Design of an Algorithm for Fast Frequency Sweep in Connecting Inmittance and Scattering Matrices for Microwave Filter Design

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(1)GRADO DE INGENIERÍA DE TECNOLOGÍAS Y SERVICIOS DE TELECOMUNICACIÓN. TRABAJO FIN DE GRADO. DESIGN OF AN ALGORITHM FOR FAST FREQUENCY SWEEP IN CONNECTING INMITTANCE AND SCATTERING MATRICES FOR MICROWAVE FILTER DESIGN. EDGAR SAAVEDRA DARRIBA 2016.

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(3) TRABAJO FIN DE GRADO TÍTULO:. DESIGN. OF AN. SWEEP. IN. ALGORITHM. FOR. CONNECTING. SCATTERING MATRICES. FOR. FAST FREQUENCY INMITTANCE. AND. MICROWAVE FILTER. DESIGN AUTOR:. EDGAR SAAVEDRA DARRIBA. TUTOR:. VALENTÍN DE LA RUBIA HERNÁNDEZ. DEPARTAMENTO: DEPARTAMENTO TECNOLOGÍAS. DE DE. MATEMÁTICA APLICADA LA. INFORMACIÓN. Y. A LAS LAS. COMUNICACIONES. TRIBUNAL PRESIDENTE:. SALVADOR JIMÉNEZ BURILLO. VOCAL:. ALBERTO PORTAL RUIZ. SECRETARIO:. JOSÉ MANUEL FERNÁNDEZ GONZÁLEZ. SUPLENTE:. FRANCISCO JOSÉ NAVARRO VALERO. FECHA DE LECTURA:. CALIFICACIÓN:. ___________________________. ________________________________. I.

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(5) UNIVERSIDAD POLITÉCNICA DE MADRID ESCUELA TÉCNICA SUPERIOR DE INGENIEROS DE TELECOMUNICACIÓN. GRADO DE INGENIERÍA DE TECNOLOGÍAS Y SERVICIOS DE TELECOMUNICACIÓN. TRABAJO FIN DE GRADO. DESIGN OF AN ALGORITHM FOR FAST FREQUENCY SWEEP IN CONNECTING INMITTANCE AND SCATTERING MATRICES FOR MICROWAVE FILTER DESIGN. EDGAR SAAVEDRA DARRIBA 2016 III.

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(7) I actually want to appreciate the help my Tutor, Valentín, brought me during the course of this Thesis, especially in the last days, even staying late in College.. V.

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(9) RESUMEN. E. ste Trabajo Fin de Grado tiene como objetivo la conexión rápida en frecuencia de las matrices inmitancia o de dispersión que caracterizan cada. uno de los bloques constitutivos de un circuito de radiofrecuencia con la finalidad de obtener la respuesta en frecuencia del dispositivo. Cada uno de los bloques constitutivos del circuito está caracterizado en frecuencia por su matriz de impedancia, admitancia o de dispersión, cuya respuesta en frecuencia se describe mediante una función racional. Tras realizar el proceso de conexión de cada uno de los bloques, se obtiene la descripción del comportamiento del circuito de radiofrecuencia, y el objetivo de este proyecto es obtener esta descripción circuital global, nuevamente, como una función racional de la frecuencia. Finalmente, se propone un algoritmo que permite realizar la conexión de estas matrices inmitancia y de dispersión de forma rápida en frecuencia para posibilitar el diseño asistido por ordenador de filtros de microondas analizados mediante métodos de onda completa.. PALABRAS CLAVE •. Inmitancia. •. Dispersión. •. Conexión de bloques microondas. •. Barrido rápido en frecuencia. •. Filtros. VII.

(10) ABSTRACT. T. his Graduation Thesis aims to fast frequency sweep in connecting inmittance and scattering matrices that describe each conforming block of a radiofrequency circuit. to obtain its whole frequency response. Each conforming block is frequency-described by its impedance, admittance or scattering matrix. This frequency response is a rational function. After the process of connecting every block, the frequency response behaviour of the circuit is achieved. The target of this project is obtaining this global response —again— as a rational function in the frequency domain. Finally, an algorithm is proposed to connect rapidly in frequency the inmittance and scattering matrices. Thus, allowing computer-aided design in microwave filters using full-wave calculation methods.. KEYWORDS •. Inmittance. •. Scattering. •. Connection of microwave blocks. •. Fast frequency sweep. •. Filters. VIII.

