OSCAR ENRIQUE LLERENA CASTRO
STUDY OF A COMPACT MICROWAVE CERAMIC COAXIAL RESONATOR FILTER
ESTUDO DE UM FILTRO COMPACTO PARA MICROONDAS FEITO COM RESSONADORES COAXIAIS CERÂMICOS
CAMPINAS 2014
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UNIVERSIDADE ESTADUAL DE CAMPINAS Faculdade de Engenharia Elétrica e de Computação
OSCAR ENRIQUE LLERENA CASTRO
STUDY OF A COMPACT MICROWAVE CERAMIC COAXIAL RESONATOR FILTER
ESTUDO DE UM FILTRO COMPACTO PARA MICROONDAS FEITO COM RESSONADORES COAXIAIS CERÂMICOS
Thesis presented to the School of Electrical and Computer Engineering of the University of Campinas for the degree of Master of Science in Electrical Engineering in the field of Telecommunication and Telematics.
Dissertação apresentada à Faculdade de Engenharia Elétrica e de Computação da Universidade Estadual de Campinas para a obtenção do título de Mestre em Engenharia Elétrica, na Área de Telecomunicações e Telemática.
Orientador: Prof. Dr. Hugo Enrique Hernández Figueroa
ESTE EXEMPLAR CORRESPONDE À VERSÃO FINAL DA DISSERTAÇÃO DEFENDIDA PELO ALUNO OSCAR ENRIQUE LLERENA CASTRO, E ORIENTADA PELO PROF. DR. HUGO ENRIQUE HERNÁNDEZ FIGUEROA
CAMPINAS 2014
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ABSTRACT
Bandpass filters made with ceramic coaxial resonators are a kind of filter technology that provides high performance features like high selectivity, high power handling, excellent rejection, low passband insertion loss, etc. and given that the resonators are made of with materials of high dielectric constant r it considerably reduces their size, consequently, all the electrical characteristics mentioned before are “compacted” in a small shielded structure which make these filters suitable for applications where small sized devices are a necessity.
The most difficult task when projecting this kind of filters is to find the correct dimensions of the structure, either the length of the resonators or the dimensions of the coupling capacitors structure. This thesis presents a study on this type of filter and propose an analytical-empirical procedure with design formulas to facilitate construction of the resonators array and the coupling capacitors structure. This procedure can be extrapolated to bandpass filters with a greater number of resonators.
The design formulas allow an easy transition from the circuital model to the electromagnetic model of the filter. This is because they easily allow to compute the length of the resonators and the value of the capacitances required for a correctly resonators coupling. The comparison between the simulations in the circuital and the electromagnetic environment show that the proposed design formulas are a good first approximation for this filter design.
RESUMO
Os filtros passa-faixa feitos com ressonadores coaxiais cerâmicos são um tipo de tecnologia de filtro que fornece características de alta performance como a alta seletividade, ótimo desempenho em aplicações de alta potência, excelente rejeição, baixa perda de inserção na banda passante, etc. , e dado que os ressonadores são feitos de materiais de alta constante dielétrica consideravelmente, consequentemente, todas as características elétricas mencionadas anteriormente são compactadas em uma estrutura protegida o qual faz que este tipo de filtros sejam adequados para aplicações onde precisam-se de dispositivos de tamanho pequeno.
A parte mais difícil no momento de projetar este tipo de filtros é encontrar as dimensões certas da estrutura, sejam as alturas dos ressonadores ou as dimensões da estrutura dos capacitores de acoplamento. Esta tese presenta um estudo feito neste tipo de filtro e propõe um procedimento analítico- empírico com fórmulas para projeção para facilitar a construção do arranjo de ressonadores e a estrutura dos capacitores de acoplamento. Este procedimento pode ser extrapolado para filtros passa-faixa com um maior número de ressonadores.
As fórmulas de projeção permitem uma transição fácil do modelo circuital ao modelo eletromagnético do filtro. Isto deve-se a que com as fórmulas é fácil calcular os comprimentos dos ressonadores e os valores das capacitâncias requeridas para acoplar corretamente os ressonadores. A comparação entre as simulações no modelo circuital e o entorno eletromagnético mostram que as formulas de projeção propostas são uma ótima primeira aproximação para a projeção deste filtro.
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SUMMARY
1 INTRODUCTION...... 1
1.1 MOTIVATION...... 1
1.2 BRIEF SUMMARY OF CHAPTERS...... 3
2 CERAMIC COAXIAL RESONATOR...... 4
2.1 PARALLEL RESONANT CIRCUIT...... 4
2.2 SHORT-CIRCUITED QUARTER-WAVELENGTH LINE...... 7
2.3 CERAMIC COAXIAL RESONATOR (CCR)...... 8
2.3.1 PROFILE OF A CERAMIC COAXIAL RESONATORS...... 11
3 BANDPASS FILTER DESIGN...... 14
3.1 THE TCHEBYSHEV FILTER RESPONSE...... 14
3.2 TCHEBYSHEV LOWPASS FILTER...... 16
3.3 TRANSFORMATION FROM THE LOWPASS FILTER TO THE BASIC BANDPASS FILTER...... 19
3.4 TRANSFORMATION OF A BASIC BANDPASS FILTER INTO A COUPLED- RESONATOR BANDPASS FILTER...... 21
4 BANDPASS FILTER OF CERAMIC COAXIAL RESONATORS WITH COUPLING CAPACITORS...... 26
4.1 INTRODUCTION OF THE CCR EQUIVALENT CIRCUIT...... 26
4.2 ANALYTICAL PART OF THE PROCEDURE TO SOLVE A BANDPASS FILTER WITH FOUR CCRs WITH CAPACITIVE COUPLING...... 28
4.2.1 SYMMETRY CHARACTERISTICS OF A BANDPASS FILTER OF FOUR RESONATORS CAPACITVELY COUPLED...... 29
4.2.2 THE CONCEPT OF NODAL RESONANCE FREQUENCY...... 30
4.3 EMPIRICAL PART OF THE PROCEDURE TO SOLVE A BANDPASS FILTER WITH FOUR CCRs WITH CAPACITIVE COUPLING...... 35
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4.3.1 EMPIRICAL EXPRESSIONS FOR THE RESONANCE FREQUENCIES OF THE CCRs...... 35
4.3.2 ITERATIVE PROCESS TO EXPRESS k1 AND k 2 IN FUNCTION OF THE FILTER FREQUENCY AND BANDWIDTH...... 36
4.4 DESIGN FORMULAS FOR A BANDPASS FILTER WITH FOUR CCRs WITH CAPACITIVE COUPLING...... 42
4.5 VALIDATION OF THE DESIGN FORMULAS USING CIRCUITAL SIMULATION...... 45
5 SIMULATIONS AND RESULTS...... 51
5.1 ELECTROMAGNETIC SIMULATION OF A CCR...... 51
5.2 VALIDATION OF DESIGN FORMULAS ON A 3D-STRUCTURE FOR ELECTROMAGNETIC SIMULATION...... 56
5.3 STRUCTURE FOR THE COUPLING CAPACITORS...... 63
5.4 EXAMPLE OF DESIGN OF A BANDPASS FILTER WITH COUPLING CAPACITORS...... 69
6 CONCLUSIONS AND FUTURE WORK...... 77
7 BIBLIOGRAPHY ...... 80
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This work is dedicated to Delia and Emma.
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SPECIAL THANKS TO:
- Prof. Dr. Hugo Enrique Hernández Figueroa for the orientation on the development of this document. The guidelines provided were always very helpful and accurate.
- Prof. Dr. Lucas Heitzmann Gabrielli for the helpful suggestions and corrections concerning with electromagnetic simulation issues.
- Dr. Luciano Prado de Oliveira for his very useful suggestions regarding electromagnetic simulation and ideas concerning the proper presentation of concepts of this study.
- M. Sc. Tadeu Pires Pasetto and M. Sc. Edson César dos Reis, both of them from BRADAR, who helped with technical suggestions regarding the reverse-engineering approach to be used.
- Prof. Dr. Leonardo Lorenzo Bravo Roger for very useful suggestions with theoretical concepts.
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LIST OF FIGURES
Figure 1.1: A bandpass filter made with coaxial resonators and capacitive coupling...... 2 Figure 2.1: A typical RLC parallel circuit...... 5 Figure 2.2: A short circuited length of lossy transmission line...... 7 Figure 2.3: Equivalence between RLC parallel circuit and short circuited quarter wavelength line...... 8 Figure 2.4: A short circuited quarter wavelength coaxial line...... 9 Figure 2.5: a) Typical ceramic coaxial resonator structure. b) Equivalent RLC circuit...... 10 Figure 2.6: Q-factor curves versus frequency for different resonator cross-sections from manufacturer Temex...... 12 Figure 3.1: Response of a Tchebyshev lowpass filter of order n = 7 and ε = 0.6446 dB...... 15 Figure 3.2: a) Lowpass filter circuit. b) Dual circuit...... 16 Figure 3.3: a) Tchebyshev lowpass filter. b) Bandpass filter with resonators in series and parallel configuration resulting from the lowpass-to-bandpass transformation...... 19 Figure 3.4: Arbitrary bandpass filter of n resonators with capacitive coupling...... 21 Figure 3.5: Coupled-resonator bandpass filter of fifth order and its frequency response corresponding to the parameter values in Table 3.2...... 24 Figure 4.1: Arbitrary bandpass filter of n-order composed by CCRs as resonators...... 28 Figure 4.2: Bandpass filter of four coupled CCRs...... 29 Figure 4.3: Coupling elements that contribute to the resonance frequency of j-node...... 31 Figure 4.4: Frequency response of the bandpass filter from Figure 4.2 whose lumped elements were computed with k1 = 2.450 and k2 = 0.850...... 38 Figure 4.5: Frequency response of the bandpass filter from Figure 4.2 whose lumped elements were computed with k1 = 3.550 and k2 = 1.055...... 38 Figure 4.6: Frequency response of the bandpass filter from Figure 4.2 whose lumped elements were computed with = 3.061 and = 0.970...... 38 Figure 4.7: Structure of a bandpass filter of four CCRs with lumped elements as coupling capacitors designed on CST Microwave Studio...... 42
Figure 4.8: Frequency response of the circuit from Figure 4.2 with lumped elements computed for f filter
= 800 MHz and BW = 5 MHz...... 45 Figure 4.9: Frequency response of the circuit from Figure 4.2 with lumped elements computed for
= 800 MHz and BW = 20 MHz...... 46
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Figure 4.10: Frequency response of the circuit from Figure 4.2 with lumped elements computed for f filter
= 800 MHz and BW = 40 MHz...... 46 Figure 4.11: Frequency response of the circuit from Figure 4.2 with lumped elements computed for
= 1400 MHz and BW = 5 MHz...... 47 Figure 4.12: Frequency response of the circuit from Figure 4.2 with lumped elements computed for
= 1400 MHz and BW = 20 MHz...... 47 Figure 4.13: Frequency response of the circuit from Figure 4.2 with lumped elements computed for
= 1400 MHz and BW = 40 MHz...... 48 Figure 4.14: Frequency response of the circuit from Figure 4.2 with lumped elements computed for
= 2000 MHz and BW = 5 MHz...... 48
Figure 4.15: Frequency response of the circuit from Figure 4.2 with lumped elements computed for
= 2000 MHz and BW = 20 MHz...... 49
Figure 4.16: Frequency response of the circuit from Figure 4.2 with lumped elements computed for
= 2000 MHz and BW = 40 MHz...... 49 Figure 5.1: Structure of a ceramic coaxial resonator on CST Microwave Studio...... 52 Figure 5.2: Setup of simulation for a coaxial resonator on CST Microwave Studio...... 52 Figure 5.3: Plot of phase curves of a single resonator...... 53 Figure 5.4: Simulation setup for a CCR with incorporated solder tab...... 54 Figure 5.5: Plot of phase curves of a single resonator with solder tab...... 55 Figure 5.6: Equivalent 3D-structure for the bandpass filter circuit from Figure 4.2...... 56 Figure 5.7: Comparison between the frequency responses of circuital and electromagnetic simulations for a bandpass filter designed with formulas from section 4.4 with specifications f filter = 800 MHz and
BW = 5 MHz...... 57 Figure 5.8: Comparison between the frequency responses of circuital and electromagnetic simulations for a bandpass filter designed with formulas from section 4.4 with specifications = 800 MHz and
= 20 MHz...... 58 Figure 5.9: Comparison between the frequency responses of circuital and electromagnetic simulations for a bandpass filter designed with formulas from section 4.4 with specifications = 800 MHz and
= 40 MHz...... 58
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Figure 5.10: Comparison between the frequency responses of circuital and electromagnetic simulations for a bandpass filter designed with formulas from section 4.4 with specifications f filter = 1400 MHz and
BW = 5 MHz...... 59 Figure 5.11: Comparison between the frequency responses of circuital and electromagnetic simulations for a bandpass filter designed with formulas from section 4.4 with specifications = 1400 MHz and
= 20 MHz...... 59 Figure 5.12: Comparison between the frequency responses of circuital and electromagnetic simulations for a bandpass filter designed with formulas from section 4.4 with specifications = 1400 MHz and
= 40 MHz...... 60 Figure 5.13: Comparison between the frequency responses of circuital and electromagnetic simulations for a bandpass filter designed with formulas from section 4.4 with specifications = 2000 MHz and
= 5 MHz...... 60 Figure 5.14: Comparison between the frequency responses of circuital and electromagnetic simulations for a bandpass filter designed with formulas from section 4.4 with specifications = 2000 MHz and
= 20 MHz...... 61 Figure 5.15: Comparison between the frequency responses of circuital and electromagnetic simulations for a bandpass filter designed with formulas from section 4.4 with specifications = 2000 MHz and
= 40 MHz...... 61 Figure 5.16: Parts of the coupling capacitors structure. a) Solder tabs. b) Bottom metallic plates. c) Dielectric substrate between bottom and top metallic plates. d) Top metallic plates and SMA connectors...... 63 Figure 5.17: Parallel plates configuration and the gap capacitors and parallel plate capacitors resulting...... 64 Figure 5.18: Equivalent circuit of the coupling capacitors structure...... 64 Figure 5.19 E-field distribution plot over the dielectric substrate for phases: ...... 65 Figure 5.20: Simplified equivalent circuit for the capacitors structure from Figure 5.17...... 66 Figure 5.21: Parameters of the capacitors structure to determine the overlapped areas...... 67 Figure 5.22: Iteration 1 for filter tunning using the structure and dimensional parameters from Figure
5.21 for a bandpass filter at f filter = 1500 MHz and = 5 MHz. The initial dimensional parameters are shown at Table 5.2 and Table 5.3 ...... 71
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Figure 5.23: Iteration 2, the dimensions of the parallel plate capacitors were reduced in order to diminish the coupling capacitances...... 71 Figure 5.24: Iteration 3, the dimensions of all parallel plate capacitors are reduced in such way that the return loss shows improvements...... 72 Figure 5.25: Iteration 4, the dimensions of all parallel plate capacitors are reduced for a second time and the return loss is now under -20 dB level and the frequency response is showing a bandwidth of 27 MHZ...... 72 Figure 5.26: Iteration 5, the lengths of the resonators were decreased in order to shift the center frequency up to 1500 MHz. The bandwidth is almost 30 MHz...... 73 Figure 5.27: Iteration 6, a final decrement is done on the dimensions of the parallel plate capacitors. The bandpass filter is now centered at the design filter frequency with a bandwidth that differs only 0.4 MHz from the specification...... 73
Figure 5.28: Filter response with f filter = 1300 MHz and BW = 20 MHz...... 74
Figure 5.29: Filter response with = 1300 MHz and BW = 20 MHz (Zoom)...... 74
Figure 5.30: Filter response with = 800 MHz and BW = 40 MHz...... 75
Figure 5.31: Filter response with = 800 MHz and BW = 40 MHz (Zoom)...... 75
Figure 5.32: Filter response with = 1030 MHz and BW = 10 MHz...... 76
Figure 5.33: Filter response with = 1030 MHz and BW = 10 MHz (Zoom)...... 76
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LIST OF TABLES
Table 2.1: Common values for W and d commercially available...... 13 Table 2.2: Best-performance frequency range (MHz) for Temex λ/4-length resonators for a given dielectric constant of the material and a given cross-section...... 13 Table 3.1: Tchebyshev lowpass filter parameters from n = 1 to n = 5 for = 0.01 dB...... 19 Table 3.2: Parameter values for fifth-order bandpass filter with filter frequency and bandwidth equal to 1300 MHz and 40 MHz, respectively...... 24
Table 4.1: Values for k1 for different f filter and BW...... 40
Table 4.2: Values for k2 for different and BW...... 41
Table 4.3: Results of the frequency responses from Figure 4.8 through Figure 4.16...... 50 Table 5.1: Comparison between design frequencies and frequencies obtained by simulation...... 54 Table 5.2: Comparison between design frequencies and frequencies obtained by simulation...... 55 Table 5.3: Center frequency obtained from circuital (Figure 4.2) and electromagnetic (Figure 5.6) simulation using design formulas from section 4.4...... 62 Table 5.4: CCR parameters computed for the example of design...... 70 Table 5.5: Parameters of the coupling capacitors structure...... 70
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LIST OF ACRONYMS AND ABBREVIATIONS
BW: Bandwidth CCR: Ceramic Coaxial Resonator PEC: Perfect Electrical Conductor RF: Radio frequency UHF: Ultra High Frequency NRF: Nodal Resonance Frequency SMD: Surface mount design RLC: Resistor - inductor - capacitor
