Ku band waveguide diplexer design for satellite communication. Implementation by additive manufacturing and experimental characterization

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(1)Universidad Politécnica de Madrid Escuela Técnica Superior de Ingenieros de Telecomunicación. TRABAJO FIN DE GRADO. Ku band waveguide diplexer design for satellite communication Implementation by additive manufacturing and experimental characterization. MADRID, 2015. IRENE ORTIZ DE SARACHO PANTOJA.

(2) Trabajo Fin de Grado Ku band waveguide diplexer design for satellite communication. Implementation by additive manufacturing and experimental characterization Autor Irene Ortiz de Saracho Pantoja Tutor José Ramón Montejo Garai Departamento Señales, Sistemas y Radiocomunicaciones. Tribunal Presidente:. D. Juan Enrique Page de la Vega. Vocal:. D. Javier Gismero Menoyo. Secretario:. D. José Ramón Montejo Garai. Suplente:. D. Mariano Barba Gea. Fecha de lectura:. Calificación:.

(3) Universidad Politécnica de Madrid Escuela Técnica Superior de Ingenieros de Telecomunicación. TRABAJO FIN DE GRADO. Ku band waveguide diplexer design for satellite communication Implementation by additive manufacturing and experimental characterization. MADRID, 2015. IRENE ORTIZ DE SARACHO PANTOJA.

(4) Abstract A great amount of telecommunication services such as television distribution or navigation systems are based on satellite communication. As it occurs in other spatial applications, there are some key resources which are severely limited on board spacecrafts, as mass or volume. In this sense, one of the most important passive devices, which allows a better use of such resources, is the diplexer of the feed antenna system. This device enables the use of one single antenna for both transmission and reception channels, resulting in an optimization of the above resources. The main goal of this work is to design a diplexer fulfilling real satellite-communication specifications. This device consists in two filtering structures joined by a three-port junction. In addition, the use of waveguide technology is imperative, due to the high power level handled. The diplexer design is accomplished by dividing the structure in separate parts, in order to make the process feasible and efficient. Firstly, different filter configurations are developed – high-, low- and bandpass responses –, even though only two of them will be diplexed. When tackling their initial design, a theoretic synthesis is performed through the use of circuit models. The filters are subsequently optimized by using full-wave CAD techniques, particularly mode matching. At this point it is essential to analyze the structures and their symmetry in order to determine which modes are actually propagating, to reduce computational effort. Finally, FEM method is used to verify the results previously obtained. Once the filter design is concluded, the three-port-junction dimensions are calculated. Eventually, the whole diplexer is optimized to fit the electric specifications. Furthermore, this work presents a brand-new added value: the physical implementation and experimental characterization of both the diplexer and the filters. This possibility, unfeasible until now because of its high cost, derives from the development of additive manufacturing techniques. The prototypes are printed in plastic (PLA) by means of a low-cost 3D-printer, and afterwards metallized. This technology entails two different limitations: the precision of the geometric dimensions (±0.2 mm) and the conductivity of the metallic paint which covers the walls of the waveguide. A comparison between simulated and measured values is included in this work, as well as an analysis of the experimental results. In summary, this work expounds a real engineering process: the problem of designing a device which satisfies real specifications, the limitations caused by a manufacturing process, the eventual experimental characterization and the inference of conclusions.. Keywords: diplexer, waveguide filter, Ku band, satellite communication, additive manufacturing.

(5) Resumen Gran cantidad de servicios de telecomunicación tales como la distribución de televisión o los sistemas de navegación están basados en comunicaciones por satélite. Del mismo modo que ocurre en otras aplicaciones espaciales, existe una serie de recursos clave severamente limitados, tales como la masa o el volumen. En este sentido, uno de los dispositivos pasivos más importantes es el diplexor del sistema de alimentación de la antena. Este dispositivo permite el uso de una única antena tanto para transmitir como para recibir, con la consiguiente optimización de recursos que eso supone. El objetivo principal de este trabajo es diseñar un diplexor que cumpla especificaciones reales de comunicaciones por satélite. El dispositivo consiste en dos estructuras filtrantes unidas por una bifurcación de tres puertas. Además, es imprescindible utilizar tecnología de guía de onda para su implementación debido a los altos niveles de potencia manejados. El diseño del diplexor se lleva a cabo dividiendo la estructura en diversas partes, con el objetivo de que todo el proceso sea factible y eficiente. En primer lugar, se han desarrollado filtros con diferentes respuestas – paso alto, paso bajo y paso banda – aunque únicamente dos de ellos formarán el diplexor. Al afrontar su diseño inicial, se lleva a cabo un proceso de síntesis teórica utilizando modelos circuitales. A continuación, los filtros se optimizan con técnicas de diseño asistido por ordenador (CAD) full-wave, en concreto mode matching. En este punto es esencial analizar las estructuras y su simetría para determinar qué modos electromagnéticos se están propagando realmente por los dispositivos, para así reducir el esfuerzo computacional asociado. Por último, se utiliza el Método de los Elementos Finitos (FEM) para verificar los resultados previamente obtenidos. Una vez que el diseño de los filtros está terminado, se calculan las dimensiones correspondientes a la bifurcación. Finalmente, el diplexor al completo se somete a un proceso de optimización para cumplir las especificaciones eléctricas requeridas. Además, este trabajo presenta un novedoso valor añadido: la implementación física y la caracterización experimental tanto del diplexor como de los filtros por separado. Esta posibilidad, impracticable hasta ahora debido a su elevado coste, se deriva del desarrollo de las técnicas de manufacturación aditiva. Los prototipos se imprimen en plástico (PLA) utilizando una impresora 3D de bajo coste y posteriormente se metalizan. El uso de esta tecnología conlleva dos limitaciones: la precisión de las dimensiones geométricas (±0.2 mm) y la conductividad de la pintura metálica que recubre las paredes internas de las guías de onda. En este trabajo se incluye una comparación entre los valores medidos y simulados, así como un análisis de los resultados experimentales. En resumen, este trabajo presenta un proceso real de ingeniería: el problema de diseñar un dispositivo que satisfaga especificaciones reales, las limitaciones causadas por el proceso de fabricación, la posterior caracterización experimental y la obtención de conclusiones.. Palabras clave: diplexor, filtro en guía de onda, banda Ku, comunicaciones por satélite, manufacturación aditiva.

(6) Para M., por cuidar de mis neuronas. Para J.R., por intentar que salieran cestos con los mimbres que había..

(7) Contents 1 Introduction and design specifications. 2. 2 Waveguides as transmission lines 2.1 Propagation in rectangular waveguides . . . . . . 2.2 Attenuation in rectangular waveguides . . . . . . 2.3 Waveguide implementation of filtering structures 2.3.1 Low-pass filters . . . . . . . . . . . . . . . 2.3.2 High-pass filters . . . . . . . . . . . . . . 2.3.3 Band-pass filters . . . . . . . . . . . . . .. . . . . . .. 4 4 6 7 7 8 9. 3 Modal analysis as full-wave design technique 3.1 Symmetry and its implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 10 11. 4 Low-pass filter. 13. 5 High-pass filter. 15. 6 Band-pass filters 6.1 Synthesis process . . . . . . . . . . . . . . . . . . . 6.1.1 Chebychev filters and low-pass response . . 6.1.2 J-inverters and band-pass transformation . 6.1.3 Slope parameter and distributed resonators 6.2 Physical dimensions . . . . . . . . . . . . . . . . . 6.3 Optimization and final response . . . . . . . . . . . 6.3.1 TX filter . . . . . . . . . . . . . . . . . . . . 6.3.2 RX filter . . . . . . . . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. 17 18 18 19 21 22 24 24 25. 7 Diplexer 7.1 Main challenges when designing a diplexer 7.2 Junction design and final optimization . . 7.3 Double bend . . . . . . . . . . . . . . . . 7.4 Diplexer . . . . . . . . . . . . . . . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 26 27 29 32 34. . . . .. . . . .. . . . .. . . . .. . . . . . .. . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. 8 Manufacture and losses 8.1 Additive Manufacturing (AM) . . . . . . . 8.2 Low-cost 3D printing and RepRap . . . . 8.3 Challenges and limitations of 3D printing 8.4 Metallization process . . . . . . . . . . . . 8.5 Conduction losses . . . . . . . . . . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 35 35 36 36 38 39. 9 Measurement and analysis of 9.1 Low-pass filter . . . . . . . 9.2 High-pass filter . . . . . . . 9.3 Band-pass reception filter . 9.4 Diplexer . . . . . . . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 41 42 43 45 47. results . . . . . . . . . . . . . . . . . . . .. . . . .. . . . .. . . . .. 10 Conclusions. 48. 11 Future work. 50. 12 Contributions. 50. References. 51. 1.