(11) RESUMO. E. ste Traballo Fin de Grao ten como propósito a conexión rápida en frecuencia das matrices inmitancia ou dispersión que caracterizan cada un dos bloques. constitutivos dun circuíto de radiofrecuencia coa finalidade de obtermos a resposta en frecuencia do dispositivo. Cada un dos bloques constitutivos do circuíto está caracterizado en frecuencia pola súa matriz de impedancia, admitancia ou dispersión. As súas respostas en frecuencia descríbense mediante unha función racional. Tras realizarmos o proceso de conexión de cada un dos bloques, obtense a descrición do comportamento do circuíto de radiofrecuencia. O obxectivo desde proxecto é obtermos esa descrición circuital global, unha vez máis, como unha función racional da frecuencia. Á fin, proponse un algoritmo que permite realiza-la conexión destas matrices inmitancia e dispersión de forma rápida en frecuencia para posibilitar o deseño asistido por ordenador de filtros de microondas analizados mediante métodos de onda completa.. ETIQUETAS •. Inmitancia. •. Dispersión. •. Conexión de bloques microondas. •. Barrido rápido en frecuencia. •. Filtros. IX.

(12) RESUMON Ĉ. i tio Tezo serĉas la konecton rapida en frekvenco do matricoj de inmitanco aŭ difuzo ke karakterizas ĉiu bloko de la cirkvito de mikroondoj, por akiri la respondon totala. de la aparato. Ĉio bloko de la cirkvito estas difinata en frekvenco per ĝia matrico de impedanco, admitanco aŭ difuzo. Funkcio racionala priskribas ĝiaj respondojn en frekvenco. Poste la konecto de ĉiu blokoj, ni akiri la priskribon de la konduto de la cirkvito de radiofrekvenco. La celo de ĉi projekto estas atingi la priskribo plena en frekvento de la cirkvito kompleta, denove, kiel funkcio racionala. Fine, ni rivelas algoritmon ke permesas la konekton de ĉi tiuj matricoj de inmitanco aŭ difuzo rapide en frekvenco per asisti la desegnon asistata per komputilo de filtriloj de mikroondoj analizata per metodoj da ondo kompleta.. ETIKEDOJN •. Inmitancon. •. Difuzon. •. Konekton do blokoj de mikroondoj. •. Skani rapide do frekvencon. •. Filtrilon. X.

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(14) TABLE OF CONTENTS I. SNEAK PEEK. 1. II. SCOPE OF STUDY. 3. 1. INTRODUCTION 2. CONNECTING CONSECUTIVE BLOCKS 3. FREQUENCY RESPONSE DESCRIPTION OF IMPEDANCE PARAMETERS. III. DEVELOPMENT. 3 6 9. 12. 1. AIM 2. POLE EXPANSION 3. ALGORITHM FOR TWO-PORT DEVICES CASCADE-CONNECTION. IV. VALIDATION. 12 13 15. 22. 1. ORDER 8 WAVEGUIDE FILTER (10,2–10,6 GHZ) 2. 9,6–10,6 GHZ DIPLEXER. 23 24. V. CONCLUSION AND FUTURE SIGHTS. 28. VI. APPENDIX. 29. A. MATLAB CODE FOR RESIDUE CALCULATION B. FREQUENCY ANALYTIC VARIATION C. INDEX OF FIGURES. VII. REFERENCES. 29 30 32. 33. XII.

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(16) I.. D. SNEAK PEEK evelopment of microwave filters is crucial nowadays, with the increasing amount of radiofrequency devices. We all carry several microwave devices every day.. However, it is not something only used in the customer end. It is also something very important in Military Defence, Outer Space, Medical and Pharmaceutical Industry, Radar, etc. Whilst radiofrequency applications grow continuously, it has become more complex to arrange efficiently the radio spectrum. This means that filters need to be more frequencyselective and more reliable. Designing microwave filters is a tough task. The more precise you go, the more iterations you need, and then, more time is spent in their development. When designing microwave filters, several methods can be used. All this methods and algorithms perform well, but they spend so much time since they need to calculate the filter response for each point of frequency. This leads to the requirement of calculating the inverse matrix thousands or millions of times. ! " = "$(& + " ( ))+, $ - a. (1). For instance, for a Ku-bandb filter with a bandwidth of 1 GHz and a precision of 1 MHz, 1000 frequency points are needed [1]. When we enter in more precise applications, precision increases rapidly and then, computing time does. The main factor which affects in computing time is inverting matrices. In order to solve this, the generalised eigenvalue problem can be used.. a b. s=j2πf, where f is frequency Ku-band asserts for 12 GHz – 18 GHz 1.