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
1. INTRODUCTION.
The electrical characteristics of a ceramic coaxial resonator (CCR) made it suitable for its use on bandpass filters, mostly for applications on microwave frequencies where power handling and narrow-band requirements are difficult issues to achieve. Besides, miniaturization is another important feature that is attributable to this kind of filters given its compact construction and the small sizes that can be achievable due to the use of high dielectric constant materials. This document provides a study on bandpass filters made with four CCRs and an analytical-empirical procedure for its structure design.
1.1. MOTIVATION.
As a request of a Brazilian company dedicated to the fabrication and design of applications on the radiofrequency and microwave spectrum range, it was required to design a bandpass filter based on CCRs for specific frequencies and bandwidths. This filter technology is already available by manufactures like Lorch Microwave or Mini-circuits but the high cost justifies the study of the “know how” on the design process. The filter to be reversed-engineering is shown in Figure 1.1. The structure consists on a row configuration of four CCRs with capacitive coupling in the form of a suspended dielectric substrate with parallel plate capacitors. The structure has a metallic enclosure to protect the ceramic resonators and the capacitors structure from external perturbations that could cause variations on the filter response, also it works as electrical ground. The slots on the top of the structure prevent short-circuit between SMD connectors and the resonators. The process of design of this kind of filter starts addressing the equivalent circuit of the bandpass filter in a circuital model. This circuital model is a circuit of four coupled resonators in the form of lumped elements. Then, using the equivalent circuit of a CCR, it allows to build the 3D-structure of the bandpass filter. This transition from the circuital model into the electromagnetic model is to perform the electromagnetic simulations of the bandpass filter structure to obtain its frequency response. The main problem of this reverse-engineering is the large number of unknowns related to the resonators structure and its composition material. In the case of the resonators
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter structure, the main unknowns are the lengths and the cross-section dimensions of the resonators and the theoretical criteria to calculate them. Regarding the composition material, the main unknowns are the relative dielectric constant and loss tangent parameters to characterize such material for simulation.
Figure 1.1: A bandpass filter made with coaxial resonators and capacitive coupling.
This thesis attempts to find a procedure to design bandpass filters made of resonators with capacitive coupling using equivalents circuital and electromagnetic models. This procedure at some point is analytical, because it uses bandpass-filter symmetry properties and other concepts to reduce the analytical equations to solve the bandpass filter parameters, and then it turns into empirical because it uses data from trial-and-error iterations to develop empirical design formulas to facilitate the calculation of the dimensions of the resonators 3D- structure.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
1.2. BRIEF SUMMARY OF CHAPTERS.
Chapter 2 presents the concepts related to the electrical characteristics of a CCR such they are the characteristic impedance and resonance frequency in function of the resonator dimensions. An equivalent lumped-element circuit for the CCRs is derived. This is an important step when incorporating them into the bandpass filter design. Chapter 3 addresses the concepts for filter design. It starts describing the Tchebyshev function to introduce the lowpass prototype filter and its parameters. Then, it explains the transformation from a lowpass into a bandpass filter using concepts of frequency mapping and impedance scaling. Finally, it presents design formulas to build arbitrary bandpass filter based on coupled resonators and capacitive coupling in the form of lumped element circuit which is the equivalent model of the bandpass filter from Figure 1.1. Chapter 4 introduces the equivalent circuit of a CCR into the circuital model of the bandpass filter based on coupled resonators from Chapter 3. An analytical procedure is proposed to allow the parametrization of the coupling capacitors in function of the elements of the CCRs equivalent circuit. Then, the empirical part of the procedure achieves to express the resonance frequencies of the CCRs as a function of the filter frequency and bandwidth of the bandpass filter. The obtained design formulas aims the easy transition from the circuital model into the 3D electromagnetic model. Chapter 5 covers the electromagnetic simulations of a single CCR to verify the relationship between its length and its resonance frequency. Also, this chapter tests the accuracy of the empirical design formulas obtained in Chapter 4 for the resonators structure of the bandpass filter. Electromagnetic simulations are performed over the 3D-structure built based on the empirical design formulas to compare the obtained frequency response with the frequency response from the simulations of circuital model from Chapter 4. Chapter 5 also proposes a procedure to design the coupling-capacitors structure. Finally, it is shown an example of design in order to explain how to project the entire bandpass filter made with four resonators and coupling capacitors, first on a circuital model and then on a 3D-structure. Chapter 6 describes the conclusions taken from this study and the proposed future work.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
2. CERAMIC COAXIAL RESONATOR.
Microwave resonators are used in many RF applications, more commonly in filters, oscillators, and tuned amplifiers. A type of microwave resonator is the ceramic coaxial resonator (CCR) which is the main component of the bandpass filter shown in Figure 1.1. In order to design those type of bandpass filters, it is necessary to understand the electrical characteristics of the CCR and its equivalent circuit. A CCR is electrically similar to quarter-wavelength transmission line that is short- circuited on one of its ends. A short-circuited quarter-wavelength transmission line, or also known as a quarter-wavelength resonator, is a representation of a resonator using a model of distributed parameters (RHEA, 2010). The current chapter focuses on finding the equivalence between the distributed-parameters model and the lumped-elements model of the CCR. A lumped-elements model of a CCR consists on a RLC (resistor-inductor-capacitor) parallel circuit (RHEA, 2010). The importance of the lumped-element representation (RLC parallel circuit) is because it will be employed in the design procedure of the bandpass filters with coupled resonators in Chapter 3.
2.1. PARALLEL RESONANT CIRCUIT.
At frequencies close to resonance, a microwave resonator can be modeled as the RLC parallel circuit shown in Figure 2.1. The elements of this circuit are a capacitor C, an inductor
L, and a resistor R in a parallel configuration with an input impedance Z in defined by the following equation (POZAR, 2011):
1 1 1 (2.1) Zin jC , R jL where j is the imaginary unit and is the angular frequency of the input signal.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
Figure 2.1: A typical RLC parallel circuit.
The complex power Pin delivered to the resonator is (POZAR, 2011):
1 2 1 j Pin V jC, (2.2) 2 R L
and the power dissipated Ploss by the resistor R is (POZAR, 2011):
2 1 V P , (2.3) loss 2 R where V is the input voltage and I is the input current of the circuit. To define the quality factor Q of a RLC parallel circuit, it is important to first define the average electrical energy We stored in the capacitor C which is (POZAR, 2011):
1 2 W V C , (2.4) e 4
and the average magnetic energy Wm stored in the inductor L which is (POZAR, 2011):
1 2 1 W V . (2.5) m 4 2 L
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
The resonance phenomenon occurs when We and Wm are equal, then equating expressions (2.4) and (2.5) derives the expression for the resonance frequency 0 as follows:
1 . (2.6) 0 LC
At frequencies close to resonance 0 , the input impedance can be simplified as follows (POZAR, 2011):
1 Z in . (2.7) 1 R 2 j C
This equation will be very useful when relating the RLC parameters with the quarter wavelength resonator parameters on next section. The quality factor Q is defined as, the ratio between the total average energy stored in the reactive elements of the resonator (inductor and capacitor) and the dissipated power
Ploss by the resistor, multiplied by the resonance frequency (POZAR, 2011):
We Wm Q 0 . (2.8) Ploss
The resonator losses are mainly due to: - Conductive losses on the metallic coating. - Losses on the dielectric material. - Losses by spontaneous radiation on the opened side of the resonator. All these losses are represented by the resistor R on the RLC parallel circuit.
Equations 2.8 deduces that lower losses on the resonator implicates higher Q, which is the desired condition for resonators when they are used on filters. This characteristic will be useful on Chapter 3 when introducing the CCR equivalent circuit into the bandpass filter
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter design given that since the currently CCRs are designed to have the lowest losses, it allows to remove the resistor R from the equivalent lumped-element circuit.
Figure 2.2: A short circuited length of lossy transmission line.
2.2. SHORT-CIRCUITED QUARTER-WAVELENGTH LINE.
A short-circuited lossy transmission line of arbitrary length is shown in Figure 2.2 where Z in is the input impedance that is related to the characteristic impedance Z 0 of the line, to the propagation constant and to the attenuation constant by the following equation (POZAR, 2011):
Zin Z 0 tanh j . (2.9)
The input impedance of a short-circuited quarter-wavelength line at frequencies close to resonance 0 , can be approximated to (POZAR, 2011):
1 Z in , (2.10) Z 0 j / 2Z 00
When 4 (electrical length θ = 90⁰), the transmission line has a type of resonance similar to a RLC parallel circuit (POZAR, 2011), being the wavelength for an arbitrary resonance frequency.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
Figure 2.3: Equivalence between RLC parallel circuit and short circuited quarter wavelength line.
Equations (2.7) and (2.10) are similar expressions of the input impedance on both models (distributed-parameters and lumped-elements models) and by equating them and then using equation (2.6), it delivers the parametric equivalence between the short-circuited quarter-wavelength resonator and the RLC parallel circuit in the following equations:
4Q Z R 0 0 , (2.11) 1 L 2 , (2.12) 0 C C . (2.13) 40Z0
2.3. CERAMIC COAXIAL RESONATOR (CCR).
One form of a short-circuited quarter-wavelength transmission line is a coaxial line of the same length that is short-circuited on one side and open-circuited on the other. Figure 2.4 shows the described configuration. As an open load produces total reflection, a pure standing wave is formed inside the coaxial line, therefore it resonates at a certain frequency
0 that depends on the resonator length . The transverse section of the coaxial line is determined by the outer diameter W and the inner diameter d.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
Figure 2.4: A short-circuited quarter-wavelength coaxial line.
The relationship between the length of the resonator and the resonant wavelength is given by equation (2.14). It indicates that the length of the resonator is an odd number times of the quarter resonant wavelength.