(8) 1. Introduction and design specifications. Satellite communication is one of the most important space applications. It encompasses many services used by millions of people every day, such as television broadcast or navigation systems. It also includes less-known services, for instance tele-medicine. Besides, the importance of mobile satellite systems to connect remote areas with the rest of the world must not be forgotten. This variety of applications makes satellite communication a very mature and developed field, with certain peculiarities. Design specifications of the passive microwave devices on board spacecrafts are practically unique, since there is a great amount of satellites, each of them working in different orbits and frequencies. Satellite frequencies range from 3 to 30 GHz (Super High Frequency). However, this wide spectrum is divided into sub bands, each of them devoted to certain applications. Power levels handled also depend on the different purposes. Besides, the architecture of each satellite launched into orbit presents constrained dimensions. Devices must respect those limitations, hence certain geometries are not allowed. As a result, designing devices for satellite communication becomes a new and different challenge every time. In addition, all designs are critical, since the malfunction of one of them could induce the whole system to be useless. As it might be expected, there is redundancy of some devices, but it is not the solution to an erroneous design. Another important point to bear in mind is the limitation of resources in outer space. A great effort is done to reduce mass and volume of all devices on board, seeking to adjust to the launching capabilities, among others. In this sense, a key device and also the subject of this work is the diplexer. It is a passive element which enables the use of one single antenna for both transmission and reception channels (Fig. 1).. Figure 1: Block diagram of a diplexer. The diplexer is composed of two filtering structures and a three-port junction. In satellite communication, its behavior must be considered in light of three aspects: isolation, power handling and insertion losses of the structure. As it occurs in any communication system, there are two different frequency bands: one for transmission and the other one for reception. The former is located at lower frequencies than the latter, since the power available in the satellite is strongly limited and lower frequencies imply less propagation losses. Once this is established, the power transmitted by the satellite is sensitively higher than the received one. This implies that the diplexer must be capable of handling different power levels, being reduced 2.

(9) losses imperative. Due to the low level of received power, high insertion losses could severely damage the signal and prevent the receiver to work properly. Besides, high isolation between both frequency bands is crucial to avoid any uncontrolled leaks. Once the characteristics of diplexers in general are expounded, the particular specifications of this design will be presented. The goal of this work is to design, physically implement and measure a diplexer which fulfills real electrical specifications for Ku band (12-18 GHz). Such requirements are specified in Table 1.. TX Channel RX Channel Rejection (over TX and RX) Return losses (TX and RX). 11.9 - 12.2 GHz 13.75 - 14 GHz 60 dB 28 dB. Table 1: Design specifications. As it was shown in the block diagram of the diplexer, the immediate idea is to use two band-pass filters to implement the device (Fig. 2a). However, the lack of specifications about rejection under the lowest and over the highest frequency permits considering a diplexer compounded by a low-pass and a high-pass filter instead (Fig. 2b). This is the reason why four different filters are presented in this work, although eventually only the band-pass responses will be diplexed. In any case, considering the working frequency of the diplexer and the aforementioned power-related issues of such devices, the most suitable technology to implement a diplexer for satellite communications is waveguide technology, due to its power-handling capability and its low losses when compared with other transmission lines.. (a) Two band-pass filters. (b) High-pass and low-pass filters. Figure 2: Different diplexer configurations. The most practical approach is to design the filters individually before joining them to obtain the diplexer. The process followed and therefore the structure of this work are in accordance with this idea. Firstly, waveguides characteristics and their behavior as transmission lines are detailed in the following section, as well as how they are used to implement filtering structures. The next section covers the basis of modal analysis and how symmetries have been used to simplify the designs. Then, the four filters are thoroughly explained, starting with low- and high-pass filters, which do not require a synthesis process. After that, the diplexer itself is expounded as a combination of both band-pass filters. Once all the structures have. 3.

(10) been designed, the manufacturing process and its limitations are described. This section represents a new approach, since it makes use of additive manufacturing techniques (Fused Filament Fabrication (FFF) with PLA) and a subsequent metallization with paint. At this point, conduction losses are taken into consideration. The final part of this work is devoted to the measurement and analysis of the results obtained.. 2. Waveguides as transmission lines. Waveguides consist on a single hollow conductor that can propagate electromagnetic energy above a certain frequency. It is a technology widely used at microwave frequencies, and there are several advantages which explain such use. Waveguides are able to handle very high power levels with reduced losses at frequencies where other transmission lines do not work properly. Furthermore, the flanges used to connect waveguide sections present less reflections than the associated with other connectors, for instance coaxial lines. However, waveguide usable bandwidth is narrow when compared with other transmission lines, and the physical structures tend to be bulky, especially at low frequencies. Finally, the manufacturing process of a waveguide device is expensive and complicated, and this is one of the reasons why such designs have not been feasible at an academic level until now.. Figure 3: Rectangular waveguide structure. There are two basic geometrics for hollow waveguides: rectangular and cylindrical. All the devices expounded through this work have been done by using rectangular waveguides, thus this geometry will be thoroughly explained. Fig. 3 shows a rectangular waveguide of width a and height b, as they are commonly referred.. 2.1. Propagation in rectangular waveguides. Modes propagating through a hollow waveguide correspond to the solutions to Maxwell’s equations when the right boundary conditions are applied. In general terms, three modes of propagation may exist in a transmission line, according to the pattern of electric and magnetic fields when solving Maxwell’s equations: TEM, TE and TM. In this case, TEM mode cannot exist since waveguides are single-conductor transmission systems. TE (Transverse Electric) modes correspond to the case when there is no E-field in the direction of propagation; whereas TM (Transverse Magnetic) modes imply that no H-field exists in that direction. After solving Maxwell’s equations for TE modes (Ez = 0), it can be seen that H-field components have a sinusoidal variation as follows,. Hx ∝ sin. mπ nπ x cos y a b 4.

(11) Hy ∝ cos. mπ nπ x sin y a b. Hz ∝ cos. mπ nπ x cos y a b. Of course, these variations are also present in the E-field components, since both fields are related. In the previous expression, two integers appear: m and n. They can take on all values from zero to infinity; therefore the solutions to the equations are infinite. Each pair of values describes a mode, i.e. a particular form of transmitting electromagnetic energy through the waveguide. Such modes are designated as T Emn , where m and n represent the number of half sine wave variations of the field component in the x and y directions respectively. The results for the TM modes (Hz = 0) are analogous. E-field components change sinusoidally, as well as the H-field ones. Similarly, TM modes are described as T Mmn , although in this case m and n cannot equal to zero. Whether any of them does, all field components result to be zero.. Ex ∝ cos. mπ nπ x sin y a b. Ey ∝ sin. mπ nπ x cos y a b. Ez ∝ sin. mπ nπ x sin y a b. Once the possible solutions are specified, it is fundamental to describe the conditions under which the fields may exist. A certain mode can propagate only when the propagation constant is real. This condition can be expressed as f > fc , i.e. the working frequency must be higher than fc , so-called cutoff frequency, which equals. c co fc = =√ λc µr r. r. m 2 n 2 + 2a 2b. Below this frequency, which depends on the dielectric material, the waveguide dimensions and the considered mode, no electromagnetic energy propagates. The signal decays drastically with distance as it attempts to travel across the guide. This reduction of the signal is not related to any dissipative losses, but with high reflection instead. As a result, waveguide behavior is profoundly similar to a high-pass filter, an intrinsic characteristic which will be used later on this work. It is worth noticing too, that all TE and TM modes have the same cutoff frequency, except from the TE modes with subscripts m or n equal zero.. 5.

