# Synthesis and design of microwave filters and duplexers with single and dual band responses

код для вставкиСкачатьSYNTHESIS AND DESIGN OF MICROWAVE FILTERS AND DUPLEXERS WITH SINGLE AND DUAL BAND RESPONSES Iman K. Mandal Thesis Prepared for the Degree of MASTER OF SCIENCE UNIVERSITY OF NORTH TEXAS August 2013 APPROVED: Hualiang Zhang, Major Professor Yan Wan, Co-Major Professor Xinrong Li, Committee Member Shengli Fu, Chair of the Department of Electrical Engineering Dr. Costas Tsatsoulis, Dean of College of Engineering Mark Wardell, Dean of the Toulouse Graduate School UMI Number: 1526852 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI 1526852 Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106 - 1346 Mandal, Iman K. Synthesis and Design of Microwave Filters and Duplexers with Single and Dual Band Responses. Master of Science (Electrical Engineering), August 2013, 61 pp., 5 tables, 28 figures, bibliography, 52 titles. In this thesis the general Chebyshev filter synthesis procedure to generate transfer and reflection polynomials and coupling matrices were described. Key concepts such as coupled resonators, non-resonant nodes have been included. This is followed by microwave duplexer synthesis. Next, a technique to design dual band filter has been described including ways to achieve desired return loss and rejection levels at specific bands by manipulating the stopbands and transmission zeros. The concept of dual band filter synthesis has been applied on the synthesis of microwave duplexer to propose a method to synthesize dual band duplexers. Finally a numerical procedure using Cauchy method has been described to estimate the filter and duplexer polynomials from measured responses. The concepts in this thesis can be used to make microwave filters and duplexers more compact, efficient and cost effective. Copyright 2013 by Iman K. Mandal ii ACKNOWLEDGEMENTS I would like to greatly express my gratitude to my advisor, Dr. Hualiang Zhang for countless academic and professional guidance. He has had enormous patience in guiding me through a research area which I was completely unacquainted with. I would also like to thank my co-advisor, Dr. Yan Wan, for her kind help and support through this process. I want to express my thanks to Dr. Xinrong Li for his support as committee member for my thesis. I would also like to thank all my professors and friends at my lab which made my research experience so much fun. I am also highly obliged to Professor G. Macchiarella, Richard Cameron and Microwave Filter community for providing assistance and advice on numerous occasions. Finally, none of what I have done or have been so far would have been possible without the support of my family, especially my brother Dr. Suman Mandal for being constant source of inspiration and support. iii TABLE OF CONTENTS ACKNOWLEGEMENTS ..................................................................................................................... iii TABLE OF CONTENTS....................................................................................................................... iv LIST OF TABLES ................................................................................................................................ vi LIST OF FIGURES ............................................................................................................................. vii CHAPTER 1 SYNTHESIS OF NARROW BAND MICROWAVE FILTERS ................................................ 1 1.1 Introduction........................................................................................................... 1 1.2 Original Contributions ........................................................................................... 2 1.3 Coupled Resonator Filters ..................................................................................... 3 1.4 Filters using Non-Resonant Nodes ........................................................................ 7 1.5 Transfer and Reflection Polynomial Synthesis .................................................... 11 1.6 Evaluation of Coupling Matrix ............................................................................. 12 1.7 Coupling Matrix Reconfiguration ........................................................................ 16 1.8 Example of Synthesis........................................................................................... 17 CHAPTER 2 SYNTHESIS OF MICROWAVE DUPLEXERS ................................................................... 20 2.1 Introduction......................................................................................................... 20 2.2 General Structure ................................................................................................ 20 2.3 Polynomial Synthesis........................................................................................... 21 2.4 Example of Synthesis........................................................................................... 23 CHAPTER 3 SYNTHESIS OF DUAL BAND FILTERS ........................................................................... 27 3.1 Introduction......................................................................................................... 27 3.2 Polynomial Synthesis........................................................................................... 27 iv Chapter 4 DESIGN OF DUAL BAND DUPLEXERS ............................................................................ 34 4.1 Introduction......................................................................................................... 34 4.2 Synthesis Procedure ............................................................................................ 34 4.3 Implementation in Cavity Type and Non-Resonating Node Type....................... 39 CHAPTER 5 MODEL ORDER REDUCTION OF MICROWAVE DUPLEXERS ....................................... 45 5.1 Introduction......................................................................................................... 45 5.2 Formulation of Cauchy Method .......................................................................... 45 5.3 Application in General Polynomial Function:...................................................... 47 5.4 Application in General Microwave Filter Function ............................................. 49 5.5 Application in Microwave Duplexers .................................................................. 51 5.6 Conclusion ........................................................................................................... 53 CHAPTER 6 CONCLUSION AND FUTURE WORK ............................................................................ 54 BIBLIOGRAPHY .............................................................................................................................. 55 v LIST OF TABLES Table 1.1 Filter Polynomials for an 8th Order Chebyshev Filter. ................................................... 17 Table 2.1 Extracted TX Filter Polynomials..................................................................................... 25 Table 2.2 Extracted RX Filter Polynomials .................................................................................... 25 Table 2.3 Duplexer Polynomials.................................................................................................... 26 Table 5.1 Low Pass Polynomial Coefficients ................................................................................. 50 vi LIST OF FIGURES Fig. 1.1 (a) Equivalent circuit of n-coupled resonators for loop equation formation (b) Its network representation. ......................................................................................................... 4 Fig. 1.2 A general coupling arrangement ........................................................................................ 8 Fig. 1.3 Filter with NRN – dotted circles represent frequency invariant susceptance; Solid circles represent capacitance in parallel to frequency invariant susceptance (normalized resonator); Black lines represent admittance inverters. ...................................................... 10 Fig. 1.4 Canonical transversal array. (a) N resonators including direct source-load coupling M SL . (b) Equivalent circuit of the k th low-pass resonator in the array. ......................................... 13 Fig. 1.5 Series type low pass prototype with inter-resonator couplings. ..................................... 13 Fig. 1.6 N + 2 canonical coupling matrix for the transversal array. The matrix is symmetric with respect to the principal diagonal. ......................................................................................... 16 Fig. 1.7 Polynomial response of the 8th order filter in normalized frequency domain. ............... 18 Fig. 1.8 Polynomial response of the 8th order filter in real frequency domain. ........................... 19 Fig. 1.9 Filter response generated from coupling matrix ............................................................. 