(17) /0 = 230. (2). 3+, /0 = 20. (3). If we rely on eigenvalues and eigenvectors intelligently, we will reach a terrific reduction of computing time since diagonal matrices will be inverted instead of general values matrices. Paying attention to the work made by Arcioni and Conciauro [2], we can see they developed a novel algorithm which takes advantage of eigenvalues as well. With this, the global response of the system is obtained without the need of frequency by frequency calculation. Therefore, a very fast algorithm can be implemented to rapidly analyse microwave structures and circuits.. Fig. 1: Comparison between using direct BI-RME analysis and the method developed in Arcioni and Conciauro’s paper [2] a. Following their investigation, our aim is applying these principles to obtain an algorithm that allows a fast frequency sweep calculation in microwave circuits, reducing drastically the amount of time spent in RF characterisation by means of impedance, admittance or scattering parameters.. a. Amplitude of the S-parameters for the component described in page 1993 of [2], Fig. 3 2.

(18) II. SCOPE OF STUDY 1. I NTRODUCTION Consider two constituting blocks, connected the way that follows:. ieI veI. I. icII. icI. vcI=vcII. II. ieII veII. Each block is described by means of its impedance matrix: Z. It’s known that Z matrix relates the voltages and currents at the ports od the device. Thus: 0 =!·5. (4). Hence, if we split the voltages and currents internal and external ones (internal voltages and currents referring to the ports to be connected): For block I:. For block II:. 7 7 067 !66 7 56 7 = ! , being ! = 7 087 587 !86 77 77 0677 !66 77 56 77 = ! , being ! = 77 0877 5877 !86. 7 !68 7 !88 77 !68 77 !88. Equations satisfying this can be found as: 7 7 7 7 067 = !66 56 + !68 58. (5). 7 7 7 7 087 = !86 56 + !88 58. (6). 77 77 77 77 0677 = !66 56 + !68 58. (7). 3.

(19) 77 77 77 77 0877 = !86 56 + !88 58. (8). If we look carefully, we will notice that, due to the connection: 087 = 0877 = 0. (9). 587 = −5877 = 5. ( 10 ). So we can write a new system based on (5), (7), and (6) and (8): 7 7 7 067 = !66 56 + !68 5 77 77 77 77 06 = !66 56 − !68 5 7 7 77 77 7 77 0 = !86 56 − !86 56 + !88 + !88 5. ( 11 ). Which we can write in matrix form as: 067 0677. =. 0. 7 !66. 0. 7 !68. 567. 0. 77 !66. 77 −!68. 5677. 7 !86. 77 −!86. 7 77 !88 + !88. 5. 067. ( 12 ). 567. 0677. =. 0. !6. !;. !;-. !<. 567. ( 13 ). 5. where: !6 =. 7 !66 0. 0 77 ; !66. !; =. 7 !68 77 ; −!68. 7 77 !8 = !88 + !88. And we obtain: 067 = !6 − !; !8+, !;0677. 4. 567 5677. ( 14 ).

(20) Here we can see an approach of what our problem will be: inverting a Zc matrix that has different values for each point of frequency. If we define our microwave block by means of impedance matrices as a function of frequency as follows: ! " = "b S + " ( T. +, b a. ( 15 ). Now we can apply eigendecomposition to not have the necessity to invert an arbitrary matrix for each point of frequency: S + A(T. +,. = ((ST +, + A ( TT +, )))+, = T +, (ST +, + A ( B)+,. C = ST +,. DE6FG;. CH = HI. T +, (HIH +, + A ( B)+,. DE6FG;. T +, (H(I + A ( B)H +, )+,. DE6FG;. b. ( 16 ). C = HIH +,. T +, (HIH +, + A ( HH +, )+,. DE6FG;. T +, H(I + A ( B)+, H +,. With this, now we can write: ! " = "bT +, H I + " ( B. +,. H +, b-. ( 17 ). As we can see, we do not need to invert complex matrices frequency-by-frequency anymore. Now we only need to invert diagonal matrices made of the adding of aigenvalues Λ and the point of frequency we are at (s). The price to pay is that we need to carry out an eigendecomposition, which is done only once, and we can then easily obtain the frequency response of the impedance matrix Z(s).. a. ~ asserts to the reduced order model variable underneath and the superscript T to the transpose matrix. b U asserts to the identity matrix. 5.

(21) 2. C ONNECTING C ONSECUTIVE B LOCKS. C. onnecting cascade blocks is trivial when speaking of Transmission Matrices —also known as ABCD parameters. This type of matrix is very convenient since many. microwave networks consist of a cascade connection of two or more two-port networks. In this case, it is useful to define a 2x2 transmission matrix for each two-port network. The resulting ABCD matrix can be easily found my multiplying the ABCD matrices of the individual blocks. [3]. Fig. 2: A two-port network (up); a cascade connection of two-port networks (down). ABCD parameters can be written in matrix form as: H, / = J, K. 3 C. H( J(. ( 18 ). And the connection shown above as: H, / = , J, K,. 3, C,. /( K(. 3( C(. HL JL. ( 19 ). This way, we can cascade as many blocks as we want by simply multiplying their transmission parameters. Notice that they should be multiplied in the correct order as long as matrix multiplication is not —generally— a commutative operation.. 6.

