high (OC). The signal finds a clear straight path to the output port. R ser1 R ser2 R ser3 R ser 4 R shu1 R shu2 R shu3 Assuming all series resonators have the same 𝑓 𝑎𝑠 (which again is not always necessarily true), when 𝑓=𝑓𝑎𝑠, the impedances of the series resonators are very high (they act as an open circuit). The signal sees a very high impedance series path and gets reflected to the input port. Figure 20. Working principle of a ladder network based on acoustic resonators a) TZ created by the shunt resonators b) behavior in the in-band region and c) TZ created by the series resonators. 2.2 Characteristic polynomials for ladder acoustic filters As outlined in section 1.2, general Chebyshev characteristic polynomials offers the flexibility to propose the type of filter responses that can be implemented by means of ladder 2 DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS | 52 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. configurations. Although it may exist other types of characteristic polynomials for this purpose, the nature of an AW resonator and the arrangement in a ladder configuration, requires characteristic polynomials with: Equal number of transmission zeros than number of resonators forming the filter The location of the transmission zeros is defined by the number of series or shunt resonators: oIn a filter of order N odd that starts with a series resonator (and therefore ends with a series resonators), the number of transmission zeros above the upper band edge are (N+1)/2, and at the lower side band are (N-1)/2. oIn a filter of order N odd that starts with a shunt resonator (and therefore ends with a shunt resonators), the number of transmission zeros below the lower band edge are (N+1)/2, and at the upper side band are (N-1)/2. oIn a filter of order N even, the number of transmission zeros in both, below the lower band edge and above the upper band edge are N/2. In this case two filter configuration exist: Starting with series resonator. Starting with shunt resonator. Those polynomials will be used to synthesize the ladder network based on AW resonators and they can define any position of transmission zeros and reflection zeros. In practice, this will offer many designs that can meet certain filter specifications and requirements, by means of proposing different sets of characteristic polynomials. As presented in previous chapter, the characteristic polynomials are defined in the lowpass frequency domain, and the equivalent lowpass circuit models of the AW resonators of Figures 15a and 15b will be extracted from the synthesis. 2.3 Synthesis procedure. Element extraction Element extraction is applied for the type of polynomials described above. This results in a recipe that can always be followed and only slight variations need to be considered depending if an even or an odd order filter is synthesized and on the location of the transmission zeros. As indicated above, this results in variations of the filter configuration and, therefore, this directly affects the extraction procedure. This section outlines the procedure for the case of an N odd order filter where the number of transmission zeros above the upper band edge are (N+1)/2, and in the lower side band are (N1)/2. Despite this particular case the concept and rules to be followed at each step are completely general. Procedure: 2 DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS | 53 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. 1) Define the characteristic polynomials according the rules of previous section. The transmission zeros above the upper band edge band are located at Ω𝑖, where i=1, 3, 5... N, and the transmission zeros below the lower band edge band are located at Ω𝑗, where j=2, 4 ... N-1. 2) Obtain the input filter impedance, ZT(s), following (4). Recall that this assumes a 2-port network loaded with a 1  normalized impedance. 𝑍𝑇(𝑠)=𝐸(𝑠)+𝐹(𝑠)𝜀𝑅 ⁄ 𝐸(𝑠)− 𝐹 (𝑠)𝜀𝑅 ⁄=𝑍𝑛(𝑠) 𝑍 𝑑(𝑠) Z T (s),Y T (s) where Zn(s) and Zd(s) represent, respectively, the numerator and denominator. 3) The first element to extract, is a susceptance jBi, which is extracted to obtain a remaining admittance Y1(s) such that exhibits a transmission zero at 1 of the upper side band (antiresonance of a series resonator). This value can be obtained by evaluating the admittance YT(s) at the normalized frequency of the transmission zero, as: 𝐵𝑖=𝑖𝑚𝑎𝑔( 𝑌 𝑇 ( 𝑗 Ω1)) Although the real part is zero, this process forces to take only the imaginary part, to avoid numerical problems. Note that flexibility exists on selecting the transmission zero to be extracted, which could be any of the ones located above the upper band edge. Selection of a different transmission zero would result in a different acoustic resonator. Then, the resulting network is: Z T (s) Y T (s) Z 1 (s) Y 1 (s) Y=jB i Where Y1(s), is obtained as: 𝑌 1(𝑠)= 𝑌 𝑇(𝑠)− 𝑗 𝐵𝑖 4) Due to the condition of 3), the resulting admittance Y1(s) exhibits a zero at 1, therefore the impedance Z1(s) could be written as [54]: 2 DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS | 54 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. 𝑍 1(𝑠)= 𝐾1 𝑠− 𝑗 Ω1+ 𝑍 2(𝑠) Which would result in the following network: Z 1 (s) Y 1 (s) Z 2 (s),Y 2 (s) Y=jB i Y=jB 1 C 1 Z T (s) Y T (s) Where the value of 𝐶1 and 𝐵1 are: 𝐶 1=1 𝐾1,𝐵 1=−Ω1 𝐾1 and K1 can be easily obtained by applying: 𝐾1= 𝑍 1(𝑠)(𝑠− 𝑗 Ω1)|𝑠= 𝑗 Ω1 5) Extract a reactive element j𝑋1 to obtain a remaining impedance with a transmission zero at  2 of the lower side band (resonance of the shunt resonator). This value can be obtained by evaluating the impedance Z2(s) at the normalized frequency of the transmission zero, as: 𝑋1=𝑖𝑚𝑎𝑔( 𝑍 2( 𝑗 Ω2)) Again, flexibility exists on selecting the transmission zero to be extracted, as long as this is located below the lower band edge. This results in the following network: 2 DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS | 61 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Figure 25. Impedance of the synthesized resonators. Blue lines correspond to series resonators, and red lines correspond to shunt resonators (above). Synthesized frequency response of the 7th order filter (below). 2.3.2 Bandpass transformation The synthesized responses presented so far correspond to the lowpass prototype evaluated at the normalized frequency domain . Those responses result from the evaluation of the characteristic polynomials or by evaluating the lowpass prototype. As occurs in conventional (non-acoustic) filters both responses overlap all over the frequency range. To obtain the bandpass responses one can evaluate the characteristic polynomials considering the frequency transformation of equation (27), at f, or by evaluating the resulting synthesized network after circuit transformation to the bandpass circuit model [24]. Recall at this point that the transformation from the lowpass prototype to the bandpass circuit model uses the circuit transformation outlined in section 1.7.5. In contrast with conventional (non-acoustic) resonators, where the response of the lowpass prototype matches the response of the bandpass prototype but a shift in frequency, in this case , certain disagreement at both in-band and outof-band response exists. As mentioned in section 1.7.5, this is because the circuit transformation from lowpass to bandpass is not fully consistent with the frequency transformation. S - p a r a m e t e r s ( d B ) Zres (dB) 2 DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS | 62 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Figure 26. Filter response of a 5th order filter for a) 50 MHz bandwidth, b) 200 MHz bandwidth, c) 400 MHz bandwidth. In order to illustrate this point, which will be observed and referred along the designs presented in this work, the figure above shows the frequency responses of three filter, whose lowpass response is the same and therefore the same lowpass prototype and characteristic polynomials, which correspond to a 5th order filter, with three transmission zeros at the upper side band and two transmission zeros at the lower side band, and 20 dB return losses. The filter responses differ on the their bandwidth, being 50 MHz, 200 MHz and 400 MHz, corresponding respectively to Figure 26.a, Figure 26.b and Figure 26.c. Red lines outline the response of the characteristic polynomials, and black line correspond to the bandpass synthesized network. The results of Figure 26 reveal that when the filter bandwidth increases, the disagreement between the ideally synthesized response and the bandpass responses increases, both at the inband and at the out-of-band frequency ranges. Despite of that, all the filter responses in all cases are still very good, which demonstrates the validity of the synthesis procedure and circuit transformations proposed. At this point, it is very important to mention that in all the responses above no restriction have been considered on the value of the coupling coefficient of the resonators, and that their values increase as the bandwidth increases. 1.8 1.82 1.84 1.86 1.88 1.9 1.92 1.94 Frequency (GHz) -60 -40 -20 0 1.6 1.7 1.8 1.9 2 2.1 Frequency (GHz) -60 -40 -20 0 1.4 1.6 1.8 2 2.2 2.4 Frequency (GHz) -60 -40 -20 0 2 DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS | 63 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. 2.4 Circuit transformation The synthesis method above does not offer the flexibility of selecting the coupling coefficient of the resonators in advance, and for a given characteristic polynomials the unique flexibility, for instance, in an odd order filter is the transmission zero assigned to each resonator. The flexibility of this method lays on the fact that a ladder filter topology can be proposed for any general Chebyshev set of polynomials, and it is well-known that there are many sets of polynomials that fulfill given filter specifications. This allows for selecting the most convenient polynomials for the implementation. Despite of that, full flexibility on the a priori selection of the resonator parameters does not exist. The present section recalls and formulates the circuit transformation applied at the resonator level in order to obtain the desired coupling coefficient. Those transformations are very well established on the design of acoustic filters and require the inclusion of external elements. However, the inclusion of external elements is uncommonly desired from a practical point of view, since increases volume, introduces losses and further complicates the design and manufacturing procedure. Regarding this latter statement then, the ultimate goal would be to provide a filter network without external elements and with desired values of coupling coefficients. Similar arguments can be exposed on the selection of the resonant frequency of the resonators. As occurs with the coupling coefficients those cannot be prescribed in advanced and result from the proposed characteristic polynomials and the way the transmission zeros are extracted. Although the manufacturing techniques allow for full flexibility on the selection of the resonant frequency, in practice, a maximum of four to six resonant frequencies might be tailored if need it in a single wafer. Note that increasing the number of the available resonant frequencies in a single chip, would complicate the fabrication process and substantially increase the cost. Additionally, the differences between all the resonant frequencies to be implemented in a single chip cannot significantly differ. Figure 27 shows the possible circuit transformations to be applied in a single resonator. Figure 27.a1 and Figure 27.a2 show how a series capacitance would be added to a BVD resonator. Figure 27.b1 and Figure 27.b2 include a shunt capacitor, Figure 27.c1 and Figure 27.c2 includes a shunt inductor, and Figure 27.d1 and Figure 27.d2 include a series inductor. 