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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES A PUBLICATION OF THE IEEE MICROWAVE THEORY AND TECHNIQUES SOCIETY JR MTT-S NOVEMBER 2018 VOLUME 66 NUMBER 11 IETMAB (ISSN 0018-9480) THIS ISSUE INCLUDES THE JOURNAL WITHIN A JOURNAL ON MICROWAVE SYSTEMS AND APPLICATIONS REGULAR PAPERS OF THE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES EM Theory and Analysis Techniques Application of Belevitch Theorem for Pole-Zero Analysis of Microwave Filters With Transmission Lines and Lumped Elements .............................................................................................. E. L. Tan and D. Y. Heh 4669 Closed-Form Solution of Rough Conductor Surface Impedance ............................................... D. N. Grujic 4677 Devices and Modeling A GaN HEMT Global Large-Signal Model Including Charge Trapping for Multibias Operation ....................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. P. Gibiino, A. Santarelli, and F Filicori 4684 15-Gb/s 50-cm Wireless Link Using a High-Power Compact III-V 84-GHz Transmitter ................................ . . ... . . .. ... .. . . .. . .. .. .. ... .... .. ... .. . .. . . ... ... . J. Wang, A. Al-Khalidi, L. Wang, R. Morariu, A. Ofiare, and E. Wasige 4698 Nonreciprocal Components Based on Switched Transmission Lines ......................................................... . ... .. .. ....... ..... .. ..... .. A. Nagulu, T. Dine, Z. Xiao, M. Tymchenko, D. L. Saunas, A. Alu, and H. Krishnaswamy 4706 Millimeter-Wave Double-Ridge Waveguide and Components ............. S. Manafi, M. Al-Tarifi, and D. S. Filipovic 4726 Design of Frequency Selective Surface-Based Hybrid Nanocomposite Absorber for Stealth Applications ............. . ..... .. .. .. ..... .... ... . . .. ... ... ... ... .... ... . V. K. Chakradhary, H. B. Baskey, R. Roshan, A. Pathik, and M. J. Akhtar 4737 Conversion Rules Between X-Parameters and Linearized Two-Port Network Parameters for Large-Signal Operating Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Essaadali, A. Jarndal, A. B. Kouki, and F M. Ghannouchi 4 7 45 Passive Circuits Theoretical Analysis of RF Pulse Termination in Nonlinear Transmission Lines .......................................... . . .. .. .. . .. .... ... .. . . ... .. ... .... .. ....... .... .. ... . . ... .......... M. Samizadeh Nikoo, S. M.-A. Hashemi, and F Farzaneh 4757 Integrated Full-Hemisphere Space-to-Frequency Mapping Antenna With CRLH Stripline Feed Network .............. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. D. Enders, J. H. Choi, and J. K. Lee 4765 Multiport ln-Phase/Antiphase Power Dividing Network With Bandpass Response Based on Dielectric Resonator .... ... .. ...... ... .. ........ ...... .. ..... .. .. .. ... .. ... ..... ........... .. ...... ... .......... .... .. .. ..... ...... ... W. Yu and J.-X. Chen 4773 Generalized Synthesized Technique for the Design of Thickness Customizable High-Order Bandpass Frequency-Selective Surface ............................................................. K. Payne, K. Xu, and J. H. Choi 4783 (Contents Continued on Page 4667) +.IEEE (Contents Continued from Front Cover) Asymmetrical Impedance Inverter for Quasi-Optical Bandpass Filters With Transmission Lines of Fixed Length .... ...................................................................................................... P. K. Loo and G. Goussetis 4794 A New Balanced Bandpass Filter With Improved Performance on Right-Angled Isosceles Triangular Patch Resonator ..................................................... Q. Liu, J. Wang, L. Zhu, G. Zhang, F. Huang, and W. Wu 4803 Systematic Evaluation of Spikes Due to Interference Between Cascaded Filters ............................................ ................... A. Morini, G. Venanzoni, P. M. Iglesias, C. Ernst, N. Sidiropoulos, A. Di Donato, and M. Farina 4814 Dual-ModeCharacteristicsofHalf-ModeSIWRectangularCavityandApplicationstoDual-BandFiltersWithWidely Separated Passbands ...................................................................... K. Zhou, C.-X. Zhou, and W. Wu 4820 Hybrid and Monolithic RF Integrated Circuits Analysis and Design of N-Path RF Bandstop Filters Using Walsh-Function-Based Sequence Mixing .................. .................................................................................................... A. Agrawal and A. Natarajan 4830 Compact Series Power Combining Using Subquarter-Wavelength Baluns in Silicon Germanium at 120 GHz ......... ............................................................................................. S. Daneshgar and J. F. Buckwalter 4844 0.3–14 and 16–28 GHz Wide-Bandwidth Cryogenic MMIC Low-Noise Amplifiers ........................................ .............. E. Cha, N. Wadefalk, P.-Å. Nilsson, J. Schleeh, G. Moschetti, A. Pourkabirian, S. Tuzi, and J. Grahn 4860 A 1.8–3.8-GHz Power Amplifier With 40% Efficiency at 8-dB Power Back-Off ........................................... ............................................................................... P. Saad, R. Hou, R. Hellberg, and B. Berglund 4870 Instrumentation and Measurement Techniques Nonuniformly Distributed Electronic Impedance Synthesizer ................... Y. Zhao, S. Hemour, T. Liu, and K. Wu 4883 Jitter Sensitivity Analysis of the Superconducting Josephson Arbitrary Waveform Synthesizer ........................... .......................................... C. A. Donnelly, J. A. Brevik, P. D. Dresselhaus, P. F. Hopkins, and S. P. Benz 4898 JOURNAL WITHIN A JOURNAL ON MICROWAVE SYSTEMS AND APPLICATIONS JOURNALWITHINAJOURNALPAPERS Wireless Communication Systems Effect of Out-of-Band Blockers on the Required Linearity, Phase Noise, and Harmonic Rejection of SDR Receivers Without Input SAW Filter .............................................................. A. Rasekh and M. Sharif Bakhtiar 4913 An Ultralow-Power RF Wireless Receiver With RF Blocker Energy Recycling for IoT Applications ................... ....................................... O. Elsayed, M. Abouzied, V. Vaidya, K. Ravichandran,and E. Sánchez-Sinencio 4927 AFullyIntegrated300-MHzChannelBandwidth256QAMTransceiverWithSelf-InterferenceSuppressioninClosely Spaced Channels at 6.5-GHz Band ............................... Y. Zhang, N. Jiang, F. Huang, X. Tang, and X. You 4943 A Real-Time Architecture for Agile and FPGA-Based Concurrent Triple-Band All-Digital RF Transmission ......... ................ D. C. Dinis, R. Ma, S. Shinjo, K. Yamanaka, K. H. Teo, P. V. Orlik, A. S. R. Oliveira, and J. Vieira 4955 A 0.4-to-4-GHz All-Digital RF Transmitter Package With a Band-Selecting Interposer Combining Three Wideband CMOS Transmitters ............................................................................................................... ................ N.-C. Kuo, B. Yang, A. Wang, L. Kong, C. Wu, V. P. Srini, E. Alon, B. Nikolic´, and A. M. Niknejad 4967 Extraction of the Third-Order 3×3 MIMO Volterra Kernel Outputs Using Multitone Signals ........................... ......................................................................... Z. A. Khan, E. Zenteno, P. Händel, and M. Isaksson 4985 InstantaneousSampleIndexedMagnitude-SelectiveAffine Function-BasedBehavioralModelfor DigitalPredistortion of RF Power Amplifiers .......................................................................... Y. Li, W. Cao, and A. Zhu 5000 CompositeNeuralNetworkDigitalPredistortionModelforJointMitigationofCrosstalk, I/Q Imbalance,Nonlinearity in MIMO Transmitters ...................................................... P. Jaraut, M. Rawat, and F. M. Ghannouchi 5011 Wireless Power Transfer and RFID Systems Increasing the Range of Wireless Power Transmission to Stretchable Electronics .......................................... ................................................................. E. Siman-Tov, V. F.-G. Tseng, S. S. Bedair, and N. Lazarus 5021 Bootstrapped Rectifier–Antenna Co-Integration for Increased Sensitivity in Wirelessly-Powered Sensors ............... .......................................................................................... J. Kang, P. Chiang, and A. Natarajan 5031 Microwave Imaging and Radar Applications A Linear Synthetic Focusing Method for Microwave Imaging of 2-D Objects .............................................. ................................................................................ T. Gholipur, M. Nakhkash, and M. Zoofaghari 5042 W-BandMIMOFMCWRadarSystemWithSimultaneousTransmissionofOrthogonalWaveformsforHigh-Resolution Imaging .... S.-Y.Jeon, M.