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Generalised direct matrix synthesis approach for lossless filters

2016; Institution of Engineering and Technology; Volume: 11; Issue: 2 Linguagem: Inglês

10.1049/iet-map.2016.0318

1751-8733

Yuxing He, Gang Wang, Liguo Sun, Gerard Rushingabigwi,

Advanced Antenna and Metasurface Technologies

IET Microwaves, Antennas & PropagationVolume 11, Issue 2 p. 158-164 Research ArticleFree Access Generalised direct matrix synthesis approach for lossless filters Yuxing He, Yuxing He Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei, 230027 People's Republic of ChinaSearch for more papers by this authorGang Wang, Gang Wang Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei, 230027 People's Republic of China Key Laboratory of Electromagnetic Space Information, Chinese Academy of Sciences, Hefei, 230027 People's Republic of ChinaSearch for more papers by this authorLiguo Sun, Corresponding Author Liguo Sun liguos@ustc.edu.cn Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei, 230027 People's Republic of ChinaSearch for more papers by this authorGerard Rushingabigwi, Gerard Rushingabigwi Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei, 230027 People's Republic of ChinaSearch for more papers by this author Yuxing He, Yuxing He Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei, 230027 People's Republic of ChinaSearch for more papers by this authorGang Wang, Gang Wang Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei, 230027 People's Republic of China Key Laboratory of Electromagnetic Space Information, Chinese Academy of Sciences, Hefei, 230027 People's Republic of ChinaSearch for more papers by this authorLiguo Sun, Corresponding Author Liguo Sun liguos@ustc.edu.cn Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei, 230027 People's Republic of ChinaSearch for more papers by this authorGerard Rushingabigwi, Gerard Rushingabigwi Department of Electronic Engineering and Information Science, University of Science and Technology of China, Hefei, 230027 People's Republic of ChinaSearch for more papers by this author First published: 01 January 2017 https://doi.org/10.1049/iet-map.2016.0318Citations: 3AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract A generalised direct matrix synthesis is proposed for microwave lossless filters with arbitrary phase shift at centre frequency and complex terminal impedances. The functions of traditional lossless filters, ideal phase shifters, and ideal impedance matching networks are thus substantially combined into one entity to provide an attractive technique for the realisation of compact narrowband microwave front-ends. The synthesis starts with a group of general S-parameters that consider both amplitude and phase responses as well as complex terminations for an Nth-order filter. A transversal array network incorporating resonators as well as non-resonating nodes is then introduced to accommodate the given S-parameters. In result, a canonical (N + 2) × (N + 2) coupling matrix with newly defined source–source and load–load couplings is generated. Some examples, including a fully canonical prototype and a mixed topology diplexer, are demonstrated to show the flexibility of the methodology. Moreover, an eighth-order coaxial cavity filter, with a −120° additional phase shift for S-parameters, is designed and fabricated. The measured results are well consistent with the synthesis ones, which validates the availability of this study in physical implementations. 1 Introduction During past few decades, a big attention has been paid to the direct synthesis of microwave filters [1-8]. Among them, a matrix-based approach, which is recommended by Cameron in [3, 4], is widely applied due to its simplicity. The theory has then been developed by many other researchers in recent years to increase its adaptability, such as direct synthesis of dissipative/lossy filters [5, 6] and filters terminated with complex loads in [7, 8]. It is currently noticed that only the amplitude of S-parameters (i.e. filtering function) is concerned in the existing filter synthesis. As a result, phase shift at the centre frequency of a filter synthesised by above approaches is fixed to specific values that correlate