2 DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS | 64 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. L s C s C p a1) b1)c1)d1) a2) b2) c2) d2) Figure 27. BVD model with the inclusion of external elements in order to modify the coupling coefficient. a) inclusion of a series capacitor, b) inclusion of a shunt capacitor, c) inclusion of a shunt inductor, and d) inclusion of a series inductor. From simple inspection of the external elements above along with the BVD circuit model (Figure 11), and also the equivalent BVD model (Figure 13), one can easily conclude that the frequency performance of a BVD circuit model with the inclusion of a series or shunt capacitance is not affected and this new circuit emulates exactly an acoustic BVD circuit performance with an equivalent coupling coefficient and different series or shunt resonance [23] [13]. This is clearly illustrated in Figure 28. Figure 28 shows the impedance frequency performance of a 4.1% coupling coefficient BVD model. This would be the value resulting from the synthesis. Since in practice the coupling coefficient of the resonator cannot be freely selected, , we would need to transform it to a 4.1 % coupling coefficient equivalent resonator, by means of suitable acoustic resonators, 6.7% for instance in this case (whose impedance in indicated in black), with an additional external element, a shunt or series capacitor..By doing so, the impedance of a 6.7% acoustic resonator is shown in dashed red line in Figure 28, where the series resonator 𝑓𝑠 remains as in the synthesized resonator and the shunt resonance 𝑓𝑝 is moved to lower frequencies. If the series capacitor is connected at this resonator, it emulates a 4.1% resonator, indicated in dotted red line. Note that this perfectly overlap the synthesized resonator response. In solid black line the required 6.7% resonator would perform with identical 𝑓𝑝 as in the synthesized resonator and 𝑓𝑠 is moved to higher frequencies. Again, when this resonator is connected to a shunt capacitor this exactly emulates the 4.1% synthesized resonator, indicated in red line. 2 DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS | 65 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. This concludes therefore that a series or shunt capacitor results on an equivalent acoustic resonator with lower coupling coefficient. Figure 28. Impedance response of BVD resonator with the inclusion of external capacitors. Legend details on each of the lines in the figure. In contrast with the previous cases, when an additional inductance is added in shunt or series to the acoustic resonator, as shown in Figure 27.c2 and Figure 27.d2, the resulting circuit emulates an acoustic resonator with larger coupling coefficient. The example to illustrate this statement assumes a target coupling coefficient of 7.5% to be implemented with acoustic resonators of 6.7% and additional inductors. Results of that are summarized in Figure 29. When a shunt inductor is added to get the required 7.5%, the overall impedance exhibits the same 𝑓𝑠 and the 𝑓𝑝 is shifted to higher frequencies. Whereas in the case of a series inductor, the impedance will have now identical 𝑓𝑝 and the 𝑓𝑠 is shifted to lower frequencies. Both cases allow to emulate a higher coupling coefficient resonator. Nevertheless, this approach is only valid in a certain frequency range. The inset of Figure 29 shows the broadband behavior of these equivalent circuits. In the case of adding a shunt inductance the impedance gives rise to a pole at lower frequencies, and in the case of adding a series inductance a zero-impedance point occurs at higher frequencies. Note that this might have important implications in the broadband response of the filter, and this fact should be considered in advance. 123456 Frequency (GHz) -20 0 20 40 60 80 100 Zresonator Zresonator + Z Cser Zresonator // Z Cshu 1.8 1.82 1.84 1.86 1.88 Frequency (GHz) -20 0 20 40 60 80 2 DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS | 66 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Figure 29. Impedance response of BVD resonator with the inclusion of external inductors. Legend details on each of the lines in the figure. Effects of the inclusion of external elements in the broadband response can be seen in the inset at the lower right side. Formulation to find the values of the equivalent circuit, this is the values of the BVD model and external components for a given targeted coupling 𝑘𝑒2target, is detailed below for the four circuit transformations above. First, we assume as initially synthesized values, 𝑓𝑠, 𝑓𝑝, and 𝐶𝑠, 𝐶𝑝, 𝐿𝑠 defining the BVD circuit, or 𝐶2, 𝐿2 and 𝐶3 for the equivalent BVD circuit. Cadd shunted 𝑓𝑠, 𝐿𝑠 and 𝐶𝑠 remain as in the synthesis 𝑓𝑝_𝑛𝑒𝑤 is obtained as: 𝑓 𝑝_𝑛𝑒𝑤= 𝑓 𝑠 0.5+√0.25−4 𝜋2𝑘𝑒2𝑡𝑎𝑟𝑔𝑒𝑡 (32) 𝑪𝒑_𝒏𝒆𝒘 is obtained as: 2 DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS | 67 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. 𝐶 𝑝_𝑛𝑒𝑤= 𝐶 𝑠 (2𝜋 𝑓 𝑝 _ 𝑛𝑒𝑤)2𝐿𝑠 𝐶 𝑠−1 (33) Cadd is obtained as: 𝐶 𝑎𝑑𝑑= 𝐶 𝑝 − 𝐶 𝑝 _ 𝑛𝑒𝑤 (34) Cadd series 𝑓𝑝, 𝐿2 and 𝐶2 remain as in the synthesis 𝑓𝑠_𝑛𝑒𝑤 is obtained as: 𝑓 𝑠_𝑛𝑒𝑤= 𝑓 𝑝(0.5+√0.25−4 𝜋2𝑘𝑒2𝑡𝑎𝑟𝑔𝑒𝑡) (35) 𝐶3_𝑛𝑒𝑤 is obtained as: 𝐶 3_𝑛𝑒𝑤=1 (2𝜋 𝑓 𝑠 _ 𝑛𝑒𝑤)2𝐿2− 𝐶 2 (36) Cadd is obtained as: 𝐶 𝑎𝑑𝑑=1 1 𝐶 3−1 𝐶 3 _ 𝑛𝑒𝑤 (37) Now we use 𝐶2, 𝐿2 and 𝐶3_𝑛𝑒𝑤 to obtain the new values of the conventional BVD resonator by applying the formulation in section 1.7.3. Ladd shunted 𝑓𝑠, 𝐿𝑠 and 𝐶𝑠 remain as in the synthesis 𝑓𝑝_𝑛𝑒𝑤 is obtained as: 2 DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS | 68 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. 𝑓 𝑝_𝑛𝑒𝑤= 𝑓 𝑠 0.5+√0.25−4 𝜋2𝑘𝑒2𝑡𝑎𝑟𝑔𝑒𝑡 (38) 𝐶𝑝_𝑛𝑒𝑤 is obtained as: 𝐶 𝑝_𝑛𝑒𝑤= 𝐶 𝑠 (2𝜋 𝑓 𝑝 _ 𝑛𝑒𝑤)2𝐿𝑠 𝐶 𝑠−1 (39) Ladd is obtained as: 𝐿𝑎𝑑𝑑=1 (2𝜋 𝑓 𝑝 _ 𝑛𝑒𝑤)2( 𝐶 𝑝 _ 𝑛𝑒𝑤− 𝐶 𝑝 ) (1) Ladd series 𝑓𝑝, 𝐿2 and 𝐶2 remain as in the synthesis 𝑓𝑠_𝑛𝑒𝑤 is obtained as: 𝑓 𝑠_𝑛𝑒𝑤= 𝑓 𝑝(0.5+√0.25−4 𝜋2𝑘𝑒2𝑡𝑎𝑟𝑔𝑒𝑡) (40) 𝐶3_𝑛𝑒𝑤 is obtained as: 𝐶 3_𝑛𝑒𝑤=1 (2𝜋 𝑓 𝑠 _ 𝑛𝑒𝑤)2𝐿2− 𝐶 2 (41) Cadd is obtained as: 𝐿𝑎𝑑𝑑=1 (2𝜋 𝑓 𝑠 _ 𝑛𝑒𝑤)2(1 𝐶 3−1 𝐶 3 _ 𝑛𝑒𝑤) (42) Now, we use 𝐶2, 𝐿2 and 𝐶3_𝑛𝑒𝑤 to obtain the new values of the conventional BVD resonator by applying the formulation in Section 1.7.3. 2 DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS | 69 2.5 Input series inductance As presented on the synthesis of ladder acoustic filters, when the filters start with a series resonator, the synthesis method forces to extract a shunt susceptance as a first component, indicated as Y=-jBi, see figure in point 3) of section 2.3, where Bi is a positive value. This results therefore in a shunt inductance as a first element of the network. This configuration results being impractical in real developments and it is more suitable to start with a series inductance instead. a) SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. 2 DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS | 70 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. this turn to fulfill ZINa=ZINa_2. Note that this assumption does not reduce the generality of the following development, since jXLP (in Figure 30.b) and -jBi_2 are added together, and the model would be valid as long as the addition matches. To this end, the set of equations below defines the value of ZINa: 2 DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS | 77 First, we apply the circuit transformation to our synthesized shunt BVD in order to obtain the equivalent BVD. In other words, from the circuit in Figure 35.a we obtain the circuit in Figure 35.b. The next step will be adding 𝐿𝑔𝑛𝑑 to the circuit (Figure 35.c). At this point, the initial conditions are set to obtain a BVD model that offers identical 𝑓𝑠 and 𝑓𝑝 than the synthesized one. This results in the following equations involving the new values, 𝐶33 and 𝐿𝑔𝑛𝑑: Where 𝑓𝑠=1(2𝜋√𝐿𝑠𝐶𝑠)⁄ . And, Therefore, if we compare the frequency dependence between the original BVD impedance response 𝑍𝐵𝑉𝐷 (47) and the 𝑍𝑇 impedance response (49), corresponding to BVD+𝐿𝑔𝑛𝑑, the differences will become more significant as long as we keep moving far from the frequency point that sets the equation (51) and also as long as the value of 𝐿𝑔𝑛𝑑 increases. From the circuit perspective, since 𝐿2, 𝐶2 in Figure 35.b define the anti-resonance of the resonator and these are not modified, we know that even if we add a ground inductance, 𝑓𝑝 will remain the same. This is consistent with the solutions obtained from the impedance expression (49). In addition, since we already set 𝑓=𝑓𝑠 we assure the impedance will achieve the same value in both singular frequency points (𝑓𝑠, 𝑓𝑝). Otherwise, for any other condition 𝑓  𝑓𝑠 , the near-out-of-band notch created by our shunt branch would appear at a location different from our prescribed transmission zeros. SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. 2 DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS | 78 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Finally we will transform the circuit in Figure 35.c to the one in Figure 35.d obtaining thus a new BVD resonator in series with the calculated 𝐿𝑔𝑛𝑑. The example we display in Figure 36 shows the performance of a 100 MHz lossless filter centered at 2.35 GHz by means of synthesizing a 7th order filter. The filter topology consists on a ladder network with four series resonators and three shunt resonators. This filter produces four transmission zeros at the upper side band at 2.4225 GHz and three transmission zeros at the lower side band at 2.2776 GHz. The initially synthesizes return losses are 18dB. Figure 36. In-band transmission and reflection coefficient of a 7th order filter with the effects of a parasitic inductance. Using this filter as an example, we evaluate the effects of having a ground inductance at only the first shunt resonator of the filter. We consider three different cases: 1) the effect of including an additional transmission zero very close to the passband, fOoB_TZ = 2.5 GHz, 2) at an arbitrary intermediate frequency fOoB_TZ = 3.25 GHz and 3) at fOoB_TZ = 4.5 GHz, thus far from the passband. Once fOoB_TZ is set we use (51) to solve (52) obtaining 𝐶33 and 𝐿𝑔𝑛𝑑. Results of these three cases are outlined in the Table 3 and Figure 36, where we can clearly identify the position of the additional transmission zeros, in red, blue and green, respectively. Dashed black lines depict the response of the resonator and the filter with no ground inductor whatsoever. Note that the position of the initial transmission zeros produced by the shunt branch at the lower band edge remains always the same as per the condition of (51) 𝑓=𝑓𝑠= 1(2𝜋√𝐿𝑠𝐶𝑠)⁄ . Even though solution values for 𝐶33 are not relevant, they are shown in the Table 3 along with solution values for 𝐿𝑔𝑛𝑑 and the other parameters involved in each step of the procedure. It is worth to pay attention to what is happening to the coupling value of this new BVD, 𝑘𝑒_𝑛𝑒𝑤 2 , due to these circuit transformations. Since the synthesized resonator is absorbing the ground inductance, it behaves as an equivalent resonator (defined by 𝐿𝑠𝑠, 𝐶𝑠𝑠, 𝐶𝑝𝑝) with a much lower 𝑘𝑒2 that makes use of the inductor to virtually increase its value up to the one achieved before the “absorption”. Now, our resonator is therefore very impractical in terms of feasibility. Fortunately, to counteract this situation the synthesis allows much flexibility to play with both, 𝑓𝑠(=𝑇𝑍𝑙𝑏) and 𝑓𝑇𝑍_𝑂𝑜𝐵 in order to reestablish a proper 𝑘𝑒2 value. 