19 Fig. 2.1 Duplexer configuration..................................................................................................... 20 Fig. 2.2 Frequency mapping: f 0 = , B f1, RX f 2, RX = f 2,TX − f1, RX ................................................ 21 Fig. 2.3 Synthesized Duplexer polynomial response .................................................................... 24 Fig. 3.1 Effect of narrowing of passbands on return loss levels. .................................................. 30 Fig. 3.2 Narrowing done around upper passband. ....................................................................... 30 Fig. 3.3 Narrowing done around lower passband edges. ............................................................. 31 Fig. 3.4 Effect of additional transmission zero at w = 1.6 ............................................................. 32 vii Fig. 3.5 A fifth order filter with three real transmission zeros (solid lines). The return loss levels are equalized with complex transmission zero pair at 0.825 ±1.6j (dashed lines)............... 33 Fig. 4.1 Frequency transformation for dual band duplexers. ....................................................... 35 Fig. 4.2 Proposed dual-band duplexer: (a) topology with non-resonating nodes, (b) topology with cavity resonators........................................................................................................... 39 Fig. 4.3 Synthesized response of the dual-band duplexer with non-resonating node topology (S21 for RX channel, S31 for TX channel).............................................................................. 41 Fig. 4.4 Insertion loss at RX channel ( S12 ). .................................................................................... 43 Fig. 4.5 Insertion loss at TX channel ( S13 )...................................................................................... 44 Fig. 4.6 Return loss at Input ( S11 ). ................................................................................................. 44 Fig. 5.1 SNR = 5 dB. ....................................................................................................................... 48 Fig. 5.2 SNR = 20 dB. ..................................................................................................................... 48 Fig. 5.3 SNR = 50 dB. ..................................................................................................................... 49 Fig. 5.4 Example of polynomial estimation for a 6th order filter. ................................................. 51 Fig. 5.5 Exact and estimated polynomial response for duplexers. ............................................... 52 viii CHAPTER 1 SYNTHESIS OF NARROW BAND MICROWAVE FILTERS 1.1 Introduction Microwave filters and duplexers are essential components of communication systems. Frequency spectrum is the most expensive resource among all and to optimize its use filters and duplexers must be designed to be compact and capable of handling high power at the same time. Filters may be classified into categories in several ways. One typical way is to classify them based on different classes of response functions, defined in terms of the location of the poles, the insertion-loss function, and the zeros within the passband. The zeros are usually spaced throughout the passband to give an equiripple or Chebyshev response since this is far more optimum and superior to the maximally flat or Butterworth response, which is rarely used. As far as the poles are concerned, the most common type of filter response has all these poles located at dc or infinity and is often described as an all-pole Chebyshev filter, or simply as a Chebyshev filter [1]. When one or more poles are introduced into the stopbands at finite frequencies, the filter is known as a generalized Chebyshev filter or as a pseudo-elliptic filter. The special case where the maximum number of poles are located at finite frequencies such that the stopbands have equal rejection level is the well-known elliptic function filter. This is now rarely used since it has problems in practical realization and is not optimum when specific stopbands are required—one seldom needs rejection up to infinite frequency. It is almost always better to place the poles where they are most needed, and also to minimize their 1 number, since each additional finite frequency pole may increase the implementation complexity and expense. The above discussion relates equally to the main categories of filters defined in terms of the general response types of low-pass, bandpass, high-pass, and bandstop. In general, pseudoelliptic filters (or generalized Chebyshev filters) as the most useful and powerful filter types are best designed using exact synthesis techniques. Several techniques were developed to reduce computational complexities [2]. The typical procedure is to synthesize a low-pass prototype, which is then resonated to form a bandpass filter. The categories considered are combline, interdigital, parallel-coupled-line bandpass and bandstop, ring and patch filters [3], and stepped-impedance filters [4]. The several media for implementation include waveguide, dielectric resonators, coaxial lines, evanescent-mode filters, and various printed circuit filters using microstrip, stripline, and suspended substrate. Also frequency tuning is another very important aspect of filter designs [5]. In this chapter the synthesis technique for cross coupled resonator bandpass filters exhibiting pseudo-elliptic filter response and the concept of non-resonant nodes has been described, which also lays foundation for the discussion in the next a few chapters. A history of early filter researches can be found in [6], [7]. A large number of filter concepts discussed in this thesis can be found in great detail in the classic book on microwave filters by Matthaei, Young and Jones [8]. Also extensive details on the newer filter synthesis techniques can be found in [9]. 1.2 Original Contributions There are two significant contributions in this thesis: 2 • Comprehensive study on synthesis of dual band filters, control of return loss levels using stopband modifications and additional transmission zeros. • Proposing a method for the synthesis of dual band duplexers, which is based on duplexer synthesis and dual band filter synthesis techniques. 1.3 Coupled Resonator Filters Extensive literature is available on the theory of coupled resonator filters [10], [11], [12], [13], [14], [15], [16], [17], [15], [18], [19], [20]. The equivalent circuit of an n coupled resonator filter network is shown in Fig. 1.1. The loop equations can be written as: 1 es R1 + jwL1 + i1 − jwL12 i2 − jwL1n in = jwC1 1 0 − jwL21i1 + jwL2 + i2 − jwL2 n in = jwC 2 (1.1) 1 0 − jwLn1i1 − jwLn 2 i2 + Rn + jwLn + in = jwCn where Lij = L ji represents the mutual inductance between resonators i and j , and all the loop currents are supposed to have the same direction as shown in Fig. 1.1. 3 Fig. 1.1 (a) Equivalent circuit of n-coupled resonators for loop equation formation (b) Its network representation. In matrix form, the set of equations can be represented as: 1 R1 + jwL1 + jwC 1 − jwL21 − jwLn1 where the n×n − jwL12 jwL2 + 1 jwC2 − jwLn 2 i1 es − jwL2 n i2 = 0 0 i 1 n Rn + jwLn + jwCn − jwL1n (1.2) matrix is the impedance matrix [ Z ] . For simplicity, let us first consider a synchronously tuned filter. In this case, all resonators resonate at the same frequency, namely the midband frequency of filter w0 = 1 LC , where L= L= L= = Ln and 1 2 C= C= C= = Cn . The impedance matrix may be expressed by 1 2 [ Z ] =w0 L ⋅ FBW ⋅ Z 4 (1.3) where FBW = ∆w w is the fractional bandwidth and Z is the normalized impedance matrix which, for synchronously tuned filter is given by: R1 w L ⋅ FBW + p 0 1 w L21 ⋅ − j Z = w0 L FBW − j w Ln1 ⋅ 1 w0 L FBW −j 1 w L12 ⋅ w0 L FBW p −j 1 w Ln 2 ⋅ w0 L FBW w L1n 1 w0 L FBW 1 w L2 n −j ⋅ w0 L FBW Rn +p w0 L ⋅ FBW −j (1.4) with = p j 1 w w0 − FBW w0 w (1.5) as the complex lowpass frequency variable. Also, Ri 1 = w0 L Qei (1.6) Qe1 and Qen are the external quality factors of the input and output resonators, respectively. The coupling coefficients are defined as: M ij = Lij L (1.7) The normalized coupling coefficients are given by: mij = M ij FBW (1.8) In case of an asynchronously tuned filter the resonant frequency of each resonator is different and may be given by w0i = 1 Li Ci , the coupling coefficient of asynchronously tuned filter is defined as: 5 Lij M ij = (1.9) Li L j From the above relations the normalized Z can be derived as: 1 q + p − jm11 e1 − jm21 Z = − jmn1 Each − jm12 p − jm22 − jmn 2 − jm2 n 1 + p − jmnn qen − jm1n (1.10) mii account for asynchronous tuning for each resonator, i.e., frequency shifts from the center frequency. Then the scattering parameters are obtained as: = S 21 2 1 ⋅ [ A]n1 −1 qe1 ⋅ qen 2 −1 S11 = ± 1 − ⋅ [ A]11 qe1 (1.11) with [ A] = [ q ] + p [U ] − j [ m] where [U ] is the n×n identity matrix, [ q ] is n×n matrix with all elements zero except the q11 = 1 qe1 and qnn = 1 qen , [ m] is the general n × n coupling matrix. The n + 2 coupling matrix also includes the external couplings from source and load to each resonator. It is very easy to transform one form of coupling matrix to another [12], [21], [22]. Till now only inductive coupling has been considered. Coupling can be capacitive too, and the corresponding coupling coefficients are called electrical coupling coefficients detailed in [11]. 