(22) Connecting consecutive blocks becomes a little bit tougher when it comes to inmitance matrices —both admittance and impedance. Impedance parameters are convenient for series connection, while admittance parameters are convenient for parallel connection, but none of them are suitable for cascade connection. The equivalence between ABCD and impedance parameters is called as: [4]. !,, =. / /C − 3K 1 C ; !,( = ; !(, = ; !(( = K K K K. /=. ( 20 ). !,, !a 1 !(( ;3= ;K= ;C= !(, !(, !(, !(,. ( 21 ). Hence, if we are willing to perform a cascade connection taking advantage of the ABCD multiplication and considering the conversions shown before:. H, = J,. 7 !,, 7 !(, 1 7 !(,. !7 7 !(, 7 !(( 7 !(,. 7 1 !,, H, = 7 77 J, !(, !(, 1. 77 !,, 77 !(, 1 77 !(,. ! 77 77 !(, 77 !(( 77 !(,. !7 7 !((. 7 77 1 !,, (!,, + 1) H, = 7 77 77 J, !(, !(, !,, + 1. 77 !,, 1. HL JL. ! 77 77 !((. DE6FG;. HL JL. ( 22 ). DE6FG;. 77 ! 7 ( ! 77 + !(( ) 7 77 77 !(( ( ! + !(( ). HL JL. If we decompose into external and internal/connection voltages and currents, and we are talking about two-port networks, then (for the first block in the cascade, being connected port 2):. a. ! = !,, !(( − !,( !(, 7.

(23) !,, ≡ !66 ; !,( ≡ !68 ; !(, ≡ !86 ; !(( ≡ !88. ( 23 ). This will be analogous for the second block, mindful of the connection: if it is between port 2-1 or port 2-2. For instance, considering a connection between the port 2 of each block results as follows: 7 77 1 !66 (!66 + 1) H67 = 7 77 7 77 !86 !68 J6 !66 + 1. 77 ! 7 ( ! 77 + !88 ) 7 77 77 !88 ( ! + !88 ). H677 J677. ( 24 ). Where impedance parameters referring to blocks I and II can be seen as well as its external and connection parameters. When using directly Z parameters, we have (12) and (13), where we can also see parameters referring to each block as well as its corresponding external and connection parameters. Thus, obtaining (14). 067 = !6 − !; !8+, !;0677. 8. 567 5677. ( prev. 14 ).

(24) 3. F REQUENCY R ESPONSE D ESCRIPTION OF I MPEDANCE P ARAMETERS. O. ur Z parameters will be obtained from b, T and S. These parameters represent the frequency-domain behaviour variation of the microwave block, letting us obtain. its impedance parameters this way (1): ! " = "b(S + " ( T)+, bO. ( prev. 1 ). Where Z(s) will have values for those parameters corresponding to Z11, Z12, Z21 and Z22 for each point of frequency. Once this is done, the impedance-defined model is achieved. Remembering from (23), it is known that this values are directly those corresponding to external and connection impedance values. Therefore, consecutive cascade blocks can be connected in order to obtain its whole behaviour. Instead of using vectors b, T and S, their corresponding ROMa vectors will be used. This permits less spending of time whilst achieving the same results in the band of interest and nearly the same results out of band. These results are valid in almost any case, and they are always useful when trying to obtain a first approach to split hairs later. Working on MORb: ! " = "b S + " ( T. +, b c. ( prev. 15 ). Parameters T and S represent the behaviour of the field inside the microwave device whilst b claims for the excitation to be projected onto the ports. We are not interested in the field solution, but in the impedance solution. This can be computed projecting its field on the ports using (15).. a. ROM asserts to Reduced Order Model. MOR asserts to Model Order Reduction. c ~ asserts to the reduced order model variable underneath. b. 9.

(25) Now, instead of inverting that complex matrices for each point of frequency, we are using the eigenvalue problem in order to reduce its computing complexity. Consequently, reducing the amount of time in calculation. From (16) we achieved (17): ! " = "bT +, H I + " ( B. +,. H +, b-. ( prev. 17 ). If we compare the results obtained between using direct application of (15) and calculation-simplified (17), we see:. Fig. 3: Comparison between both ways of calculation; circles represent the direct way and continuous lines the eigenvalues-simplified way. 10.

(26) If we take a closer look to a certain interval of frequencies:. Fig. 4: Comparison between both ways of calculation; closer look. We can see both methods perform the same way. With the second we obtain a significant enhance in computing time since complex matrices are only inverted once.. 11.