2 DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS | 79 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. fOoB_TZ Figure 35.a) Figure 35.b) Figure 35.c) Figure 35.d) 𝑘𝑒2 =5.53%, 𝑓 𝑠=2.2776GHz 𝐿2=0.12nH 𝐶2=39pF 2.5GHz 𝐿𝑠=58.07nH 𝐶𝑠=0.084pF 𝐶𝑝=1.77pF 𝐶3=1.85pF 𝐶33=0.242pF 𝐿𝑠𝑠= 3.14uH, 𝐶𝑠𝑠= 1.49fF 𝐶 𝑝 𝑝=0.24pF 𝐿𝑔𝑛𝑑=17.54nH 𝑘𝑒 _ 𝑛𝑒𝑤 2=0.7596%, 𝑓 𝑠 _ 𝐵𝑉𝐷𝑛𝑒𝑤=2.3239GHz 3.25GHz 𝐶33=0.9pF 𝐿𝑠𝑠s=0.235uH, 𝐶𝑠𝑠=0.02pF 𝐶 𝑝 𝑝=0.88pF 𝐿𝑔𝑛𝑑 = 2.79nH 𝑘𝑒 _ 𝑛𝑒𝑤 2=2.79%, 𝑓 𝑠 _ 𝐵𝑉𝐷𝑛𝑒𝑤=2.3046GHz 5GHz 𝐶33=1.45pF 𝐿𝑠𝑠=92.9nH, 𝐶𝑠𝑠=0.05pF 𝐶 𝑝 𝑝=1.4pF 𝐿𝑔𝑛𝑑=0.73nH 𝑘𝑒 _ 𝑛𝑒𝑤 2=4.3834%, 𝑓 𝑠 _ 𝐵𝑉𝐷𝑛𝑒𝑤=2.2889GHz Table 3: Summary of the circuit parameters for the synthesis responses of Figure 36. Figure 37 demonstrates that we can counteract the in-band degradation by means of relocating our transmission zero, 𝑻𝒁𝒍𝒃 to a new place and recover the initial coupling value. We run the example with the worst case scenario simulated before and set 𝒇𝑻𝒁_𝑶𝒐𝑩 at 2.5GHz. Figure 37: Filter response for two different location of the transmission zeros prescribed in the synthesis 1.6 1.8 2 2.2 2.4 2.6 2. 8 Frequency (GHz) 109 -80 -60 -40 -20 0 20 40 60 80 100 2 DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS | 80 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. In conclusion, the procedure outlined creates a new transmission zero integrating the ground inductor to the original BVD. It provides with very good results unless the selected 𝑓𝑂𝑜𝐵_𝑇𝑍 was very close to the passband, in which case, it may degrade the in-band response spoiling the return losses at the upper band edge. In addition, by inspection of (53) we can see that in order to keep fulfilling the equality, if we place 𝑓𝑂𝑜𝐵_𝑇𝑍 far from the band, 𝐿𝑔𝑛𝑑 will be necessarily small. 𝐿𝑔𝑛𝑑=1 1 𝐿2−(2𝜋 𝑓 𝑂𝑜𝐵 _ 𝑇𝑍)2 𝐶 2+1 (2𝜋 𝑓 𝑂𝑜𝐵 _ 𝑇𝑍)2 𝐶 33 (53) Therefore, it is also clear that the closer we set fOoB_TZ to the passband, the higher is the value of the ground inductor needed, and the lower is the 𝑘𝑒2 of the resultant BVD (Figure 35.d). This situation can be improved by changing the frequency value used to set (51). For this particular case, several different frequency values have been tested to this purpose as indicated in the legend of Figure 38. Differences on the in-band response can be observed. Note that this affects to the position of the transmission zero initially synthesized to 2.2776 GHz. These results foresee the idea of avoiding major degradation in the in-band frequency response and try to remark the flexibility provided by the synthesis. Figure 38. Transmission and reflection coefficient of a 7th order filter with the effects of including a ground inductor in the first shunt resonator. The additional transmission zero is fixed to 2.5 GHz. 2.7 General modeling of parasitic effects The synthesis procedure presented in this chapter provides a quick extraction of a filter network that fulfills the system requirements. This fast procedure allows the integration of this method into a more complex system analysis that for instance accounts for the parasitic effects. 2.2 2.3 2.4 2.5 -40 -20 0 20 40 60 80 2.25 2.3 2.35 2.4 2.45 2.5 -70 -60 -50 -40 -30 -20 -10 0 Insertion Loss (dB) 2.25 2.3 2.35 2.4 2.45 2.5 -50 -40 -30 -20 -10 0 Return Loss (dB) Frequency (GHz) Frequency (GHz) Impedance(dB) Frequency (GHz) 2 DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS | 81 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Figure 39 outlines the generalized case in which parasitic effects might affect the filter network. In this figure each node is connected to an external parasitic network. From the practical point of view this network could be defined as S parameters through a .snp file format. For the node and resonator disposition defined here, the nodal admittance matrix [Y] can be written as: ⌈𝑌⌉= ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ −𝑌1−𝑌1000⋯ −𝑌1𝑌1+𝑌2+𝑌3−𝑌2−𝑌30⋯ 0−𝑌2−𝑌200⋯ 0 0−𝑌30𝑌3+𝑌4+𝑌5−𝑌4⋯ 000−𝑌4−𝑌4⋯ ⋮⋮⋮⋮⋮⋱ ⋮ ⋮ ⋮ ⋮⋮ ⋯𝑌𝑁−5+𝑌𝑁−4+𝑌𝑁−3 −𝑌𝑁−4 −𝑌𝑁−3 00 ⋯−𝑌𝑁−4 −𝑌𝑁−4 000 0⋯−𝑌𝑁−3 0𝑌𝑁−3+𝑌𝑁−2+𝑌𝑁−1 −𝑌𝑁−2 −𝑌𝑁−1 ⋯0 0−𝑌𝑁−2 −𝑌𝑁−2 0 ⋯0 0− 𝑌 𝑁−1 0− 𝑌 𝑁−1 ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ (54) Where the network has N nodes, N-1 resonators andYi is the admittance corresponding to the acoustic resonator i. R 1 R 3 R N-3 R N-1 R 2 R 4 R N-2 External parasitic network External parasitic network 1N-2 24 35N-1 N Figure 39. Ladder filter network where each node is connected to an external parasitic network. Figure 40 depicts a 5th order ladder filter connected to a parasitic network that includes undesired cross couplings between nodes 1 and 3, and between nodes 1 and 4. Those cross couplings are modeled by means of admittance inverters [56]. The nodes connected to the laminate have not been considered just for the sake of simplicity, however it is likely that parasitic effects might appear between resonators and the laminate and between ground inductors as well. 2 DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS | 82 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Y ser1 Y ser2 Y ser3 Y shu1 Y shu2 J 2,4 J 1,4 14 23 Figure 40. Ladder network of a 5-th order filter with parasitic cross couplings between nodes 1 and 4, and between nodes 2 and 4. Connections of the shunt resonators with the laminate have not been considered here. By analogy to the general nodal admittance matrix above (54), and considering the different nodal definition of the example, we can write down the new matrix that describes the ladder network of Figure 40, without the inclusion of parasitic effects neither the ground inductors, as: [𝑌]= ⎣ ⎢ ⎢ ⎡ − 𝑌 𝑠𝑒𝑟1 − 𝑌 𝑠𝑒𝑟1 − 𝑌 𝑠𝑒𝑟1 𝑌 𝑠𝑒𝑟1+ 𝑌 𝑠ℎ𝑢1+ 𝑌 𝑠𝑒𝑟2 00 − 𝑌 𝑠𝑒𝑟2 0 0 − 𝑌 𝑠𝑒𝑟2 0 0 𝑌 𝑠𝑒𝑟2+ 𝑌 𝑠ℎ𝑢2+ 𝑌 𝑠𝑒𝑟3 − 𝑌 𝑠𝑒𝑟3 − 𝑌 𝑠𝑒𝑟3 − 𝑌 𝑠𝑒𝑟3 ⎦ ⎥ ⎥ ⎤ (55) And a corresponding admittance matrix describing the parasitic network of Figure 40 like (56): [𝑌𝑝𝑎𝑟𝑎𝑠𝑖𝑡𝑖𝑐]= ⎣ ⎢ ⎢ ⎡ 000− 𝑗 𝐽14 000−𝑗𝐽 24 0000 − 𝑗 𝐽14 − 𝑗 𝐽24 00 ⎦ ⎥ ⎥ ⎤ (56) Then both admittance matrix can be added together to evaluate the filter performance under the existence of parasitic effects, which are modeled as ideal frequency invariant admittance inverters: 2 DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS | 83 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. [𝑌𝑡𝑜𝑡𝑎𝑙]=[𝑌]+[𝑌𝑝𝑎𝑟𝑎𝑠𝑖𝑡𝑖𝑐]=⎣ ⎢ ⎢ ⎡ − 𝑌 𝑠𝑒𝑟1 − 𝑌 𝑠𝑒𝑟1 − 𝑌 𝑠𝑒𝑟1 𝑌 𝑠𝑒𝑟1+ 𝑌 𝑠ℎ𝑢1+ 𝑌 𝑠𝑒𝑟2 0 − 𝑗 𝐽14 − 𝑌 𝑠𝑒𝑟2 − 𝑗 𝐽24 0 − 𝑌 𝑠𝑒𝑟2 − 𝑗 𝐽14 − 𝑗 𝐽24 𝑌 𝑠𝑒𝑟2+ 𝑌 𝑠ℎ𝑢2+ 𝑌 𝑠𝑒𝑟3 − 𝑌 𝑠𝑒𝑟3 − 𝑌 𝑠𝑒𝑟3 − 𝑌 𝑠𝑒𝑟3⎦ ⎥ ⎥ ⎤ (57) The approach above, which is based on conventional circuit analysis, is used to illustrate several cases. The first one consists in a 5th order lossless filter centered at 1.95 GHz, with 75 MHz bandwidth, and syntesized on its ladder configuration. In this case, only a cross coupling between the input and the output is evaluated by means of including an admittance inverter, J14, as in Figure 40. The filter is evaluated for a -80 dB cross coupling. This effect is evaluated when the filter is synthesized for two different position of transmission zeros. Results for each of those cases are detailed in Figure 41.a and Figure 41.b, respectively. Blue line corresponds to the synthesized response and black line outlines the filter performance under parasitic effects. For these two cases the parasitic effects do not degrade the in-band response, this can be seen on the reflection coefficient (pink lines). From this example, we can conclude that the effects of identical parasitic effects might be very different depending on the initial filter network. Under this consideration, it becomes very usefull to have a procedure that can directly evaluate the parasitic effects. a) b) Figure 41. a) and b) Filter responses for different position of transmission zeros and the effects of a given parasitic network. To further illustrate this capability, the following example considers a 7th order lossless filter evaluated under two different cross-couplings. Both cross-couplings are fix to -80dB. In one case, Figure 42.a, the parasitic cross coupling is set between node 1 (source) and 4 (just before the last series resonator). The other case, Figure 42.b, considers the cross-coupling between node 2 (just after the first series resonator) and node 4. As occurred in previous example the effects in the in-band are negligible, and these parasitic mainly affects the position of the transmission zeros. This example allows to conclude that the position of the cross-coupling, and 1.8 1.85 1.9 1.95 22.05 2.1 2.1 5 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 Frequency response with parasitic effects Frequency (GHz) S parameters (dB) 1.8 1.85 1.9 1.95 22.05 2.1 2.15 2.2 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 Frequency response with parasitic effects Frequency (GHz) S parameters (dB) 2 DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS | 84 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. therefore the arrangement of the resonators on the chip, might be an important issue for the final filter response. Again, the ability to have a fast synthesis procedure might be very useful to determine in advance the most robust response to parasitic effects and help on the definition of the filter layout. a) b) Figure 42. a) and b) Filter responses for different position of the parasitic cross-coupling. The last example considers the filter above where now the parasitic cross coupling is set between nodes 2 (just after the first series resonator) and 3 (after the second series resonator). In this case two different values of cross-coupling are evaluated. Figure 43.a shows the effects when a -80 dB cross-coupling is inserted, and Figure 43.b when a -50 dB cross-coupling is inserted. In the first case the effects of the parasitic are negligible, whereas in the second case this affects both the in-band and out-of-band response. However, even in the latter case, the filter response is not much degraded, being therefore, this particular filter significantly robust to undesired parasitic effects (within the tested range of values) between nodes 2 and 3. This might conclude that those two resonators could be arranged physically close in the layout. 1.8 1.85 1.9 1.95 22.05 2.1 2.15 2.2 -110 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 Frequency response with parasitic effects Frequency (GHz) S parameters (dB) 1.8 1.85 1.9 1.95 22.05 2.1 2.15 2.2 -110 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 Frequency response with parasitic effects Frequency (GHz) S parameters (dB) 2 DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS | 85 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. a) b) Figure 43. Filter performance for a) -80 dB cross-coupling, b) Filter performance for -50 dB crosscoupling. 2.8 Synthesis of a filter for the Band 40 The last section of this chapter presents the synthesis of a filter following real requirements. The result of the synthesized filter is then compared to the final manufactured filter. Both synthesized and implemented filters follow the same topology and with very similar values on their components. This example therefore validates the utility of the synthesis procedure as a tool to highly contribute in the process of acoustic filter development. The filter requirements correspond to the receiver band 40, and are listed in the Table 4. 