6 1.4 Filters using Non-Resonant Nodes Using non-resonant nodes with admittance inverters provides a useful approach to filter design, which is often convenient for microstrip implementation. Several literatures are available on non-resonant nodes [23], [24], [25], [26], [27], [28]. Overall, four types of components are used in the low pass prototype in this approach. 1) Resonators: These are represented by unit capacitors in parallel with the frequency-invariant reactances jbi which account for the frequency shifts in their resonant frequencies from the center frequency. 2) Admittance Inverters J i : These are identical to the coupling coefficients between the nodes. 3) Non-resonating Nodes: These are internal nodes connected to ground by frequency-invariant reactance jBi . These are not in parallel with any capacitor. 4) Input (source) and Output (load): These are normalized conductances, G= G= 1. S L A resonator that is responsible for an attenuation pole at a normalized frequency si = jwi is represented by a unit capacitor in parallel with a constant reactance jbi = − jwi . Such a dangling resonator is only connected to an NRN. For a filter of order N with attenuation poles at finite real frequencies, there are N z number of dangling resonators and N - N z resonators along the inline path between the input and the output. When choice of arrangement is not unique, but flexible and for the designer to decide. 7 Nz N z < N the The general topology of a filter with non-resonating node is shown in Fig. 1.2. There are three kinds of possible couplings here: Resonant-Resonant ( J 34 ), Resonant-Non Resonant ( J13 ) and Nonresonant-Nonresonant ( J12 ). A suitable topology is shown in Fig. 1.3. Fig. 1.2 A general coupling arrangement The generalized coupling coefficient is defined as: ki , j = Ji, j Bi ⋅ B j (1.12) where BNR ,i non-resonant susceptance Bi = Beq ,i resonant susceptance (1.13) where BNR ,i is the non-resonant susceptance and Beq,i is the resonant susceptance. Beq,i is defined as Beq ,i = 1 ∂Bris ,i 2 ∂w 8 (1.14) w = w0 Bris ,i ( w ) represents the total susceptance of the i -th resonator. In case of coupling with the external loads, a generalized external Q is similarly defined: QEXT ,i = where Bi J 2 0,i (1.15) G0 Bi is still given by (1.13) and G0 is the external conductance. The parameters k and QEXT have the same dimensions. The generalized coupling coefficients are tabulated in [29]. Let M i , j be the admittance inverter parameters, bk the frequency-invariant susceptances, and ck be the capacitances of the filter prototype. Then a resonant node is defined by the parameters ( bk , ck ) and NRN is associated with ( bk , 0 ) . Assuming Bn = B f 0 is the filter fractional bandwidth, the novel generalized parameters are evaluated as follows: • Resonant-resonant coupling: ki , j = B n • ci , c j (1.16) Resonant-nonresonant coupling: ki , j = Bn • M i, j M i, j ci ⋅ b j (1.17) Nonresonant-nonresonant Coupling: ki , j = M i, j bi ⋅ b j QEXT ,i can be evaluated as: 9 (1.18) QEXT ,i = QEXT ,i = Also, the resonant frequencies ci Bn , node i resonant M 0,2 i bi M 0,2 i , node i nonresonant (1.19) (1.20) f k and sign of susceptance bk of NRN are related by: 2 Bn ⋅ bk f 0 Bn ⋅ bk fk = − + + 4 2 ck c k (1.21) Once these are obtained, the microstrip implementation can be done with procedure outlined in [30]. Fig. 1.3 Filter with NRN – dotted circles represent frequency invariant susceptance; Solid circles represent capacitance in parallel to frequency invariant susceptance (normalized resonator); Black lines represent admittance inverters. Synthesis procedures involving non-resonating nodes were found in several literatures such as [31], [32], [33]. 10 1.5 Transfer and Reflection Polynomial Synthesis Synthesis of these polynomials is outlined in [9], [34], [35]. For any two-port lossless filter network composed of a series of N intercoupled resonators, the transfer and reflection functions (scattering parameters) (definitions can be found in [36]) may be expressed as a ratio of two N th degree polynomials: S11 ( w ) = FN ( w ) ε R EN ( w ) (1.22) PN ( w) ε EN ( w) (1.23) S 21 ( w ) = where s by w is the real frequency variable related to the more familiar complex frequency variable s = jw . For a Chebyshev Filtering Function, ε is a constant normalizing S21 ( w) to the equiripple level at w = ±1 as follows: = ε PN ( w ) 10 RL /10 − 1 FN ( w ) w=1 1 ⋅ (1.24) where RL is the prescribed return loss level in decibels and is assumed that all the polynomials have been normalized such that their highest degree coefficients are unity. or ε R = ε ε −1 2 if the function is fully canonical. εR =1 S11 ( w ) and S21 ( w ) share a common denominator EN ( w ) and the polynomial PN ( w ) contains n fz transfer function finite position transmission zeros. 2 2 1 , along with Using the law of energy conservation for a lossless network, S11 + S 21 = (1.22) and (1.23) we have, 11 = S 212 ( w ) 1 = 2 2 1 + ε CN ( w ) 1 (1 + jε C ( w) ) (1 − jε C ( w) ) N (1.25) N where CN ( w ) = FN ( w ) PN ( w ) (1.26) CN ( w ) is known as filtering function of degree N and has a form for the general Chebyshev characteristic: N CN ( w ) = cosh ∑ cosh −1 ( xn ) n =1 (1.27) where xn = and w − 1 wn 1 − w wn jwn = sn is the position of the n fz number of finite position transmission zeros in the complex s plane and the remaining N − n fz transmission zeros at w = ±∞ . For a prescribed set of TZs that make up the polynomial P ( w ) and a given equiripple return loss level, the reflection numerator polynomial F ( w ) may be built using efficient recursive technique [1] and then E ( w ) may be found using the Conservation of Energy principle for lossless networks. 1.6 Evaluation of Coupling Matrix The second step of the synthesis procedure is to calculate the values of coupling elements of a canonical coupling matrix from the transfer and reflection polynomials. The coupling matrix is a very special matrix that is extremely common in the literature on microwave filters. A coupling matrix, all by itself, can characterize a low pass prototype filter 12 network. Also, a coupling matrix can be modified using similarity transform, which is a purely mathematical technique, to obtain different configurations which are easy to realize with a practical circuit. Fig. 1.4 Canonical transversal array. (a) N resonators including direct source-load coupling M SL . (b) Equivalent circuit of the k th low-pass resonator in the array. Fig. 1.5 Series type low pass prototype with inter-resonator couplings. 13 A general coupling matrix is called transversal coupling matrix, which is shown in Fig. 1.4. The transversal coupling matrix comprises a series of N individual first-degree low pass sections, connected in parallel between the source and load terminations but not to each other. The direct source load coupling inverter M SL is included to allow fully canonical transfer functions to be realized (according to the minimum path rule, i.e., n fz max , the maximum number of finite position transmission zeros that may be realized by the network = N − nmin where nmin is the number of resonator nodes in the shortest route through the couplings in the network between the source and load terminations). In a fully canonical network, nmin = 0 and so n fz max = N , which is the degree of the network. Each of the N low-pass sections comprises one parallel-connected capacitor Ck and one frequency invariant susceptance Bk , connected through admittance inverters of characteristic admittances M Sk and M Lk to the source and load terminations, respectively (the values of all these parameters will be extracted through the synthesis procedure). The circuit of the k th lowpass section is shown in Fig. 1.4(b). Now the admittance parameter matrix [YN ] is derived in two ways. One is from the scattering parameters and the other is from the circuit elements of the transversal array network. By comparing them, elements of the coupling matrix can be derived in terms of the coefficients of the S11 ( w ) and S 21 ( w ) polynomials. 