(27) III. DEVELOPMENT 1. A IM. N. ow it is known how to reduce the amount of time whilst achieving the same precise results, it is time to use it to connect different microwave devices and obtaining its. whole frequency behaviour. Cascade-connecting microwave devices means forcing their voltages in the connecting ports to be equal, as well as forcing their currents in the connecting ports to be equal but with opposite sign. Our aim is to obtain the complete frequency performance of the connected blocks the same way their individual behaviour is described, i.e. with a rational function: (1).. 12.

(28) 2. P OLE E XPANSION Let us consider we can define our microwave device in the form of pole expansion: S. !(") = " ET,. QE " ( − RE. ( 25 ). where s denotes the frequency, Ri the matrix residues and pi the corresponding pole. The poles represent the N resonating modes. The accuracy of the representation (25) increases with N [5]–[7]. This expression can be related with a rational function already known (15), (17) as: !(") = "K(Λ + " ( B)+, K -. ( 26 ). where the superscript T denotes the transpose matrix, U denotes the identity matrix and Λ denotes the diagonal matrix of minus the poles. If R is low-rank, coefficients C can be obtained from the residues as long asa:. Q=. Q,, Q(,. Q,( K = , Q(( K(. K,. K(. ( 27 ). Remembering from (15) and (17), we are working with b, T and S, after some manipulations, residues can be obtained asb: QEV = (WEV H)- ∘ (H +, ) +, $). a b. ( 28 ). Notice that we are considering reciprocal radiofrequency devices, namely: X12=X21. For a more precise reasoning see Appendix A. 13.

(29) where the operator ○ asserts for an element-by-element matrices multiplication, Fij define some matrix parameters of the device and V is a full matrix whose columns are the corresponding eigenvectors so that: /H = HC,. being / = ) +, &. ( 29 ). Comparing results obtained by direct pole expansion application (25) and the rational function that has been deduced (26), there can be seen both perform the same waya:. Fig. 5: Comparison between S parameters obtained from direct application of pole expansion and the rational function related.. a. Indeed, if the mean error is calculated, it is about –320 dB, i.e. there is no error. This difference is surely due to Matlab calculation error and approximations. 14.

(30) 3. A LGORITHM FOR T WO -P ORT D EVICES C ASCADE -C ONNECTION. T. he aim of this project is obtaining the complete frequency-behaviour of a junction among two two-port reciprocal microwave devices in the form of a rational. function; i.e. the same way as the individual frequency behaviour of each two-port device was defined. This makes possible to run the algorithm two-by-two devices as many times as we need with no change since every result is obtained in the same form as the original definitions. Taking into account what Arcioni and Conciauro did in [2], we can follow their steps in order to make some manipulations with our C parameters, eigenvalues and eigenvectors. This let us achieve some new C parameters that will describe the frequency-behaviour of the new two-port block made of the junction of two devices. To make this feasible, working with the solutions related to the generalised eigenproblem is needed. Doing this will entail making some changes to Λ achieving a new diagonal matrix of poles modified to satisfy the new eigenproblem —this will be clearer once we get into the formulation. Let Cc be the connection-corresponding C parameters and Ce the externals. Let refer to the first two-port device with the superscript I and the other with II and arrange them as follows: K67 K6 = 0. 0 K677. K8 = K87. K877. ( 30 ). Remembering from (27), we can deduce C1/e and C2/c from residues. These deduction is valid when calculating the corresponding Z11 and Z22. However, they are not valid when calculating Z12 and Z21 because this decomposition is only correct when R is a rank-1 matrix. Generally, this will be true, but not for all the resonant modes so we must perform another type of decomposition such as Takagi’s [9]: ^ = _`_ where L is a real nonnegative diagonal matrix and W is unitary. 15. ( 31 ).

(31) Hence, applying Takagi’s factorisation to the matrix residues, residue-by-residue, will result in a new vector equivalent to the matrix residues when multiplied by the scalar also given by Takagi’s factorisation. If the matrix residues is low-rank, then only one scalar and one Takagi’s vector will be necessary for each residue. However, when this matrix is not low-rank —this means it is rank-2—, two will be needed. Namely:. Q=. Q,, Q(,. Q,( = _, Q((. _(. a, 0. 0 a(. _, _(. ( 32 ). where W is, in our case, a 2x1 vector (referring to the 2x2 matrix residue; maximum rank2). Deciding when it is necessary the second vector and the second scalar is a compromise between simplicity and accuracy when the second one is not exactly zero. Our criterion is to neglect it when there are more than three magnitude orders of difference between them. Once we have Takagi’s vectors, we can calculate Ce and Cc such as: K6 0. 0 = _`_ K8. ( 33 ). Notice that, since second vector of poles will have the same impact in both Ce and Cc, they both — Ce and Cc — are the same length. Let D be the diagonal matrix made of ΛI and ΛIIa: 7 C= Λ 0. 0 Λ77. a. ( 34 ). It is important to note that poles should be arranged consequently to the Takagi’s factorisation. 16.