1.8 1.85 1.9 1.95 22.05 2.1 2.15 2.2 -110 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 Frequency response with parasitic effects Frequency (GHz) S parameters (dB) 1.8 1.85 1.9 1.95 22.05 2.1 2.15 2.2 -110 -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 Frequenc y response with parasitic effects Frequency (GHz) S parameters (dB) 2 DEVELOPMENT OF LADDER TOPOLOGIES IN ACOUSTIC FILTERS | 86 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Frequency (MHz) Min (dB) Typ (dB) Max (dB) Parameter Notes 2300 – 2400 -1.2 -1.8 S(2,1) Insertion Loss 2300 – 2400 9.5 S(1,1) S ( 2 , 2 ) Return Los s ( VSWR 2:1 ) 1 – 2215 20 S(2,1) Out-of-Band Attenuation 2215 – 2240 10 2240 – 2275 5 2428 – 2471 30 2471 - 2481 30 40 2481 - 12500 20 Table 4. RX band 40 specifications. To meet the specification above we propose the following filter parameters:  Order: 7  Bandwidth: 100 MHz  Central frequency: 2350 MHz  Return losses: 15 dB  Transmission zeros at the upper band (4) in MHz: 2447, 2429, 2429, 2447  Transmission zeros at the lower band (3) in MHz: 2266, 2266, 2266 The synthesis procedure is applied with these filter parameters. As mentioned in the synthesis procedure, even for a given input filter parameters, flexibility exists on the order in which the transmission zeros are extracted. For the current synthesis the extraction of the transmission zeros is performed as they are listed above. This results in the following network, whose circuit parameters are detailed in Table 5. L in R ser1 R ser2 R ser3 R ser4 R shunt1 R shunt2 R shunt3 L out Figure 44. Filter network of the synthesized topology 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 93 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Figure 48. Outline of a transversal network, a) without direct source-load coupling, b) with direct sourceload coupling. A well-known significant conclusion is that any transfer function can be synthesized as a transversal topology:  For any number of transmission zeros  For complex transmission zeros (for instance in case of phase equalization)  For any position of transmission zeros  For multiband filter responses Note that the features listed above are not complaint with a conventional ladder topology based on acoustic wave filter, where the number of transmission zeros is equal to the number of resonators and their positions are basically defined by the material coupling coefficient. Additionally, in a ladder acoustic wave filter topology the achievable bandwidth is limited by the coupling coefficient of the resonators and performing dual band, for not saying multiband, are not trivial. 3.2 Transformation to Transversal Topology Based on acoustic wave resonators The following section describes the steps and circuit transformations applied to a transversal network as the one in Figure 48, to result in a network based on acoustic wave resonators. Let start considering a single signal path, this is input port, coupling, resonator, coupling and output port, which would correspond to one branch of the outlined transversal network above. Details on the single signal path are in Figure 49. a) SL Conventional Resonator Conventional Coupling b) SL Conventional Resonator Conventional Coupling Direct coupling Source-Load 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 94 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. a) J Si J iL C i jB i b) 11 C ni jB ni Figure 49. Single path of a conventional transversal network. a) non-normalized values, b) scaling transformation has been applied to obtain unitary coupling coefficients. The values of 𝐽𝑆𝑖, 𝐽𝑖𝐿, 𝐶𝑖 and 𝐵𝑖 can be related to the values resulting in (59)),(see (11), (12)). Note that 𝐽𝑆𝑖 and 𝐽𝑖𝐿, correspond to the coupling from the source to the resonator and to the resonator to the load by means of an admittance inverter. 𝐶𝑖 and Bi are the lowpass capacitance and frequency independent reactance, which set the lowpass resonant frequency of the corresponding branch. In a general Chebyshev type response with equal input and output impedances, 𝐽𝑆𝑖 and 𝐽𝑖𝐿 are equal in magnitude and can differ only on 180 degrees on phase. Scaling factors into the resulting coupling matrix, this is multiplying a given row and column by the same value, can be applied without affecting the final filter response. By doing this on each resonator node, the equivalent circuit of Figure 49.b is obtained, where the admittance transformers are set to 1 (or -1 in the case of a 180º). For the sake of clarity, we present the concept of going from a conventional transversal topology (see Figure 48) to a transversal topology based on acoustic wave resonators, in a symmetric 2nd order filter without transmission zeros. In that case the initial conventional transversal topology would consist only on two branches without any direct coupling between the source and the load. Additionally, due to properties of the general Chebyshev polynomials, these two branches will be detailed as in Figure 50. a) 11 C ni jB ni b) 1-1 C ni -jB ni 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 95 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Figure 50. Outlined of the two paths existing in a 2-port filter with symmetric response and without transmission zeros. Figure 50 reveals that in these conditions the impedance of the resonator of each branch (defined by C𝑛𝑖) are equal, and the mutual couplings [27], defined by the shunt reactance term, are also equal in magnitude and have opposite phase. Figure 50 also shows that one of the inverters has opposite phase. It is worth to mention, that these conditions and properties fulfill, in path pairs, as long as the frequency response is symmetric, the order of the filter is even and the number of transmission zeros is lower that the order of the filter. For an odd order filter this also fulfills with an additional branch whose resonance frequency equals to the central frequency of the filter (=0, in the low pass prototype). Taking therefore the branches of Figure 50, we can add at each branch an admittance inverter of opposite sing, without affecting the frequency response, as detailed in the figure below Figure 51. Note that, since the branches are connected in parallel they simply add, therefore cancelling the effect of the inserted inverters. a) 1 Cn i jBn i J SLi 1 b) -1 Cn i -jBn i -J SLi 1 Figure 51. Outlined of the two paths existing in a two-port filter with symmetric response and without transmission zeros, with additional inverters included. Each of the branches sketched in Figure 51 can be transformed into individual lowpass prototypes of acoustic resonators, detailed in Figure 15 in Chapter 1. Although detailed in [57], [25], below we outline this procedure. The first step would consist on transforming the shunt admittance (defined by 𝐶𝑛𝑖 and 𝐵𝑛𝑖) into series impedances (defined by 𝐿𝑆𝑖 and 𝑋𝑆𝑖), where 𝐿𝑆𝑖=𝐶𝑛𝑖 and 𝑋𝑆𝑖=𝐵𝑛𝑖. The resulting circuits from such transformation are outlined in Figure 52. The term T, in the circuit of Figure 52.b corresponds to a transformer of -1 (or a phase shift of 180 degrees). From Figure 52, and using the -network circuit model defining an admittance inverter [27], is then straightforward to obtain the equivalent circuits of Figure 53. Details on such transformation can be found in Appendix 1. 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 96 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. a) JSLi LsijXsi b) -J SLi T=-1 Ls i -jXs i Figure 52. Outlined of the two paths, with additional inverters included, after initial transformation a) Ls i Z=jXs i Y=jJ SLi Y=-jJ SLi Y=-jJ SLi b) Ls i Z=-jXs i Y=jJ SLi Y=-jJ SLi Y=-jJ SLi T=-1 Figure 53. Outlined of the two paths, with additional inverters included, after initial transformation For this particular case the two branches result on identical resonators, with the same electroacoustic coupling coefficient and impedance, and they only differ on their resonant frequencies, which are distributed along the bandwidth. The section above detailed the steps to go from a conventional transversal network based on single resonant frequency resonators into a transversal network based on acoustic wave resonators. Note that in the procedure above no conditions have been set on the values of 𝐽𝑆𝐿𝑖. This value can be arbitrary selected, as long as the summation of all new introduced admittance 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 97 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. inverters is zero, or equal to the initially defined by the synthesis (for the case of a canonical synthesis, Figure 48.b. The section bellow details on how to select the proper inserted admittance inverter to obtain the desired coupling coefficient. 3.3 Procedure to extract the required cross coupling for a given coupling coefficient 𝒌𝒆 𝟐. The process starts by recalling the lowpass prototype of a BVD model of Figure 15b, in Figure 54. Note that in this case we use a different notation on the circuit parameters to be consistent with the notation in section 3.2. Below we reproduce part of the formulation in section 1.7.5, using the notation of Figure 54. Figure 54. Lowpass prototype of the BVD model From the circuit model we can identify the series resonant frequency, Ω𝑠, of the lowpass prototype when the impedance of the acoustic branch 𝛧𝑆 equals 0: 𝑍 𝑠= 𝑗 Ω𝑠𝐿𝑠+ 𝑗 𝑋𝑠=0 (60) From this, we can set the following relation between the acoustic inductance and the acoustic reactance: 𝑋𝑆𝑖=−Ω𝑠𝐿𝑆𝑖 (61) Now we can obtain the parallel resonant frequency, Ω𝑝, of the lowpass prototype when the admittance of the resonator Y𝐵𝑉𝐷_𝐿𝑃, equals 0 𝑌 𝐵𝑉𝐷_𝐿 𝑃 = 𝑗 𝐽 𝑆𝐿𝑖+1 𝑗 Ω 𝑝 𝐿𝑆𝑖+ 𝑗 𝑋𝑆𝑖=0 (62) Ls i Z=jXs i Y=jJ SLi 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 98 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Now by substituting (61) into (62), results in the following relation: 𝐽 𝑆𝐿𝑖=−1 (Ω𝑠−Ω 𝑝 )𝐿𝑆𝑖 (63) From the conventional frequency transformation lowpass-to-bandpass, we can also write down: Ω𝑠=1 FBW( 𝑓 𝑠 𝑓 0− 𝑓 0 𝑓 𝑠) Ω 𝑝 =1 FBW( 𝑓 𝑝 𝑓 0− 𝑓 0 𝑓 𝑝 )(64) and Ω𝑠−Ω 𝑝 =1 FBW[( 𝑓 𝑠 𝑓 0− 𝑓 0 𝑓 𝑠)−( 𝑓 𝑝 𝑓 0− 𝑓 0 𝑓 𝑝 )] (65) which can be re-written as: Ω𝑠−Ω 𝑝 =1 FBW( 𝑓 𝑠− 𝑓 𝑝 𝑓 0− 𝑓 0 𝑓 𝑝 − 𝑓 0 𝑓 𝑠 𝑓 𝑠 𝑓 𝑝 )= 2 FBW( 𝑓 𝑠− 𝑓 𝑝 𝑓 0) (66) where we have considered that 𝑓0=√𝑓𝑠𝑓𝑝,and then 𝑓0𝑓𝑝−𝑓0𝑓𝑠 𝑓𝑠𝑓𝑝=𝑓𝑝−𝑓𝑠 𝑓0. Now, from the expression of the coupling coefficient 𝑘𝑒2 [53] (also in (19)), 𝑓𝑠 𝑓𝑝≅1−4 𝜋2𝑘𝑒2(1+4 𝜋2𝑘𝑒2), we can obtain the following relation: 𝑓 𝑝 − 𝑓 𝑠 𝑓 𝑝 ≅4 𝜋2𝑘𝑒2(1+4 𝜋2𝑘𝑒2)(67) which can be approximated, as: 𝑓 𝑝 − 𝑓 𝑠 𝑓 0≅4 𝜋2𝑘𝑒2(1+4 𝜋2𝑘𝑒2)(68) 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 99 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Now, by substituting (68) into (66), we obtain: which in turn can be considered in (63) The resulting equation (70) outlines the required cross-coupling in the lowpass prototype to be added into a conventional resonator for a given coupling coefficient of the acoustic wave resonator. At this point is worth to mention that this value is positive, which is consistent with the definition of the equivalent circuit of Figure 54, which results in a positive admittance value, which gives rise into a capacitance as expected in an AW resonator (and its BVD model). From this we may conclude that we can select the desired coupling coefficient to perform the synthesized filter response. Although this results from the initial assumption of filter with symmetric response and with an even order of the filter, we have observed that by selecting the 𝐽 value using (70), cancellation between all 𝐽 values of the filter also occurs in asymmetric responses and odd order filters. This is due to the nature of the characteristic polynomials and the resulting synthesized 𝐿 value. 