(12) Another important characteristic of waveguides arises when considering the characteristic impedance (Zo ), which is the voltage-current ratio for travelling waves. In waveguides, it cannot be uniquely defined, since voltage can be determined in several ways for a given field pattern. However, this is unimportant when performing filter calculations, as the results will not differ regardless of the characteristic impedance chosen, as long as it is kept equal through the process. Finally, there is one more aspect about propagating modes which should be mentioned. The dominant or fundamental mode of a waveguide is the one with the lowest cutoff frequency, whereas the others are referred to as higher modes. In a rectangular waveguide, the dominant mode results to be T E10 , and its cutoff frequency can be written as. fc =. c 2a. (1). where an empty guide has already been considered (r = 1). In waveguide transmission, it is desirable that only the dominant T E10 mode propagates along the guide, and the range of frequencies where this occurs is called the single-mode bandwidth. Its lower limit is settled by the cutoff frequency of the dominant mode, while the higher corresponds to the cutoff frequency of the first higher mode. Notwithstanding these theoretical limits, the real usable frequency range of a waveguide is smaller to ensure a suitable behavior. Dimensions of rectangular waveguides are standardized The ratio a/b equals almost 2, as a trade-off between the desire of low attenuation and the avoidance of higher modes to propagate too soon. Considering the working band of the diplexer, the most suitable standard waveguide to work with is WR-75, whose characteristics are detailed in Table 2.. Inner dimensions a x b (inches) 0.750 x 0.375. Inner dimensions a x b (mm) 19.050 x 9.525. Usable Frequency Range (GHz) 10 - 15. T E10 Cutoff frequency (GHz) 7.87. Table 2: WR-75. 2.2. Attenuation in rectangular waveguides. The phenomenon of attenuation in waveguides must be treated differently whether the working frequency is above or below cutoff. Below the cutoff frequency, modes are not able to propagate since they are evanescent, i.e. highly attenuated. In this case, the attenuation constant α can be expressed as. s α=. 2π λc. 2 − ω 2 µ. (N p/m). (2). Above the cutoff frequency, there would not be any attenuation in a waveguide made of a perfect conductor. However, such conductors do not exist, therefore some losses must be considered. This is 6.

(13) accomplished by introducing the attenuation constant αc . In order to calculate its value, a new electromagnetic problem must be solved with the proper boundary conditions. Owing to the difficulty of this task, a perturbational method is used instead. As a result, the attenuation constant for conduction or ohmic losses for a T Emn mode results to be. (αc )mn. 2Rs = s 2 fc bη 1 − f. (. b 1+ a. . fc f. 2. b + a. 0n − 2. . fc f. 2 !. m2 ab + n2 a2 m2 b2 + m2 a2. ) (N p/m) (3). r. ωµo ),which depends on the conductivity of the material where Rs is the surface resistance (Rs = 2σ r µ σ(S/m), η = is the intrinsic impedance and 0n corresponds to 1 (n = 0) or 2 (n 6= 0). Particularizing the previous expression for T E10 may be convenient, since it is the dominant mode.. ". 2 # 2b fc 1+ a f Rs s αc (T E10 ) = 2 bη fc 1− f. (N p/m). (4). All devices designed in this work are made with air-filled waveguides, but if there was a lossy dielectric instead, another attenuation term should be added to reflect dielectric losses.. 2.3. Waveguide implementation of filtering structures. Once the main characteristics of waveguides are expounded, this technology must be used to implement three different filtering structures: low-, high- and band-pass. Basic grounds are briefly explained here, whereas a deep analysis will be conducted later on this work, applied to the particular specifications of these filters.. 2.3.1. Low-pass filters. First of all, it is important to notice that due to the intrinsic high-pass behavior of waveguides and in spite of their name, low-pass filters do not work below cutoff frequency. Once this is established, the idea underlying waveguide low-pass filters is similar to the stepped impedance filters commonly used with microstrip or stripline technologies. If short-section lossless transmission lines are considered (βl << π/2), a section of the transmission line can be represented by the equivalent circuit shown in Fig. 4. When a high-impedance transmission line is used (Zo = ZH ), the behavior of the line is practically 7.

(14) Figure 4: Equivalent circuit of a lossless short transmission line (βl << π/2). inductive (B ≈ 0). On the contrary, low-impedance transmission lines (Z0 = ZL ) cause an almost capacitive response (X ≈ 0). This fact allows the replacement of the inductors and capacitors which form a low-pass filter prototype by a cascade of transmission lines with high and low impedances. This idea cannot be directly applied to waveguide technology, fundamentally because of the absence of a unique characteristic impedance. However, corrugated waveguide filters bear a resemblance to stepped impedance ones, since they shape a low-pass response by raising and lowering the waveguide height, while the width is maintained uniform. The ridge surfaces created across the structure are known as corrugations, and make the structure look similar to the schema depicted in Fig. 5.. Figure 5: Low-pass corrugated waveguide structure, side view. 2.3.2. High-pass filters. Implementing waveguide high-pass filters makes use of the aforementioned high-pass intrinsic behavior. The basic idea is to use a section of waveguide with the appropriate dimensions to allow the fundamental mode to propagate only above the desired frequency. As it was previously mentioned, when working with T E10 as the dominant mode the only dimension that influences the cutoff frequency is the width (Eq. 1), thus it will be the key parameter whereas the height will be kept constant in the whole structure. Besides, a stepped taper will be needed in order to match the input and output ports. Fig. 6 shows the top view of a high-pass waveguide structure.. 8.

(15) Figure 6: High-pass waveguide structure, top view. 2.3.3. Band-pass filters. Band-pass filters make use of the phenomenon of resonance. In resonant circuits the average energy stored in electric and magnetic fields is the same at a certain frequency, called resonant frequency. Such circuits are widely used in electronic networks and systems, and they are implemented by using lumped elements at low frequencies. They√consist of a L-C shunt or series configuration, equaling in both cases the resonant frequency ωr = 2π/ LC. However, as it occurs with other circuits, this implementation is not feasible at microwave frequencies. Several transmission-line techniques are used to overcome such limitation in order to work with resonant circuits at high frequencies. The cavity resonator is the most important to this work, since it results to be the waveguide implementation of these circuits. Before describing cavity resonators and the role they play in band-pass filters, there is an essential factor in any resonant circuit that must be introduced: the quality factor, Q. It can be defined for any resonance phenomena in terms of the stored energy and the power lost at the resonant frequency, giving an indication of how good a resonator is. There are three Q factors that can be defined, depending on which kind of dissipative loss is being considered. The unloaded Q (QU ) is a measure of the resonator itself. An infinite value means that the circuit is dissipationless. If only the losses over the external circuit are considered, the external Q (QE ) is defined. Finally, the loaded Q (QL ) is a function of both the quality of the resonant circuit and its coupling to the external circuitry. The relation between them can be expressed as. 1 1 1 = + QL QU QE Cavity resonators are hollow metallic enclosures which present a resonant behavior. They can be seen as a section of waveguide shorted at both ends. Resonance happens when the length of the cavity is a multiple of λg /2, and the number of modes able to resonate inside is infinite. Besides, very high values of unloaded Q can be reached. However, in order to excite a particular mode, a cavity resonator must be coupled to the circuit, i.e. an aperture must be opened in the waveguide so that the fields can be excited. This coupling is made in several ways, for instance with inductive, capacitive or resonant irises. In all cases, it represents a discontinuity in the waveguide. Band-pass filters are compounded by several coupled resonators. Couplings could be from any nature, but inductive irises (Fig. 7) have been used in this work, as they present several advantages in terms of manufacturing and insertion losses provided. The number of cavities will be determined after a synthesis process.. 9.