with the filtering function. Despite a phase shift is particularly discussed in [9], the effect is considered as an offset for the complex termination, and the shift value is thus still not selectable. Therefore, additional phase shifters are required to get the desired phase shift, which is inconvenient for compact front-ends realisation in modern communication systems. To combine the traditional filters in the literatures [1-9] and ideal phase shifter into one entity, in this paper, a novel generalised matrix synthesis approach for filters considering both amplitude and phase responses is proposed, which allows the phase shift of a filter at the centre frequency to be arbitrary without any influence on the filtering function or requirement on the termination impedances. As a result, the functions of traditional lossless filters, ideal phase shifters, and ideal impedance matching networks are substantially combined into one entity by this approach, providing an interesting technique to design combined two-port networks for compact narrowband microwave front-ends realisation. As for the synthesis, a group of general S-parameters is presented, adapted to which a reasonable transversal array network incorporating both non-resonating nodes (NRNs) and resonators is introduced. Then, two main developments to construct the canonical coupling matrix associated with the transversal array have been made, including: (i) the coupling elements at the top and bottom rows are obtained in a new way; and (ii) pure real source–source and load–load couplings are newly defined and displayed at the top left and bottom right corners of the matrix. Demonstrated as a filter with specific phase shift at the centre frequency in this work, it should be mentioned that this approach recommends a simple and valuable matrix-based solution for synthesising mixed topology filters/diplexers [10] as well. Basic theory of this approach is detailed in Section 2. Some examples, including fully canonical cases [i.e. filter order equals to the number of transmission zeroes (TZs)] and mixed topology cases, are illustrated in Section 3 to show flexibility of the approach. An eighth-order filter is synthesised, fabricated, and tested to give further validation of the approach in physical implementations in Section 4, and Section 5 concludes. 2 Synthesis theory The synthesis is done in the low-pass frequency domain. An Nth-order lossless low-pass prototype with arbitrary phase shift at the centre frequency and complex terminal impedances (Z1 and Z2) is recommended in Fig. 1. Besides a set of characteristic polynomials E, F, P and coefficients ɛ, ɛr which have been detailed in [3, 4] to construct the amplitude response of S-parameters, additional phase shifts −θ11, −θ22, and −θ0 are introduced for S11, S22, and S21, separately, to get desired phase responses. As a result, S-parameters for the prototype in Fig. 1 result in the following equation (1)where S110, S220, and S210 are the S-parameters defined by traditional filter synthesis in [3, 4]. Note that polynomials E, F, and P are normalised to their respective highest degree coefficients and E* (or F*) is defined as the polynomial whose roots are symmetric to the corresponding roots of E (or F) about the imaginary axis. Since lossless and reciprocal case is considered, the unitary condition of S-parameters is still satisfied, which yields the relations (2a) (2b) Fig. 1Open in figure viewerPowerPoint Nth-order lossless low-pass prototype with arbitrary phase shift at the centre frequency and complex terminations Promoted by using power wave renormalisation theory in [7, 11], the Y-parameters for the prototype can be transferred from (1), which are listed as follows (3) (4)With partial fraction expansions, the Y-parameters in (3) and (4) are re-expressed by the roots of polynomial yd, i.e. jλ1, jλ2, …, jλN, together with the relevant residues, i.e. r11k, r21k, r22k (k = 1, 2, …, N) (5)Note that jK11 and jK22 in (5) represent as the remainders of polynomials y11 and y22, respectively, which are not considered in a traditional synthesis [1-8] (K11 = K22 = 0 is always valid under the condition θ11 = θ22 = θ0 = 0). In this work, however, θ11 and θ22 are raised to be arbitrary, and the