2m 1 m 0,1,2,.... (2.14) 4
A ceramic coaxial resonator is a coaxial structure filled with a material of relative dielectric constant r . The CCRs available on current microwave devices market are presented in two configurations, one is a λ/4 CCR and the other is a λ/2 CCR. The current section focuses on λ/4 CCRs given that, for obvious reason, is smaller than the other and also because it is the configuration used in the bandpass filter shown in Figure 1.1. Figure 2.5 shows the typical structure for a λ/4 CCR and its RLC parallel circuit equivalence. The majority of the CCRs commercially available have a transverse section that on the outer side is squared with rounded corners and on the inner side is cylindrical. The fact that the outer side is not entirely cylindrical introduces a geometric factor g on the typical expression for the characteristic impedance Z 0 of a coaxial line resulting in (TUSONIX, 2013), (TRANS-TECH, 2013a):
60 W Z0 ln g , (2.15) r d
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter where g = 1.079. Parameters W and d correspond to the outer side and the inner diameter respectively.
Figure 2.5: a) Typical ceramic coaxial resonator structure. b) Equivalent RLC circuit.
The CCR length res at its resonance frequency f res for a quarter-wavelength configuration is (CHEN et al., 2004):
res g 4 , (2.16)
where g is the wavelength of the guided TEM mode excited in the resonator defined by following equation (TRANS-TECH, 2013a):
c g , (2.17) f res r
where c is the speed of light equal to 3e8 m/s or 3e11 mm/s. Substituting equation (2.16) into (2.17), the length of a λ/4 CCR is defined as follows:
c res , (2.18) 4 f res r
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
The λ/4 CCR can be modeled as a RLC parallel circuit using equations (2.11), (2.12),
(2.13), (2.18) and 0 2 f 0 , where f 0 f res , to obtain the expressions from (2.19) to (2.21) for the RLC equivalent circuit (TRANS-TECH, 2013b) shown in Figure 2.5.
8Z L 0 r (2.19) 2 c
C r (2.20) 2c Z 0 4Q Z R 0 0 (2.21)
From equation (2.18), the CCR length diminishes squarely respect to r , then the resonator is significantly reduced when using ceramic materials of a high . However, there is a limitation over the resonator length, it should not be lower than the outer side W of the transverse section because it can lead into the excitation of not-desired modes and also when lowering the resonator length, the Q-factor gets reduced which deteriorates the performance as a resonator because radiation of electromagnetic field increments on the opened side (VENDELIN; PAVIO; ROHDE, 2005).
2.3.1. PROFILE OF A CERAMIC COAXIAL RESONATORS.
The term profile for a CCR defines the dimensions of its transverse section, those are the parameters W and d corresponding to the outer square side and the inner diameter respectively. The CCRs that are commercially available have standardized profiles with some variations between manufacturers. Table 2.1 shows common values for W and d for some CCR profiles from some vendors (Skyworks, Tusonix, Temex, Trans-Tech, etc.). Table 2.2 presents, as a reference information, the frequency range for best performance from manufacturer Temex for CCRs cross-section from Table 2.1 for three different dielectric constants very common on market. It is important to quote that Table 2.2 refers to λ/4 CCRs because there also exists a λ/2 CCR version available on market with different frequency ranges (TEMEX, 2004).
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
Figure 2.6: Q-factor curves versus frequency for different resonator cross-sections from manufacturer Temex.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
Table 2.1: Common values for W and d commercially available.
W 2.0 3.0 4.0 6.0 12.0 (mm) ± 0.2 ± 0.2 ± 0.2 ± 0.2 ± 0.2
d 0.65 0.95 1.5 2.0 3.55 (mm) ± 0.1 ± 0.1 ± 0.1 ± 0.1 ± 0.1
Table 2.2: Best-performance frequency range (MHz) for Temex λ/4-length resonators for a given dielectric constant of the material and a given cross-section.
W/d (mm) 2.0/0.65 3.0/0.95 4.0/1.50 6.0/2.0 12/3.55
Εr = 21 2000 – 4000 1500 – 4000 1000 – 4000 600 – 2500 600 – 1250
Er = 37 1500 – 3000 1500 – 3000 800 – 3000 500 - 2000 450 – 1000
Er = 90 900 – 2000 650 – 2000 450 – 2000 450 – 1000 300 – 650
Manufacturers of CCRs also provide charts that describe the resonators Q-factor versus frequency for different dielectric constants and cross-sections (profiles) as it is shown on Figure 2.6.
13
Study of a Compact Microwave Ceramic Coaxial Resonator Filter
3. BANDPASS FILTER DESIGN.
The current chapter addresses the concepts for bandpass filter design. The sequence of the sections starts with a brief revision of basic concepts as they are the Tchebyshev response and the lowpass filter that synthesizes it. Then, it is explained the transformation from the lowpass filter into the basic bandpass filter using the concepts of frequency mapping and impedance scaling. And finally it is presented the design procedure to transform the basic bandpass filter into a coupled-resonator bandpass filter to be employed in Chapter 4 to represent the equivalent lumped-element circuit of the bandpass filter from Figure 1.1.
3.1. THE TCHEBYSHEV FILTER RESPONSE.
In 1930, Stephen Butterworth published his work "On the Theory of Filter Amplifiers" describing the theory to design a lowpass filter with flat frequency response over a determined passband. His contribution established the basis for the subsequent investigation on filters. Years later, Russian mathematician Pafnuty Tchebyshev developed the nth-order Tchebyshev polynomials Cn of the first kind which mathematical characteristics were applied to synthesize networks to develop lowpass filters with equal- ripple response in the passband, also known as the Tchebyshev lowpass filters. When comparing both, the Tchebyshev lowpass filters have a steeper roll-off response but also ripple in the passband while on the other hand the Butterworth lowpass filters have a total flat response in the passband but smoother roll-off response on the stopband. The steeper roll-off is the desired characteristic for applications where faster attenuation is needed out of the passband. The transfer function of the Tchebyshev filter is defined by the following equation (CHEN, 1986):
2 H 0 H j 2 2 , (3.1) 1 Cn c
14
Study of a Compact Microwave Ceramic Coaxial Resonator Filter
|H(j)|2
H 1 0
H /(1+2) 0.8 0
0.6
0.4
0.2
0 0 0.5 1 1.5 2 Figure 3.1: Response of a Tchebyshev lowpass filter of order n = 7 and ε = 0.6446 dB.
2 where H0 is an arbitrary constant in such way that H 0 1. The parameter n corresponds to the order of the Tchebyshev polynomial Cn and also it is the filter order. The parameter
represents the frequency domain and c is the cut-off frequency. The Tchebyshev polynomials Cn of nth-order are defined by the following recurrence formula (CHEN, 1986):
Cn1 2Cn Cn1 , (3.2)
where C0 1 and C1 . The ripple parameter determines the minimum and maximum ripple over the passband and is usually specified in decibels (dB). Since it fits the condition 2 1, the maximum value is one or 0 dB, this quantity is related to the ripple parameter through the following equation (CHEN, 1986):
dB 10log 2 1. (3.3)
Figure 3.1 shows a lowpass filter of order n = 7 with the cutoff frequency c 1 and a ripple factor = 0.6446 dB. Here, the ripple factor was exaggerated for purposes of
15
Study of a Compact Microwave Ceramic Coaxial Resonator Filter visualization in the graph but generally it is desired 0.1dB in order to have a flatter response as possible on the passband.
Figure 3.2: a) Lowpass filter circuit. b) Dual circuit.
3.2. TCHEBYSHEV LOWPASS FILTER.
The preceding section explained the Tchebyshev function and its relationship with the lowpass filter response. The current section introduces a topology that synthesizes the equation (3.1) into a network of lumped elements known as the Tchebyshev lowpass filter that for future quoting will be denoted as the lowpass filter. The theoretical background on the statement of this topology corresponds to the contribution of engineer Wilhelm Cauer (CAUER, 1958) who developed the theory for the realization of a topology with lumped elements that reproduces any arbitrary Tchebyshev response. Parting from the Tchebyshev transfer function, Cauer deduced the input impedance function (CHEN, 1976) of an arbitrary network whose topology is a ladder network of inductors (L) and capacitors (C) resulting in the Tchebyshev lowpass filter. Reference (VALKENBURG, 1974) and (BELEVITCH, 1952) explain more deeply the procedure employed by Cauer. Figure 3.2 shows the two canonical Cauer’s topologies for the lowpass filter, one being the dual of the other. The network in Figure 3.2a starts the sequence of elements with
16
Study of a Compact Microwave Ceramic Coaxial Resonator Filter
' ' ' a parallel capacitor C1 after the source load R0 followed by a series inductor L2 and so on
' ' until a last capacitor Cn which is parallel to the end load Rn1 whereas in Figure 3.2b the sequence of elements is totally inverted. The parallel-capacitor stages are replaced by series- inductor stages and vice versa. The network elements for both topologies, as described on Figure 3.2, are defined as follows:
' R0 g0 , (3.4)
' ' Li Ci gi , (3.5)
' Rn1 gn1 . (3.6)
The values for the Tchebyshev parameters g i are computed as follows (MATTHAE; YOUNG; JONES, 1964):
g0 1, (3.7) 2a g 1 , (3.8) 1
4ak1 ak gk ; k 2,3,...,n. (3.9) bk1 gk1
gn1 1 for n odd, (3.10) 2 gn1 coth / 4 for n even, where n is the order of the lowpass filter. From equation (3.10), it can be noticed that depending on the order of the filter, the termination loads can be different. The coefficients ak , bk , and are calculated from the following equations (MATTHAE; YOUNG; JONES, 1964):
17
Study of a Compact Microwave Ceramic Coaxial Resonator Filter
2k 1 ak sin k 1,2,...,n , (3.11) 2n
2 2 k bk sin k 1,2,...,n , (3.12) n sinh , (3.13) 2n
lncoth , (3.14) 17.37 where is the passband ripple parameter in dB from equation (3.3).
Table 3.1 shows the values for the parameters g i based on formulas from (3.7) to (3.14) for n = 1 to n = 5. The lowpass filter elements are all normalized to make the source
' impedance R0 g0 1 and the cutoff frequency c 1. In order to scale into other impedance levels and frequencies, it must be applied the following transformations (MATTHAE; YOUNG; JONES, 1964):
R R 0 R, For resistors ' (3.15) R0
R L 0 L , for inductors ' (3.16) R0
' R0 for capacitors C C , (3.17) R0
The primed quantities stand for the normalized parameters and the unprimed for the corresponding scaled-circuit parameters. Therefore, and are arbitrary frequencies that correspond to the normalized lowpass filter and to the scaled lowpass filter, respectively.
18
Study of a Compact Microwave Ceramic Coaxial Resonator Filter
Table 3.1: Tchebyshev lowpass filter parameters from n = 1 to n = 5 for = 0.01 dB.
n g1 g 2 g 3 g 4 g5 g 6 1 0.0960 1.0000 - - - - 2 0.4488 0.4077 1.1007 - - - 3 0.6291 0.9702 0.6291 1.0000 - - 4 0.7128 1.2003 1.3212 0.6476 1.1007 - 5 0.7563 1.3049 1.5773 1.3049 0.7563 1.0000
Figure 3.3: a) Tchebyshev lowpass filter. b) Bandpass filter with resonators in series and parallel configuration resulting from the lowpass-to-bandpass transformation.
3.3. TRANSFORMATION FROM THE LOWPASS FILTER TO THE BASIC BANDPASS FILTER.
The transformation from lowpass to basic bandpass filter is possible through a frequency mapping. Reference (CHEN, 1986) explains the concepts of frequency transformation to map the desired passband and stopband -of the desired bandpass filter- into the passband and stopband of the lowpass filter. The frequency mapping used for the lowpass-to-bandpass transformation is defined on equations (3.18) to (3.20) and it is used for narrow bandpass filters where the bandwidth is less than 5% of the filter center frequency (MATTHAE; YOUNG; JONES, 1964), more detailed theory can be found on references (CHEN, 1986), (CHEN, 1976), and (COLLIN, 2001).
19
Study of a Compact Microwave Ceramic Coaxial Resonator Filter
c 0 , (3.18) w 0
0 1 2 , (3.19) w 2 1 . (3.20) 0 Where: = bandpass filter frequency resulting from the mapping, = lowpass filter frequency to be mapped,
c = cutoff frequency of the lowpass filter,
0 = central frequency of the lowpass filter,
1 = lower frequency of the passband of the bandpass filter,
2 = upper frequency of the passband of the bandpass filter, w = relative bandwidth of the bandpass filter.
Figure 3.3b shows the ladder network for a bandpass filter obtained from the lowpass filter from Figure 3.3a. The frequency mapping results in that each inductor is replaced by a series inductor-capacitor configuration and each capacitor by the parallel configuration, respectively. Given that the elements of the lowpass filter are normalized, it is necessary an impedance scaling that is referred in equations (3.21) and (3.22) to obtain the respective inductors Li and capacitors Ci for the bandpass filter (MATTHAE; YOUNG; JONES, 1964).
gi Z0 w Li ; Ci ; i 2,4,6,... (3.21) w g Z 0 i 0 0 ,
w Z0 gi Li ; Ci ; i 1,3,5,... (3.22) g w Z i 0 0 0 ,
20
Study of a Compact Microwave Ceramic Coaxial Resonator Filter
where the parameter g i corresponds to the i-th lowpass filter element and parameters 0 and w are defined on equations (3.21) and (3.22) respectively.
3.4. TRANSFORMATION OF A BASIC BANDPASS FILTER INTO A COUPLED-RESONATOR BANDPASS FILTER.
Figure 3.4: Arbitrary bandpass filter of n resonators with capacitive coupling.