3.4 From lowpass prototype to bandpass prototype To evaluate the practical values of the resulting filters, we need to apply frequency and element transformation to go from the lowpass prototype to the bandpass prototype. The process starts by using the lowpass prototype of Figure 54, where the acoustic branch is defined by 𝐿𝑆𝑖 and 𝑗𝑋𝑆𝑖. These values are then used to calculate the resonant frequency of the acoustic branch in the lowpass prototype, as: (71) The shunt reactive element −𝑗𝐽𝑆𝑁, defines then the anti-resonant frequency 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 100 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Which in turn can be related with the coupling coefficient [53], as: 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 101 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. 𝐶 𝑝 𝐶 𝑠=1 (1−4 𝜋2𝑘2𝑒(1+4 𝜋2𝑘2𝑒))−2−1=𝑘 (78) We call this value k, and it will be extensively used on the design procedures defined above. Note that k is constant for a given value of coupling coefficient (for a certain operation frequency). From (78), we may conclude that the ratio between the static and acoustic capacitances is fixed by the synthesis value and the prescribed coupling coefficient 𝑘𝑒2. For values of 𝑘𝑒2≈6.8% we would obtain 𝐶𝑝 𝐶𝑠≈17, no matter the fractional bandwidth of the filter. Equation (78) also indicates that the ratio 𝐶𝑝/𝐶𝑠 would increase for smaller 𝑘𝑒2. This is illustrated in Figure 55, below. Figure 55. Value of k as a function of 𝒌𝒆 𝟐 Once all the circuit parameters of the BVD model of the acoustic wave resonators are fixed, its impedance is also defined. Following the conventional definition [53] and using the transformation from lowpass prototype and pass band prototype the resulting impedance might be written as: 𝑍 0=1 ωs·( 𝐶 𝑝 + 𝐶 𝑠)=𝐿𝑆𝑖 𝐹 𝐵𝑊(𝑘+1) (79) This expression reveals that the resulting impedance of each resonator is defined by the synthesized lowpass values, 𝐿𝑠𝑖 (so the characteristic polynomials), the prescribed coupling coefficient (which is related with k) and the fractional bandwidth of the filter FBW. 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 102 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. The formulation described in this section might be used to provide a transversal topology based on acoustic wave resonators, for a given prescribed 𝑘𝑒2 coefficient. This allows to conclude that a transversal topology based on acoustic wave resonators might be used to obtain: The filter bandwidth is not limited by the 𝑘𝑒2 of the resonators The 𝑘𝑒2 could be prescribed by the designer The filter transmission zeros are not defined by 𝑓𝑠 of 𝑓𝑝 of the resonators It is possible to place transmission zeros from the filter bandwidth without the need of including additional inductances The number of transmission zeros might be different than the number of acoustic resonators It is possible to synthesize any transfer function: oMultiband responses oResponses with equalization zeros The resulting figure configuration would be: Lshunted BALUN Lshunted Figure 56. Transversal filter topology based on acoustic wave resonators. The topology of Figure 56 shows the need of including a BALUN at the output (or input) of the filter to account for the desired 180 phase shift required in some branches (see Figure 53.b). Figure 53 also reveals the need of having shunt reactive elements at the input and at the output, which results in shunt inductances (Lshunted) at the input and output ports of the filter, as we will see in the following sections. In spite of the significant advantages of this procedure over the conventional ladder designs we would also mention the potential drawbacks: Need of including a BALUN/Transformer at the input or output of the filter. The resulting values of the resonators (𝐿𝑠 , 𝐶𝑠 and 𝐶𝑝) are in part defined by the initially synthesized parameters, 𝐿𝑆𝑖 and 𝑋𝑆𝑖. This results in very high impedance resonators. This statement will be illustrated in the following example. The series frequency 𝑓𝑠 of the resonators are all different. 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 109 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. By following the above transformations, the resulting filter topology of Figure 56, will be modified to the new filter topology outlined in Figure 59. BALUN L shunted L shunted C residual Figure 59. Transversal filter topology based on acoustic BAW resonators, after the transformation outlined in this section. In addition to the direct transformation from Figure 58.a to Figure 58.e, the resulting topology of Figure 59 also exhibits an additional capacitance transversal to all the resonators. Existence of this capacitance is not always required and its value would depend on the synthesized network. Details on that will be found in the following section. We will call this capacitance as residual capacitances, 𝐶𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙. 3.7 Synthesis Procedure: Approach II. Prescribed Impedance. The following approach uses the formulation in previous section to transform from the equivalent circuit of Figure 58.a to the equivalent circuit of Figure 58.d, where the coupling of each resonators and their impedances, a therefore size of the resonator are set by the filter designer. The steps for the corresponding approach are: 1. Apply a conventional synthesis procedure - as outlined in (1), (58) and (59) to a general Chebyshev type response, to obtain a conventional transversal network, see Figure 48. 2. Apply circuit transformation on Figure 51, branches, to obtain the equivalent circuit of Figure 53 for each branch. This is 𝐿𝑆𝑖 and XSi for each resonator. 3. Apply element and frequency transformation from lowpass to bandpass following the set of equations (71)-(76), in order to obtain 𝐿𝑠, 𝐶𝑠, for each resonator. 4. Set the desired impedance of each resonator through equation (94), and the desired coupling (recall that it is related with the k1 value (usually k1=17, for 𝑘𝑒2=6.8%), see 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 110 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. equation (78) from equation (88). This defines a set of two equations (94) and (88) and two unknowns 𝐶6 and 𝐶7. (96) (97) (98) (99) (100) 6. From the value of  we can obtain the values of 𝐿3 and 𝐶5 of the circuit by applying: (101) where 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 111 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. 𝑘(102) 7. The last step is to calculate the residual capacitances 𝐶𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙, if any. The procedure is defined on the following steps: a. Calculate the value of 𝐶, Figure 58.a, that would be needed in case of the synthesis Approach I: i. (103) (104) b. From the initial values of 𝐶𝑝, we can calculate the required value of the matching inductances 𝐿𝑠ℎ𝑢𝑛𝑡𝑒𝑑 by following step 5 of the procedure described in section 3.5. c. From the initial, conventional synthesized network, it is clear that the summation of all 𝐶𝑝 of one branch of the BALUN, should compensate all the 𝐶𝑝 on the other branch of the BALUN. If this is not happening we need to include some additional capacitance (𝐶𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙) in one of the branches, to perfectly match the synthesized response. Note as well, that since this additional 𝐶𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙 is introduced in the lowpass prototype, as a -network admittance inverter from the source to the load additional shunt inductors should be added at the input and output ports. We call these inductances as residual inductances, 𝐿𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙. This value can be directly calculated as: 𝐿𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙=1 2𝜋 𝑓 0 𝐶 𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙 (105) which is shunt connected to 𝐿𝑠ℎ𝑢𝑛𝑡𝑒𝑑. Strictly speaking only, a 𝐶𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙 is required in one of the branches, nevertheless in practical application, having additional capacitances at each 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 112 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. branch might help to compensate the out-of-band performance. Details of that will be seen along the examples shown at the end of this document. 3.7.1 Example Approach II The following example synthesizes the same frequency response that has been synthesized on the synthesis Approach I. The filter frequency responses are detailed in Figure 60.a. Red and blue correspond to the filter response described by the characteristic polynomials, whereas in thick dash blue and red, it is shown the response of the resulting filter topology. Note that both responses are in very good agreement. Figure 60.b shows the impedance of each individual resonator. This figure reveals that all resonators have the same coupling coefficient (6.8%) and also the same impedance. In this example the impedance of each resonator has been set to 60 . Can also be observed that the 𝑓𝑠 of each resonator are different. a) b) Figure 60. a) Filter response, b) impedance of each individual resonator. 3.8 Synthesis Procedure Approach III. Prescribed Resonant Frequency. The following approach also uses the formulation in Section 3.6 to transform from the equivalent circuit of Figure 58.a to the equivalent circuit of Figure 58.d, where the coupling of each resonator and its series resonant frequency is set by the filter designer. At this point, it is worth to mention that a single resonant frequency for all the resonators is not possible (otherwise it would need a very wide range of impedances) when the filter needs to cover a wide passband. In that case, a set or groups of several series resonant frequencies will be selected. 1.7 1.8 1.9 22.1 2.2 2.3 2.4 2.5 -5 0 5 10 15 20 25 30 35 40 45 Z (dB) Frequency (GHz) All resonators equal Impedance: 60  1.85 1.9 1.95 22.05 2.1 2.15 2.2 -100 -80 -60 -40 -20 0 Frequency (GHz) 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 113 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. To clearly explain the process, we will start by considering a unique resonant frequency, which will be referred as Approach III.1. Then, we will extend this method to consider several groups of resonant frequencies, which will be referred as Approach III.2. Note that this latter approach is very oriented to a practical implementation. 3.8.1 Approach III.1 1. Apply a conventional synthesis procedure - as outlined in (1), (58) and (59) to a general Chebyshev type response, to obtain a conventional transversal network, see Figure 48. 2. Apply circuit transformation on Figure 51 (branches) to obtain the equivalent circuit of Figure 53 for each branch. This is 𝐿𝑆𝑖 and XSi for each resonator. 3. Apply element and frequency transformation from lowpass to bandpass following the set of equations (71)-(76), in order to obtain 𝐿𝑠, 𝐶𝑠, for each resonator. 