14 From the derived coefficients the eigenvalues λk and the associated residues r22k and r21k for k = 1, 2,..., N can be calculated using partial fraction expansion. Thus the following expression for [YN ] is obtained: [YN ] = 0 j K∞ K∞ N r 1 +∑ ⋅ 11k 0 k =1 ( s − jλk ) r21k r12 k r22 k (1.28) Now using ABCD matrices, converting the elements of low-pass resonator prototypes to individual y - parameter matrices, and then adding them together to form the complete [YN ] matrix the second expression is obtained: 0 [YN ] = j M SL M SL N M Sk2 1 + ⋅ ∑ 0 k =1 ( sCk + jBk ) M Sk M Lk M Sk M Lk 2 M Lk (1.29) This leads to the following relations: Ck = 1 , Bk = M kk = −λk , M SL = K ∞ , M Lk2 = r22 k and M Sk M Lk = r21k Therefore, = M Lk r21= k , M Sk r21k = , k 1, 2,..., N r22 k (1.30) The capacitors Ck are all unity and the frequency-invariant susceptances Bk ( = representing the self-couplings −λk , M 11 → M NN ), the input couplings M Sk , the output couplings M Lk , and the direct source-load couplings M SL are all defined, thus completing the reciprocal N + 2 transversal coupling matrix M representing the network. With this coupling matrix, M Sk are the N input couplings and they occupy the first row and column of the matrix from positions 1 15 to N . Similarly, M Lk are the N output couplings and they occupy the last row and column of M . All other entries are zero. The resulting coupling matrix is illustrated in Fig. 1.6. Fig. 1.6 N + 2 canonical coupling matrix for the transversal array. The matrix is symmetric with respect to the principal diagonal. 1.7 Coupling Matrix Reconfiguration Once the coupling matrix is obtained, series of similarity transformations can be applied to it to obtain different filter topologies, without affecting the filter response. This is extremely useful because this allows filter designers to conveniently change the topology to fit with practical realization. In [35], [37], several configurations such as folded form, arrow canonical form, wheel form and cul-de-sac form and ways to convert from one form to another have been explained. 16 1.8 Example of Synthesis To demonstrate, an 8th order bandpass filter with cutoff frequencies at 885MHz and 934MHz is synthesized. The synthesized filter polynomials (normalized to the unit frequency) are listed in Table 1.1 with decreasing order of frequency. Table 1.1 Filter Polynomials for an 8th Order Chebyshev Filter. ε 5.5793 FN ( w ) PN ( w ) EN ( w ) 1.0000 1.0000 1.0000 -2.5508 -4.6509 -2.5508 + 1.9856i 0.8225 8.0518 -1.1489 - 5.1480i 2.6956 -6.1556 7.8891 + 1.2723i -2.2980 1.7547 -5.1496 + 6.8045i -0.1900 -3.1443 - 6.0988i 0.6775 4.1799 + 0.2094i -0.1495 -1.0236 + 1.2853i -0.0073 -0.0517 - 0.3103i The polynomial response is shown in Fig. 1.7 (note: this is the response at the normalized low-pass frequency range). The response in bandpass real frequency is obtained by the following transformation: f = wB + ( wB ) 2 2 17 + 4 f 02 (1.31) where f is the real frequency and B is the bandwidth. The polynomial response in real frequency is shown in Fig. 1.8. Also the filter response generated from synthesized coupling matrix is shown in Fig. 1.9. Correspondingly, the N + 2 coupling matrix is obtained as the following: -0.3546 0.5022 -0.4767 0.3871 -0.2372 0.0858 -0.2952 0.2949 0 -0.3546 1.2494 0 0 0 0 0 0 0 0.5022 0 0.8472 0 0 0 0 0 0 0 0 0.0426 0 0 0 0 0 -0.4767 0.3871 0 0 0 -0.6102 0 0 0 0 0 0 0 0 -0.9184 0 0 0 -0.2372 0.0858 0 0 0 0 0 0 0 -0.9966 0 0 0 0 0 0 -1.0819 0 -0.2952 0.2949 0 0 0 0 0 0 0 -1.0830 0 0 0 0 0 0 0 0 0 10 0 0.3546 0.5022 0.4767 0.3871 0.2372 0.0858 0.2952 0.2949 0 S21 0 S11 Reflection, Rejection Loss -10 -20 -30 -40 -50 -60 -70 -80 -90 -100 -3 -2 -1 0 Frequency 1 2 3 Fig. 1.7 Polynomial response of the 8th order filter in normalized frequency domain. 18 10 S21 0 S11 -10 Reflection, Rejection Loss -20 -30 -40 -50 -60 -70 -80 -90 -100 8 8.5 9 Frequency 9.5 10 8 x 10 Fig. 1.8 Polynomial response of the 8th order filter in real frequency domain. 0 S21 S11 Reflection, Rejection Loss -20 -40 -60 -80 -100 -120 -3 -2 -1 0 Frequency 1 2 Fig. 1.9 Filter response generated from coupling matrix 19 3 CHAPTER 2 SYNTHESIS OF MICROWAVE DUPLEXERS 2.1 Introduction In addition to microwave filters, microwave duplexers are indispensable microwave components for communication systems. They are typically used to connect the RX and TX filters of a transceiver to a single antenna through a suitable three-port junction. The rapid development of mobile communication systems over the past decade has stimulated the need for duplexers with compact size as well as high selectivity. To meet these stringent requirements, different synthesis techniques have been proposed. Specifically a very convenient synthesis method was discussed in [38]. Here the general procedure of it is described. 2.2 General Structure A common duplexer configuration is shown below. A TX and an RX filter are connected via a transformer with a turn ratio of n :1 and a susceptance b0 . Fig. 2.1 Duplexer configuration 20 2.3 Polynomial Synthesis A duplexer is a lossless three port network and four polynomials are required to define its scattering parameters in the low pass normalized domain. = S11 n0 N ( s ) = , S 21 D (s) p0t Pt ( s ) = , S31 D (s) In (2.1), the highest degree coefficients of together with constants p0 r Pr ( s ) D (s) (2.1) N , D, Pt , Pr are imposed to be equal to 1 no , p0t , p0 r respectively. The roots of D ( s ) represent poles of the network and roots of N ( s ) represent the transmission zeros in the complex plane. The synthesis of the duplexer is carried out in a normalized frequency domain defined = Ω by the usual low pass to band pass frequency transformation Fig. 2.2 Frequency mapping: f 0 = , B f1, RX f 2, RX = The frequency mapping and definitions of ( f0 B ) ( f f 0 − f0 f ) . f 2,TX − f1, RX . f 0 and B are shown in Fig. 2.2. The passband limits of RX filter are f1,RX and f 2,RX while those of the TX filter are represented by f1,TX , f 2,TX respectively. The two low pass prototype RX and TX filters are characterized through their characteristic polynomials which are related to their scattering parameters, which are given in (2.2). 21 S11TX = PTX ( s ) = ETX ( s ) TX = S 21 p0TX PTXn ( s ) ETX ( s ) F (s) = RX ERX ( s ) RX 11 S S 21RX = In (2.2), the polynomials FTX ( s ) ETX ( s ) PRX ( s ) = ERX ( s ) (2.2) p0 RX PRXn ( s ) ERX ( s ) FTX and ETX have degree npTX (order of TX filter) and FRX , ERX have degree npRX (order of RX filter). The highest degree coefficients of these polynomials are equal to 1. The polynomials given by PTX and PRX have the highest degree coefficients p0TX and p0RX which determine the return loss level at passband limits. The TX and RX transmission zeros then define the normalized polynomials PTXn and PRXn . By analyzing the three port network and computing the admittances, Ny jb S S + DTX DRX + DRX STX 2 = yin n= n 2 0 TX RX Dy STX S RX where (2.3) STX , S RX , DTX , DRX are given by ETX + FTX 2 E + FRX S RX = RX 2 E − FTX DTX = TX 2 E − FRX DRX = RX 2 STX = Finally we get the following expressions: 22 (2.4) D D +S S = N ( s ) STX S RX − n 2 TX RX 2 TX RX 1 − jn b0 2 1 − jn b0 n0 = 1 + jn 2b0 = D ( s ) STX S RX + n 2 DTX DRX + STX S RX 1 + jn 2b0 (2.5) n = Pt ( s ) P= p0TX TXn S RX , p0 t 2 1 + jn b0 n = Pr ( s ) P= p0 RX RXn STX , p0 r 2 1 + jn b0 Using the transmission zeros of TX and RX filters and the reflection zeros at the input port of the duplexer, the duplexer characteristic polynomials are generated. Then using an iterative method by using lossless conditions, the polynomials PTX , coefficients PRX and also their p0TX and p0RX are obtained. In this iterative method the goal is to satisfy lossless condition preserving real transmission zeros and reflection zeros. Once these polynomials are obtained, the TX and RX filters polynomials are extracted using final values of STX , S RX , DTX , DRX and polynomial fitting. Once the characteristic polynomials are obtained the individual TX and RX filters and the junction can be realized using Waveguides, Cavity resonators or Non-Resonant nodes. For the cavity type design, coupling matrices need to be obtained. While for the non-resonant node realization, admittance parameters and dangling resonator frequencies are to be determined. 2.4 Example of Synthesis A duplexer with RX passband from 880MHz to 915MHz and TX passband from 925MHz to 960MHz is synthesized. RX Filter is of order seven and TX Filter is of order eight. RX finite 23 transmission zeros are taken at 850, 925 and 913 MHz while TX finite transmission zeros have been selected at 898, 911.4, 915 and 969MHz. The response of the synthesized duplexer polynomial is shown in Fig. 2.3. Synthesized duplexer polynomials are listed in Table 2.3. Also, the extracted TX and RX Filter polynomials are given in Table 2.1 and Table 2.2. 