values of K11, K22, and K0 should all be reconsidered. Keeping Z1 = a1 + jb1 and Z2 = a2 + jb2 in mind, and assuming the number of TZs of the prototype as nz, it is observed that (6)To accommodate the required Y-parameters, a new transversal array network is settled for the Nth-order prototype as demonstrated in Fig. 2. In addition to elements (N first-order low-pass resonators and source–load direct coupling MSL) in [4], the proposed array network utilises two more NRNs at the source and load, respectively, i.e. the shunt frequency invariant susceptances MSS and MLL. It should be noted that the attached NRNs are allowed to achieve multiple couplings with the low-pass resonant sections. Fig. 2Open in figure viewerPowerPoint Generalised canonical transversal array for Nth-order low-pass prototype a Ideal parallel-connected transversal array b Equivalent circuit of the kth first-order low-pass resonant section presented in [4] Therefore, the Y-parameters for the overall transversal array network in Fig. 2 are obtained by adding the admittance matrices for the N individual low-pass resonant sections and the shadow area network together, yielding (7)Comparing (7) with (5) to conform the Y-parameters, circuit parameters of the transversal array network can then be related to the acquired λk, r21k, r11k, r22k and K11, K22, K0 according to the following definitions (8)It should be mentioned that the coupling elements at the top and bottom rows, i.e. MSk and MLk, are obtained in a new way comparing with [4, eq. (16)]. The new definitions in (8) are important to avoid the zero value for r22k (under this case extracted-pole structures are introduced in the filter topology; this special condition will be detailed in Section 3). After getting the circuit parameters, canonical (N + 2) × (N + 2) coupling matrix [M], which associates with the proposed new transversal array, is determined in Fig. 3. It is noticed that pure real source–source and load–load couplings appear at the top left and bottom right corners of the matrix. These corner couplings are redefined as the shunt NRNs at the source (i.e. MSS) and at the load (i.e. MLL) terminations in Fig. 2, which are indispensable for determining the arbitrary phase shift of the proposed low-pass prototype. In addition, it is found that the coupling matrix can be further simplified by using matrix rotations [12] and optimisations [13]. The S-parameters associated with the derived coupling matrix are computed by [8] (9)where [A] = ω[U] − j[O] + [M]. Note that [U] represents an (N + 2) × (N + 2) unitary matrix with the exception of [U]1,1 = [U]N+2,N+2 = 0, while [O] is an (N + 2) × (N + 2) zero matrix with the exception of [O]1,1 = 1/Z1 and [O]N+2,N+2 = 1/Z2. Fig. 3Open in figure viewerPowerPoint Canonical coupling matrix for an Nth-order proposed prototype, containing pure real elements MSS and MLL at the top left and bottom right corners, separately After the above derivations, it is observed that the functions of traditional filters, ideal phase shifters, and ideal impedance matching networks are described by a single (N + 2) × (N + 2) coupling matrix, which can be constructed in one circuit entity in further implementations. 3 Illustrative examples and discussions As detailed in Section 2, a generalised matrix-based filter synthesis that is applicable for arbitrary phase shift at the centre frequency as well as complex terminal impedance conditions are provided. In this section, the obtained results are verified by two synthesis examples, which demonstrate the flexibility of this work for synthesising fully canonical filters as well as mixed topology filters/diplexers. 