The basic bandpass filter studied on section 3.3 is a network that contains both types of resonators (series and parallel configurations). In the current section, it will be studied the procedure to transform an arbitrary basic bandpass filter of n resonators into a bandpass filter that contains only one type of resonator (parallel configuration) with capacitive coupling (MATTHAE; YOUNG; JONES, 1964) as it shown on Figure 3.4. It is important to quote that inductive coupling is possible but is not addressed on current study. The sequence of transformations will be first to pass from the lowpass filter to the basic bandpass filter using frequency mapping and impedance scaling equations from section 3.3 and then from the basic bandpass filter to the coupled-resonator bandpass filter using the admittance-inverter concept (COLLIN, 2001). The following procedure transforms a basic bandpass filter into a coupled-resonator bandpass filter with capacitive coupling. All the following parameters are referred to the lumped elements from Figure 3.4. The termination loads Ra and Rb are 50 ohms assuming perfect impedance matching. The sequence of steps is as follows:
* * * 1. Choose the initial values for the parallel capacitors Cres 1 , Cres 2 , …, and Cres n . The fact that the parallel capacitors are initially denoted with asterisks is because they will be corrected on later steps.
21
Study of a Compact Microwave Ceramic Coaxial Resonator Filter
2. Compute the parallel resonator inductors:
1 Lres j * 2 , for j = 1 to n, (3.23) Cres j 0
3. Compute the admittance inverters:
* 0 Cres 1 w J 0,1 , (3.24) Ra g0 g1 c
C* C* w 0 res j res ( j1) J j, j1 , for j = 1 to n (3.25) c g j g j1
* 0 Cres n w J n, n1 . (3.26) Ra gn gn1 c
Where g j corresponds to the lowpass-filter parameters from Table 3.1. The parameters 0 and w are defined on equations (3.19) and (3.20) respectively. The parameter
c corresponds to lowpass-filter cutoff frequency and is always equal to 1 when considering that the elements of the lowpass filter are normalized.
4. Using the admittance inverters proceed to calculate the coupling capacitances:
J 0,1 C0,1 , 2 (3.27) 0 1 J 0,1 Ra
J j, j1 C j, j1 , for j = 1 to n (3.28) 0
J n, n1 Cn, n1 . 2 (3.29) 0 1 J n, n1 Rb
22
Study of a Compact Microwave Ceramic Coaxial Resonator Filter
5. With the coupling capacitances, proceed to calculate the net shunt capacitances:
* e Cres1 Cres1 C0,1 C1, 2 , (3.30)
* Cres j Cres j C j1, j C j, j1 , for j = 2 to n-1 (3.31)
* e Cres n Cres n Cn1, n Cn, n1 , (3.32)
e e where C0,1 , and Cn, n1 are the corrected values of C0,1 , and Cn, n1 , respectively:
C e 0,1 , C0,1 2 (3.33) 1 0 C0,1 Ra C e n, n1 . Cn, n1 2 (3.34) 1 0 Cn, n1 Rb
* Note that the initial capacitances Cres j , are useful only to determine the final values for the parallel resonator capacitances Cres j . Equations (3.30) through (3.32) can be rewritten as follows in order to introduce the concept of nodal capacitance (see section 4.2.2):
* e Cres 1 Cres 1 C0, 1 C1, 2 , (3.35)
* Cres j Cres j C j1, j C j, j1 , for j = 2 to n-1 (3.36)
* e Cres n Cres n Cn1, n Cn, n1 . (3.37)
23
Study of a Compact Microwave Ceramic Coaxial Resonator Filter
Figure 3.5: Coupled-resonator bandpass filter of fifth order and its frequency response corresponding to the parameter values in Table 3.2.
Table 3.2: Parameter values for fifth-order bandpass filter with filter frequency and bandwidth equal to 1300 MHz and 40 MHz, respectively.
* Cres j Lres j C j , j1 Cres j j J j , j1 [pF] [nH] [pF] [pF] 0 - - 0.002578 0.3183 - 1 1.00 149.92 0.000358 0.0438 0.6432 2 2.00 74.96 0.000429 0.0525 1.9036 3 3.00 49.97 0.000607 0.0743 2.8732 4 4.00 37.48 0.001131 0.1385 3.7872 5 5.00 29.98 0.005765 0.7371 4.1856
24
Study of a Compact Microwave Ceramic Coaxial Resonator Filter
* From equations (3.35) to (3.37), the capacitance Cres j are the net capacitance of an arbitrary j-node when all the other nodes are short-circuited. This is the concept of nodal capacitance which is related to the nodal resonance frequency that will be covered on section 4.2.2. As a demonstration of this procedure, Table 3.2 contains the lumped-element values for a coupled-resonator bandpass filter of fifth order (n = 5). The filter frequency f filter and bandwidth BW are set to be 1300 MHz and 40 MHz, respectively. The lowpass filter elements gi are computed for a bandpass ripple equal to 0.01 dB, in such way that the return loss (S11) is guaranteed to be below the -20dB level. Figure 3.5 shows the network of the bandpass filter and its frequency response. The markers 1 and 2 at -20dB level, for frequencies 1279.6 MHz and 1321 MHz respectively, limit the bandpass and that results on a bandwidth of approximately BW = 41.40 MHz. This procedure allows to design a bandpass filter of n-order with only one type of resonators (parallel LC resonators). It is important to point out that the initial values for the parallel resonators Cres j are arbitrary as it is shown on Table 3.2. Chapter 4 studies the same bandpass filter with the exception that the LC-resonators correspond to the equivalent circuits of CCRs, therefore the values of parallel inductances Lres j and parallel capacitances Cres j , besides being related by the equation (3.23), will also be related by equations (2.19) and (2.20) depending on the CCRs length.
25
Study of a Compact Microwave Ceramic Coaxial Resonator Filter
4. BANDPASS FILTER OF CERAMIC COAXIAL RESONATORS WITH COUPLING CAPACITORS.
Section 3.4 introduced the bandpass filter with parallel LC-resonators and capacitive coupling. The inductor and capacitor of the LC-resonators are arbitrary values related only by equation (3.23) for a determined resonance frequency 0 of the resonator. In the current chapter, those LC-resonators are replaced by the equivalent circuits of the CCRs. This adds equations (2.19) and (2.20) to equation (3.23) as requirements for the LC-resonator values. Continuing the development of the chapter, a method is presented to solve the bandpass filter elements. The method consists in two parts, one analytical and one empirical. The analytical part uses the concept of nodal resonance frequency to modify equations from (3.23) through (3.37) such that the coupling capacitors are easily expressed in terms of the inductors and capacitors of the CCRs equivalent circuits. Finally, the empirical part explains the method to express the resonance frequencies of the CCRs in function of the filter frequency and bandwidth which are known parameters once the filter specifications are defined. Once the resonance frequencies of the CCRs are expressed in function of known parameters, with equations (2.18) through (2.20), the inductors and capacitors of the CCRs equivalent circuits are computable, and with that the bandpass filter is solved for any filter frequency and bandwidth from the frequency range studied.
4.1. INTRODUCTION OF THE CCR EQUIVALENT CIRCUIT.
On section 2.3, the RLC equivalent circuit for a CCR was described and it was explained that the resistance R in Figure 2.7b represents the CCR losses. On the current market of microwave and radio-frequency devices, a typical CCR is made of a dielectric material with high r (most common values are 21 ± 2, 38 ± 2, and 90 ± 2), which in terms of radiation means that very little energy is radiated through the CCR opened side. Besides, the loss tangents of the typical dielectric materials are very low, less than 0.0005 according
26
Study of a Compact Microwave Ceramic Coaxial Resonator Filter to (TRANS-TECH, 2013a) and (TEMEX, 2004). This means that the losses on the dielectric are also very low. And finally, the silver coating around the CCR provides very low conductor losses. All these facts lead to assume that the radiation and ohmic losses of the CCR can be neglected and this can be interpreted as if the resistor R from the CCR equivalent circuit is infinite. An infinite parallel resistor on a parallel RLC configuration is the same as an opened circuit, therefore, the resistor R can be remove for practical purposes. Besides, the interest about using equivalent circuits of the CCRs in the form of a parallel LC configuration is on its contribution to the resonance frequency of the entire bandpass filter when resonators are correctly coupled. The exclusion of any resistor R from all the CCRs equivalent circuits will only set the insertion loss of the bandpass filter on a 0dB level instead of a lower level which is the case when losses are considered, but the resonance frequency of the entire filter is not affected at all. Figure 4.1 shows a bandpass filter of n resonators where the CCRs equivalent circuits, composed by the parallel inductors Lres j and capacitors Cres j , are coupled by the coupling capacitors C j1, j . The lumped elements of this bandpass filter meet the equations for a coupled-resonator bandpass filter from section 3.4. The difference is that the inductors and capacitors of the resonators now also comply with equations (2.19) and (2.20) for a given CCR length. From section 3.4, it was concluded that the elements of any arbitrary coupled- resonator bandpass filter can be computable if it is assumed that the inductors and capacitors are known values. This is deduced in the first two steps of the procedure where the arbitrary initial values for the parallel capacitors are assumed and then the parallel inductors are computed from equation (3.23).
Continuing the explanation of the reasoning for the resolution of the bandpass filter from Figure 4.1, according to equations (2.19) and (2.20), in order to know the values of the inductor and capacitor of the equivalent circuit of a CCR, it is necessary first to know the
CCR length as well as the dielectric constant r of the CCR material and the cross-section parameters W and d. Assuming that all the CCRs are made of the same dielectric material
27
Study of a Compact Microwave Ceramic Coaxial Resonator Filter and have the same cross-section, then the only variable for equations (2.19) and (2.20) would be the CCRs length. Therefore, it appears that the coupled-resonator bandpass filter can be solved by knowing the correct length, or what is the same given equation (2.18), by knowing the correct resonance frequency of all the CCRs of the bandpass filter for a certain filter frequency and bandwidth.
Figure 4.1: Arbitrary bandpass filter of n-order composed by CCRs as resonators.
The main problem, therefore, is to find the correct length or resonance frequency of the CCRs and this will probably take several iterative trials. However, as it will be explained in section 4.2 and section 4.3, it is possible to express the resonance frequency of the CCRs in function of the filter frequency and bandwidth. The procedure for solving the actual bandpass filter of four CCRs consists of an analytical and also an empirical part. The analytical part is developed in section 4.2 and aims to express the value of the coupling capacitors as function of the elements of the equivalent circuit of the resonators. The empirical part developed in section 4.3 defines the resonance frequencies of the CCRs with empirical formulas.
4.2. ANALYTICAL PART OF THE PROCEDURE TO SOLVE A BANDPASS FILTER WITH FOUR CCRs WITH CAPACITIVE COUPLING.
This analytical part takes advantage of the circuit symmetry properties of the bandpass filter to simplify and reduce variables. Then, an analytic resolution of equations is performed using the concept of nodal resonance frequency and equations from section 3.4 to
28
Study of a Compact Microwave Ceramic Coaxial Resonator Filter express the value of the coupling capacitors as function of the lumped elements of the equivalent circuits of the CCR 1 and CCR 2.
Figure 4.2: Bandpass filter of four coupled CCRs.
4.2.1. SYMMETRY CHARACTERISTICS OF A BANDPASS FILTER OF FOUR RESONATORS CAPACITVELY COUPLED.
The procedure starts using the symmetry properties of the circuit from Figure 4.2 to minimize the number of variables. The symmetry of the circuit is referenced to the coupling capacitor C2,3 . What is to the right side of C2,3 is reflected symmetrically to the left. For instance, the equivalent circuits of CCR 1 and CCR 4 are the same and the same condition is applied for CCR 2 and CCR 3. This is expressed in the following equivalences:
Lres 1 Lres 4 , (4.1)
Lres 2 Lres 3 , (4.2)
Cres 1 Cres 4 , (4.3)
Cres 2 Cres 3 . (4.4)
The symmetry applied to the coupling capacitors delivers the following equivalences:
C0,1 C4,5 , (4.5)
C1,2 C3,4 . (4.6)
29
Study of a Compact Microwave Ceramic Coaxial Resonator Filter
And the same for the nodal inductances and nodal capacitances:
Lnode 1 Lnode 4 , (4.7)
Lnode 2 Lnode 3 , (4.8)
Cnode 1 Cnode 4 , (4.9)
Cnode 2 Cnode 3 . (4.10)
Equations (4.1) through (4.10) are used in order to work only with elements that are at the right side of C2,3 , that is the lumped elements of CCR 1, CCR 2, and coupling capacitors C0,1 , C1,2 , and .
4.2.2. THE CONCEPT OF NODAL RESONANCE FREQUENCY.
In order to formulate equations to express the coupling capacitors as function of the lumped elements of CCR 1 and CCR 2 is necessary to introduce the concept of nodal resonance frequency.
NODAL RESONANCE FREQUENCY.
Figure 4.3 shows an arbitrary j-node of a bandpass filter network. The nodal resonance frequency is the frequency at which the j-node resonates and is given by the following equation:
1 f node j (4.11) 2 Lnode j Cnode j
The lumped elements which contribute to the resonance frequency of the j-node are the LC-resonator (composed by Lres j and Cres j ) and the immediate coupling elements of the j-th resonator, which in Figure 4.3 are the coupling capacitors C j1, j and C j, j1 . In order to
30
Study of a Compact Microwave Ceramic Coaxial Resonator Filter
determine Lnode j and Cnode j from equation (4.11), all nodes in the circuit except of j-node must be short-circuited to ground, then it places the following relations:
Lnode j Lres j , (4.12)
Cnode j Cres j C j1, j C j, j1 , (4.13)
The concept of nodal resonance frequency indicates that if all resonators of a bandpass filter are correctly coupled, then all the nodes contribute to that particular resonance. In other words, all nodal resonance frequencies f node j must be equal to the filter frequency f filter when the resonators are correctly coupled.
Figure 4.3: Coupling elements that contribute to the resonance frequency of j-node.
ANALYTICAL RESOLUTION OF EQUATIONS.
The concept of nodal resonance frequency states that all nodes resonate at the same frequency if all CCRs are correctly coupled. Applying this concept for the bandpass filter from Figure 4.2 facilitates the manipulation of equations (3.23) through (3.37).