20 0 S11 Scattering Parameters in dB -20 S21 S31 -40 -60 -80 -100 -120 -140 -160 700 750 800 850 900 Frequency 950 1000 Fig. 2.3 Synthesized Duplexer polynomial response 24 1050 1100 Table 2.1 Extracted TX Filter Polynomials F (s) p0 2.5729e-004 E (s) 1.0000 1.0000 P (s) 1.0000 0.7476 - 4.4213i 0.1086 - 4.4213i 0 - 0.3826i -7.8915 - 2.8652i -8.1651 - 0.4299i 0.8315 -4.3972 + 7.2898i -0.6960 + 8.1757i 0 + 0.2068i 3.7094 + 3.4598i 4.8206 + 0.5930i -0.0130 1.4865 - 1.0193i 0.2845 - 1.7009i -0.1355 - 0.3441i -0.3481 - 0.0760i -0.0394 + 0.0059i -0.0103 + 0.0376i -0.0001 + 0.0017i 0.0016 + 0.0005i Table 2.2 Extracted RX Filter Polynomials 7.3187e-004 E (s) F (s) p0 P (s) 1.0000 1.0000 1.0000 0.1186 + 3.6918i 0.7922 + 3.6918i 0 + 1.3574i -5.4723 + 0.3845i -5.1655 + 2.4742i 0.7503 -0.4868 - 4.1685i -2.9047 - 3.3865i 0 - 0.0776i 1.7339 - 0.3034i 1.0221 - 1.5925i 0.0960 + 0.3859i 0.4132 + 0.1065i -0.0416 + 0.0141i 0.0038 + 0.0459i -0.0007 - 0.0017i -0.0017 + 0.0007i 25 Table 2.3 Duplexer Polynomials N (s) D (s) Pt ( s ) Pr ( s ) 1.0000 1.0000 1.0000 1.0000 -1.5000 + 0.7267i 3.2670 + 0.7267i 0.4554 - 3.3091i 0.4281 + 3.0639i 2.6305 - 1.0900i 6.8423 + 2.3874i -3.0748 - 1.2551i -1.2767 + 1.0664i -3.9458 + 1.8946i 11.1661 + 4.9989i -0.7701 - 1.5344i 0.0110 + 6.5595i 2.5074 - 2.8418i 13.1200 + 8.1490i -5.2668 - 0.9836i -12.5978 + 2.6996i -3.7611 + 1.7976i 13.2342 + 9.5834i -1.8198 + 4.5704i -3.9035 -10.8536i 1.0509 - 2.6964i 10.0267 + 9.6647i 2.0718 + 1.2250i 5.4041 - 2.7098i -1.5763 + 0.7615i 6.4497 + 7.3658i 0.4402 - 0.5458i 1.0818 + 1.6577i 0.1869 - 1.1422i 3.0628 + 4.7883i -0.0848 - 0.0905i -0.3157 + 0.2589i -0.2804 + 0.1442i 1.1545 + 2.3448i -0.0105 + 0.0075i -0.0365 - 0.0361i 0.0113 - 0.2163i 0.2804 + 0.9428i 0.0003 + 0.0006i 0.0023 - 0.0028i -0.0169 + 0.0112i 0.0391 + 0.2720i 0.0000 - 0.0000i 0.0001 + 0.0001i 0.0001 - 0.0169i -0.0025 + 0.0611i -0.0001 + 0.0003i -0.0019 + 0.0090i -0.0000 - 0.0005i -0.0004 + 0.0010i 0.0000 + 0.0000i -0.0000 + 0.0001i -0.0000 - 0.0000i -0.0000 + 0.0000i Once the extracted filter polynomials are obtained, coupling matrices can be evaluated and the duplexer can be implemented with cavity resonators or waveguides [38], [39], [40], [41]. 26 CHAPTER 3 SYNTHESIS OF DUAL BAND FILTERS 3.1 Introduction In modern telecommunication systems, the need for devices that can work at multiple frequency bands simultaneously is becoming increasingly important in order to reduce size and power requirement. For example, the employment of a dual band filter eliminates the requirement of using two filters working at different bands by taking care of both bands, which leads to significant cost reductions. Also, being one unit they are generally smaller than two filters combined together. Several literatures are available on dual band filter synthesis and design such as [42], [43], [44], [45], [46], [47]. In this chapter, the procedure to synthesize a dual band filter will be discussed. 3.2 Polynomial Synthesis The characteristic polynomials for a dual band filter are defined similar to that of a single passband filter. However, there exists no direct iterative method to determine the polynomials given the transmission zeros, passbands and return loss levels in either bands. Therefore, an alternative method is initially described in [48] and is further studied in this chapter. The procedure is summarized below: • • First an initial set of values of poles and zeros is assigned in the passbands and stopbands. Next initial filter function is constructed: N A ( w) C= ( w) = B ( w) ∏(w − p ) i =1 M i ∏(w − z ) i =1 27 i (3.1) where p and • z denote the initial poles and zeros while N , M denote their numbers. The roots of the following expression are obtained: dC ( w ) = B ( w ) A′ ( w ) − A ( w ) B′ ( w ) dw • (3.2) The complex roots are discarded and the real roots are arranged including the passband and stopband edges. Let the roots in the passbands be α and the roots in the stopband be β . Each zero (pole) should lie between two roots or stopband (passband) edge. • The zeros and poles are now updated using the following equations: pl′ = pl α l −1C0 (α l −1 ) + α l C0 (α l ) − α l −1α l C0 (α l −1 ) + C0 (α l ) pl C0 (α l −1 ) + C0 (α l ) − α l C0 (α l −1 ) + α l −1C0 (α l ) zl β l −1C0 ( β l ) + β l C0 ( β l −1 ) − β l −1β l C0 ( β l −1 ) + C0 ( β l ) zl′ = zl C0 ( β l −1 ) + C0 ( β l ) − β l −1C0 ( β l −1 ) + β l C0 ( β l ) (3.3) • This procedure is continued until the values of poles and zeros converge. • Stopband edges are moved in order to bring equal return loss in lower and higher passbands (this will be explained later). • Also complex transmission-zeros can be introduced to control return loss levels. • Once the poles and zeros are found, F= (s) N ∏ ( s − jp ) i i =1 P= (s) M ∏ ( s − jz ) k k =1 • (3.4) E ( s ) is obtained by taking the roots of ε R P ( s ) + ε F ( s ) and mapping right half plane roots to the left half plane. 28 Once the filter coefficients are obtained, the coupling matrix synthesis is similar to that of a single band filter. It is noted that the return loss level in the passband depends heavily on the width of the passbands. Any dual band filter with unequal passbands tends to have a different return loss level in the two bands. Through our study, this challenge is overcome by placing suitable transmission zeros in the stopbands or by using complex transmission zeros in the passband if necessary. These techniques are illustrated below. Narrowing of Passbands Effect of narrowing transmission zeros towards passbands on return loss levels is illustrated in Fig. 3.1. It is to be noted here that in only one of the band edge, value of ε [34] is computed. Any asymmetry in the filter causes the other band to have a return loss other than the specified one. The band in which ε is computed is not affected since ε automatically compensates it. Each of the stopband edge has its own effect (not equal, generally) on how much it changes the return loss level, and the positions of stopband edges (and transmission zeros) are not unique in order to get equal return loss responses. Therefore this process is analogous to a tuning process to get the desired return loss level as well as the steepness of passband roll-off. Fig. 3.2 and Fig. 3.3 further explain this effect. The positions of stopband edges (and transmission zeros) are not unique in order to get equal return loss levels. 29 Fig. 3.1 Effect of narrowing of passbands on return loss levels. Fig. 3.2 Narrowing done around upper passband. 30 Fig. 3.3 Narrowing done around lower passband edges. Effect of narrowing towards the passband on the higher frequency band is shown above. In Fig. 3.2 the stopband edges have been narrowed towards the upper passband, which pushes the return loss levels up in the upper passband. On the other hand, in Fig. 3.3 the stopband edges have been pushed towards lower passband, which should push up the return loss levels in the lower passband. However it is to be noted that during the synthesis procedure ε has been evaluated in the lower passband and hence return loss level is fixed in lower passband. Therefore, the change is again observed in the upper passband return loss level, which is not restricted by the value of ε and is pushed down. Additional Transmission Zeros Addition of a finite transmission zero to a dual band filter with equal return loss levels in the two bands will make it unequal. In Fig. 3.4 an addition transmission zero at w = 1.6 has been imposed. The two filtering bands have the same order. As we can see, the higher filter 31 band having transmission zero at 1.6 has higher level of return loss in the passband. It can be concluded the ability to place transmission zeros at finite frequencies gives designers great flexibility to design filters that have certain rejection level at a pre-specified frequency band. Fig. 3.4 Effect of additional transmission zero at w = 1.6 Complex Transmission Zeros Finally we find that a transmission zero at any point brings down the value of the polynomial P as given in (3.4) at that point. Therefore a complex conjugate transmission zero pair can be introduced whose real part lies in the passband to adjust the return loss level of that passband. A smaller imaginary part has more effect while a large imaginary part of the complex transmission zero has little effect on it. This is illustrated in Fig. 3.5. 32 Fig. 3.5 A fifth order filter with three real transmission zeros (solid lines). The return loss levels are equalized with complex transmission zero pair at 0.825 ±1.6j (dashed lines). 33 CHAPTER 4 DESIGN OF DUAL BAND DUPLEXERS 4.1 Introduction In this chapter we propose a method to design a Microwave Duplexer that can work in two different frequency bands simultaneously [44]. The discussion will involve design of dual band filters followed by the combination of the dual band filters to form a Dual Band Duplexer. 