(16) Finally, since resonance is a phenomenon which involves a high stored-energy value, the insertion losses in band-pass filters are higher than in low- and high-pass filters.. Figure 7: Symmetrical inductive iris. 3. Modal analysis as full-wave design technique. Understanding the behavior of electric and magnetic fields in a structure requires to solve Maxwell’s equations under specific conditions. In order to accomplish it, full-wave CAD software has become a very powerful tool. Electromagnetic solvers can find the solution to Maxwell’s equations through different means: analytically or numerically. Each has limited suitability: they are very useful to evaluate certain problems but present disadvantages when facing others. Two different commercial software tools have been used to conduct this work:µWave Wizard (Mician) and Microwave Studio (CST). Both are full-wave software, but their solving of Maxwell’s equations differs. The former uses the analytical technique of mode matching, which consists in obtaining and combining the expression of each of the modes propagating through the structure, whereas the latter follows a numerical approach based on the Finite Element Method (FEM), among other kind of solvers. µWave Wizard presents two relevant limitations. Firstly, calculating the analytical expressions of the fields in the structures entails a restriction, since there are possible geometries which do not have an analytical solution. As a result, the variety of available structures is limited, although it includes an enormous amount of different geometries. The other problem is the number of modes considered when calculating a device response. Theoretically, in order to calculate the fields in the discontinuities of a structure, all modes propagating should be pondered. However, the number of solutions or modes taken into account must be truncated, due to the impossibility of handling infinite equations and variables. Therefore, the modes which are imperative to consider are those which modify the response, i.e. adding them causes a perceptible change in it. At this point, the response of the device can be assured to be convergent . In structures with a lack of symmetry, the search of convergence in the response may lead to an extraordinary computational effort, in terms of time and resources. The importance of symmetry to control this last point is thoroughly explained in the next section. On the contrary, Microwave Studio does not present any limitations regarding the possible geometries of the structures. It uses the Finite Element Method, which discretises the problem in small parts - finite elements - and solves them separately. It allows an accurate representation of complex geometries, and it does not distinguish the different modes propagating through the structure. As it was previously mentioned, both methods have been used in this work. The initial design and. 10.

(17) optimization of the filters and the diplexer has been done with µWave Wizard and subsequently verified with Microwave Studio. Besides, the latter has been mainly used to include the losses into the analysis.. 3.1. Symmetry and its implications. A priori, the number of modes that could propagate through a waveguide is infinite. However, some of them cannot be excited due to the symmetry of the structure. This fact has a positive effect on calculating the filter or diplexer response with mode matching techniques: the modes that do not propagate have no influence on the response and therefore do not have to be considered. Numerical analysis may be simplified as well, since the existence of a symmetry plane permits to consider half of the structure when computing the response. The benefits of two symmetry planes are even more significant, as only the response of a quarter of the waveguide device must be calculated. In short, the computational effort is significantly reduced, so bearing symmetries in mind is imperative when designing a waveguide structure. In first place, the definitions of electric and magnetic walls must be explained. A particular structure presents an electric wall when there is a plane to which the E-field is perpendicular. The definition is equivalent for H-field and magnetic wall. Whether considering a rectangular waveguide, there are two planes susceptible to present electric or magnetic walls: vertical and horizontal. As it can be concluded from Fig.8, the fundamental mode T E10 presents an electric wall (EW) in the horizontal plane and a magnetic wall (MW) in the vertical plane.. (a) T E10 E-field, front view. (b) T E10 H-field, front view. Figure 8: T E10 E- and H- field in a rectangular waveguide. This matches the situation depicted in Fig. 9a. The crucial point is that only the modes with the same symmetry as the one which is exciting the waveguide (T E10 ) will be generated and therefore will propagate through the waveguide. It implies that some higher modes with a different distribution of electric and magnetic walls will not exist in the guide, in spite of working above their cutoff frequency. The set of modes which actually propagate involves all TE and TM modes with subscripts m=odd, n=even; including T E10 itself. The opposite situation would have taken place if the mode used to excite the waveguide had been T E01 . Since it is the T E10 turned 90 degrees, the symmetry walls are magnetic in the horizontal plane and electric in the vertical one (Fig. 9b). Thus the propagating modes would have been TE/TM with evenodd subscripts.. 11.

(18) (a) Vertical: MW, Horizontal: EW TE/TM Subscripts(m,n): Odd, Even. (b) Vertical: EW, Horizontal: MW TE/TM Subscripts(m,n): Even, Odd. (c) Vertical: No, Horizontal: EW TE/TM subscripts(m,n): All, Even. (d) Vertical: MW, Horizontal: No TE/TM subscripts(m,n): Odd, All. Figure 9: Different configurations of symmetry planes. A breaking in the planes of symmetry of a structure would cause the immediate excitation of higher modes previously unconsidered. For instance, in a structure excited by T E10 , if the magnetic wall in the vertical plane disappears (Fig. 9c), the remaining modes propagating present n=even, but are not restricted as far as the subscript m is concerned. On the contrary, the absence of the electric wall (Fig. 9d) removes the restriction in the subscript n: all TE/TM modes with m=odd will be excited. The filtering structures previously presented – low-, high- and band-pass – maintain the same symmetry planes as T E10 , therefore the set of modes is reduced at the minimum. Nevertheless, diplexing the filter will break a symmetry plane, causing an increase in the number of modes propagating and complicating the design.. (a) H-plane structure (b) E-plane structure Figure 10: Structures. 12.

(19) In addition to the symmetry, waveguide structures may be divided in two types as well: H-plane and E-plane structures. The former does not present any changes in height (Fig. 10a) , whereas the latter does not experience any variations in width (Fig. 10b). This affects the propagating modes too: in a H-plane waveguide structure excited by T E10 , all modes will have the subscript n equal 0. If the structure was E-plane instead, the modes propagating would maintain m equal 1. Combining both criteria – symmetries and variations in height and width – the number of modes can be drastically reduced. Table 3 summarizes all the aforementioned considerations for the three waveguide filters explained in the previous section. The structure of the diplexer will be analyzed in detail in the pertinent section.. Symmetries (Vertical-Horizontal) MW-EW MW-EW MW-EW. Low-pass High-pass Band-pass. Structure. Modes (m,n). E-plane H-plane H-plane. 1, even odd, 0 odd, 0. Table 3: Modes propagating through filtering structures. 4. Low-pass filter As it was mentioned within the introduction, the low-pass filter for transmission band is part of one of the possible implementations of the diplexer. Its most important advantage when compared with a band-pass filter is the reduced insertion losses. However, diplexing it results a significant challenge, due to all the spurious responses excited. The immediate higher modes excited when diplexing the low-pass filter are T E11 and T M11 (fc = 17.59GHz for WR-75), unconsidered when designing the filter itself due to symmetries. Once the template of the filter is defined (Fig. 11), the design process must be conducted bearing in mind the main criteria that influence the filter response. A circuital synthesis has not been followed, on the contrary of what occurs with the band-pass filter, thus the process is mainly heuris-. Figure 11: TX-filter template. tic. Two essential elements condition the filter rejection: the number of corrugations (higher sections of the filter) and the height of the filter (lower sections). The losses of a waveguide structure are inversely proportional to its height, hence it would be important to maintain the low-pass filter as high as possible,. 13.