1. Using the concept of nodal resonance frequency, all nodes have the same resonance:
f node1 f node2 , (4.14) where the nodal resonance frequencies for node 1 and node 2 due to equation (4.11) are:
31
Study of a Compact Microwave Ceramic Coaxial Resonator Filter
1 f node 1 , (4.15) 2 Lnode 1 Cnode 1
1 f node 2 . (4.16) 2 Lnode2 Cnode 2
The nodal inductances Lnode j and nodal capacitances Cnode j are defined from equations (4.12) and (4.13):
Lnode 1 Lres 1 , (4.17)
Lnode 2 Lres 2 , (4.18)
e Cnode 1 Cres 1 C0,1 C1,2 , (4.19)
Cnode2 Cres 2 C1,2 C2,3 , (4.20)
e where C0 1 is defined in equation (3.33).
2. The coupling capacitors in function of the admittance inverters are:
J C 0,1 0,1 (4.21) 1 J R 2 0 0,1 A ,
J 1,2 C1,2 (4.22) 0 ,
J 2,3 C2,3 (4.23) 0 .
Expressions (4.22) and (4.23) are defined based on equation (3.28). The admittance inverters are defined based on equations (3.24) and (3.25) but substituting the capacitances
32
Study of a Compact Microwave Ceramic Coaxial Resonator Filter
* * * Cres1 , Cres 2 , and Cres 3 for the nodal capacitances Cnode1 , Cnode2 , and Cnode3 . Expression
(4.24) uses (4.14), (4.15), and (4.16) to express the Cnode1 in function of Cnode2 .
L 0 w res 2 J 0,1 Cnode2 , (4.24) g0 g1 Ra Lres 1
C C w 0 node1 node2 J 1,2 , (4.25) c g1 g2
C C w 0 node2 node3 J 2,3 . (4.26) c g2 g3
The cutoff frequency c for a normalized lowpass prototype filter is equal to 1 and the parameters g 0 , g1 , g 2 , g 3 are the lowpass filter elements computed for n = 4 and bandpass ripple equal to 0.01 dB from Table 3.1. Also the parameters w and 0 are the relative bandwidth and central filter frequency, respectively, from the lowpass-to-bandpass filter transformation from section 3.3 and are defined as follows:
- Passband lower frequency (radians): 1 2 f1 .
- Passband upper frequency (radians): 2 2 f2 .
- Passband lower frequency (Hz): f1 f filter BW 2 .
- Passband upper frequency (Hz): f2 f filter BW 2 .
- Filter frequency (radians): 0 2 f filter . - Relative Bandwidth: w 2 1 . 0
3. Finally, the analytic resolution of equations is as follows: Replacing expressions (4.15) and (4.16) on (4.14) and using (4.17) and (4.18) gives:
33
Study of a Compact Microwave Ceramic Coaxial Resonator Filter
Lres 2 Cnode 1 Cnode 2 , (4.27) Lres 1
Combining equations (4.20), (4.22), (4.25) and (4.27), it is obtained the following relation:
1 C1,2 Cres 2 C2,3 , (4.28) 11 Where
1 Lres 2 1 w , (4.29) g1 g 2 Lres 1
Combining equations (4.10), (4.20), (4.23) and (4.26), it is obtained the following relation:
2 C2,3 Cres 2 C1,2 (4.30) 1 2 , Where
1 2 w . (4.31) g2 g3
Resolving the system of equations between (4.28) and (4.30), it is obtained the following relations for coupling capacitances C1,2 and C2,3 :
1 C1,2 Cres 2 , (4.32) 1 1 2
2 C2,3 Cres 2 , (4.33) 1 1 2
where parameters 1 and 2 are defined in equations (4.29) and (4.31) respectively.
34
Study of a Compact Microwave Ceramic Coaxial Resonator Filter
In conclusion, this procedure achieved to express the coupling capacitors as a function of the lumped elements of the equivalent circuits of CCR 1 and CCR 2. The same procedure of applying the symmetry properties and the concept of nodal resonance frequency can be applied for bandpass filters with more number of resonators.
4.3. EMPIRICAL PART OF THE PROCEDURE TO SOLVE A BANDPASS FILTER WITH FOUR CCRs WITH CAPACITIVE COUPLING.
4.3.1. EMPIRICAL EXPRESSIONS FOR THE RESONANCE FREQUENCIES OF THE CCRs.
The following explanation is important to understand the empirical definition of the resonance frequency of an arbitrary CCR in a correctly coupled-resonator bandpass filter for a given filter frequency f filter and bandwidth BW. When the bandpass filter made of CCRs is correctly coupled, it is found that the length of any of the CCRs corresponds to a resonance frequency that is slightly greater than the filter frequency. This is due to the contribution of the immediate coupling capacitors to the nodal resonance frequency that makes the resonance frequency of the CCR decreases to equalize the filter frequency. Therefore, it was found convenient to express the resonance frequency of any arbitrary CCR (in a bandpass filter) as the filter frequency plus the bandwidth multiplied by a k j -factor, as described on equation (4.34). In this way, the resonance frequencies, and the lengths indirectly, of the CCRs are expressed in function of the filter specifications.
f res j f filter k j BW , (4.34)
the k j -factor is characteristic on each resonator and will be addressed with more details on section 4.3.2. From the assumption in equation (4.34), the resonance frequencies for CCR1 and CCR2 are:
35
Study of a Compact Microwave Ceramic Coaxial Resonator Filter
For CCR 1: f res 1 f filter k1 BW , (4.35)
for CCR 2: f res 2 f filter k2 BW . (4.36)
Using equations (2.18), (2.19), and (2.20) to define the lumped elements from the equivalent circuits of CCR 1 and CCR 2 in function of their respective resonance frequencies:
2Z 0 , For CCR 1: Lres 1 2 (4.37) f res 1 1 Cres 1 , (4.38) 8Z 0 f res 1
2 Z 0 for CCR 2: Lres 2 2 , (4.39) f res 2 1 Cres 2 . (4.40) 8Z0 fres 2
On the next section, it is presented a method to find expressions for the k j -factor in function of the filter frequency f filter and bandwidth BW and with equations from (4.37) through (4.40) the elements of the equivalent circuit of CCR 1 and CCR 2 are solved
(considering that the characteristic impedance Z 0 is known). Therefore, the coupling capacitors C0,1 , C1,2 , and C2,3 are solved due to equations (4.21), (4.32) and (4.33) and with this, the bandpass filter of four CCRs with coupling capacitors is solved in the circuital model.
4.3.2. ITERATIVE PROCESS TO EXPRESS k1 AND k 2 IN FUNCTION OF THE FILTER FREQUENCY AND BANDWIDTH.
Given that the filter frequency f filter and bandwidth BW are known parameters, then the attention focuses on finding the correct values for and . It is necessary to perform a trial-and-error process in which the values of k1 and k 2 are iterated until they are considered
36
Study of a Compact Microwave Ceramic Coaxial Resonator Filter
correct and that is when the nodal resonance frequencies f node1 and f node2 from equations
(4.15) and (4.16) are equal to the filter frequency f filter . In this way, it is assure that the bandpass filter resonates correctly at a certain filter frequency and bandwidth BW for
an unique set of values of k1 and k 2 . The iterative process is better explained with the following steps:
1) Choose a filter frequency f filter in the range between 800 MHz and 2000 MHz (which is the frequency range chosen for this study) and bandwidth BW between 5 MHz and 40 MHz to achieve narrowband filter condition.
2) Choose any arbitrary values for k1 and k 2 , then calculate the resonators frequencies with equations (4.35) and (4.36). 3) With equations (4.21), (4.32) and (4.33) calculate the coupling capacitors and with equations (4.37) through (4.40), the CCRs equivalent circuit elements respectively. 4) Finally, using equations (4.15) and (4.16), calculate the nodal resonance frequencies
and repeat the process from step 2 until and are equal to f filter .
Figure 4.2 shows the circuit of the four-CCR bandpass filter projected on CST Design Studio. The CST Design Studio allows to set parameterization on all lumped elements, so all the design equations from steps 2, 3 and 4 were programmed on each circuit element respectively, leaving only k1 and k2 as variables. Then, using the “Tune” tool, k1 and k2 are iterated (or tuned) manually until the condition from step 4 is acquired.
Figures 4.4 through 4.6 show three iterations to find the correct values for k1 and k 2 for a bandpass filter with f filter = 1200 MHz and BW = 20 MHz as example of the procedure. It can be noticed that the best filter response occurs only when the nodal resonance frequencies f node1 and f node2 are equal to the filter frequency f filter = 1200 MHz and that resonance happens for specific values for k1 and k2 .
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
ITERATION 1
k1 2.450
k 2 0.850
f res 1 1249.00 MHz
f res 2 1217.00 MHz f node1 1189.06 MHz f node2 1197.69 MHz Figure 4.4: Frequency response of the bandpass filter from Figure 4.2 whose lumped
elements were computed with k1 = 2.450 and k2 = 0.850.
ITERATION 2 3.550 1.055 1271.00 MHz
1221.10 MHz
1208.73 MHz
1201.64 MHz Figure 4.5: Frequency response of the bandpass filter from Figure 4.2 whose lumped
elements were computed with k1 = 3.550 and k2 = 1.055.
ITERATION 3 3.061 0.970 1261.22 MHz
1219.40 MHz
1200.00 MHz
1200.03 MHz Figure 4.6: Frequency response of the bandpass filter from Figure 4.2 whose lumped elements were computed with = 3.061 and = 0.970.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
The same procedure can be applied for different filter frequencies and bandwidths to create curves for k1 and k2 , then, using curve fitting techniques, and can be expressed empirically as equations in function of f filter and BW. In this way the bandpass filter will be solved, in theory, by only knowing the filter frequency and bandwidth. Table 4.1 and Table 4.2 shows the correct values for and respectively, obtained from applying the described procedure. The frequency range employed for the iterations was from 800 MHz to 2000 MHz and the bandwidth range was from 5 MHz to 40 MHz. As stated in section 3.4, the frequency mapping used for the lowpass-to-bandpass filter transformation is only accurate for narrow bandpass filters, which means that the bandwidth BW should be lower than 5% of the filter frequency . It is important to quote that the CCR profile used on the procedure has the following specifications: W = 6mm, d = 2mm and r = 38. According to Table 2.2, this profile has a good performance from 500 MHz to 2000 MHz.
The formulas derived from this process for k1 and k 2 are as follows:
B f k 2 filter k1 Ak1 , (4.41) BW BW Ak1 0.16624 exp 0.58017 , (4.42) 10.76263 BW Bk1 0.06662 exp 0.40534 , (4.43) 11.81403
B f k 2 filter k2 Ak 2 , (4.44) BW
Ak2 0.97052 0.00543BW , (4.45)
Bk2 0.00238 0.0012 BW , (4.46)
where f filter and BW are already expressed in MHz.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
Table 4.1: Values for k1 for different f filter and BW.
f filter BW [MHz] 5 10 15 20 25 30 35 40 800 4.652 3.453 2.916 2.591 2.366 2.196 2.063 1.952 850 4.779 3.543 2.990 2.656 2.425 2.252 2.114 2.002 900 4.900 3.630 3.061 2.719 2.482 2.304 2.164 2.050 950 5.019 3.715 3.131 2.780 2.537 2.356 2.213 2.095 1000 5.134 3.796 3.200 2.839 2.591 2.406 2.262 2.140 1050 5.246 3.876 3.265 2.897 2.643 2.454 2.305 2.183 1100 5.355 3.955 3.329 2.953 2.695 2.500 2.349 2.226 1150 5.462 4.031 3.392 3.008 2.743 2.546 2.391 2.265 1200 5.567 4.106 3.453 3.061 2.792 2.591 2.433 2.305 1250 5.670 4.178 3.513 3.114 2.839 2.635 2.474 2.343 1300 5.771 4.250 3.572 3.165 2.885 2.677 2.514 2.381 1350 5.869 4.320 3.630 3.215 2.930 2.719 2.553 2.418 1400 5.966 4.389 3.686 3.265 2.975 2.760 2.591 2.454 1450 6.061 4.457 3.742 3.313 3.018 2.800 2.628 2.489 1500 6.155 4.523 3.796 3.361 3.061 2.839 2.665 2.524 1550 6.248 4.588 3.850 3.407 3.104 2.878 2.701 2.558 1600 6.338 4.652 3.903 3.453 3.145 2.915 2.737 2.591 1650 6.427 4.716 3.954 3.499 3.186 2.953 2.771 2.624 1700 6.513 4.778 4.006 3.543 3.225 2.990 2.806 2.656 1750 6.599 4.839 4.056 3.587 3.265 3.026 2.839 2.688 1800 6.686 4.900 4.105 3.630 3.304 3.062 2.872 2.719 1850 6.769 4.959 4.154 3.672 3.342 3.097 2.905 2.750 1900 6.851 5.018 4.202 3.714 3.379 3.131 2.937 2.780 1950 6.933 5.076 4.250 3.756 3.417 3.165 2.969 2.810 2000 7.014 5.133 4.297 3.796 3.453 3.199 3.000 2.839
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
Table 4.2: Values for k2 for different f filter and BW.
f filter BW [MHz] 5 10 15 20 25 30 35 40 800 0.953 0.963 0.973 0.983 0.993 1.003 1.013 1.023 850 0.953 0.962 0.972 0.981 0.990 1.000 1.009 1.018 900 0.952 0.961 0.970 0.979 0.988 0.997 1.006 1.014 950 0.951 0.960 0.969 0.977 0.985 0.994 1.003 1.010 1000 0.950 0.959 0.968 0.975 0.984 0.991 1.001 1.007 1050 0.950 0.958 0.966 0.974 0.981 0.989 0.997 1.004 1100 0.950 0.958 0.965 0.972 0.980 0.987 0.994 1.003 1150 0.949 0.957 0.964 0.971 0.978 0.985 0.992 0.999 1200 0.949 0.957 0.963 0.970 0.977 0.983 0.990 0.997 1250 0.949 0.956 0.962 0.969 0.976 0.982 0.988 0.995 1300 0.948 0.955 0.962 0.968 0.975 0.981 0.986 0.993 1350 0.948 0.955 0.961 0.967 0.973 0.979 0.985 0.991 1400 0.948 0.954 0.960 0.966 0.972 0.977 0.983 0.989 1450 0.947 0.954 0.960 0.965 0.971 0.977 0.982 0.988 1500 0.947 0.953 0.960 0.964 0.970 0.976 0.981 0.986 1550 0.947 0.953 0.959 0.964 0.969 0.975 0.980 0.985 1600 0.947 0.952 0.958 0.963 0.968 0.974 0.978 0.984 1650 0.947 0.952 0.958 0.963 0.968 0.973 0.978 0.983 1700 0.947 0.952 0.957 0.962 0.967 0.972 0.976 0.981 1750 0.946 0.952 0.957 0.962 0.966 0.971 0.976 0.98 1800 0.946 0.951 0.956 0.961 0.966 0.97 0.975 0.979 1850 0.946 0.951 0.956 0.961 0.965 0.970 0.974 0.978 1900 0.946 0.951 0.955 0.960 0.965 0.969 0.973 0.977 1950 0.946 0.951 0.955 0.960 0.964 0.968 0.972 0.977 2000 0.945 0.950 0.955 0.959 0.963 0.968 0.971 0.976
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
4.4. DESIGN FORMULAS FOR A BANDPASS FILTER WITH FOUR CCRs WITH CAPACITIVE COUPLING.
The following are the design formulas for a bandpass filter composed of four resonators with capacitive coupling. The important parameters to build the 3D-structure from Figure 4.7 for electromagnetic simulation are the CCRs lengths and the coupling capacitances. This formulas are guarantee for CCR with cross-section parameters W = 6mm and d = 2 mm and for dielectric material r = 38. The design formulas for other cases CCRs cross-section and dielectric material can be obtained applying the same analytical-empirical procedure from section 4.2 and 4.3.