4. Select the desired resonant frequency 𝑓𝑠 of all resonators. Note that the configuration of the new BVD (Figure 58.d,e) only allows to move the initial resonant frequency – set by the initially synthesized 𝐿𝑠 and 𝐶𝑠 – to lower frequency. For that reason, a good practical application is to take as a desired 𝑓𝑠 the lowest one (𝑓𝑠𝑚𝑖𝑛) of the already synthesized resonators. 𝑓 𝑠𝑚𝑖𝑛 =1 2𝜋 √ 𝐶 𝑠𝐿𝑠(106) 5. Since all resonators would have the same 𝑓𝑠=𝑓𝑠𝑚𝑖𝑛, and we want all resonators to have the same coupling coefficient – namely 6.8% or other set by the designer-, so a given value of k1, all resonators also need to have equal 𝑓𝑝. The value of the desired 𝑓𝑝 can be found from: 𝑓 𝑝 = 𝑓 𝑠𝑚𝑖𝑛√1+𝑘1 𝑘1(107) 6. Recall here that 𝑓𝑝 cannot be changed from equivalent circuit of Figure 58.a to the equivalent circuit of Figure 58.d,e. So that the 𝑓 of each resonator should be set on the initial step, Figure 58.a. To do that we can obtain the coupling set on Figure 58.a circuit, by: 𝑘= 1 (𝑓𝑠 𝑓 𝑝 )2−1 (108) 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 114 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Where 𝑓𝑠 is the one defined by 𝐶𝑠 and 𝐿𝑠 and 𝑓𝑝 is equal for all resonators (107). Note that the resulting k values are all different. 7. Now k is used to find the 𝐶𝑝 values of each resonator, and the designing parameter  as: (109) At this point, we have all the information to find the values of the new resonators, this is 𝐶6, 𝐶7, 𝐿3 and 𝐶5. 8. From the initial values of 𝐶𝑝, we can calculate the required value of the matching inductances 𝐿𝑠ℎ𝑢𝑛𝑡𝑒𝑑 by following step 5 of the procedure described in section 3.5. 9. As in previous case some additional 𝐶𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙 and 𝐿𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙 values might be required to completely match the synthesized response. To do that we need to perform step 7 of Approach II. 3.8.2 Approach III.2 The first three - 1), 2) and 3) - steps are equals to the ones in Approach III.1. 4. Then several frequencies 𝑓𝑠 are selected. In practice, a good choice is to select the targeted frequencies from the synthesized ones, as in (106), resulting in 𝑓𝑠𝑚𝑖𝑛,1, 𝑓𝑠𝑚𝑖𝑛,2, …, 𝑓𝑠𝑚𝑖𝑛,𝑁. 5. Then each resonator with an 𝑓𝑠, initially synthesized, between 𝑓𝑠𝑚𝑖𝑛,𝐼 and 𝑓𝑠𝑚𝑖𝑛,𝐼+1 , has been modified using steps 5-9, in Approach III.1, to have a 𝑓𝑠=𝑓𝑠𝑚𝑖𝑛,𝐼. 3.8.3 Example Approach III.1 The following example synthesizes the same frequency response that has been synthesized on the synthesis Approaches I and II. The filter frequency responses are detailed in Figure 61.a. Red and blue correspond to the filter response described by the characteristic polynomials, whereas in thick dash black and magenta is shown the response of the resulting filter topology. Note that both responses are in very good agreement. Figure 61.b shows the impedance of each individual resonator. This figure reveals that all resonators have the same coupling coefficient (6.8%) and also the same resonant frequencies. It is important to notice here that the values of the impedances of the resonators are very different, and might result in a very impractical implementation. 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 115 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. a) b) Figure 61. a) Filter response, b) impedance of each individual resonator. 3.8.4 Example Approach III.2 The following example synthesizes a filter response with wider bandwidth, 150 MHz of bandwidth instead of 100 MHz, by means of a 8th order filter. The normalized positions of the transmission zeros are maintained. The filter frequency responses are detailed in Figure 62.a. Red and blue correspond to the filter response described by the characteristic polynomials, whereas in thick dash green and cyan is shown the response of the resulting filter topology. Note that both responses are in very good agreement. Figure 62.b shows the impedance of each individual resonator. Results in Figure 62.b reveals that all resonators have the same coupling coefficient (6.8%) and the resonant frequencies correspond to a prescribed set. a) b) Figure 62. a) Filter response, b) impedance of each individual resonator. 1.8 1.85 1.9 1.95 22.05 2.1 2.15 2.2 Z(dB) -20 -10 0 10 20 30 40 50 Z (dB) 1.85 1.9 1.95 22.05 2.1 2.15 2.2 1.85 1.9 1.95 22.05 2.1 2.15 2.2 -100 -80 -60 -40 -20 0 Frequency (GHz) Frequency (GHz) 0 -20 -40 -60 -80 -100 1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2 Frequency (GHz) -10 0 10 20 30 40 50 Frequency (GHz) 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 116 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. 3.9 Synthesis Procedure: Approach IV. Trade-off. The following synthesis approach takes advantage of the two latter approaches presented on section 3.7 and 3.8, by using the flexibility of the formulation above (section 3.6) to have control on the impedance of the resonators forming the filter as well as on the resonant frequency of the resonators. Note that in for practical purposes regarding manufacturing viability a designer would aim to find a solution with all resonators impedances (or sizes) within a certain range. Same happens with the set of different resonant frequencies which must be limited to a finite number (usually no more than 4 or 6). To illustrate this approach, we will use the 100 MHz 6th order filter example evaluated in previous sections. The process outlines as: 1) Propose a range of suitable impedances a) For the current example we select a range between 20 and 120. 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 117 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. 2) For each resonator of the filter we evaluate the corresponding resonant frequency for the whole range of impedances. This uses the formulation of Approach II, and obtains 𝑓𝑠 for each impedance. a) Figure 63, shows the 𝑓𝑠 as a function of the prescribed impedance for each resonators. b) Four families of 𝑓𝑠 frequencies have been defined, 𝑓𝑠,1, 𝑓𝑠,2, 𝑓𝑠,3 and 𝑓𝑠,4. 3) From the selected resonant frequency families we obtain the required impedance of each resonator. a) In this case the selected impedances are: 115 , 35 , 60 , 60 , 50  and 50 . b) Note that a lot of flexibility would exist on families 2, 3 and 4 on the selection of the desired impedances. 4) Once the desired impedances are prescribed, the procedure of Approach II needs to be applied (see section 3.7) a) Figure 64, shows the frequency dependence of the impedance of each resonator, where we can clearly identify each family. Figure 64. Impedance of each resonator. To conclude with the example, Figure 65 below shows the frequency response of the synthesized transversal topology (cyan and green), along with the synthesized response. 1.85 1.9 1.95 22.05 2.1 2.15 2.2 -10 0 10 20 30 40 50All resonators equal Impedance: 115 35 60 60 50 50 Ohms Frequency (GHz) 10*log10(Impedance) fs,1 fs,2 fs,3 fs,4 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 118 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Figure 65. Filter response. At this point, it is worth to mention that examples with much wider bandwidth response (200 MHz) have been performed following the same approach. The results for this case also yield to at least four necessary families of 𝑓𝑠, and the values of the required impedances are: 125  and 25  for family 𝑓𝑠,1, 60  for family 𝑓𝑠,2, 60  for family 𝑓𝑠,3 and 65 , for family 𝑓𝑠,4. 3.10 Advanced filter performances This section illustrates two examples with advanced filtering responses. Those are responses that usually cannot be achieved (unless a way more complex circuits are used) with conventional ladder topologies. Both examples result from a synthesis response following any of the approaches outlined above, with a uniform coupling coefficient of 6.8%. Recall here that the coupling coefficient is prescribed by the designer, and by setting a different coupling coefficient the results would be very similar (in fact identical in the lowpass prototype). The first example corresponds to a filter with very wideband response and with an arbitrary position of the transmission zeros. The filter bandwidth has been set to 200 MHz, and has been covered using only six resonators (note that this is not possible with a ladder configuration and uniform coupling coefficient of 6.8%). The transmission zeros have been located asymmetrically on both bands without the need of additional inductances. 1.85 1.9 1.95 22.05 2.1 2.15 2.2 -100 -80 -60 -40 -20 0 Operating Frequency in GHz S-parameters 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 125 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Figure 71. In-band details of the responses of Figure 70. Figure 72. Frequency response considering a uniform variation of 1% and 5% of only the impedance of the resonators. Legends are used to identify each response. The blue lines correspond to the characteristic polynomial response with none deviation of the impedances. Partial conclusion conclusions of this analysis are: When unbalancing effects are produced at the branches due to the deviation of the impedance, as the case of topologies following Figure 56 configuration, out-of-band S-Parameters(dB) 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 126 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. rejection is reduced and transmission zeros might be lost. Note however, that for 1% variation of the impedance this effects do not spoil the filter performance. The response can be recovered by applying the same variation factor to the external elements; these are C5 capacitors and Lshunted inductance. This also allows to conclude that although the transversal configuration might be very sensitive, it is also very flexible and trimming process can be applied to recover the response. It is also worth to recall here that the external network, including 𝐶, 𝐶 and 𝐿, might be potentially use to recovered the synthesized response by tailoring the contribution of each signal path. 3.12.2 Effects on the variation of the resonant frequency As mentioned above the resonant frequency of each resonator forming the filter might deviate randomly in a +/- 1 MHz range. The figures below evaluate this effect. Instead of using a statistical analysis where hundreds of simulations are performed by means of applying random variation of the resonator frequency at the given +/-1 MHz range. We perform simulation where we apply + 1 MHz or -1MHz deviation at several given resonators. This simple analysis allows to see the effects of having extreme variations at the resonant frequency of given resonators for unveil the worst cases. Figure 73 to Figure 77 show different simulations. The variations applied at each simulation are indicated in the label of the figure by means of a vector. Each position of the vector corresponds to one resonator, and the value 1 means a deviation of 1 MHz, the value -1 means a deviation of -1 MHz and the value 0 means no deviation. As in all previous cases, the analysis considers the topologies outlined in Figure 57 (red) and Figure 60 (black). Blue lines correspond to the characteristic polynomial response and they are used as a benchmark for comparison. The first important conclusion from the simulations is that no differences can be observed between the two topologies under analysis (Figure 57 and 60), and both are equally affected by deviations on the resonant frequencies. A second conclusion is that the in-band performance is barely affected. The simulations below also clearly show that the major effect occurs out-of-band, near the band edges on the position and deepness of the transmission zeros. When the position of transmission zeros changes, the rejection and sidelobe rejection accordingly changes. It is also worth to note that the effects observed on Figure 74 are very similar to the effects occurring in Figure 76, but on the opposite sideband. Note that this is consistent with the fact that in a transversal configuration the resonators are distributed along the bandwidth, being the 1st resonator the one resonating closer to the lower band edge and the 6th resonator the one resonating closer to the upper band edge. Since the filter response is symmetric, the effects 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 127 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. of shifting the first and last frequencies up in frequency are equivalent dual to the effects of shifting the first and last frequencies down in frequency. Figure 73.