4.2 Synthesis Procedure The proposed dual-band microwave duplexer is composed of two dual-band ﬁlters with the two input ports connected through a three-port junction. The two dual-band filters (RX and TX) can be characterized separately from the duplexer through several techniques [48] [43] [42]. In general, the dual-band filter employed in the analysis can be seen as a single-band filter with some of its transmission zeros falling in the passband, separating the passband into two bands. In our analysis the passband limits of the RX ﬁlter are represented by f1RX , f 2RX , f3RX , f 4RX while those of the TX ﬁlter are f1TX , f 2TX , f3TX , f 4TX . The duplexer is synthesized in a normalized frequency domain with the suitable lowpass = Ω ↔ bandpass frequency transformation ( f0 B )( f f 0 − f 0 f ) as shown in Fig. 4.1, where f1RX f 4TX (4.1) f 0 and B are defined as follows: f0 = = B f 4TX − f1RX 34 Fig. 4.1 Frequency transformation for dual band duplexers. Once the lowpass band limits are obtained, the procedure described in [3] was used to obtain the filter polynomials for both the TX and RX filters. The two lowpass prototype ﬁlters in the normalized frequency domain can be characterized separately from the duplexer through their characteristic polynomials. The characteristic polynomials of the TX and RX filters are related to their scattering parameters as follows: = S11TX FTX ( s ) TX PTX ( s ) p0TX PTXn ( s ) = , S 21 = ETX ( s ) ETX ( s ) ETX ( s ) FRX ( s ) RX PRX ( s ) p0 RX PRXn ( s ) S = , S 21 = = ERX ( s ) ERX ( s ) ERX ( s ) (4.2) RX 11 where ETX ( s ) , ERX ( s ) are polynomials of degree npTX , npRX respectively. All of these polynomials have unity coefficient for the highest degree. The major difficulty in evaluating the polynomials of dual-band duplexer is to make the responses of the dual-band RX/TX filters equal ripple in both of their passbands. As we know, the return loss level in the passbands depends heavily on the bandwidth of the passbands. A narrower bandwidth generally gives lower return loss level, and vice versa. Therefore, the dual band filters with unequal passbands tend to have different return loss level in their two bands. This challenge can be overcome by adjusting the transmission zeros positions located in the 35 stopbands or adding complex transmission zeros within the passbands as discussed in Chapter 3. • Evaluation of Duplexer Polynomials The derivation of duplexer polynomials N ( s ) , D ( s ) , Pt ( s ) , Pr ( s ) is done using the reflection zeros at the input port of the duplexer and the transmission zeros of the TX and RX Filters. Assuming lossless overall duplexer and unitary condition of the scattering matrix we have, D (= s ) D* ( − s ) n0 N ( s ) N * ( − s ) + p0 r Pr ( s ) Pr * ( − s ) + p0t Pt ( s ) Pt * ( − s ) 2 2 2 (4.3) N ( s ) N * ( − s ) depends only on the imposed reflection zeros with n0 = 1 . Now the evaluation is done in the following iterative steps: 1) Initialization: The RX and TX filters are synthesized independently of the duplexer 0 0 0 with general Chebyshev characteristics.( i.e., the polynomials FTX , ETX , PTX and FRX , ERX , PRX are generated given the number of poles ( npTX , 0 0 0 npRX ), the return loss in the two channels, and the transmission zeros of the two filters. For junctions causing additional zero, an approximate zero has to be added. An initial estimate of STX and S RX are available from the above polynomials. 2) Iteration begin: Pt and Pr are evaluated by polynomial convolution Pt = conv ( PTXn , S RX ) , Pr = conv ( PRXn , STX ) . 36 3) Evaluation of p0t and p0r : The required return loss in the two channels ( RLTX and RLRX ) is imposed at the normalized frequencies, N ( j) 2 2 2 N ( j ) + p0 r ⋅ Pr ( j ) + p0t ⋅ Pt ( j ) 2 N (− j) 2 2 2 = 10− RLTX (4.4) 10 2 2 N ( − j ) + p0 r ⋅ Pr ( − j ) + p0t ⋅ Pt ( − j ) 2 s = ±j. 2 2 = 10− RLRX 10 (4.5) p0t and p0r are obtained by solving the above system of linear equations. Applying spectral factorization technique to (4.3) D ( s ) is evaluated from its roots. 4) New estimation ( aN ( s ) + bD ( s ) ) of STX and S RX : The roots of the polynomial 1 − jn 2 ⋅ b0 and 2= STX ( s ) ⋅ S RX ( s ) are computed with a = b= 1 + jn 2 ⋅ b0 . The roots are arranged in ascending order of imaginary parts. Let npRX _ LB , npRX _ UB be the number of poles in lower and upper bands of RX filter respectively. Similarly, let npTX _ LB , npTX _ UB be the number of poles in lower and upper bands of TX filter. Then First npRX _ LB roots are assigned to S RX . Next npTX _ LB roots are assigned to STX . Next npRX _ UB roots are assigned to S RX and next npTX _ UB roots are assigned to 5) Once the new STX . STX and S RX are obtained, step 2 to step 4 are iterated until convergence is achieve to a high degree. 37 Computation of RX and TX Filter Polynomials p0TX p0 RX Now the polynomials [38]. Let 1 + jn 2 b0 = p0t n 2 1 + jn b0 = p0 r n (4.6) DTX and DRX defined similarly as in (2.4) are evaluated same as zSTX and zS RX be the roots of STX and S RX respectively. The values of DTX ( zSTX ) and DRX ( zS RX ) are computed from the following expressions: D ( zS ) D ( zSTX ) = A ⋅ DTX ( zSTX ) S RX ( zSTX ) ⇒ DTX ( zSTX ) =TX A ⋅ S RX ( zSTX ) D ( zS ) D ( zS RX ) = A ⋅ DRX ( zS RX ) STX ( zS RX ) ⇒ DRX ( zS RX ) =RX A ⋅ STX ( zS RX ) (4.7) where A is given by n 2 (1 + jn 2 b0 ) . Now from the knowledge of the degree of DRX and DTX one can find DRX and DTX using polynomial interpolation. = DTX ( s ) polyfit ( zSTX , DTX ( zSTX ) , npTX − 1) = DRX ( s ) polyfit ( zS RX , DRX ( zS RX ) , npRX − 1) (4.8) npRX npRX , LB + npRX ,UB and= npTX npTX , LB + npTX ,UB . where= Finally, the filter polynomials are obtained as: F= STX ( s ) − DTX ( s ) TX ( s ) E= STX ( s ) + DTX ( s ) TX ( s ) F= S RX ( s ) − DRX ( s ) RX ( s ) E= S RX ( s ) + DRX ( s ) TX ( s ) 38 (4.9) 4.3 Implementation in Cavity type and Non-Resonating node type To verify the dual-band duplexer synthesis technique presented above, a prototype duplexer is synthesized using both the non-resonating node topology and the cavity-resonator topology [38]. The specifications of the duplexer are listed as the following: RX Passbands: 800MHz to 807MHz, 850MHz to 857MHz TX Passbands: 815MHz to 822MHz, 865MHz to 872MHz Center Frequency: 835.22 MHz Fig. 4.2 Proposed dual-band duplexer: (a) topology with non-resonating nodes, (b) topology with cavity resonators. 39 • Dual-band Duplexer with non-resonating node topology The topology of the dual-band duplexer with non-resonating nodes is shown in Fig. 4.2(a). Both the RX and TX filters are fourth-orders with two attenuation poles. Following the proposed synthesis procedure, the duplexer characteristic polynomials are given below. The polynomial coefficients are reported in descending order. N = [1, -0.08029i, 2.0807, -0.1405i, 1.4401, -0.07284i, 0.3786, 0.01046i, 0.03211]; D = [1, 1.1634-0.0803i, 2.7574-0.1083i, 1.9599-0.2122i, 2.1502-0.1511i, 0.90040.1306i, 0.5544-0.0504i, 0.0999-0.0204i, 0.0327-0.0029i]; Pr = [1, 0.2866 - 0.6026i, 1.1758 -0.1140i, 0.1921-0.3167i, 0.4208-0.0278i, 0.0192 + 0.0047i, 0.0214 - 0.0034i] Pt = [1, 0.2950+0.4282i, 1.2800+0.0602i, 0.2089+0.1365i, 0.4556-0.0048i, 0.01770.0543i, 0.0159+0.0018i]; p0r = 0.2439; p0r = 0.2712 The S-parameters of the synthesized duplexer are shown in Fig. 4.3. The corresponding extracted J (admittance inverter parameter) and B (susceptance) values are listed below. A) RX Filter: Inverter J values: 0.5429, 1.0000, 0.5277, 1.0000, 1.6988, 3.4385 and 0.5431. Inline Susceptance: 1.4937, 0.7449, -8.6256, -0.9618 Susceptance of dangling resonators: 0.5918, -0.2178 Resonator Frequency (in MHz): 783.2, 841.9, 843.1, 870.06 B) TX Filter: 40 Inverter J values: 0.5357, 1.0000, 0.4973, 1.0000, 1.9835, 3.9943 and 0.5354. Inline Susceptance: -1.5686, -0.7128, 11.7735, 1.0060 Susceptance of dangling resonators: -0.6052, -0.1371 Resonator Frequency (in MHz): 893.6, 857.3, 830.3, 799.79 With the above extracted parameters, the duplexer can be easily implemented using transmission lines. Scattering Parameters (dB) 20 0 -20 -40 S11 -60 S31 -80 -100 700 S21 750 800 850 900 Frequency 950 1000 Fig. 4.3 Synthesized response of the dual-band duplexer with non-resonating node topology (S21 for RX channel, S31 for TX channel). Dual-band Duplexer with cavity resonator topology Similarly, the dual-band duplexer with cavity resonator as given in Fig. 4.2 (b) is also synthesized following the procedure described previously. The duplexer characteristic polynomials are given below. Junction capacitance is chosen as 0.7. 41 N = [1, -0.7000 - 0.0803i, 2.0807 + 0.0562i, -1.4565 - 0.1405i, 1.4401 + 0.0983i, 1.0081 - 0.0728i, 0.3786 + 0.0510i, -0.2650 - 0.0105i, 0.0321 + 0.0073i, -0.0225 - 0.0000i]; D = [1, 1.8199 - 0.0803i, 3.4917 - 0.1615i, 3.7966 - 0.2817i, 3.3984 - 0.2985i, 2.3483 - 0.2298i, 1.1349 - 0.1422i, 0.4870 - 0.0541i, 0.1005 - 0.0178i, 0.0232 - 0.0021i]; Pr = [1, 0.2764 - 0.5418i, 1.0522 - 0.0820i, 0.1614 - 0.2621i, 0.3347 - 0.0193i, 0.0135 + 0.0001i, 0.0161 - 0.0021i]; Pt = [1, 0.2836 + 0.3613i, 1.1371 + 0.0271i, 0.1702 + 0.0974i, 0.3582 - 0.0081i, 0.0117 - 0.0390i, 0.0119 + 0.0010i]; p0t = 0.2780; p0r = 0.3134 The extracted TX and RX Filters have the following N+2 (including the source and the load) coupling matrices respectively. M TX 0.6574 0 0 0 0 0 0.6574 − 0.1630 0.5744 − 0.0259 0.4022 0 0 0.5744 − 0.3174 − 0.1296 0 0 = 0.0259 − 0.1296 − 0.1903 0.6158 0 0 0 0.4022 0 0.6158 − 0.2450 0.5257 0 0 0 0.5257 0 0 M RX 0 0.6716 0 = 0 0 0 0.6716 0 0 0 0.1723 0.5720 0.0124 0.4381 0.5720 0.2630 − 0.1648 0 0.0124 − 0.1648 0.1283 0.6086 0.4381 0 0 0 0.6086 0 42 0.2658 0.5325 0 0 0 0 0.5325 0 To validate the proposed synthesis method for dual-band duplexers, the cavityresonator based duplexer presented above is implemented. As shown in Fig. 4.4 and Fig. 4.5 both the RX and TX filters of the duplexer are fourth-orders. The transmission zeros in the RX/TX channels are obtained with a quadruplet. The RX/TX junction at the input is realized by adding an extra resonator. Fig. 4.4, Fig. 4.5, Fig. 4.6 show the simulated insertion losses ( S12 for the RX channel and S13 for the TX channel) and return loss ( S11 at the input) of the cavity resonator based duplexer with optimized coupling coefficients. For comparison, the synthesized responses of the duplexer are also plotted in this figure. The simulated results agree well with the synthesis as expected, confirming the feasibility of the proposed synthesis technique for dual-band duplexers. 