Figure 4.7: Structure of a bandpass filter of four CCRs with lumped elements as coupling capacitors designed on CST Microwave Studio.
1. Specify the filter frequency f filter and bandwidth BW. 2. Compute the CCRs resonance frequencies and lengths:
f res 1 f filter k1 BW , (4.47)
f res 2 f filter k2 BW , (4.48)
Where k1 and k2 are defined by expressions (4.41) through (4.46).
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
3. Compute the CCRs lengths (for electromagnetic simulations):
c res 1 4 f (4.49) res 1 r , c res 2 4 f (4.50) res 2 r . c = 3e11 mm/s.
r = dielectric constant of the resonator material.
4. Compute the lumped elements for the CCRs equivalent circuit (for circuital simulations):
2Z 0 , For CCR 1: Lres 1 2 (4.51) f res 1 1 Cres 1 , (4.52) 8Z 0 f res 1
2 Z 0 for CCR 2: Lres 2 2 , (4.53) f res 2 1 Cres 2 , (4.54) 8Z0 fres 2 where 60 W Z0 ln g , r d W, d are the cross-section parameters of the resonator.
5. Compute the value for the coupling capacitors:
J 0,1 C0,1 , 2 (4.55) 0 1 J 0,1 RA
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
1 C1,2 Cres 2 , (4.56) 1 1 2
2 C2,3 Cres 2 . (4.57) 1 1 2 Where
L 0 w res 2 J 0,1 Cnode2 , g0 g1 Ra Lres 1
1 Lres 2 1 w , g1 g 2 Lres 1
1 2 w , g2 g3
Cnode2 Cres 2 C1,2 C2,3 ,
c 1, w 2 1 , 0
0 2 f filter ,
1 2 f1 ,
2 2 f2 ,
f1 f filter BW 2 ,
f2 f filter BW 2 ,
g 0 = 1 ,
g1 = 0.7128,
g 2 = 1.2003,
g 3 = 1.3212,
Ra = 50 ohm.
44
Study of a Compact Microwave Ceramic Coaxial Resonator Filter
4.5. VALIDATION OF THE DESIGN FORMULAS USING CIRCUITAL SIMULATION.
This section presents circuital simulations for nine bandpass filter specifications in order to validate the accuracy of the design formulas from section 4.4. The frequency range used for the design frequencies is from 800 MHz to 2000 MHz, therefore the filter frequencies chosen are 800 MHz, 1400 MHz, and 2000 MHz and the bandwidths 5 MHz, 20 MHz, and 40 MHz. The simulations were performed on CST Design Studio using the circuit from Figure 4.2. The value of the lumped elements and the plots of the frequency responses are shown from Figure 4.8 through Figure 4.16.
FILTER 1
Lres 1 2.8144 nH
Lres 2 2.8789 nH
Cres 1 13.2809 pF
Cres 2 13.5852 pF
C0,1 0.7115 pF
C1,2 0.0939 pF
C2,3 0.0682 pF
Figure 4.8: Frequency response of the circuit from Figure 4.2 with lumped elements
computed for f filter = 800 MHz and BW = 5 MHz.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
FILTER 2
Lres 1 2.7201 nH
Lres 2 2.8269 nH
Cres 1 12.8363 pF
Cres 2 13.3399 pF
C0,1 1.5265 pF
C1,2 0.3859 pF
C2,3 0.2780 pF
Figure 4.9: Frequency response of the circuit from Figure 4.2 with lumped elements
computed for f filter = 800 MHz and BW = 20 MHz.
FILTER 3
2.6384 nH
2.7554 nH
12.4503 pF
13.0024 pF
2.3869 pF
0.7939 pF
0.5706 pF
Figure 4.10: Frequency response of the circuit from Figure 4.2 with lumped elements computed for = 800 MHz and BW = 40 MHz.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
FILTER 4
Lres 1 1.6203 nH
Lres 2 1.6493 nH
Cres 1 7.6463 pF
Cres 2 7.7829 pF
C0,1 0.3041 pF
C1,2 0.0305 pF
C2,3 0.0222 pF
Figure 4.11: Frequency response of the circuit from Figure 4.2 with lumped elements
computed for f filter = 1400 MHz and BW = 5 MHz.
FILTER 5
1.5811 nH
1.6324 nH
7.4611 pF
7.7032 pF
0.6336 pF
0.1242 pF
0.0898 pF
Figure 4.12: Frequency response of the circuit from Figure 4.2 with lumped elements computed for = 1400 MHz and BW = 20 MHz.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
FILTER 6
Lres 1 1.5465 nH
Lres 2 1.6092 nH
Cres 1 7.2977 pF
Cres 2 7.5939 pF
C0,1 0.9451 pF
C1,2 0.2530 pF
C2,3 0.1822 pF
Figure 4.13: Frequency response of the circuit from Figure 4.2 with lumped elements
computed for f filter = 1400 MHz and BW = 40 MHz.
FILTER 7
1.1383 nH
1.1557 nH
5.3724 pF
5.4536 pF
0.1773 pF
0.0149 pF
0.0109 pF
Figure 4.14: Frequency response of the circuit from Figure 4.2 with lumped elements computed for = 2000 MHz and BW = 5 MHz.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
FILTER 8
Lres 1 1.1161 nH
Lres 2 1.1475 nH
Cres 1 5.2668 pF
Cres 2 5.4148 pF
C0,1 0.3652 pF
C1,2 0.0605 pF
C2,3 0.0438 pF Figure 4.15: Frequency response of the circuit from Figure 4.2 with lumped elements
computed for f filter = 2000 MHz and BW = 20 MHz.
FILTER 9
1.0962 nH
1.1363 nH
5.1729 pF
5.3620 pF
0.5360 pF
0.1227 pF
0.0885 pF Figure 4.16: Frequency response of the circuit from Figure 4.2 with lumped elements computed for = 2000 MHz and BW = 40 MHz.
49
Study of a Compact Microwave Ceramic Coaxial Resonator Filter
Table 4.3: Results of the frequency responses from Figure 4.8 through Figure 4.16. Design Specifications Circuital Simulation Relative Error Filter Center Center Filter Bandwidth Bandwidth Bandwidth Frequency Frequency Frequency [MHz] [MHz] [%] [MHz] [%] (%) Filter 1 800 5 800.0 5.29 0 5.48 Filter 2 800 20 800.0 21.05 0 4.99 Filter 3 800 40 800.0 41.64 0 3.94 Filter 4 1400 5 1400.0 5.30 0 5.66 Filter 5 1400 20 1400.0 21.20 0 5.66 Filter 6 1400 40 1399.7 42.10 0.02 4.99 Filter 7 2000 5 2000.0 5.30 0 5.66 Filter 8 2000 20 1999.9 21.20 0.005 5.66 Filter 9 2000 40 2000.2 42.10 0.01 4.99
Table 4.3 presents of results of the frequency responses corresponding to Figure 4.8 through 4.16. When comparing the filter center frequency obtained through circuital simulation with respect to the center frequency at which the bandpass filter was designed, the relative error is practically zero. This concludes that the design formulas, when applied to design a bandpass filter in the form of a lumped elements circuit, they have excellent precision regarding the resonance frequency. Regarding the bandwidth, the results obtained by circuital simulation are always greater than the bandwidth at which the bandpass filter was designed. The relative error is between 3.94% and 5.66% being greater the error when the design bandwidth is lower for the same design center frequency. However, the design formulas continues to be also a good approximation when concerning about the bandwidth.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
5. SIMULATIONS AND RESULTS.
This chapter starts addressing the simulations of a single ceramic coaxial resonator in order to study the accuracy of equation 2.18 and to validate the simulation setup employed. Then, the CCR is simulated introducing a solder tab placed over its opened side. This addition causes variations in the resonance frequency to be considered for simulations on the entire filter. Next, it is presented comparisons between the frequency responses of the circuital simulations and electromagnetic simulations for the bandpass filters from Table 4.3. The circuital simulations responses are already available from Figure 4.8 through Figure 4.16. For the electromagnetic simulations, it was employed simulation setup shown in Figure 4.7. Then, it is presented the structure for the coupling capacitors with the respective considerations for proper design.
5.1. ELECTROMAGNETIC SIMULATION OF A CCR.
The CCR structure for electromagnetic simulation on CST Microwave Studio is shown in Figure 5.1. The CCR is a coaxial structure in which the inner conductor is cylindrical and the outer conductor has a square format. Both conductors have a thickness of 0.1 mm and the assigned material is PEC (Perfect Electrical Conductor) in order to disregard any conductor losses. The ceramic material is defined by the dielectric constant r and the loss tangent tan . The geometrical structure is defined by the resonator length and the coaxial section, which is determined by parameters W (outer conductor side) and d (inner conductor diameter). The setup for simulation is shown in Figure 5.2. The boundary conditions is PEC for all the walls. The excitation is a waveguide port placed on the opened- side of the resonator. The background material is set up to be vacuum ( r = 1). The dielectric constant of the ceramic material is = 38, the loss tangent is 0.0005 and the coaxial section for this project is W = 6 mm and d = 2 mm. According to Table 2.2, the optimal operating frequency range for this CCR is approximately between 500 MHz to 2000 MHz.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
Figure 5.1: Structure of a ceramic coaxial resonator on CST Microwave Studio.
Figure 5.2: Setup of simulation for a coaxial resonator on CST Microwave Studio.
Table 5.1 presents simulation data of the CCR from Figure 5.1. For a given design frequency (column 1), the CCR length (column 2) is computed using equation (2.18) and then proceed to simulate the structure configured for that length. Given that the model has only one excitation port, the single S-parameter is the return loss S11. The resonance frequency of the CCR occurs when the S11-phase is zero. The resonance frequency measured from simulation not necessary is equal to the design frequency. The simulation setup introduces not desired parasitic effects, e.g. the undesired capacitance created between the PEC walls and the opened side of the resonator. If this capacitance is very large, then the resonance frequency displaces lower that the actual value and the same for other parasitic effects.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
Column 4 from Table 5.1 presents the relative error between the resonance frequency measured from simulation and the design frequencies. The relative error is defined as follows:
design frequency res. frequency from simulation Relative error (%) = 100% design frequency
If the relative error between the simulation data and the design frequency is negligible, i.e. less than < 0.1%, then the simulation setup is considered correct. Figure 5.3 shows the phase curves for the eleven simulations from Table 5.1. The curve markers are placed at the frequencies where the S11-phase is zero.
Figure 5.3: Plot of phase curves of a single resonator.
The performed simulations used a CCR structure without solder tab. In practice, all the resonators have a solder tab over the inner conductor over the opened side of the resonator as it is shown in Figure 5.4. The soldering serves as support for the suspense dielectric
53
Study of a Compact Microwave Ceramic Coaxial Resonator Filter substrate when implementing the coupling capacitance scheme. The result of the simulations performed with this new structure are shown on Figure 4.5 and Table 4.2.
Table 5.1: Comparison between design frequencies and frequencies obtained by simulation. Design Resonator Resonance Frequency Relative Frequency length from Simulation Error [MHz] [mm] [MHz] [%] 1000 12.17 999.3 0.07 1100 11.06 1099.3 0.06 1200 10.14 1199. 0.08 1300 9.36 1299.1 0.07 1400 8.69 1399 0.07 1500 8.11 1499 0.07 1600 7.60 1598.9 0.07 1700 7.16 1698.8 0.07 1800 6.76 1798.8 0.07 1900 6.40 1898.8 0.06 2000 6.08 1998.6 0.07
Figure 5.4: Simulation setup for a CCR with incorporated solder tab.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
Figure 5.5: Plot of phase curves of a single resonator with solder tab.