(32) Remembering (30) and [2], we can write the connection of the two devices as: 0 =" 0. K6 K8. C + "(B. 0 Υ =" ) 0 Σ. Υ=. KJf ΛJ + "2 B. Σ Γ. 5 , 58. K8-. JJ 2 KJJ f Λ +" B. 5 58. ( 35 ). being:. 0 JJ 2 KJJ f Λ +" B. KJf ΛJ + "2 B. Γ = K87 Λ7 + " ( B. K6-. −1 J ) Kf. 0. Σ=. +,. −1 J ) K< −1 JJ ) K<. −1 JJ ) Kf. ; ( 36 ). ;. +,. K87 - + K877 Λ77 + " ( B. +,. K877 -. 0 = "K6 C + " ( B. +,. K6- 5 + "K6 C + " ( B. +,. K8- 58. ( 37 ). 0 = "K8 C + " ( B. +,. K6- 5 + "K8 C + " ( B. +,. K8- 58. ( 38 ). With this we can rewrite (35) as:. Let us define an auxiliary variable yc:. h8 =. C + "(B. +,. K6-. K8-. 5 58. ( 39 ). So (35), and therefore (37) and (38), turns into: 0 = "K6 h8. ( 40 ). 0 = "K8 h8. ( 41 ). We can expand the dimension of this system in order to get something like this:. 17.

(33) 58 h8. ( 42 ). 0 = "K8 h8 + "058. ( 43 ). 0 = 0 "K6. Having done these changes has altered nothing numerically, but now we have ic in both equations, and ic is what we want to solve for. (39) can be also written as:. C + " ( B h8 = K6-. K8-. 5 58. DE6FG;. C + " ( B h8 − K8- 58 = K6- 5. ( 44 ). Now, we do not have an inverse matrix. With (43) and (44) a linear system like the one that follows can be made: 0 = −K8 h8 − 058 a C + " ( B h8 − K8- 58 = K6- 5. ( 45 ). In matrix form: 0 −K8-. −K8 C + "(B. 58 0 = h8 K6 5. ( 46 ). The first 2x2 matrix can be arranged such a way that: 0 −K8-. −K8 0 0 + "( C 0 B. 58 0 = h8 K6 5. ( 47 ). And, consequently, like:. a. Notice that the first equation is not affected when dividing by s and multiplying by –1. 18.

(34) 0 0 0 + "( −K80 C. −K8 B. 0 58 /" ( = K6 5 h8. ( 48 ). If we set Ω like:. Ω=. 0 0 0 + "( −K80 C. −K8 B. ( 49 ). and remembering from (42), then our system can be defined as: 0 58 /" ( = Ω+, K h8 65. ( 50 ). 58 h8. 0 = 0 "K6. ( 51 ). From (50) and (51):. 0 = 0 "K6 Ω+,. 0 K6- 5. ( 52 ). Let us work with Ω–1 expression, which is the one that involves frequency s as a difficult computation:. Ω. +,. =. 0 0 0 + "( −K80 C. −K8 B. +,. ( 53 ). Let us name two matrices like:. /=. 3=. 0 0 0 C. 0 −K8-. −K8 B. 19. ( 54 ). ( 55 ).

(35) Since (54) and (55) are positive definite (see [2], pp. 1995), they can be simultaneously diagonalised by the matrix having as columns the eigenvectors of the generalised eigenvalue problem [8]. When applying the generalised eigenvalue problem, we are getting a diagonal matrix k of generalised eigenvalues and a full matrix V whose columns are the corresponding eigenvectors so that: 3+, /H = Hk. ( 56 ). Ω+, = (HkH +, + " ( B)+, 3+, = H(k − " ( B)+, H +, 3+,. ( 57 ). Thus, we have:. Now, we can write:. 0 = 0 "K6 H(k + " ( B)+, H +, 3+,. 0 K6- 5. ( 58 ). Defining a new variable: l = 0 K6 H. l′- = H +, 3+,. 0 K6-. ( 59 ). ( 60 ). And remembering (4) and (52), we finally have: n o = op(q + or s)+t p′u. ( 61 ). As we can see, we have reached our aim of being able to connect two two-port microwave devices by means of inmitance matrices without the necessity of inverting matrices for each frequency point.. 20.

(36) We only need to invert complex matrices once. The other inversions are those of simple diagonal matrices and it means nearly no computational cost. Furthermore, we get our inmitance result, after carrying out the connection of the two building circuits, in the same way we had the initial devices defined: by a rational function. This has so many implications because we can perform this algorithm as many times as we want. Hence, we can obtain the complete frequency response of a complex device performing this algorithm in junctions of two blocks of the main device as many times as necessary. It is worth noting that the analytic frequency variation TE and TM modes come under with is not a problem. This is demonstrated in Appendix B.. 21.