- Filter responses when a) + 1MHz variation is applied to the 1st and 6th resonator. Legends are used to identify each response. Figure 74. Filter responses when a) +1MHz variation is applied to the 1st and -1MHz variation is applied to the 6th resonator Legends are used to identify each response. 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 128 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Figure 75. Filter responses when a) -1 MHz variation is applied to the 1st and 6th resonator. Legends are used to identify each response. Figure 76. Filter responses when a) +1MHz variation is applied to the 3rd and 4th resonator, and a - 1MHz variation is applied to the 1st and 6th resonator Legends are used to identify each response. 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 129 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Figure 77. Filter responses when a) +1MHz variation is applied to the 2nd, 3th and 4th resonator and a -1MHz variation is applied to the 1st, 5th and 6th resonator. Legends are used to identify each response. 3.13 Effects of a non-ideal BALUN stage on the overall filter As early noticed in the presentation of such novel topology, the existence of a transformer or BALUN stage at one end of the filter is mandatory to combine the contribution of the difference transversal branches for the final filter response. Indeed, this stage introduces a new degree of complexity for the implementation of such topologies, which could also affect the final performance of a real implementation. Ideally, we would expect or need an ideal BALUN, this is a very wide band transformer fully balanced all over the frequency range and a phase shift of 180º. In practice this might become a real bottle neck for the full development of those topologies. Despite of that, the concept based on transversal topologies on acoustic wave filters can be applied on the different techniques used to develop acoustic wave resonator, this is in BAW, SAW, CRF and Stacked Crystal Filters (SCF) configurations. This latter statement might help to find the right niche for the development of transversal acoustic filters. For instance, SAW technology allows to obtain transformers with indeed acceptable performance BALUNs. On the other hand, the possibility to apply this concept into CRF configuration, where each resonator step produces a 180º degree shift, also opens the possibility to exploit the transversal configuration, or even on SCF where phase shift can be obtained by inverse polarization of the piezoelectric material. 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 130 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Effects of a non-ideal BALUN are analyzed by means of the following equivalent circuit of Figure 78, where the parameters T1 and T2 refer to the coupling of each branch to the output port. An ideal BALUN results in T1=T2=0.5. The analysis will consider to different simple cases. First when a non-unitary coupling exists, this is T1=T2≠0.5, and second when a non-unitary signal ratio exists, this is T1≠T2. T1 T2 1 Figure 78. Equivalent circuit of the BALUN 3.13.1 Coupling (kb) sensitivity The coupling value is defined here as kb as kb=T1/0.5. The following figures evaluate the cases for kb=0.95, 0.8 and 0.75. Note that all these scenarios consider a fully balanced BALUN. This analysis has been performed with the circuit analysis software ADS (Advanced Design Systems) [58]. The results below reveal that the out-of-band performance is barely affected by the nonunitary coupling of the BALUN. This fulfils as well for the position and deepness of the transmission zeros. On the other hand, the in-band responses (or the return losses) in Figure 80, demonstrate the existence of a mismatch effect. From this figure, we can identify the different cases: the blue line corresponds to the case with kb=1, the pink line kb=0.95, the green line kb=0.8, and the brown line kb=0.75. This effect can be easily recovered by changing the output matching impedance, which needs to be scaled by the value of kb2 [25]. This statement is demonstrated at the Figure 81 where all responses perfectly overlap. This analysis has been replicated for the filter of order four with identical outcomes. The responses of the filter performance are shown in Figure 79 and 80, whereas Figure 81 outlines the results after applying the scaling factor kb2 to the output matching impedance. 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 131 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Figure 79. Response of a 6th order filter for coupling coefficients of kb=1, 0.95, 0.8 and 0.75. Figure 80. In-band details of the filter of Figure 80. 1.683 1.867 2.050 2.233 2.4171.500 2.600 -80 -60 -40 -20 -100 0 freq, GHz S-Parameters 1.933 1.967 2.000 2.033 2.0671.900 2.100 -40 -30 -20 -10 -50 0 freq, GHz S-Parameters 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 132 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Figure 81.- In-band details of the filter when matching output impedance has been changed. 3.13.2 Signal ratio ( a ) sensitivity The ideal value of balanced signal ratio is a =T1/T2=1. For this analysis we will perform simulation for a =0.98, 0.96, 0.94. Such effects are outlined in Figure 82-83. This analysis has been performed with the circuit analysis software ADS (Advanced Design Systems) [58]. In contrast with previous subsection analysis, the unbalancing evaluated in this case strongly affects the out of band filter performance, due to the change on how each branch contributes to the output signal. This statement can be clearly observed in Figure 82 where the out-of-band rejection rises up to almost 20 dB for the different unbalanced cases. The caption of the figure indicates the corresponding a value. However, Figure 83 reveals that the unbalance effect barely affects the in-band response and return losses remains quite stable. As in the previous analysis of the kb-sensitivity, one might wonder if it is possible to recover the filter performance when unbalancing effects in the BALUN exist. The response is that the filter performance can be partially recovered. To illustrate that, Figures 84 and 85 show the responses of the filters of order six and four respectively, for the case of higher unbalancing effects, i.e., a= 0.95, 0.9, 0.75. As outlined in figures above, this would create and important reduction of the out-of-band rejection, however, this effect can be compensated by scaling the impedance of the resonators of the unbalanced branches by the factor a. By doing so, Figures 84 and 85 demonstrate that the out-of-band performance can be recovered at expenses of a degradation of the in-band performance. The transversal topologies demonstrate that although being very sensitive to the balanced effect of each branch, they are also very flexible and the response can be somehow partially recovered, to still offer a good filter performance. 1.933 1.967 2.000 2.033 2.0671.900 2.100 -40 -30 -20 -10 -50 0 freq, GHz S-Parameters 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 133 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Figure 82.- Response of a sixth order filter for balance signal ratio a= 1, 0.98, 0.96, 0.94. The corresponding lines for the transmission coefficient are red, cyan, purple and dark green, respectively. Figure 83. Return losses details for the responses of Figure 83. 1.683 1.867 2.050 2.233 2.4171.500 2.600 -80 -60 -40 -20 -100 0 freq, GHz S-Parameters 1.933 1.967 2.000 2.033 2.0671.900 2.100 -40 -30 -20 -10 -50 0 freq, GHz S-Parameters 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 134 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Figure 84.- Response of a sixth order filter for unbalanced signal ratio a= 1, 0.95, 0.9, 0.75. Figure 85.- Return losses details for the responses of Figure 85. The corresponding lines for the reflection coefficient are blue (a=1), pink (a=0.95), green (a=0.9) and brown (a=0.75). 1.683 1.867 2.050 2.233 2.4171.500 2.600 -80 -60 -40 -20 -100 0 freq, GHz S-Parameters 1.933 1.967 2.000 2.033 2.0671.900 2.100 -40 -30 -20 -10 -50 0 freq, GHz S-Parameters 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 141 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Figure 92. 𝐋𝐬𝐡𝐮𝐧𝐭𝐞𝐝 value as a function of the impedance of the resonators (assumes a uniform impedance). 3.16 Conclusions This chapter detailed on the most innovative part of this thesis by proposing and developing the synthesis procedure for a new topology of acoustic wave filters. The proposed topology is referred as transversal topology and can be applied to BAW and SAW filter configurations. Although not developed in this chapter the flexibility of this method certainly opens up the opportunity to be extended to the CRF configurations. This novel topology proposes a new way of connecting the AW resonators to perform the filter response, which consists on electrically connecting all resonators directly from the input to the output, in such a way that the contribution of each transversal path is added at the output to offer the filter response. This new concept on acoustic filters allows to obtain filter responses not achievable with other filter configurations based on acoustic filters, such as: Very wideband filter response Multiband responses Responses with self-equalized in phase. In addition to the advantages on the achievable responses, this new approach also offers some promising benefits from the implementation point of view. Those benefits are: The coupling coefficient of each resonator can be selected in advanced by the designer (prescribed). Allows flexibility on the impedance of the resonators. Allows flexibility on the resonant frequency of each resonator. Lshunted (nH) 3 DEVELOPMENT OF TRANSVERSAL FILTERS IN BAW TECHNOLOGY | 142 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. At this point is also important to outline some of the drawback or eventual limitations: Those topologies need a BALUN stage. Are sensitive to non-uniform deviation of the parameters defining each branch of the transversal configuration, such as resonant frequency and impedance of the resonators. As it will be outlined in the last chapter of conclusions and future research lines, some modifications of the topology presented in this chapter can help to overcome some of the previous limitations. At this point, it is also important to conclude that this new concept to synthesize acoustic wave filter may enable the use of less typical materials and lead to the use of new manufacturing techniques for their development, since constrains on the coupling coefficient do not affect the achievable responses, this in turn might give rise to the development of acoustic filters at other frequencies and therefore contributing to the expansion of this technology to other applications. Next chapter completes some of the details on the proposed topologies by performing three case studies based on real filter electrical requirements and also accounting with some of the technological constrains. 