3.1 Illustrative example I: synthesis for fully canonical prototype For the first example, two fully canonical fourth-order prototypes (prototype A terminated with Z1 = Z2 = 1; prototype B terminated with Z1 = 0.4 + 0.6j, Z2 = 0.5 − 0.5j) are discussed with 22 dB return loss and four TZs along the imaginary axis of the complex plane, separately at −3.7431j, −1.8051j, 1.5699j, and 6.1910j. The extra phases of θ11 = 40° and θ22 = 120° are selected optionally, from which θ0 is found to be 80° according to (2). Thus, prototype A and the matrix in [4, Fig. 6a] (or prototype B and the matrix in [8, Fig. 2a]) should share the same amplitude responses, except an extra −40°/−120°/−80° is added in the phase response of S11/S22/S21. The characteristic polynomials P, E, and F are first confirmed from [3, 4]. The Y-parameters can then be evaluated by (3) and (4), and are expressed in partial fraction expansions by (5) and (6). The corresponding coupling matrices, possessing source–source, load–loading, and source–load couplings for the two proposed prototypes, are determined by using definitions in (8), and are reduced to a folded topology depicted in Fig. 4. Fig. 4Open in figure viewerPowerPoint Routing schematic diagram and coupling matrices for fourth-order fully canonical prototypes with extra phase shifts a Routing and coupling schematic b Reduced coupling matrix for prototype A (terminated with loads Z1 = Z2 = 1) c Reduced coupling matrix for prototype B (terminated with loads Z1 = 0.4 + 0.6j, Z2 = 0.5–0.5j) It is noted from Fig. 4a that the topology for prototypes A and B is exactly constructed with NRNs and resonators connected by multiple inner cross-couplings. Comparing Figs. 4a and b with [4, Fig. 6a] and [8, Fig. 2a], respectively, it is found that the additional phase shifts and complex load impedances mainly affect the termination-related couplings, i.e. MSk, MLk, MSS, and MLL, as well as the self-couplings of resonators, i.e. Mkk, while absolute values of the other couplings remain almost unchanged. Despite some of the couplings may become quite large here (such as M44, M4L, ML4, and MLL for prototype B), they can be rescaled by using general coupling coefficients [14] to ensure the realisability in further implementations. The synthesis responses for the matrices associated with Figs. 4b and c are derived by (9). In Fig. 5, the obtained S-parameters are compared with those for the matrix in [4, Fig. 6a] and [8, Fig. 2a], separately. It can be seen that the amplitude responses for the proposed prototype A (or B) and the involved prototypes in [4] (or [8]) are hardly distinguishable while the phase responses for prototype A (or B) performs desired differences with respect to the corresponding matrix in [4, Fig. 6a] (or in [8, Fig. 2a]), which perfectly validates the proposed approach. Fig. 5Open in figure viewerPowerPoint Synthesis responses for the matrix of prototype A, the matrix of prototype B, the matrix in [4, Fig. 6a], and the matrix in [8, Fig. 2a](with A, B, T1, and T2 superscripted, separately) a S-parameter amplitude responses b S-parameter phase responses for prototype A and the matrix in [4, Fig. 6a] c S-parameter phase responses for prototype B and the matrix in [8, Fig. 2a] d Phase differences between relevant S-parameters. (For phase consistence, a −360° has been added when (·)A–(·)T1 or (·)B–(·)T2 is positive.) Fig. 6Open in figure viewerPowerPoint Transversal array for Nth-order mixed topology prototype with extracted poles generated at terminations 3.2 Illustrative example II: synthesis for prototype with mixed topology Most recently, direct synthesis for mixed topology filters, i.e. filters containing both cross-coupled and extracted-pole structures, is becoming more and more popular [10, 15, 16]. To create the terminal extracted-pole structures, it is noticed that specific additional phase shifts for S-parameters are needed [15, 16]. For this reason, a mixed topology prototype shown in Fig. 6 is considered as a special case of the proposed transversal array in Fig. 2, as long as specific values of −θ11, −θ22, and −θ0 are satisfied to make MSN/MNS and ML1/M1L equal to zero [17]. However, as none of previous works raise the effect of the complex loads on the required values of additional phase shifts −θ11, −θ22, and −θ0, the whole synthesis should be reconsidered. Taking the derived Y-parameters into consideration again, the expressions in (4) are rewritten as follows (10)Note that, V1* (or V2*) represents the polynomial whose roots are symmetric to the corresponding roots of V1 (or V2) about the imaginary axis. As it is desired that MSN = MNS = ML1 = M1L = 0, it can be observed from (5) and (6) that residues r11N, r221, r21N, and r211 should be zero. In result, the first TZs for the prototype (named jωex1) should belong to a root of both polynomials yd