0 Syntehsized Simulated -20 S12 (dB) -40 -60 -80 -100 700 750 800 850 900 Frequency (MHz) 950 Fig. 4.4 Insertion loss at RX channel ( S12 ). 43 1000 0 Synthesized Simulated S13 (dB) -20 -40 -60 -80 700 750 900 800 850 Frequency (MHz) 950 1000 Fig. 4.5 Insertion loss at TX channel ( S13 ). 0 Synthesized Simulated S11 (dB) -20 -40 -60 -80 -100 700 750 950 900 850 800 Frequency (MHz) Fig. 4.6 Return loss at Input ( S11 ). 44 1000 CHAPTER 5 MODEL ORDER REDUCTION OF MICROWAVE DUPLEXERS 5.1 Introduction Often to meet specific requirements and lack of flexibilities in microwave duplexer designs lead to very high order of microwave filters. In practice, it is convenient to reduce the order of such duplexers that matches the original high order duplexer to a high degree in a specific band of concern. One method to obtain coupling matrix elements from filter response by genetic algorithm was described in [49]. The Cauchy method deals with estimation of a function by a ratio of two polynomials. Given the values of the function and its derivatives at a few points the order of the polynomials and their coefficients can be evaluated. Once the coefficients of the two polynomials have been estimated, they can be used to generate the parameter over the entire band of interest. It is particularly useful when the values of the function have been measured and a mathematical model for the function has to be estimated. However any measurement is accompanied by addition of noise. It is shown that for low level of noise, performance of Cauchy method is very good over the entire sample space. 5.2 Formulation of Cauchy Method The Cauchy method approximated a system function H(s) with a ratio of two polynomials. The procedure was described in [50] and is discussed below. P H (s) ≈ A( s) ∑a s k k k =0 = Q B (s) ∑ bk s k k =0 45 (5.1) Here the information is assumed to be N measured values of the function (H) at frequency points s j , j = 1, 2,3...N . In such case, the Cauchy Problem is, Given H ( s j ) for j=1, 2, 3…N, find P, Q, {ak, k = 0, 1, 2,..P} and {bk, k=0, 1, 2..Q} Then from (5.1), A(s j ) = H (s j ) B (s j ) A(s j ) − H (s j ) B (s j ) = 0 (5.2) Now using the polynomial expansions for A ( s ) and B ( s ) , a0 + a1 s j + a2 s 2j + + aP s Pj − H j b0 − H j b1 s1j − − H j bQ s Qj = 0 (5.3) for j = 1, 2, 3 N . Writing in matrix form, a =0 b [C ] (5.4) where 1 s1 1 s C = [ ] 2 1 sN s1P s2P sNP − H1 −H2 −H N − H1 s1 − H 2 s2 − H N sN − H1 s1Q − H 2 s2Q − H N sNQ (5.5) where [ a ] = [ a1 , a2 , a3 , , aP ] T (5.6) T (5.7) and [b] = b1 , b2 , b3 , , bQ 46 The order of matrix C is N × ( P + Q + 2 ) . A singular value decomposition (SVD) gives us the required coefficients for [U ][Σ][V ] T 0 = a and b . (5.8) According to the Theory of Total Least Squares, a b = ( const.) * [V ]P +Q + 2 (5.9) That is, the elements in the last column of V give us the solution. 5.3 Application in General Polynomial Function: Cauchy method has been applied on a function as below: 4 H (s) = ∑ ks k (5.10) k =0 5 ∑ (k + 1)s k k =0 This ratio is evaluated to find the exact values at 21 points in the range of s=2 to s=4. Different levels of noise are added to this exact value to see its effect on the estimation. The estimation accuracy is observed in the following figures. It can be seen that with low noise levels Cauchy method gives excellent estimation of this kind of functions. 47 0.5 Reference H Estimated H without noise Estimated H with Noise 0.45 0.4 Value of H 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 2 2.2 2.4 2.6 2.8 3 3.2 Frequency 3.4 3.6 3.8 4 Fig. 5.1 SNR = 5 0.5 Reference H Estimated H without noise Estimated H with Noise 0.45 0.4 Value of H 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 2 2.2 2.4 2.6 2.8 3 3.2 Frequency Fig. 5.2 SNR = 20 48 3.4 3.6 3.8 4 0.5 Reference H Estimated H without noise Estimated H with Noise 0.45 0.4 Value of H 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 2 2.2 2.6 2.4 2.8 3.2 3 Frequency 3.4 3.6 3.8 4 Fig. 5.3 SNR = 50 dB. 5.4 Application in General Microwave Filter function As mentioned in previous chapters, a general two port network can be characterized by a set of two parameters of its scattering matrix, namely the transmission and reflection coefficients, S11 and S21 . These parameter share the same poles. Therefore to generate the polynomial models for this case it is necessary to ensure the use of a common denominator. The filter response is then modeled as, n n S11 ( s ) ∑ a1k s k ∑ bk s k ≈ S11 ( s ) = = k 0= k 0 (5.11) nZ n = S21 ( s ) ∑ a2 k s k ∑ bk s k = k 0= k 0 (5.12) where n is the filter order and ≈ S 21 ( s ) nZ is the number of finite transmission zeros. This can be represented in a matrix equation: 49 a − S11Vn 1 = a2 VnZ − S 21Vn b S11 = diag {S11 ( si )} Vn 0 N ×n 0 N ×nZ a b X ] [ 0] [= (5.13) S 21 = diag {S 21 ( si )} V is a Vandermonde matrix defined as, s1m m s [Vm ] = 2 m sN s1m −1 s12 s1 s2m −1 s22 s2 sNm −1 sN2 sN 1 1 1 (5.14) The Exact polynomial model used for simulation is tabulated below and the exact polynomial response and estimated response are shown in Fig. 5.4. More details on the procedure can be found in [51]. Table 5.1 Low Pass Polynomial Coefficients ak( 2) k ak(1) 0 - 0.0244 – 0.0110j 0.1123 – 0.1158j 1 - 0.0453 – 0.2532j 0.6002 – 0.5472j 2 - 0.0745 – 0.1796j 1.3800 – 1.3009j 3 -0.1619 – 1.5168j 4 0.6856 – 0.2242j 2.4082 – 1.5359j 5 -0.0139 – 1.3188j 1.9356 – 1.3007j 6 1.0000 1.0000 0.1133 + 0.0545j 50 bk 2.4933 – 1.9912j Filter Scattering Parameter Extraction 10 |S | 11 Estimated |S | 11 0 |S | Scattering Parameters in dB 21 Estimated |S21| -10 -20 -30 -40 -50 -60 -2 -1.5 -1 -0.5 0 Frequency 0.5 1 1.5 2 Fig. 5.4 Example of polynomial estimation for a 6th order filter. 5.5 Application in Microwave Duplexers Application of the Cauchy Method on Microwave Duplexer was detailed in [52]. The parameters S11 , S21 and S31 can be measured from Lab Setups for a diplexer. After measurement of the scattering parameters, the filters can be approximated using Cauchy Method. ∑ ∑ S (s) ∑ K= (s) = S (s) ∑ nTX + nRX S11 ( s ) K= s = TX ( ) S 21 ( s ) RX k =0 mTX + mRX k =0 11 nTX + nRX k =0 mTX + mRX 31 k =0 Equation (5.15) can be written in matrix form as: 51 ak s k ck(1) s k ak s k ck( 2) s k (5.15) S21VnTX + nRX S31VnTX + nRX − S11VmTX + nRX 0 a (1) = c − S11VmRX + nTX ( 2) c a M ] c(1) 0 [= c( 2) 0 (5.16) where V is the Vandermonde Matrix defined as: s1m m s [Vm ] = 2 m sN s1m −1 s2m −1 sNm −1 s12 s22 sN2 s1 s2 sN 1 1 1 (5.17) As an example, the exact and estimated polynomial responses are shown in Fig. 5.5, which agree with each other very well. Scattering Parameters in dB 20 Diplexer Scattering Parameter Extraction 0 -20 -40 -60 -80 |S | 11 Estimated |S | 11 |S21| -100 Estimated |S21| |S31| Estimated |S31| -120 -2 -1.5 -1 -0.5 0 Frequency 0.5 1 Fig. 5.5 Exact and estimated polynomial response for duplexers. 52 1.5 2 5.6 Conclusion As we can see, Cauchy Method is an excellent mathematical procedure that uses the concept of Total Least Squares to estimate the same or reduced order polynomial models of measured parameters. Even with noisy data, the models obtained do not deviate far from exact models. 53 CHAPTER 6 CONCLUSION AND FUTURE WORK In this work methods to synthesize microwave filters and duplexers have been discussed. Especially, a method to synthesize a dual band microwave duplexers is proposed. In addition to using dual band filters to synthesize dual band duplexer, a multiband filter can be used to synthesize a multi band duplexer. However this will add the complexity of the synthesis and require high degree of optimization at later stages of design, which can be studied in future work. Also, waveguide is another way to design microwave filters and duplexers. Synthesized dual band duplexers can be designed using waveguides which can sustain high power. Once the designs of filters and duplexers are complete, measurements can be done on the prototypes and the scattering parameters can be measured with vector network analyzer at different frequencies. These measurements can be used to obtain a lower order polynomial model using Cauchy method in order to obtain filters of desired orders that roughly behave the same as the original ones. 54 BIBLIOGRAPHY [1] R. E. Collin, Foundations for Microwave Engineering, New York: IEEE Press Series on Electromagnetic Theory; Wiley-Interscience, 2001. [2] H. J. Orchard and G. Temes, "Filter Design Using Transformed Variables," Circuit Theory, IEEE Transactions on, vol. 15, no. 4, pp. 385-408, December 1968. [3] R. Gomez-Garcia and M. Sanchez-Renedo, "Microwave Dual-Band Bandpass Planar Filters Based on Generalized Branch-Line Hybrids," Microwave Theory and Techniques, IEEE Transactions on, vol. 58, no. 12, pp. 3760-3769, December 2010. [4] H. Zhang and K. Chen, "Miniaturized Coplanar Waveguide Bandpass Filters using Multisection