(37) IV. VALIDATION. I. n this section, some microwave circuits will be analysed and connected to prove the algorithm developed in this Thesis works. One filter and one diplexer will be used.. We are showing a comparison between classical method of connecting Z matrices (pointby-point) and our new algorithm. Old method results will be displayed with dashed lines and new ones with solid lines.. Fig. 6: Filter to be analysed. Fig. 7: Diplexer to be analysed. 22.

(38) 1. O RDER 8 W AVEGUIDE F ILTER (10,2–10,6 GH Z ). S. tarting with a simple example, we are going to analyse one of the filters used in the diplexer about to analyse next. In this case, we have the filter itself divided in two parts we are connecting. We have 3 modes in the connection and 1 mode in the external ports.. Fig. 8: Z parameters of the filter. Fig. 9: S parameters of the filter. 23.

(39) 2. 9,6–10,6 GH Z D IPLEXER In this case we are analysing a diplexer made out of two band-pass filters and one T junction. This is a more complex circuit. We have the response of the filters and the response of the T junction. With these, we are calculating the complete response of the diplexer with our new algorithm.. Fig. 10: Channel A filter: ~10,2–10,6 GHz. Fig. 11: Channel B filter: ~9,6–10 GHz. 24.

(40) Fig. 12: T-Junction. We will see the results obtained using fast frequency sweep are very similar to those obtained from point-by-point calculation. The differences we may appreciate are due, surely, to Matlab approximations and the Takagi’s factorisation we have implemented. At any rate, in the band of interest both responses are equal. In order not to have a large amount of plots —and having checked our circuit is symmetric—, only essential parameters are shown. Port 1 is the common in the TJunction, port 2 is the corresponding to Channel A filter and port 3 is the corresponding to Channel B filter. In this case we have 1 mode in the three external ports and 3 in each connecting port.. 25.

(41) Fig. 13: Z parameters of the diplexer. 26.

(42) Fig. 14: S parameters of the diplexer. 27.

(43) V. CONCLUSION AND FUTURE SIGHTS. I. n this Graduation Thesis, a brand new method for fast frequency sweep in connecting inmitance parameters for microwave filter design has been described. It has been. proved this algorithm works and performs correctly and reliably. All tests were made using Matlab, and in the cases it does not achieve exactly the same result as the conventional method is probably due to the calculation error and approximations of Matlaba and other functions implemented like Takagi’s Factorisation. This algorithm may have very interesting implications and might be able to reduce significantly the time of calculation when simulating electromagnetic structures. It could be implemented in software like CST or HFSS. This means it can be applied when resolving a complex filter structure dividing it in smaller ones and applying iteratively this algorithm. In the future, this algorithm can be implemented in another programming language such as CUDA C/LAPACK to enhance its preciseness and speed. It could also be implemented to use it with other type of input parameters.. Notice that we are working with values in the order of magnitude between 1023 and 100 at the same time. a. 28.

(44) VI. APPENDIX A. M ATLAB C ODE FOR R ESIDUE C ALCULATION […] cellPseudoZ = cell(1, dimRom); for n=1:dimRom fName = [name, '.dim', num2str(n), '.PseudoZ']; Matrix = readMatrixFile(fName); cellPseudoZ(n) = {Matrix}; end TTildeInv = TTilde^-1; A = TTildeInv*STilde; b = TTildeInv*BTilde; [V,D] = eig(A); VInv = V^-1; dim = length(cell2mat(cellPseudoZ(1,1))); eigenvalues = diag(D); g = VInv * b; F11 F21 F12 F22. = = = =. []; []; []; [];. for n = 1:dimRom MF = cell2mat(cellPseudoZ(1,n)); F11 = [F11, MF(1,1)]; F21 = [F21, MF(2,1)]; F12 = [F12, MF(1,2)]; F22 = [F22, MF(2,2)]; end f11 f21 f12 f22. = = = =. F11 F21 F12 F22. residueH11 residueH21 residueH12 residueH22. * * * *. V; V; V; V; = = = =. transpose(f11) transpose(f21) transpose(f12) transpose(f22). .* .* .* .*. g; g; g; g;. […]. 29.