(20) but it has an important drawback: the rejection of the filter is not favored by high structures. A possibility to counteract this lack of rejection is to increase the number of corrugations, although it implies a more complex and bigger geometry, with more losses associated.. a b thickness l1 l2 l3 h1 h2 h3 h4 Figure 12: TX Low-pass filter, side view. 19.05 9.525 2.5 4.3 7.63 3.27 20.25 19.84 21.3 20.14. Table 4: TX Low-pass filter dimensions (mm). As it can be seen, designing a low-pass filter means a trade-off between rejection and insertion losses, physically represented by the corrugations and the height of the structure. The first step which needs to be taken is to set both parameters, as well as the thickness of the corrugations. They all will have different heights, but their thickness is influenced by the manufacturing process. Wider corrugations will be printed more easily due to the accuracy of the 3D-printer, and their metalization will be easier too. However, very wide corrugations make specifications difficult to achieve. As a result, thickness is set to 2.5 mm. The starting point is to fix feasible values for the two key parameters. In this case, µWave Wizard was used to evaluate if three corrugations and half the WR-75 height were enough to obtain a response which adjusted to the template. It was not, thus the next natural step was to increase the number of corrugations. A filtering structure with four corrugations had an adequate response, therefore the following goal was to rise the height as much as possible in order to diminish losses, being the limit the WR-75 height. This cut-and-try process may seem imprecise, but it is a practical way to take one of the most important decisions in a waveguide low-pass filter. Then, the remaining degrees of freedom of the structure, i.e. height of the corrugations and distance between them, must be used to obtain a response which fits the template perfectly. This is performed by using an optimizer taking all relevant modes into consideration, according to the symmetry issues mentioned in the previous section. Figure 13: TX Low-pass filter. The final appearance and dimensions of the low-pass filter are shown in Table 4 and Fig. 12 and 13. It has been possible to rise the height of the filter and equal b, hence insertion losses will be the lowest possible. This causes the degrees of freedom to be seven instead of nine: the height of the four corrugations and the separation between them. Filter response can be seen in Fig. 14. In addition, fields inside the filter can be simulated to verify that there is electromagnetic energy travelling across the structure (Fig. 15a) at frequencies within the 14.

(21) Figure 14: TX Low-pass filter response. (a) E-field at f=12.05 GHz. (b) E-field at f=13.75 GHz. Figure 15: E-field in the TX low-pass filter, side view. transmission band. On the contary, outside the passband, particularly at one frequency located in the reception band, there is energy at the input port but it does not propagate through the filter (Fig. 15b).. 5. High-pass filter. The complementary of the low-pass filter is a high-pass one, which can be used to fulfill the requirements of the reception channel of the diplexer. A high-pass filter may seem to have a simple geometry due to its resemblance to a regular waveguide and its behavior. However, there are several issues that must be considered during the design process, involving rejection and attenuation. In this case, the template of the filter is depicted in Fig. 16. As it was previously mentioned, the parameter which determines the cutoff frequency of a rectangular waveguide for the fundamental mode T E10 is the width a (Eq. 1). Its value must be chosen considering the effective attenuation inside – insertion losses – and outside – rejection – the passband . This trade-off is illustrated in Fig. 17. The rejection of the filter is caused by the attenuation of the mode below cutoff (Eq. 2, red dashed line in Fig. 17). The width of the guide must be chosen so that the specifications over transmission band are fulfilled. In this sense, a higher cutoff frequency would be desirable, since it would guarantee enough rejection. However, if the cutoff is too close to the inferior frequency of the passband, problems with. 15.

(22) Figure 16: RX filter template. Figure 17: High-pass filter losses. insertion losses arise. Attenuation above cutoff is caused due to conduction losses (Eq. 4, blue continuous line in Fig. 17). Considering a good conductor (σ = 4.7 · 108 S/m), it can be seen that attenuation is not uniform near the cutoff frequency. If such frequency is very close to the lowest one of the filter, attenuation and therefore insertion losses vary over the passband, which is an unwanted effect since it difficulties equalization. In the previous disquisition another important parameter has been forgotten: length. When examining attenuation, it is accounted for a constant in dB/m, thus length plays an important role too. For the same width, rejection will be higher in a longer filter, which would be considered to be positive, but insertion losses will be higher too. Besides, the final purpose of these design must not be forgotten: the diplexer is made for spatial applications, therefore length and weight are key factors. Taking all this into consideration, Fig. 17 shows a reasonable trade off in terms of dimensions: 11.1 mm width and 57 mm length. Nevertheless, these dimensions are just the basis to begin to optimize the structure: the width is set but the length is one of the degrees of freedom. Several steps and discontinuities must be placed in both sides of the filter in order to match the ports to the waveguide specified, WR-75. Deciding the number of steps is heuristic, keeping in mind that the priority is to design a structure as short as possible. The result can be seen in Fig. 18, remembering that high-pass filters are H-plane structures and therefore b equals the height of the WR-75.. Figure 18: RX High-pass filter, top view. Although there was no intention of designing a symmetric filter, it ended in that way. Steps in both sides have dimensions similar enough to consider them the same, which represents an advantage in terms of 16.

(23) optimizing parameters and computational effort.. a_WR75 a_intermediate a b length connection. 19.05 12.19 11.1 9.525 48.33 10.38. Table 5: RX High-pass filter dimensions (mm). Figure 19: RX High-pass filter response. (a) E-field at f=13.9 GHz. (b) E-field at f=12.2 GHz Figure 21: E-field in the RX high-pass filter, side view. Figure 20: RX High-pass filter. The final dimensions and the filter response can be seen in Table 5 and Fig. 19 respectively, as well as its three-dimensional appearance (Fig. 20). As it was done with the low-pass filter, fields can be simulated in the structure. At frequencies over passband (Fig. 21a) the filter lets electromagnetic energy pass, whereas at frequencies over rejection band (Fig. 21b) the opposite occurs.. 6. Band-pass filters. The first set of filters which could be used to implement the diplexer have already been explained. The low- and high-pass filters have been developed in first place due to their heuristical design and simplicity. On the contrary, a band-pass filter requires a synthesis process to establish the number of resonant cavities, the couplings and all the geometrical dimensions. Although the diplexer requires two filters to be designed, the synthesis process is the same, thus both structures will be expounded at the same time within this work. Once the theoretical dimensions are fixed, an optimization process will be conducted to determine the final geometry of the filters.. 17.

(24) 6.1 6.1.1. Synthesis process Chebychev filters and low-pass response. Chebychev characteristic is extensively used when designing filters. These designs take that name because their response approximates an analytical function, in this case Chebychev polynomials. The response is also called "equal-ripple", since there is an enclosed ripple, related to return losses, over the passband. The edge of the passband of the low-pass prototype corresponds to ω = 1, which is the last frequency with ripple. These filters present a sharper response when compared with other characteristics, for instance Butterworth filters.. (a) TX filter. (b) RX filter Figure 22: Different Chebychev filter orders. Given a set of specifications, the first parameter to determine is the order of the Chebychev response which matches such specifications. It has been done by programming in Matlab the whole process which is being described here, and then applying it while varying the filter order. For the sake of clarity, different responses with order N equal 3, 4 and 5 are depicted in Fig. 22 for both transmission and reception filters. The goal is to choose the lowest order which fulfills the requirements, which results in N=4 for both filters.. Figure 23: Chebychev low-pass prototype. Once the order is decided, the aspect of the low-pass prototype is established (Fig. 23). The cutoff frequency corresponds to ω = 1 and the element values can be easily computed with some recursive formulas. However, it is imperative to determine ripple in first place. There is a relationship between return losses, which are given in the specifications and are the same for both filters, and ripple. The latter Ptx ). equals insertion losses, which are the ratio between transmitted and incident power ( = IL = Pinc Knowing that Ptx = Pinc − Pref , ripple can be written as 18.