4 CASE STUDIES | 143 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. CASE STUDIES This chapter details on some case studies corresponding to filter and multiplexer designs using the synthesis procedures described in chapters 2 and 3, for ladder filter topologies and transversal topologies, respectively. For the particular case of ladder topologies, a software tool has been developed with the aim to be used as a first step in a filter design process, by providing several topologies (solution space) for given filter electrical specifications. Most of the examples presented below are based on real specifications of commercial filters or multiplexers. The presented designs do not only account for the initial mathematical response but also consider the implementation aspects, as the coupling coefficient of the resonators, impedances and resonant frequencies of the resonators. Evaluation of the losses are also considered in some cases. 4.1 Software for the synthesis of ladder filters The tool includes all the mathematical formulation presented in Chapter 2 for the synthesis of ladder filters and it has been proposed as a user friendly interface where the designer would have the opportunity to obtain fast initial solutions of several synthesized networks. The aim of this subsection is then to present the main features of this tool. Graphic User Interface – Ladder AW filter The usual first step for a BAW/SAW filter designer would be to find a preliminary topology that meets some given specifications. A schematic ladder type filter will be drawn in some electronic design automation software giving initial values to the basic parameters of the elements placed on the circuit. For a BAW filter these usually will be: 𝑓𝑠, the resonant frequency of the acoustic resonators. 𝐶0, the static capacitance of the acoustic resonators. CHAPTER 4 4 CASE STUDIES | 144 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. 𝐿𝑔𝑛𝑑, ground inductances that connects the shunt resonators to the common ground in the laminate. 𝑘𝑒2, electro-acoustic coupling coefficient. Additional external element values, - i.e. capacitors and inductances -, used to virtually modify the 𝑘𝑒2 and/or to adapt the filter to the input/output port. Several optimization iterations will be run in order to find the values of the parameters described above that makes the electrical response of the filter to meet the specifications. These values have been previously constrained within a certain range to make feasible the manufacturability of the filter. After the preliminary topology is obtained, that is, the optimized values of the filter parameters provide a good solution, the designer may proceed to the next step. This should be to create an equivalent layout based on the optimized schematic that will give the initial appearance to the die. Along with the die a 3D model with all the layered filter structure, including pads, vias, laminate, etc. has to be created in order to perform accurate EM simulations of the overall device, including everything except, maybe, the packaging. But at this point the designer is already dealing with many issues that are out of the scope of the filter synthesis. Unlike this typical methodology, the synthesis procedure manipulates a slightly different set of parameters due to its own mathematical nature. That is, the designer optimizes a ladder topology that will very unlikely provide with a frequency response that looks like a Chebyshev polynomial. This happens because the intrinsic relationships between all the parameters in the filter have to comply with a very particular set of conditions, which have been stated previously in this thesis, in order to reproduce a characteristic polynomial filtering shape. So far, we have seen how varying the values of certain main parameters of the filter – order, transmission zeros, return losses, bandwidth, center frequency – we obtain a different topology for each new combination of all those values. One of the purposes of this work is to offer synthesized network ladder topology that can meet a set of given specifications. To this end a GUI tool has been created. An overall view of one of the newest versions of the GUI is shown in the Figure 93. The GUI has been created using the MATLAB® platform. Different parts of the GUI have been numbered from 1 to 4, whose features are outlined latter along the text. The GUI has grown up in size and functionalities little by little to be able to model more and more accurately the acoustic wave filter behavior up to the point that many features that are not related to the synthesis itself were added (losses of the BVD considering resonator size-dependent performance, optimization engine and more). Even a secondary window that allows to visualize the response of a group of multiplexed filters previously stored was created. 4 CASE STUDIES | 145 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Figure 93. General Overview of the GUI Main Window 1 42 3 4 CASE STUDIES | 146 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Settings Section. It involves the most significant input parameters that take part on the synthesis. That is the order of the filter, Return Losses value, Bandwidth, Center Frequency and position of the prescribed Transmission Zeros. Since the topology used is always ladder-type, the Transmission Zeros below the Lower Band Edge will be originated by the resonant frequency of the shunt resonators and the TZ above the Upper Band Edge will be originated by the anti-resonance of the series resonators. It is therefore necessary to specify exactly which TZ corresponds to each resonator. Also for the particular case of the odd order filters it is necessary to state the number of shunt resonators and series resonators that we want to place in the topology. Additionally and also related with the synthesis, it is possible to modify here the values of the Reflection Zeros (RZ), which are the roots of the F(s) characteristic polynomial. This may be convenient in order to fine-tuning the Return Losses (RL) or even the 𝑘 𝑒2 values of the resonator. This section has other built-in features like setting the values of the out-of-band Transmission Zeros. As we stated previously, these are created by the inclusion of ground inductances between the shunt resonators and the common ground of the filter. In this section we choose the frequency where we want to include a notch most likely in order to improve the out-of-band rejection or the Inter-Band isolation if the purpose of the filter is to be included in a multiplexer. Finally, it is possible to set the values of the parameters related to the losses of the filter, like the quality factor Q of the resonators and their input resistance, 𝑅 𝑠 , for both series and shunt resonators. Figure 94 Settings Section. 1 | 147 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Sections 2 and 3 are devoted to the plotting of the results. Section 2 allows to depict the frequency response of the synthesized filter in both rectangular and Smith Chart i Figure 95. Frequency Response (right) Section 3 on its side shows the schematic circuit of the synthesized topology and all the related output parameters. It also highlights the most significant parameters for a BAW filter designer. Figure 96. Topology Section (above) 2 3 | 148 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. 4 Figure 97. Screen shot for the input parameters of Section 4 Section 4 offers post-processing features for the synthesized networks. These allow for the evaluation of losses, lateral model and computation of the non-linearities. Additionally this section allows for selecting the proposed candidates (different topologies that result from the synthesis), inclusion of the filter specifications through a filter mask, and even to read previous designs save as .s2p files. 4.2 Case studies of ladder filters The following subsection presents several case studies where ladder topologies have been synthesized to meet real filtering stages specifications. In particular three standalone filters have been synthesized and evaluated. A real triplexer scenario has been also considered, which latter has been extended into a quadplexer, in order to verify the suitability of the proposed approach. The synthesized values, these are impedances, series resonant frequencies and coupling coefficients of the resonators, are not always revealed for confidential issues, nevertheless the synthesized networks do not only meet the electrical requirement but also the technological requirements. This is, range of feasible impedances, frequencies and coupling coefficients. 4.2.1 Band 39 LTE The first case study shows a very wideband filter covering the band 39 LTE. In particular, a high order filter N=11 in ladder configuration, synthesized by means of the element extraction technique and the software tool presented in previous section. The synthesized network has been evaluated by the software of previous section along with its implementation in AWR Microwave Office®, an RF & MW circuit design software powered by National Instruments, making use of the exact same parameters that the synthesis provided. However, in AWR a fairly more sophisticated model – regarding the SMR stack and the BVD model - is used for each resonator. 4 CASE STUDIES | 149 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. This model is QorvoTM proprietary and includes considerations related to non-linearities [59] [60], lateral propagation modes [44] and extended BVD model [53], therefore additional losses, parasitic effects and the quality factor dependence on the size and geometry of the resonators are taken into account. Table 7. Specifications for the B39 Wideband filter and simulated values for the resonator stack and ground inductor losses. Table 7 lists the specifications for the filter and the simulated values for the most representative parameters of the resonators as well as the losses considered for the external inductors. Figure 99 shows one topology that meets the specifications. This same topology has been replicated in AWR to make the comparison shown in Figure 98. 4 CASE STUDIES | 150 SYNTHESIS OF ACOUSTIC WAVE FILTERS. LADDER AND TRANSVERSAL TOPOLOGIES. Figure 98. In-band frequency response (above) and Wide-Band response (below) of the synthesized B39 Wideband filter (solid red and blue) compared to the AWR simulation including the in-house Qorvo resonator model (dashed orange and purple) and also with the same model with the inac and qdeg variables disabled (dotted green and black) [53]. 1.8 1.85 1.9 Frequency (GHz) -40 -30 -20 -10 0 1 2 3 4 5 6 Frequency (GHz) -100 -80 -60 -40 -20 0 |S11|(dB) SYNTHESIS + LOSSES |S21|(dB) SYNTHESIS + LOSSES |S11|(dB) BVD LVL2 |S21|(dB) BVD LVL2 |S11|(dB) BVD LVL2 (inac & Qdeg OFF) |S21|(dB) BVD LVL2 (inac & Qdeg OFF) S-Parameters ( dB ) S-Parameters ( dB )