(45) B. A NALYTIC F REQUENCY V ARIATON. W. hen calculating the inmitance response of a microwave circuit, the inmitance parameters should be alterated by a matrix containing the analytic frequency. variation of the mode. TEM modes are not affected, but TE and TM modes are [10], [11]. 1, :;< , 1/:;< ,. for PWC functions, TEM and spherical modes for TE modes for TM modes. ( A-1 ). where. G. :;< (?) =. E. ?BC ?D ?BC ?. −1 E. ( A-2 ). −1. with kci being the cutoff wavenumber of mode i, k0 being a specific wavenumber and k being the wavenumber of the frequency itself. We can get this variation as a diagonal matrix we will call P. Then, once we have obtained the impedance (or admitance) parameters for each point of frequency, the real —taking into account this variation— impedance comes from: H = IHI. ( A-3 ). When connecting two microwave blocks, the mode propagating to the next block in the connection has to be the same. Defining the analytic frequency variation as a function depending on the current: JB ′ = :(?)JB. ( A-4 ). JL = :(?)JMNOP. ( A-5 ). 30.

(46) And, due to the connection itself, we know that the connecting currents are forced to be contrary in the connecting ports. It means that: Q QQ JMNOP :(?) = −JMNOP :(?). JBQ = −JBQQ ;. JLQ = −JLQQ. ( A-6 ) ( A-7 ). As we can see, the analytic frequency variation has no effect on the connecting ports. For the external ports, it can be applied the same way it is applied for one isolated block as told in (A-3).. 31.

(47) C. I NDEX OF F IGURES Fig. 1: Comparison between using direct BI-RME analysis and the method developed in Arcioni and Conciauro’s paper [2]. 2. Fig. 2: A two-port network (up); a cascade connection of two-port networks (down). 6. Fig. 3: Comparison between both ways of calculation; circles represent the direct way and continuous lines the eigenvalues-simplified way Fig. 4: Comparison between both ways of calculation; closer look. 10 11. Fig. 5: Comparison between S parameters obtained from direct application of pole expansion and the rational function related.. 14. Fig. 6: Filter to be analysed. 22. Fig. 7: Diplexer to be analysed. 22. Fig. 8: Z parameters of the filter. 23. Fig. 9: S parameters of the filter. 23. Fig. 10: Channel A filter: ~10,2–10,6 GHz. 24. Fig. 11: Channel B filter: ~9,6–10 GHz. 24. Fig. 12: T-Junction. 25. Fig. 13: Z parameters of the diplexer. 26. Fig. 14: S parameters of the diplexer. 27. 32.

(48) VII. REFERENCES [1]. Ku-Band order 6 TE-mode filter, Matlab analysis. [2]. Paolo Arcioni, Giuseppe Conciauro, “Combination of Generalized Admittance Matrices in the Form of Pole Expansions”, IEEE Trans. Microwave Theory Tech., vol. 47, pp. 1990-1996, October 1999. [3]. David M. Pozar, “Microwave Engineering, 4th ed.”, Wiley 2012, chapter 4. [4]. David M. Pozar, “Microwave Engineering, 4th ed.”, Wiley 2012, page 192. [5]. G. Conciauro, P. Arcioni, M. Bressan, and L. Perregrini, “Wide-band modeling of arbitrarily shaped-plane components by the boundary integral—Resonant mode expansion method,” IEEE Trans. Microwave Theory Tech., vol. 44, pp. 1057– 1066, July 1996. [6]. P. Arcioni, M. Bressan, G. Conciauro, and L. Perregrini, “Wide-band modeling of arbitrarily shaped -plane components by the boundary integral—Resonant mode expansion method,” IEEE Trans. Microwave Theory Tech., vol. MTT-44, no. 11, pp. 2083–2092, Nov. 1996. [7]. P. Arcioni, M. Bressan, G. Conciauro, and L. Perregrini, “A fast algorithm for the wideband analysis of 3-D waveguide junctions,” in Proc. 2nd Int. Conf. Comput. Electromag., Nottingham, U.K., Apr. 12–14, 1994, pp. 311–314. [8]. J. N. Franklin, Matrix Theory. Englewood Cliffs, NJ: Prentice-Hall, 1968, pp. 106. [9]. https://en.wikipedia.org/wiki/Matrix_decomposition. [10]. J. Abdulnour and L. Marildon, “Boundary elements and analytic expansions applied to H-plane waveguide junctions”, IEEE Trans. Microwave Theory Tech., vol. 42, pp. 1038–1045, June 1994. [11]. Valentín de la Rubia and Juan Zapata, “Microwave Circuit Design by Means of Direct Decomposition in the Finite-Element Method”, IEEE Trans. Microwave Theory and Tech., vol. 55, pp. 1520-1530, July 2007. [12]. Valentín de la Rubia, Ulrich Razafison and Yvon Maday, “Reliable Fast Frequency Sweep for Microwave Devices via the Reduced-Basis Method”, IEEE Trans. Microwave Theory Tech., vol. 57, pp. 2923–2937, December 200. 33.

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