Stepped-Impedance Resonators," Microwave Theory and Techniques, IEEE Transactions on, vol. 54, no. 3, pp. 1090-1095, March 2006. [5] R. Zhou, I. Mandal and H. Zhang, "Microwave Bandpass Filters with Tunable Center Frequencies and Reconfigurable Transmission Zeros," Microwave and Optical Technology Letters, vol. 55, no. 7, pp. 1526-1531, July 2013. [6] R. Levy and S. B. Cohn, "A History of Microwave Filter Research, Design, and Development," Microwave Theory and Techniques, IEEE Transactions on, vol. 32, no. 9, pp. 1055-1067, September 1984. [7] R. Levy, R. Snyder and G. Matthei, "Design of Microwave Filters," Microwave Theory and techniques, IEEE Transactions on, vol. 50, no. 3, pp. 783-793, March 2002. [8] G. L. Matthaei, L. Young and E. M. T. Jones, Microwave Filters, Impedance-Matching 55 networks and Coupling Structures, Boston: Artech House, 1980. [9] R. J. Cameron, R. Mansour and C. M. Kudsia, Microwave Filters for Communication Systems : Fundamentals, Design and Applications, Wiley, 2007. [10] I. Bahl, Lumped Elements for RF and Microwave Circuits, Boston: Artech House, 2003. [11] S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications, Wiley, 2001. [12] J. Thomas, "Cross-Coupling in Coaxial Cavity Filters - A Tutorial Overview," Microwave Theory and Techniques, IEEE Transactions on, vol. 51, no. 4, pp. 1368-1376, April 2003. [13] I. Hunter, Theory and Design of Microwave Filters, London: The institution of Engineering and Technology, 2001. [14] B. Easter, "Direct-coupled Resonator Filters with Improved Selectivity," Electronics Letters, vol. 4, no. 19, pp. 415-416, 20 September 1968. [15] W. Meng and K.-L. Wu, "A Hybrid Approach to Synthesis of Microwave Coupled-Resonator Filters," Microwave Symposium Digest, 2006. IEEE MTT-S International, pp. 119-122, June 2006. [16] Z. Jun, Z. Hualiang and Z. Qi, "A novel High-Performance Narrow-Band Bandpass Filter," Antennas and Propagation Society International Symposium, 2003. IEEE, vol. 3, pp. 977980, June 2003. [17] E. D. Silva, High Frequency and Microwave Engineering, London: The Open University, 2001. [18] W. Meng and K.-L. Wu, "A Hybrid Synthesis Technique for N-Tuplets Microwave Filters 56 Cascaded by Resonator," Microwave Conference, 2006. APMC 2006. Asia-Pacific, pp. 11701173, December 2006. [19] J. Ni, J. Wang, Y.-X. Guo and W. Wu, "Design of Compact Microstrip Dual-Mode Filter with Source-Load Coupling," Microwave and Millimeter Wave Technology (ICMMT), 2010 International Conference on, pp. 882-884, May 2010. [20] G. Pfitzenmaier, "Synthesis and Realization of Narrow-Band Canonical Microwave Bandpass Filters Exhibiting Linear Phase and Transmission Zeros," Microwave Theory and Techniques, IEEE Transactions on, vol. 30, no. 9, pp. 1300-1311, September 1982. [21] V. V. Tyurnev, "Coupling Coefficients of Resonators in Microwave Filter Theory," Progress In Electromagnetics Research B, vol. 21, pp. 47-67, 2010. [22] G. Macchiarella, "Synthesis Of Prototype Filters with Triplet Sections Starting From Source And Load," Microwave and Wireless Components Letters, IEEE, vol. 12, no. 2, pp. 42-44, February 2002. [23] S. Amari and G. Macchiarella, "Synthesis of Inline Filters with Arbitrarily Placed Attenuation Poles by using Nonresonating Nodes," Microwave Theory and Techniques, IEEE Transactions on, vol. 53, no. 10, pp. 3075 - 3081, 2005. [24] S. Amari and U. Rosenberg, "New In-Line Dual- And Triple-Mode Cavity Filters with Nonresonating Nodes," Microwave Theory and Techniques, IEEE Transactions on, vol. 53, no. 4, pp. 1272-1279, April 2005. [25] O. Glubokov and D. Budimir, "Extraction of Generalized Coupling Coefficients for Inline Extracted Pole Filters With Nonresonating Nodes," Microwave Theory and Techniques, IEEE 57 Transactions on, vol. 59, no. 12, pp. 3023-3029, December 2011. [26] S. Amari and U. Rosenberg, "New In-Line Dual- And Triple-Mode Cavity Filters With Nonresonating Nodes," Microwave Theory and Techniques, IEEE Transactions on, vol. 53, no. 4, pp. 1272-1279, April 2005. [27] G. Macchiarella, "A Powerful Tool for the Synthesis of Prototype Filters with Arbitrary Topology," Microwave Symposium Digest, 2003 IEEE MTT-S International, vol. 3, pp. 14671470, June 2003. [28] G. Dai, "Design of compact bandpass filter with improved selectivity using source-load coupling," Electronics Letters, vol. 46, no. 7, pp. 505-506, April 2010. [29] G. Macchiarella, "Generalized Coupling Coefficient for Filters with Nonresonant Nodes," Microwave and Wireless Components Letters, IEEE, vol. 18, no. 12, pp. 773-775, December 2008. [30] G. Macchiarella and S. Tamiazzo, "Synthesis of Microwave Duplexers using Fully Canonical Microstrip Filters," Microwave Symposium Digest, 2009. MTT '09. IEEE MTT-S International, pp. 721-724, June 2009. [31] S. Amari, "Direct Synthesis of Cascaded Singlets and Triplets by Non-Resonating Node Suppression," Microwave Symposium Digest, 2006. IEEE MTT-S International, pp. 123-126, June 2006. [32] S. Amari, "Direct Synthesis Of Folded Symmetric Resonator Filters with Source-Load Coupling," Microwave and Wireless Components Letters, IEEE, vol. 11, no. 6, pp. 264-266, June 2001. 58 [33] G. Macchiarella and S. Amari, "Direct Synthesis of Prototype Filters with Non-Resonating Nodes," Microwave Conference, 2004. 34th European, vol. 1, pp. 305-308, 14 October 2004. [34] R. J. Cameron, "Advanced Coupling Matrix Synthesis Techniques for Microwave Filters," Microwave Theory And Techniques, IEEE Transactions on, vol. 51, no. 1, pp. 1-10, January 2003. [35] R. J. Cameron, "General Coupling Matrix Synthesis Methods for Chebyshev Filtering Functions," Microwave Theory and Techniques, IEEE Transactions on, vol. 47, no. 4, pp. 433 - 442, 1999. [36] D. M. Pozar, Microwave Engineering, Wiley, 1989. [37] S. Tamiazzo and G. Macchiarella, "An Analytical Technique For The Synthesis Of Cascaded N-Tuplets Cross-Coupled Resonators Microwave Filters Using Matrix Rotations," Microwave Theory and Techniques, IEEE Transactions on, vol. 53, no. 5, pp. 1693-1698, May 2005. [38] G. Macchiarella and S. Tamiazzo, "Novel Approach to the Synthesis of Microwave Diplexers," Microwave Theory and Techniques, IEEE Transactions on, vol. 54, no. 12, pp. 4281 - 4290, 2006. [39] J.-S. Hong and M. Lancaster, "Recent Advances in Microstrip Filters for Communications and other Applications," Advances in Passive Microwave Components (Digest No.: 1997/154), IEE Colloquium on, pp. 2/1-2/6, 22 May 1997. [40] X. Liu, L. P. B. Katehi and D. Peroulis, "Novel Dual-Band Microwave Filter Using Dual- 59 Capacitively-Loaded Cavity Resonators," Microwave and Wireless Components Letters, IEEE, vol. 20, no. 10, pp. 610-612, November 2010. [41] G. Macchiarella, "Accurate Synthesis of Inline Prototype Filters using Cascaded Triplet and Quadruplet Sections," Microwave Theory and Techniques, IEEE Transactions on, vol. 50, no. 7, pp. 1779-1783, July 2002. [42] G. Macchiarella and S. Tamiazzo, "Design Techniques for Dual-Passband Filters," Microwave Theory and Techniques, IEEE Transactions on, vol. 53, no. 11, pp. 3265 - 3271, 2005. [43] G. Macchiarella and S. Tamiazzo, "A Design Technique for Symmetric Dualband Filters," Microwave Symposium Digest, 2005 IEEE MTT-S International, 2005. [44] I. Mandal, R. Zhou, H. Ren, Y. Wan and H. Zhang, "Synthesis of Dual-Band Microwave Duplexers," in 2013 Texas Symposium on Wireless & Microwave Circuits & Systems, Waco, Texas, April 2013. [45] M. Sánchez-Renedo and R. Gómez-García, "Microwave Dual-Band Bandpass Planar Filter using Double-Coupled Resonating Feeding Sections," Microwave Conference, 2009. EuMC 2009. European, pp. 101-104, September 2009. [46] R. Gomez-Garcia, M. Sanchez-Renedo, B. Jarry, J. Lintignat and B. Barelaud, "Microwave Multi-Path Dual-Passband Filters for Wide-Band Applications," Microwave Conference, 2009. EuMC 2009. European, pp. 109-112, September 2009. [47] H. -Y. A. Yim, F.-L. Wong and K. -K. M. Cheng, "A New Synthesis Method for Dual-Band Microwave Filter Design with Controllable Bandwidth," Microwave Conference, 2007. 60 APMC 2007. Asia-Pacific, pp. 1-4, December 2006. [48] Y. Zhang, K. Zaki, J. Ruiz-Cruz and A. Atia, "Analytical Synthesis of Generalized Multi-band Microwave Filters," Microwave Symposium, 2007. IEEE/MTT-S International, pp. 1273 1276, 2007. [49] H. B. S. a. C. K. J. Zhang, "Efficient Parameter Extraction of Microwave Coupled-Resonator Filter Using Genetic Algorithms," Int. Journal of RF and Microwave Comp Aid Eng., vol. 21, no. 2, pp. 137-144, March 2011. [50] R. Adve and T. Sarkar, "The effect of Noise in the Data on the Cauchy Method," Microwave and Optical Technilogy Letters, vol. 7, no. 5, pp. 242-247, April 1994. [51] A. G. Lamperez, T. K. Sarkar and M. S. Palma, "Generation of Accurate Rational Models of Lossy Systems Using the Cauchy Method," Microwave and Wireless Component Letters, IEEE, vol. 14, no. 10, pp. 490-492, October 2004. [52] D. Traina, G. Macchiarella and T. Sarkar, "Robust Formulations of the Cauchy Method Suitable for Microwave Duplexers Modeling," Microwave Theory and Techniques, IEEE Transactions on, vol. 55, no. 5, pp. 974-982, May 2007. 61

1/--страниц