and y22n, while the last TZs (named jωex2) should be a root of both polynomials yd and y11n. To satisfy these conditions, it can be easily observed from (10) that (11)and thus (12)Hence, to construct a mixed topology in Fig. 6, a filter terminated with complex loads Z1 and Z2 should add extra phase shifts −θ11, −θ22, and −θ0 decided according to (12) for corresponding S-parameters. Note that, the techniques in [12, 13] are still available here to further simplify the inner cross-coupled structure. Besides, TZs jωex1 and jωex2 turn into the two extracted poles generated at source and load separately. On the basis of above discussions, two mixed topology prototypes performing 22 dB return loss are demonstrated. The first example (prototype C), is of fourth-order, terminated with complex loads Z1 = 0.7 −0.3j, Z2 = 1, and with two TZs at 2.5j, −3.8j. The second one, (prototype D), is of fifth-order, terminated with complex loads Z1 = 1.2 + 1j, Z2 = 1, and with two TZs at −2.2j, −3.1j. Particularly, 2.5j (or −2.2j) is generated by an extracted-pole section at the source of prototype C (or D), while −3.8j (or −3.1j) is realised by a triplet. Since only one extracted pole generated at the source is needed, θ22 is decided directly to be zero so that no extra phase shifts for S22 is added. After the characteristic polynomials E, F, and P obtained, θ11 = 190.7389°, and 196.7588° is determined from (12) for prototypes C and D, separately. As θ22 = 0 is assumed, there exists θ0 = θ11/2 for both prototypes. The synthesis continues with the same procedure for prototypes A and B. Ultimately, the reduced routing schematic diagrams, coupling values, and the synthesis of S-parameter amplitude responses for prototypes C and D are provided in Fig. 7. Fig. 7Open in figure viewerPowerPoint Routing schematic diagrams, coupling values, and synthesis responses for prototypes C and D (with C and D superscripted, separately). Value with solid box: self-coupling for resonator or NRN; and value without box: cross-coupling a Routing schematic diagram and S-parameter amplitude response for prototype C b Routing schematic diagram and S-parameter amplitude response for prototype D To show flexibility of the proposed approach, a mixed topology diplexer is designed as Fig. 8 depicts, where two channel filters are, respectively, converted from the low-pass prototypes C and D and combined by a star-junction. It is decided that the low channel filter operates at fL = 2.535 GHz while the high channel filter at fH = 2.655 GHz, and the 70 MHz passband is for both channel filters. Besides, terminal impedance of Port 1/2/3 is decided as 1 Ω. Fig. 8Open in figure viewerPowerPoint Routing schematic, general coupling coefficients, and resonant frequencies for corresponding elements of the diplexer. Value with solid box: self-coupling for NRN; value with dotted block: resonant frequency for resonator; and value without box: cross-coupling Utilising the technique proposed in [18], the diplexer is determined by an iterative process in which the low and high channel filters are updated alternatively using the synthesis in this work. Noting that for each iteration, θ11 for the high/low channel filter should be re-decided from (12) according to the renewed load Z1 that derived from the latest iteration. The de-normalised coupling coefficients and corresponding resonant frequencies of the mixed topology diplexer, as shown in Fig. 8, are derived after twice iterations for each channel filter. The diplexer is verified by a circuit model in ADS (version 2011.10) with simulation results illustrated in Fig. 9. It can be seen that the diplexer demonstrates desired performances, which validates the flexibility of this work. Fig. 9Open in figure viewerPowerPoint Circuit model simulation results for the mixed topology diplexer 4 Experiment validation The proposed approach is experimentally validated by an eighth-order coaxial cavity filter, operating at 1.9925 GHz with 65 MHz bandwidth, 24 dB return loss, and two TZs at 1.946 and 2.037 GHz. To facilitate the synthesis, it is assumed that Z1/Z0 = Z2/Z0 = 1 while Z0 is the characteristic impedance equalling to 50 Ω. Compared with a design of the same amplitude response [4], a −120° extra phase shift