(25) =1−. Pref (W ) Pinc (W ). (dB) = −10 log10 (1 − 10. =⇒. −RL(dB) 10. ). Now the low-pass filter coefficients can be calculated. g0 is a resistance, therefore its value is in ohms, but g5 occurs to be a conductance, thus the calculated value is in siemens. The values of the other elements are expressed in farads or henrys depending on whether they are capacitors or inductors..   coth2 β g5 = 4  1. 2a1 g1 = γ. g0 = 1. gk =. 4ak−1 ak bk−1 gk−1. N even N odd. k = 2, 3, 4. β, γ, ak and bk correspond to. β = ln(coth. . . ) 17.37. γ = sinh. β 2N. . ak = sin. (2k − 1)π 2N. . bk = γ 2 + sin2. . kπ N. . where N and are the aforementioned order of the filter and ripple in dB respectively.. 6.1.2. J-inverters and band-pass transformation. The next step is turning the low-pass prototype into a band-pass one, with the proper central frequency and bandwidth. When doing that, capacitors and inductors will be transformed into shunt and series resonators respectively. However, implementing the filter will only involve identical shunt resonators, thus a new and very important element must be added to the circuit: J-inverters.. [ABCD] =. 0 jJ. j/J 0. . Figure 24: J-inverter and its ABCD matrix. Figure 25: J-inverter equivalences. 19.

(26) J-inverters can be defined with their ABCD matrix (Fig. 24). Since the parameter J does not vary with frequency, they are supposed to be invariant, i.e. their response is the same in the whole band considered. This is not possible when implementing them physically, and it will cause modifications in the filters response. Inverters are used to transform the elements of the circuit and their values without changing the response. Different equivalences are used through the synthesis process to normalize and fix the values of all elements involved. They are captured in Fig. 25 and 26, and their correctness can be easily proved by concatenating ABCD matrices.. r. Yt Y. r. Yt Y. r. Gt G. J3 = J1 J4 = J2 (a). J2 = J1. (b) Figure 26: J-inverters transformations. Figure 27: Low-pass prototype normalized. Once the inverters have been placed so that there are only shunt capacitors in the low-pass prototype (CLP ), it is time to transform the response into a band-pass one. Previously, the values of the capacitors and the resistance in both ends have been normalized to 1 (Fig. 27). The band-pass transformation is not lineal, therefore the response obtained is more accurate for filters with small bandwidth. If ω1 and ω2 denote the edges of the passband (ω1 < ω2 ), the following substitution leads to a band-pass response:. ω. −→. ω k2 − , k1 ω. k1 = ω2 − ω1 , k2 =. ωo2 ω2 − ω1. ωo is the center frequency and can be expressed as the geometric mean of the edges of the passband 20.

(27) √ (ωo = ω1 ω2 ). The application of such transformation in the capacitors transforms each of them into a shunt resonator (Fig. 28), where the values of LBP and CBP equal. LBP =. ω2 − ω1 ωo2 CLP. CBP =. CLP ω2 − ω1. Figure 28: Band-pass prototype with lumped elements. 6.1.3. Slope parameter and distributed resonators. The band-pass filter designed so far is not finished yet. Lumped elements which form the resonators do not work properly at high frequencies, hence they must be replaced. As it was discussed in previous sections, transmission lines are necessary. In this case, all shunt resonators will be replaced by transmission lines of length λgo /2, i.e. they measure λg /2 at the center frequency of the filter, which is the resonant frequency. Until now, lumped resonators have been handled according to the values of the capacitor and inductor which form them. However, resonant circuits in general are not formed by such elements. As a result, the best way to define a resonator is by using its resonant frequency and slope parameter (SP). The latter provides a convenient means for relating the resonant properties of any circuit, for instance transmission lines, to a simple lumped equivalent circuit. For a shunt resonator,. ωo = √. r. 1. SP =. LBP CBP. CBP CLP (ω2 − ω1 ) = LBP ωo. It can be seen more clearly now the usefulness of having normalized the capacitors in the low-pass prototype: the slope parameter of the lumped resonators is easily calculated with the specifications of the passband and is the same for all resonators. The transmission lines which will replace them do have their own slope parameter too,. SPltx. π = 2Zo. . λgo λo. 2. being Zo the characteristic impedance of the line. In this case, the line represents a waveguide, particularly a cavity resonator, thus the distinction between λg and λ does have sense. For other transmission lines which do not have cutoff frequencies, such as stripline, λgo equals λo .. 21.

(28) Figure 29: Band-pass prototype with distributed elements. Now both types of resonators are perfectly defined. According to the transformation seen in Fig. 26a, the substitution of the lumped elements by the transmission lines affects the values of all J-inverters. In the end, the final aspect of the band pass prototype is shown in Fig. 29. The values of the J-inverters after all transformations are. s J01 =. SPltx Zo (SP )g0 g1. s J40 =. SPltx Zo (SP )g4 g5. Ji,k =. SPltx 1 √ SP gi gk. i, k = 1...4. To summarize, the low-pass prototype has suffered two main transformations. The first one has turned the low-pass into a band-pass prototype, and has added some distortion to the response due to the bandwidth of the filter. The second one has replaced lumped resonators for distributed elements. Besides, due to the λg /2 periodicity of transmission lines, the transformation from lumped to distributed elements originates a replica of the filter response at higher frequencies. It is shown in Fig. 30a for the transmission filter, but the same occurs with the reception one. The existence of this undesired response causes a lack of rejection, since the slope of the filter skirt is not as steep as it would be if the attenuation became infinite (Fig. 30b).. (a) TX Band-pass filter response. (b) TX and RX Band-pass filter response - Skirt detail. Figure 30: Effects of the transformation from lumped to distributed elements. 6.2. Physical dimensions. The synthesis process provides the circuit parameters of the band-pass filters. The next step consists in using those parameters to determine the geometrical dimensions of the filter before optimizing it to. 22.

(29) J01 = J40 J12 = J34 J23 λgo /2 (mm). TX filter (11.9-12.2 GHz) 0.3193 0.0771 0.0561 16.446. RX filter (13.75-14 GHz) 0.2497 0.0472 0.0343 13.13. Table 6: Synthesis results. obtain the final response. The results from the synthesis are presented in Table 6 for both filters. The values of the J-inverters have been calculated with the previous expressions and are symmetrical. There are two different elements in the filter that must be physically implemented: the resonators and the inverters. The former have already been designed: the length determined by the synthesis is the one of the waveguide resonator. The latter represents the couplings between resonators, i.e. the inductive irises presented in section 2.3.3 (Fig. 7). The height and thickness of the irises are fixed. As it was previously discussed, band-pass filters are H-plane structures, therefore the height is the same as the WR-75: b=9.525 mm. Thickness must be chosen following the same criteria used to decide the thickness of corrugations in the low-pass filter: manufacturing issues. In this case, irises have a thickness of 2 mm. The width of the irises will be adjusted so that the coupling corresponds with the value of the J-inverter. The most effective way of doing it is by calculating the S-parameters of the inverters and designing irises with the same response. The scattering parameters follow the expressions:. |S11 | = 20 log10. 1 − J2 1 + J2. |S21 | = 20 log10. (dB). 2J 1 + J2. (dB). Here the J-inverters are being considered non-ideal for the first time, since S-parameters of the iris will match the ones of the inverter only at a certain frequency or at a reduced bandwidth. The widths obtained (Table 7) are not very accurate, but it is not very important since they are only the basis for the optimization.. J01 = J40 J12 = J34 J23. TX filter (11.9-12.2 GHz) 9.75 6.25 5.75. RX filter (13.75-14 GHz) 8.25 5.1 4.75. Table 7: Width of the coupling irises (mm). 23.