Table 5.2: Comparison between design frequencies and frequencies obtained by simulation. Resonance Frequency Design Frequency Relative Error from Simulation [MHz] [MHz] [%] 1000 994.7 0.53 1100 1093.6 0.58 1200 1192.6 0.62 1300 1291.4 0.66 1400 1390.0 0.71 1500 1488.7 0.75 1600 1587.3 0.79 1700 1685.8 0.84 1800 1784.2 0.88 1900 1882.5 0.92 2000 1980.7 0.97
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
It is notable that the simulation of the CCR with a solder tab decreased the resonance frequency. The cause is the parasitic effects that the solder tab introduced to the model. The solder tab can be considered as small increment on the inductance of the CCR equivalent circuit, therefore the resonance frequency will slightly decrease. According to data on the relative error column in Table 4.2, the effect is not linear in relation to the frequency. The greater the frequency, the greater the decrement on the resonance frequency of the CCR caused by the solder tab. This effect should be taken into account when performing electromagnetic simulations for the entire four-resonator array.
5.2. VALIDATION OF DESIGN FORMULAS ON A 3D-STRUCTURE FOR ELECTROMAGNETIC SIMULATION.
The structure from Figure 5.6 is composed of four CCRs and coupling capacitances in the form of lumped elements. The CCRs lengths and coupling capacitances are computable from the equations of section 4.4 once the filter frequency and bandwidth are known. The idea of using lumped elements as coupling capacitances is that these elements do not introduce parasitic effects, so it is the most suitable simulation setup to validate the design formulas before implementing the structure for the coupling capacitors.
Figure 5.6: Equivalent 3D-structure for the bandpass filter circuit from Figure 4.2.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
The following plots show comparisons between the frequency responses from circuital simulations performed on section 4.5 and the frequency responses from electromagnetic simulations to be performed using the simulation setup from Figure 4.7 for the same bandpass filter specifications ( f filter and BW) from section 4.5. The plots contain data concerning to the resonance frequencies of the CCRs, their lengths and the coupling capacitors C0,1 , C1,2 , and C2,3 . This validation aims to test the accuracy of the design formulas to provide a fast and easy transition from the circuital design to the 3D-model design for electromagnetic simulations.
FILTER 1
f res1 823.22 MHz
f res 2 804.78 MHz
Length 1 14.78 mm
Length 2 15.12 mm
C0,1 0.7115 pF
C1,2 0.0939 pF
C2,3 0.0682 pF Figure 5.7: Comparison between the frequency responses of circuital and electromagnetic simulations for a bandpass filter designed with formulas from section 4.4 with
specifications f filter = 800 MHz and BW = 5 MHz.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
FILTER 2
f res1 851.73 MHz
f res 2 819.58 MHz
Length 1 14.28 mm
Length 2 14.84 mm
C0,1 1.5265 pF
C1,2 0.3859 pF
C2,3 0.2780 pF Figure 5.8: Comparison between the frequency responses of circuital and electromagnetic simulations for a bandpass filter designed with formulas from section 4.4 with
specifications f filter = 800 MHz and BW = 20 MHz.
FILTER 3
878.14 MHz
840.85 MHz
Length 1 13.85 mm
Length 2 14.47 mm
2.3869 pF
0.7939 pF
0.5706 pF
Figure 5.9: Comparison between the frequency responses of circuital and electromagnetic simulations for a bandpass filter designed with formulas from section 4.4 with specifications = 800 MHz and = 40 MHz.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
FILTER 4
f res1 1429.85 MHz
f res 2 1404.76 MHz
Length 1 8.51 mm
Length 2 8.66 mm
C0,1 0.3041 pF
C1,2 0.0305 pF
C2,3 0.0222 pF Figure 5.10: Comparison between the frequency responses of circuital and electromagnetic simulations for a bandpass filter designed with formulas from section 4.4 with
specifications f filter = 1400 MHz and BW = 5 MHz.
FILTER 5
1465.35 MHz
1419.29 MHz
Length 1 8.30 mm
Length 2 8.57 mm
0.6336 pF
0.1242 pF
0.0898 pF
Figure 5.11: Comparison between the frequency responses of circuital and electromagnetic simulations for a bandpass filter designed with formulas from section 4.4 with specifications = 1400 MHz and = 20 MHz.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
FILTER 6
f res1 1498.16 MHz
f res 2 1439.72 MHz
Length 1 8.12 mm
Length 2 8.45 mm
C0,1 0.9451 pF
C1,2 0.2530 pF
C2,3 0.1822 pF Figure 5.12: Comparison between the frequency responses of circuital and electromagnetic simulations for a bandpass filter designed with formulas from section 4.4 with
specifications f filter = 1400 MHz and BW = 40 MHz.
FILTER 7
2035.04 MHz
2004.74 MHz
Length 1 5.98 mm
Length 2 6.07 mm
0.1773 pF
0.0149 pF
0.0109 pF
Figure 5.13: Comparison between the frequency responses of circuital and electromagnetic simulations for a bandpass filter designed with formulas from section 4.4 with specifications = 2000 MHz and = 5 MHz.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
FILTER 8
f res1 2075.85 MHz
f res 2 2019.11 MHz
Length 1 5.86 mm
Length 2 6.03 mm
C0,1 0.3652 pF
C1,2 0.0605 pF
C2,3 0.0438 pF Figure 5.14: Comparison between the frequency responses of circuital and electromagnetic simulations for a bandpass filter designed with formulas from section 4.4 with
specifications f filter = 2000 MHz and BW = 20 MHz.
FILTER 9
2113.52 MHz
2039.01 MHz
Length 1 5.76 mm
Length 2 5.97 mm
0.5360 pF
0.1227 pF
0.0885 pF
Figure 5.15: Comparison between the frequency responses of circuital and electromagnetic simulations for a bandpass filter designed with formulas from section 4.4 with specifications = 2000 MHz and = 40 MHz.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
Table 5.3: Center frequency obtained from circuital (Figure 4.2) and electromagnetic (Figure 5.6) simulation using design formulas from section 4.4. Center frequency [MHz] Filter Relative Error [%] Circuital Electromagnetic Filter 1 800 796.1 0.49 Filter 2 800 793.4 0.83 Filter 3 800 793.0 0.88 Filter 4 1400 1387.2 0.91 Filter 5 1400 1384.4 1.11 Filter 6 1399.7 1382.4 1.24 Filter 7 2000 1977 1.15 Filter 8 1999.9 1971.6 1.42 Filter 9 2000.2 1968.3 1.59
A conclusion from analyzing and comparing the frequency responses is that in all cases the center frequency of the bandpass filter in the electromagnetic simulation is slightly lesser than the center frequency from circuit simulation, i.e. the frequency response presents a frequency shifting “to the left”. From Table 5.1, the relative-error column shows that the frequency shifting increases as the design filter frequency increases. Also, a clear decoupling between the resonators is appreciated which is seen from the deterioration of the return loss S11 for all filter cases. These problems can be attributed to the “not-considered” effects in the transition from the circuital model to the electromagnetic model, e.g. the appearance of parasitic reactances which are characteristic on every 3D-structure. Another cause might be the metallic solder tabs placed over the resonators and not considered in the circuital model. The solder tabs slightly increase the “electrical length” of each resonator and consequently increase the inductances and capacitances of the equivalent circuit. Therefore, extra capacitances and inductances exist in the electromagnetic simulation that are not being considered in the circuital model. This increment on the reactances increases the nodal capacitances and nodal inductances, and by equation (4.11), the nodal resonance frequency decreases (shifts to the
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter left) and the coupling capacitances in the lumped elements, whose values were computed based on the circuital model, no longer correctly couples the “electrically” larger resonators.
5.3. STRUCTURE FOR THE COUPLING CAPACITORS.
The studied bandpass filter (Figure 1.1) has a suspended dielectric substrate placed over the resonators solder tabs. The same structure format is reproduced to create the coupling capacitors between the resonators. Figure 5.16 describes the parts of the coupling capacitors structure. Figure 5.16a shows the solder tabs placed on the opened side of the resonators. The suspended dielectric substrate is soldered to the metallic tabs. Figures 5.16b, 5.16c and 5.16d shows the bottom metallic plates, dielectric substrate and top metallic plates of the structure. There are two SMA connectors placed on the top plates for the coupling with external connectors.
Figure 5.16: Parts of the coupling capacitors structure. a) Solder tabs. b) Bottom metallic plates. c) Dielectric substrate between bottom and top metallic plates. d) Top metallic plates and SMA connectors.
The capacitors generated by the coupling structure are shown in Figure 5.17. The dielectric substrate was rendered invisible for a better appreciation of the configuration of the parallel plates. On this structure, there are two types of capacitors: the gap capacitors and
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter the parallel plate capacitors (ppc). Figure 5.18 shows an approximated equivalent circuit for the gap capacitors and parallel plate capacitors configuration.
Figure 5.17: Parallel plates configuration and the gap capacitors and parallel plate capacitors resulting.
Figure 5.18: Equivalent circuit of the coupling capacitors structure.
Figure 5.19 shows the E-field distribution plots for three different phases (0, 15, and 30 degrees) over the coupling region between CCR 1 and CCR 2 in order to deduce which is the dominant capacitor. This plot belongs to a bandpass filter correctly coupled. It can be seen that the E-field distribution is more dense and intense on the regions of the parallel plate capacitors (red ellipses) rather than the regions of the gap capacitors (blue ellipses). This fact is used to neglect the contribution of the gap capacitors to the resonators coupling. Then, the
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter equivalent circuit from Figure 5.18 results into the simplified equivalent circuit shown in Figure 5.20.
Figure 5.19 E-field distribution plot over the dielectric substrate for phases: a) 0 degrees, b) 15 degrees and c) 30 degrees.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
Figure 5.20: Simplified equivalent circuit for the capacitors structure from Figure 5.17.
The capacitor C0,1 from the circuital model is associated to the parallel plate configuration ppc1. The coupling capacitor C1,2 is associated to the series configuration between ppc2 and ppc3 as well as C2,3 that is associated to the series configuration between ppc4 and ppc5. The capacitance C of a parallel plate capacitor is defined by the following formula:
A C r 0 , (5.1) d where:
r : dielectric constant of the substrate material.
12 0 : 8.8541810 F /m. A: parallel plate overlapped area. d: dielectric substrate thickness.
Figure 5.21 shows the length parameters that define the overlapped areas between the bottom and top plates. Again, as it was indicated on the circuital model, the symmetry properties can be applied, therefore the parallel plate configuration of interest are ppc1, ppc2, ppc3 and ppc4 and their overlapped areas are define as follows:
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
Figure 5.21: Parameters of the capacitors structure to determine the overlapped areas.
Appc 1 cap _ y cap _ x01, (5.2)
Appc 3 cap _ y cap _ x12 , (5.3)
Appc 4 cap _ y cap _ x23. (5.4)
Using equation (5.1) to define the capacitances of the capacitors created by the respective parallel plate configuration:
r 0 Appc1 C (5.5) ppc1 d
r 0 Appc2 C (5.6) ppc2 d
r 0 Appc4 C (5.7) ppc4 d
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
From the simplified equivalent circuit in Figure 5.20, the coupling capacitors from the circuit model are related to the coupling capacitors from the parallel plate configuration in the following way:
C0,1 Cppc 1 (5.8)
C ppc2 C (5.9) 1,2 2
C ppc4 C (5.10) 2,3 2
Then using equations from (5.2) through (5.10) to obtain the length parameters cap_x01, cap_x12, cap_x23 once the parameter cap_y is chosen a value:
C d cap _ x01 0,1 (5.11) r 0 cap _ y
2 C1,2 d cap _ x12 (5.12) r 0 cap _ y 2 C d cap _ x23 2,3 (5.13) r 0 cap _ y
Equations (5.11), (5.12), and (5.13) give the initial values to start the iteration process of the length parameters of the coupling capacitor structure. These initial values should decrease as iteration process goes on because of the following reasons:
- The Fringe-effect capacitance has not being considered on the equations, so the actual capacitance created by the parallel plate configuration should be greater than the estimated just with equation (5.1). - The gap capacitors were neglected at starting the analysis given that the parallel plate capacitors effect was dominant. Then, the net coupling capacitance between resonators is being underestimated.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
The dielectric substrates to be used on the coupling capacitor structure are commercially available and their thicknesses are standardized values (25 mil, 30 mil, 62 mil,
etc.) as well as the relative permittivity values ( r = 2.2, 4, 10, etc.). Therefore, Equation (5.1) is a good first-approximation to calculate the approximated overlapped areas of the parallel plate configurations. Still, there are other external and undesired capacitive effects that appeared because of the metallic enclosure. These parasitic effects causes variations on the frequency response of the filter. For instance, in most simulations the filter resonance frequency is shifted to lower values. The greater the effects of parasitic capacitances are, the greater the nodal capacitance will be and therefore the resonance frequency of the filter will be lower. For this reason there is not an exact method to find the correct values of the overlapped areas. This part of the tunning process is entirely iterative.
The resonators lengths calculated from section 4.4 are a good approximation to the actual lengths. The effects of parasitic capacitances and inductances will cause over-coupling between resonators causing, especially on the region between the resonators and the coupling structure. Most of the time, a very slight decrease on the resonators length will be necessary so the filter frequency returns to the desired value.
5.4. EXAMPLE OF DESIGN OF A BANDPASS FILTER WITH COUPLING CAPACITORS.
Following it will be presented an example of design of a bandpass filter with four CCRs with capacitive coupling. The CCRs have a cross-section with parameters W = 6 mm
and d = 2 mm and the dielectric constant of the ceramic is r = 38.
1. Filter specifications: f filter = 1500 MHz and BW = 30 MHz.
2. For CCR 1 and CCR 2, compute the resonances frequencies with expressions (4.47) and (4.48) and the respective lengths with expressions (4.49) and (4.50). The computed values are shown in Table 5.4.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
3. Calculate the equivalent circuit elements for CCR 1 and CCR 2. The computed values are shown in Table 5.4.
4. Calculate the coupling capacitances C0,1 , C1,2 , and C2,3 with expressions (4.55), (4.56), and (4.57). The computed values are shown in Table 5.5.