is added for S11, S22, and S21. In other words, it is stipulated that −θ0 = −θ11 = −θ22 = −120°, and thus θ0 = θ11 = θ22 = 120°. The synthesis is done by the given approach in the low-pass domain. After frequency normalised to the low-pass domain, the locations of TZs turn to ±1.4j. The design procedure is in line with the examples in Section 3. According to the existing approach in [4], characteristic polynomials E, F, and P can be easily obtained. Thereafter, eigenvalues and residues for the required Y-parameters are evaluated by (3) and (4) [see Table 1]. The routing schematic diagram with coupling values for the corresponding low-pass prototype and a photograph for the final fabricated filter are shown in Fig. 10, where the proposed filter contains not only resonators but also two NRNs at respective terminations. It is pointed out that both resonators and NRNs are realised by the metal rods, but with and without screws, separately. Table 1. Eigenvalues, residues, and remainders for the eighth-order fabricated filter with extra phase shifts k Eigenvalues Residues λk r21k r11k r22k 1 −2.3239 1.9187 1.9187 1.9187 2 −2.3260 −1.9131 1.9131 1.9131 3 1.0710 −0.0309 0.0309 0.0309 4 −0.9958 0.0528 0.0528 0.0528 5 0.9056 0.0935 0.0935 0.0935 6 −0.6524 −0.1620 0.1620 0.1620 7 0.4878 −0.1595 0.1595 0.1595 8 −0.0902 0.2005 0.2005 0.2005 K0 0 K11 1.7321 K22 1.7321 Fig. 10Open in figure viewerPowerPoint Routing schematic diagram and photograph for the eighth-order filtera Routing schematic diagram with coupling values for the corresponding low-pass prototype. Value with solid box: self-coupling for resonator or NRN; and value without box: cross-coupling b Photograph for the fabricated filter, with top lid removed Fig. 11 reports the comparison between measured results for the proposed filter and the polynomial model results for its corresponding traditional filter [4]. As desired, the amplitude responses of S-parameters shown in Fig. 11a are found almost the same. In addition, phase responses of S21 for the proposed filter (named as φ21) and the traditional one (named as φ210) are provided in Fig. 11b. It is revealed that the phase difference (φ210−φ21) is 123.263° at the centre frequency 1.9925 GHz, and varies from 116° to 125° while the frequency from 1.98 to 2.0 GHz, which is very close to the desired value (i.e. θ0 = 120°). In general, the measured results agree with desired synthesis results very well, which validates the availability of this work in further physical implementations. Fig. 11Open in figure viewerPowerPoint Measured S-parameters for the eighth-order filter (solid lines) and polynomial model S-parameters for its corresponding traditional filter [4] (dashed lines) a Amplitude response b Phase response 5 Conclusion In this work, a novel generalised direct matrix synthesis approach has been proposed for lossless filters with more general phase responses and complex terminal impedances. A new transversal array network incorporating resonators and NRNs is proposed as the basis for the synthesis. The canonical coupling matrix is determined by new definitions for the coupling elements, which implies that the functions of traditional filters, ideal phase shifters, and ideal matching networks can be constructed in one circuit entity. Using this approach, fully canonical as well as mixed topology cases are discussed to show its flexibility, and a coaxial cavity filter with extra phase shift is designed and fabricated to validate its effectiveness in physical implementations. The proposed approach is meaningful as it deals with filtering function, phase shift, and complex terminal impedances all at once, expanding the overall adaptability for direct filter synthesis, and providing an attractive technique to realise compact narrowband microwave front-ends. 6 Acknowledgment This work was supported in part by the National Natural Science Foundation of China under grant 61272471. 7 References 1Atia, A., Williams, A.: 'Narrow-bandpass waveguide filters', IEEE Trans. Microw. Theory Tech., 1972, 20, (4), pp. 258– 265 (doi: 10.1109/TMTT.1972.1127732) 2Atia, A., Williams, A., Newcomb, R.: 'Narrow-band multiple-coupled cavity synthesis', IEEE Trans. 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