(30) 6.3. Optimization and final response. Theoretical dimensions of both waveguide filters have already been calculated. However, the final dimensions will widely differ. The reason is that resonators do not measure λgo /2 at the center frequency of the passband because they are coupled. The apertures – irises – used to connect the cavities with each other and to excite the electromagnetic modes modify their length. Resonators are electrically longer now and therefore resonance occurs at lower frequencies. To counteract such effect, the physical length of the cavities must be reduced. As a result, the real dimensions of all waveguide resonators will be significantly shorter than the theoretical result. Not all resonators are disturbed in the same way: central cavities are the least modified. Their couplings are smaller and therefore they affect less to the resonance frequency. As the ends of the filter are reached, coupling irises become wider, i.e. resonators differ more from ideal closed cavities. Consequently, the variation in length is more significant. These variations do not have to be symmetrical, although the theoretical filters were. At the beginning of the optimization process, all lengths and irises must be considered independent. In this case, however, after optimizing the structures counting with all variables, the dimensions of the irises in both filters resulted almost symmetrical. Thus, they have been forced to be the same so that the number of variables is smaller. The final dimensions and responses of both filters are expounded below, in comparison with the theoretical response previously obtained.. 6.3.1. TX filter. The final dimensions and the side view of the transmission filter are shown in Table 8 and Fig. 31 respectively. As it can be seen in Fig. 32, the final response of the filter once the optimization has been performed is very similar to the theoretical one, at least in terms of reflection (S11 ). However, the aforementioned effect of lack of rejection caused by the existence of spurious bands at higher frequencies is much more significant than it was expected. As a result, the specifications of rejection over reception band are barely fulfilled. Nevertheless, diplexing filters usually increases rejection, thus this drawback may not be very critical.. Figure 31: Band-pass filter, top view. Finally, the three-dimensional appearance of the filter, as well as the simulation of the electric field inside 24.

(31) the structure, are represented in Fig. 33 and 34. At frequencies over the passband of the filter, the energy travels across the structure, whereas over the rejection band there is no transmission.. a b thickness l1 l2 w1 w2 w3. 19.05 9.525 2 13.29 14.93 9.73 6.37 5.83. Table 8: TX Band-pass filter dimensions (mm). Figure 32: TX Band-pass filter response vs theoretical synthesis. (a) E-field at f=12.05 GHz. (b) E-field at f=13.75 GHz Figure 33: TX Band-pass filter (CST). 6.3.2. Figure 34: E-field in the TX band-pass filter, top view. RX filter. The reception filter has an analogous design, but the existence of spurious in higher frequencies is not a negative feature, like it was in the transmission filter. The template the filter must fulfill involves frequencies below the passband, whereas the lack of rejection affects higher frequencies. Besides, there already was a margin between the rejection needed and the theoretical response, thus rejection specifications are more than enough satisfied. As a result, the passband of the filter can be made wider. It presents a fundamental advantage: insertion losses are reduced, as it will be seen in the pertinent section. This process is made heuristically with the help of the optimizer, thus the final response of the filter is not similar to the result of the theoretical synthesis (Fig. 35). In spite of it, the theoretical design of the filter is imperative and cannot be omitted. It provides the basis to obtain a better response for the specifications given. The physical appearance of the filter, as well as its dimensions are shown in Fig. 37 and Table 9. The 25.

(32) a b thickness l1 l2 w1 w2 w3. Figure 35: RX Band-pass filter response vs theoretical synthesis. 19.05 9.525 2 10.24 11.7 9.19 6.13 5.6. Table 9: RX Band-pass filter dimensions (mm). parameters of the latter are the same that the ones of the transmission filter (Fig. 31). The fields inside the structure at frequencies in and outside the passband are depicted in Fig. 36. Their behavior is the same as in previous cases: electromagnetic energy propagates within the specified band, but it is rejected at other frequencies.. (a) E-field at f=13.9 GHz. (b) E-field at f=12.2 GHz Figure 36: E-field in the RX band-pass filter, top view. 7. Figure 37: RX Band-pass filter. Diplexer. Diplexers are the most reduced version of multiplexers, which are a combination of several devices: Tjunctions and filters. They are widely used for communication satellite applications, even though their main disadvantage is the difficulty to scale them, i.e. to add channels to an existing multiplexer. Nmultiplexers combine N structures, needing a N+1-port junction. Consequently, diplexers are made out of two filters joined together by a three-port junction. The specifications involve the filtering structures, although joining them may be as hard as the design of the filters themselves. The idea is that, at the passband of one of the filters, the second one projects a virtual short circuit in the junction, so that all the electromagnetic energy propagates in the desired way.. 26.

(33) Although four filters have been designed to cover all possibilities of fulfilling specifications, only the two band-pass filters will be diplexed. The reason is that diplexing the other two filters is a more difficult task due to all the spurious responses excited, specially by the low-pass filter, and it is left for a subsequent work. There are two possibilities when deciding the configuration of the diplexer: using H-plane or E-plane junctions (Fig. 38). The filters used in both cases are the same: the only thing that changes is how they are placed in the diplexer. The choice is made considering once again the cutoff frequencies of the different modes of propagation. In a H-plane junction (Fig. 38b), the width is at least twice the WR-75 width. Since cutoff frequency of T E10 is inversely proportional to a (Eq. 1), the fundamental mode starts propagating at lower frequencies and so do higher modes. As a result, at the input port of the filters there is a great amount of undesired modes. With E-plane structures (Fig. 38a), however, the cutoff frequency of the fundamental mode is kept the same, since the width of the junction does not differ from the WR-75. As a result, an E-plane junction is used in the diplexer.. (b) H-plane (a) E-plane Figure 38: Three-port junction configurations. The most adequate approach is to design the junction separately before joining the filters to obtain the whole structure. Before doing that, the main difficulties or challenges of the diplexer design, putting filters aside, will be explained so that the effort required to obtain a satisfactory response for the diplexer can be contextualized.. 7.1. Main challenges when designing a diplexer. First of all, it must be repeated that this section explains the problems of diplexing the filters once they have been designed and have an adequate response. As it has been expounded in previous sections, the process followed to obtain such responses may be cumbersome and should be under no circumstance underestimated. Once this is established, there are two main problems intrinsically related which must be taken into consideration. The first problem is derived from the use of an E-plane junction. As the name indicates, there is no variation in width. However, the structures which are being diplexed, i.e. the band-pass filters, are H-plane structures, meaning that width does vary, and height is the constant dimension. As a result, the diplexer cannot be categorized following this classification.. 27.