5. The dielectric substrate to be used for the coupling capacitors structure has r = 4 and a thickness H = 30 mil (0.762 mm).
6. Calculate the initial length parameters from Figure 5.21 with expressions (5.11) to (5.13). It is convenient first to choose a value for cap_y that is less than 5 mm because it is desired for any metallic plate (of the capacitors) not to be closed to the PEC walls of the structure. On this case it will be used cap_y = 4.5 mm.
Table 5.4: CCR parameters computed for the example of design.
f res j length Cres j Lres j
[MHz] [mm] [pF] [nH] CCR 1 1585.22 7.67 6.6228 1.4034 CCR 2 1529.26 7.96 6.8744 1.4568
Table 5.5: Parameters of the coupling capacitors structure.
C [pF] 0.7146 C [pF] 0.1636 C [pF] 0.1180 0,1 1,2 2,3 A [mm2] 15.4 A [mm2] 3.5 A [mm2] 2.5 ppc1 ppc2 ppc4 cap_x01 [pF] 3.42 cap_x12 [pF] 1.5 cap_x23 [pF] 1.1
The following simulations are performed using values presented on Table 5.4 and Table 5.5 for the dimensional parameters from Figure 5.21:
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
ITERATION 1 length 1 7.67 mm length 2 7.96 mm cap_x01 3.42 mm cap_x12 1.50 mm cap_x23 1.10 mm gap_sup1 0.50 mm gap_sup2 1.00 mm gap_inf1 1.00 mm gap_inf2 3.00 mm Figure 5.22: Iteration 1 for filter tunning using the structure and dimensional parameters from Figure 5.21 for a bandpass filter at f filter = 1500 MHz and BW = 5 MHz. The initial dimensional parameters are shown at Table 5.2 and Table 5.3
ITERATION 2 length 1 7.67 mm length 2 7.96 mm cap_x01 3.10 mm cap_x12 1.30 mm cap_x23 0.95 mm gap_sup1 0.80 mm gap_sup2 1.30 mm gap_inf1 0.60 mm gap_inf2 3.00 mm Figure 5.23: Iteration 2, the dimensions of the parallel plate capacitors were reduced in order to diminish the coupling capacitances.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
ITERATION 3 length 1 7.67 mm length 2 7.96 mm cap_x01 2.65 mm cap_x12 1.00 mm cap_x23 0.80 mm gap_sup1 1.00 mm gap_sup2 1.50 mm gap_inf1 1.125 mm gap_inf2 3.10 mm Figure 5.24: Iteration 3, the dimensions of all parallel plate capacitors are reduced in such way that the return loss shows improvements.
ITERATION 4 length 1 7.67 mm length 2 7.96 mm cap_x01 2.48 mm cap_x12 1.06 mm cap_x23 0.85 mm gap_sup1 1.00 mm gap_sup2 1.50 mm gap_inf1 1.09 mm gap_inf2 3.00 mm Figure 5.25: Iteration 4, the dimensions of all parallel plate capacitors are reduced for a second time and the return loss is now under -20 dB level and the frequency response is showing a bandwidth of 27 MHZ.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
ITERATION 5 length 1 7.42 mm length 2 7.70 mm cap_x01 2.48 mm cap_x12 1.06 mm cap_x23 0.85 mm gap_sup1 1.00 mm gap_sup2 1.60 mm gap_inf1 1.09 mm gap_inf2 3.00 mm Figure 5.26: Iteration 5, the lengths of the resonators were decreased in order to shift the center frequency up to 1500 MHz. The bandwidth is almost 30 MHz.
ITERATION 6 length 1 7.42 mm length 2 7.70 mm cap_x01 2.47 mm cap_x12 1.05 mm cap_x23 0.84 mm gap_sup1 1.00 mm gap_sup2 1.60 mm gap_inf1 1.09 mm gap_inf2 3.00 mm Figure 5.27: Iteration 6, a final decrement is done on the dimensions of the parallel plate capacitors. The bandpass filter is now centered at the design filter frequency with a bandwidth that differs only 0.4 MHz from the specification.
For the porpuse of demostrating the realizability of this procedure, it was projected and simulated bandpass filters for other filters specifications:
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
BANDPASS FILTER ON 1300 MHz WITH 20 MHz-BANDWIDTH length 1 8.80 mm length 2 9.91 mm cap_x01 2.90 mm cap_x12 1.20 mm cap_x23 0.80 mm gap_sup1 1.00 mm gap_sup2 1.70 mm gap_inf1 1.20 mm gap_inf2 1.80 mm cap_y 4.00 mm Er(resonator) 38 Er(substrate) 4
Figure 5.28: Filter response with f filter = 1300 MHz and BW = 20 MHz.
BANDPASS FILTER ON 1300 MHz WITH 20 MHz-BANDWIDTH (ZOOM) length 1 8.80 mm length 2 9.91 mm cap_x01 2.90 mm cap_x12 1.20 mm cap_x23 0.80 mm gap_sup1 1.00 mm gap_sup2 1.70 mm gap_inf1 1.20 mm gap_inf2 1.80 mm cap_y 4.00 mm Er(resonator) 38 Er(substrate) 4 Figure 5.29: Filter response with = 1300 MHz and BW = 20 MHz (Zoom).
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
BANDPASS FILTER ON 800 MHz WITH 40 MHz-BANDWIDTH length 1 13.70 mm length 2 14.30 mm cap_x01 2.90 mm cap_x12 1.80 mm cap_x23 1.30 mm gap_sup1 0.80 mm gap_sup2 1.20 mm gap_inf1 0.90 mm gap_inf2 1.20 mm cap_y 4.90 mm Er(resonator) 38 Er(substrate) 10
Figure 5.30: Filter response with f filter = 800 MHz and BW = 40 MHz.
BANDPASS FILTER ON 800 MHz WITH 40 MHz-BANDWIDTH (ZOOM) length 1 13.70 mm length 2 14.30 mm cap_x01 2.90 mm cap_x12 1.80 mm cap_x23 1.30 mm gap_sup1 0.80 mm gap_sup2 1.20 mm gap_inf1 0.90 mm gap_inf2 1.20 mm cap_y 4.90 mm Er(resonator) 38 Er(substrate) 10 Figure 5.31: Filter response with = 800 MHz and BW = 40 MHz (Zoom).
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
BANDPASS FILTER ON 1030 MHz WITH 10 MHz-BANDWIDTH length 1 11.10 mm length 2 11.50 mm cap_x01 4.10 mm cap_x12 1.40 mm cap_x23 1.20 mm gap_sup1 0.80 mm gap_sup2 1.20 mm gap_inf1 1.10 mm gap_inf2 1.50 mm cap_y 5.20 mm Er(resonator) 38 Er(substrate) 2.2
Figure 5.32: Filter response with f filter = 1030 MHz and BW = 10 MHz.
BANDPASS FILTER ON 1030 MHz WITH 10 MHz-BANDWIDTH (ZOOM) length 1 11.10 mm length 2 11.50 mm cap_x01 4.10 mm cap_x12 1.40 mm cap_x23 1.20 mm gap_sup1 0.80 mm gap_sup2 1.20 mm gap_inf1 1.10 mm gap_inf2 1.50 mm cap_y 5.20 mm Er(resonator) 38 Er(substrate) 2.2 Figure 5.33: Filter response with = 1030 MHz and BW = 10 MHz (Zoom).
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
6. CONCLUSIONS AND FUTURE WORK.
The objective of this thesis is to provide and conduct a systematic reverse-engineering procedure of the bandpass filter shown in Figure 1.1. For that, it is identified three important concepts for the construction of this filter: the ceramic coaxial resonator, the design of a bandpass filter with coupled resonators and the design of the capacitive coupling structure. It is clear that this study involves two distinct and sequential stages. The first stage is the study of the bandpass filter on a circuital model, i.e. using lumped circuit elements to represent the resonators and coupling capacitances. After having succeeded in understanding the behavior of the bandpass filter over a circuital environment, the next step is the study of the electrical behavior of the 3D structure that is the actual structure of the filter, using an electromagnetic environment. At the stage of the circuital modeling, the study first addresses the equivalent circuit of the ceramic coaxial resonator and its features. Then, it proceeds to study the procedure for designing bandpass filters with coupled resonators, so that by introducing the equivalent circuit of the CCR as the resonator, it is obtained the circuit equivalent model for the bandpass filter in Figure 1.1. The design formulas obtained with the referred procedure are an excellent method to solve the coupled-resonators bandpass filter in the circuital environment just by knowing the desired center frequency for the filter and its bandwidth. The results in Section 4.5 show excellent frequency responses for all projected bandpass filters. Therefore, the proposed design formulas work optimally in the circuital model environment. The next step is the study of the 3D electromagnetic model to perform simulations for verifying the precision of the design formulas when building the bandpass filter structure. The design formulas are a transition from the circuital model to the electromagnetic model, given that it allows to know the lengths of the coaxial resonators and the values of the coupling capacitances necessary for the structure to resonate at the desired frequency over a determined bandwidth. The results of electromagnetic simulations presented in Section 5.2, based on the structure of Figure 5.6, remark that the frequency response of all projected filters have a center frequency shift towards lower values. Importantly, these electromagnetic simulations
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter were performed on a structure where the coupling capacitors were modeled in the form of lumped elements, which minimizes the existence of other parasitic reactances. In other words, even in the simulation environment closest to the ideal, there is a frequency offset and a notable decoupling between resonators given the form of the return losses for all projected filters. The frequency shift, according to Table 5.1, is up to 1.59% of the center frequency at which the structure should resonate. The explanation for this difference in accuracy between the frequency responses of the circuital model and the electromagnetic model is that there are external factors that are not being considered in the circuital model such as various parasitic reactances own to the 3D structure. In addition, as mentioned in Section 5.2, the metallic tabs soldered to the coaxial resonators slightly decrease their resonance frequency. The effect caused by the metallic tabs are also not considered in the design formulas, therefore, the transition from the circuital model to the electromagnetic model presents an uncoupling effect between resonators due to the fact that the values of the coupling capacitances are computed for resonators without metallic tabs. The greatness of the coupling capacitances are in the range of 0.1 - 1 pF, so any small variation in the "electrical size" of the resonator will mean a variation on the values of the resonator equivalent circuit and a decoupling in the filter resonance. Considering that the most ideal simulation model already have unwanted effects, the next step is to build the structure for coupling capacitors in the form of an array of parallel metallic plates on a suspended dielectric substrate welded to the metallic tabs of the resonators as was showed in Figure 5.21. A simple method is provided for finding the initial values for the dimensions of the metallic plates and then based on trial-and-error iterations the structure is tunned. At the beginning it is very likely that the structure resonates with a bad return loss at a filter center frequency shifted "to the left" (lower frequencies). Following the recommendations from Section 5.4, it proceeds to tune the structure to improve return loss without yet correcting the frequency at which the filter is resonating. Then, when a good return loss is achieved, proceed to shift the filter center frequency towards right (greater frequencies) decreasing the resonators lengths keeping the length difference between them without changing the dimensions of the structure of the coupling capacitors. This analytical-empirical procedure is intended as a guide to solving bandpass filters of the same type but with a greater number of resonators for applications that need a faster
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter rejection outside the passband. As a final conclusion it can be pointed out that the provided method together with the proposed design formulas are a good first approximation for building the bandpass filter structure to resonate at a determined frequency and bandwidth. This is considering that without a systematic method, there is no assured way to find out the correct lengths of the resonators and coupling capacitances for a certain design, unless it is performed several trial-and-error iterations which implicates great amounts of simulation time. However, there is still some details pending to be explored in order to optimize the filter tunning. For example, for future work may be mentioned the optimization of the exposed procedure for correcting the design formulas in order to consider the variations of the resonator reactances due to the implementation of a metal tab. Thus, it would be expected the improvement of the frequency response in the transition from circuital model to the electromagnetic model. Another important point that escaped this thesis is a study of the sensitivity of this type of filter. The study could try to estimate the change in the frequency response due to some undesired variations, either on the dimensions of the structure such as manufacturing errors affecting the lengths of the resonators or the dimensions of the parallel plates, or small variations in the dielectric constant of the ceramic of the resonator or the dielectric substrate that is part of the structure of capacitors. This study was done strictly for ceramic coaxial resonators with a dielectric constant equal to 38 and resonator cross-section parameters W = 6 mm and d = 2 mm. From the above, it would be good continuity to the study of these filters the fact of applying the proposed procedure for other ceramic materials (different dielectric constants) and other cross- sections. Finally, a future study can be the implementation of a procedure that facilitates the design of the coupling capacitors structure. This study should consider the effects of fringing fields of the parallel plate capacitors and the contributions of the gap capacitors to the resonators coupling.
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Study of a Compact Microwave Ceramic Coaxial Resonator Filter
7. BIBLIOGRAPHY
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COLLIN, Robert E. Foundations for Microwave Engineering, 2nd Edition. New Delhi: Wiley India Pvt, 2001.
MATTHAE, G. L.; YOUNG, Leo and JONES, E. M. T. Microwave Filters, Impedance Matching Networks and Coupling Structures, Volume 1. New York: McGraw-Hill, 1964.
POZAR, David M. Microwave Engineering. 4th. ed. New York: John Wiley & Sons, 2011.
RHEA, Randall W. Discrete Oscillator Design: Linear, Nonlinear, Transient, and Noise Domains. Norwood: Artech House, 2010.
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VALKENBURG, Mac Elwyn Van. Network analysis. 3rd. ed. New Jersey: Prentice-Hall, 1974.
VENDELIN, George D.; PAVIO, Anthony M. and ROHDE, Ulrich L. Microwave Circuit Design Using Linear and Nonlinear Techniques. 2nd. ed. New Jersey: John Wiley & Sons, 2005.
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