(34) Along the same lines, there is a breakdown in one of the symmetry planes when looking at the symmetries of T E10 . It is very clear that the junction only presents symmetry in the vertical plane, and so does the diplexer. The use of an H-plane junction instead would have caused the disappearance of symmetry in the vertical plane. As a result, the diplexer is neither an E-plane nor a H-plane structure and only presents one symmetry plane (magnetic wall in the vertical plane, like in Fig. 9d), thus the number of modes propagating is significantly higher than the ones in the filters themselves. It implies that the computational effort of simulating or optimizing the diplexer response is important in terms of time and resources. This is the main reason why the junction is firstly designed separately, so that the computation of the whole structure is avoided as much as possible. The modes considered in the filters, the junction and the diplexer are reflected in Table 10.. Band-pass filters Junction Diplexer. Symmetries (Vertical-Horizontal) MW-EW MW-No MW-No. Structure. Modes (m,n). H-plane E-plane No. odd, 0 1,all odd, all. Table 10: Modes propagating through diplexer elements. The second problem is related to the existence of more modes propagating in the structure. So far, filters have been designed and simulated assuming that the only excitation at the input port was the fundamental mode T E10 , but adding the junction changes it. The higher modes excited in the junction are not attenuated enough when they reach the input port of the filters, thus they become part of the excitation. Higher modes have different wavelengths than that for T E10 , therefore the pass and stop bands for energy occur at different frequencies, causing a disruption in the response. The first immediate countermeasure would be to connect the filters with the junction by using long waveguide sections in order to attenuate higher modes. However, it would affect the fundamental mode too, and diplexers and multiplexers in general must be as short and light as possible in order to avoid very high losses or an excessive impact of ambient conditions in the electric response. The most feasible solution and the one which has been used is finding a trade-off between the length of the connections and a retouch of filters dimensions, beginning with the cavities and irises closer to the input port. As it has been explained, diplexing the filters entails several complications related with higher modes and their propagation. Besides, the increase of the computational effort makes finding a solution for these difficulties a hard task. All approximations or shortcuts which help to get close to the final solution without much computational calculations are desired and necessary.. 28.

(35) 7.2. Junction design and final optimization. The basic design chosen for the three-port junction consists of a common port with WR-75 dimensions and several steps to make the structure high enough to attach both filters. The output ports are waveguide sections with dimensions equal WR-75 so that filters can be correctly joined. The number of steps and their length, as well as the dimensions of the output sections, are determined heuristically with the help of EM software. The junction has been claimed to have a separate design, but it may not be clear the way to proceed, since the specifications of the diplexer involve the filters response, not the junction itself. The idea is to build an intermediate one-port device, as Fig. 39 shows, characterized by the reflection coefficient, i.e. S11 . The S11 of each filter has already been calculated, thus it can be saved in a file and the devices can be treated as black boxes with a known response. These black boxes are simple mismatched loads which are connected to the output ports of the junction, resulting in a one-port structure whose S-parameter must be computed. The goal is to design the junction so that the S11 of the one-port structure is a combination of the reflection coefficients of both filters.. Figure 39: One-port equivalent junction. This scheme is quite easy to deal with and fast from a computational perspective, since the filters are characterized by a stored response, meaning that it does not have to be calculated at each iteration of the optimizer. Besides, reflection coefficient is the most difficult parameter to adjust, since diplexed filters do not usually present problems to fulfill rejection specifications. As a result, this hybrid solution to design the junction is valid, although it does not care about the transmission coefficient. Now that the way to evaluate the adequacy of the junction is clear, it is time to explain how to establish the starting dimensions, so that the optimizer works correctly. The first part to address is the length of the linking sections. As it was stated in the previous section, one of the main problems when designing a diplexer is that the higher modes excited in the junction will not reach the filters attenuated enough so that they do not disturb the original response. The real attenuation of such modes can be approximated in order to estimate the minimum desirable length for the output connections. Fig. 40 shows the attenuation below cutoff of the three immediate higher modes. As it was expected, they all have "odd-all" subscripts, due to the symmetry considerations mentioned above. The attenuation is computed for a 10-mm WR-75 waveguide, the cutoff frequencies are marked with dashed lines and the passband of the filters appear with dot-dashed lines. 29.

(36) Figure 41: T E11 attenuation for different lengths in a WR-75. Figure 40: Attenuation below cutoff for several modes in a 10-mm WR-75. The mode which influences the response the most is T E11 , although its cutoff frequency (fc = 17.59GHz) is almost 4 GHz above the superior edge of the working band of the diplexer . This is the most affected frequency, with a T E11 attenuation of barely 20 dB. Fig. 41 shows T E11 only, this time varying the length of the waveguide section. The edge of the reception band-pass filter, f=14 GHz, is marked with a wider dashed line to see the attenuation more clearly. Obviously, the longer the connecting section, the more attenuated the mode is. It is remarkable how the higher mode is barely 10 dB rejected in a 5-mm waveguide section. Taking these ideas into consideration, output sections should have a length around 10 mm so that the attenuation of higher modes is mitigated to a certain degree. The rest of the structure dimensions cannot be estimated in the same way. The best method to design it is to use the minimum degrees of freedom possible, and see if the response is good enough. If it is not, new steps must be added, but always in reduced amounts. If the number of variables to optimize was too high, the optimization algorithm would not work properly, although the dimensions were close to optimum. Once that optimization process is completed, a three-port junction has been designed. Besides, since the filters responses have already been attached, it could be reasonable to think that the diplexer is complete. However, this is far from the truth. At the moment the black-box loads with their stored response are replaced by the real filters, the response of the diplexer changes remarkably and does not fit the template. The reason why that occurs has already been established. In spite of having chosen the lengths of the connections long enough to partly attenuate higher modes, T E10 is not the only mode present at the input of the filter, which was the premise under which filters were designed in previous sections. As a result, there are two filters whose performance is satisfactory when measured separately, and a threeport junction which joins ideal responses. Now, the final optimization of the whole structure cannot longer be avoided. It will affect both the junction and the filters. Thus, the black boxes must be replaced by. 30.

(37) the real filters, and the optimization must be re-started, leaving all dimensions of the junction as well as some of the parameters of filters as degrees of freedom. The wisest approach is to add variables carefully, starting with the width and length of the first iris and cavity respectively, and incorporating more progressively if the specifications are not fulfilled.. length_0 length_1 h0 h1 link_f1 link_f2. 6.05 3.09 20.96 26.03 9.43 10.18. Table 11: Final dimensions of the optimized junction (mm). Figure 42: Optimized junction, side view. The final appearance of the junction is shown in Fig. 42. Its retouched and definitive dimensions, as well as the ones of the diplexed filters, appear in Table 11 and 12. Both filters have the same variables, which correspond to Fig. 43. In Table 12, the first column corresponds to the original dimensions, whereas the second contains the dimensions of the diplexed filters, with the variation in brackets. It can be seen that symmetry has been lost as well. The response of the diplexer will be detailed next, but to illustrate how the definitive junction differs from the one calculated with the ideal responses of the filters, Fig. 44 has been plotted. It shows the response of the final junction when the non-diplexed filters are attached, and it is clear how the reflection coefficient does not respect the template.. Figure 43: Generic diplexed band-pass filter, side view. 31.

(38) TX filter (11.9-12.2 GHz) a b thickness. RX filter (13.75-14 GHz) 19.05 9.525 2. l1 l2 l3 l4. 13.29 14.93 14.93 13.29. 13.24 (-0.05) 14.93 14.93 13.29. 10.24 11.7 11.7 10.24. 10.26 (+0.02) 11.72 (+0.02) 11.71 (+0.01) 10.27 (+0.03). w1 w2 w3 w4 w5. 9.73 6.37 5.83 6.37 9.73. 9.95 6.38 5.84 6.38 9.74. 9.19 6.13 5.6 6.13 9.19. 9.2 (+0.01) 6.13 5.59 (-0.01) 6.11 (-0.02) 9.15 (-0.04). (+0.22) (+0.01) (+0.01) (+0.01) (+0.01). Table 12: Dimensions of the diplexed filters(mm). Figure 44: Response of the optimized junction with non-optimized filters attached. 7.3. Double bend. At this point, the diplexer is already completed but it is not ready to be manufactured. If it is intended for measuring, flanges must be attached. They are waveguide connectors, and their dimensions obey some standards. In this case, the flange used for a WR-75 waveguide is a cover flange with six holes and outer dimensions 38.1 x 38.1 mm (Fig. 45). These flanges have to be attached at the three ports of the diplexer. As a result, the distance between the center of the two output ports of the filters must be at least 38.1 mm. With the current dimensions, there is a distance of 16.505 mm, clearly insufficient to attach the flanges. Therefore, a new element must be added: the double bend.. 32.