-H. Ka, S. Shin, M. Kim, S. Kim, S. Kim, J. Kim, A. Dewantari, J. Kim, and H.Chung 5051 (Contents Continued on Page 4668) (Contents Continued from Page 4667) An Efficient Algorithm for MIMO Cylindrical Millimeter-Wave Holographic 3-D Imaging .............................. ................................................................................. J. Gao, B. Deng, Y. Qin, H. Wang, and X. Li 5065 A Portable K-Band 3-D MIMO Radar With Nonuniformly Spaced Array for Short-Range Localization ............... .................................................................................................................. Z. Peng and C. Li 5075 Integration of SPDT Antenna Switch With CMOS Power Amplifier and LNA for FMICW Radar Front End ......... ...................................................................................... B. Kim, J. Jang, C.-Y. Kim, and S. Hong 5087 A C-Band FMCW SAR Transmitter With 2-GHz Bandwidth Using Injection-Locking and Synthetic Bandwidth Techniques .......................................................... S. Balon, K. Mouthaan, C.-H. Heng, and Z. N. Chen 5095 A Fundamental-and-Harmonic Dual-Frequency Doppler Radar System for Vital Signs Detection Enabling Radar Movement Self-Cancellation .................................................................. F. Zhu, K. Wang, and K. Wu 5106 Microwave Sensors and Biomedical Applications Novel Microwave Tomography System Using a Phased-Array Antenna ...................................................... ........................... Y. Abo Rahama, O. Al Aryani, U. Ahmed Din, M. Al Awar, A. Zakaria, and N. Qaddoumi 5119 An RF-Powered Crystal-Less Double-Mixing Receiver for Miniaturized Biomedical Implants ........................... ................................................................................... M. Cai, Z. Wang, Y. Luo, and S. 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IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES 1 Application of Belevitch Theorem for Pole-Zero Analysis of Microwave Filters With Transmission Lines and Lumped Elements Eng Leong Tan , Senior Member, IEEE, and Ding Yu Heh , Member, IEEE Abstract—This paper presents the application of Belevitch change considerably.The numbers of poles and zeros as well theorem for pole-zero analysis of microwave filters synthesized as their locations for a synthesized filter may be different with transmission linesand lumpedelements. Thescattering (S) from the originally specified or designed ones. For instance, matrix determinant ((cid:2)) based on the Belevitch theorem, aptly the poles of a synthesized Chebyshev filter may be deviated called Belevitch determinant,comprises polesandzeros that are separated in different half-plane regions. Using the Belevitch from the reference Chebyshev ellipse after realization. It is determinant, the poles and zeros of filter transfer functions to be shown later that the number of poles of a synthesized can be determined separately with certainty, e.g., by applying classical coupled-line filter [1] turns out to be more. Hence, the contour integration method based on argument principle. the number of poles and zeros should be ascertained after Note that the contour integration can be evaluated numerically synthesis,whiletheirlocationsshouldbeverifiedandchecked without requiring complicated overall analytical expressions. The proposed method is able to solve the poles and zeros for for further tuning wherever needed, especially for filters with filterssynthesizedwithnoncommensuratetransmissionlinesand transmission lines transformed from lumped element proto- lumpedelements,wherethetransformmethodandtheeigenvalue types. Since poles and zeros control the amplitude response, approach are inapplicable. Several applications are discussed phase, and group delay of a filter directly, they need to be to demonstrate the use of Belevitch theorem and the contour analyzedforbetterunderstandingandmanipulationofthefilter integration method to determine the poles and zeros of various microwave filters on the complex plane. response. Unfortunately, for microwave filters synthesized from IndexTerms—Argumentprinciple,Belevitchtheorem,contour transmission-line structures, it is often difficult to determine integration,microwavefilters,numericalmethod,polesandzeros, S matrix determinant. the poles and zeros since the overall analytical expressions maybe verycomplicateddueto the involvementof(nonpoly- nomial) transcendental functions. The difficulties in solving I. INTRODUCTION poles and zeros are further exacerbated when the filters con- MICROWAVE filters have been the subject of much sist of both transmission lines and lumped elements. Their research due to their wide applications in wireless resultant scattering (S) parameters’ expressions, if one could communication.Variousfilter synthesismethods,designs,and ever derive, would comprise combinations of various integer transformations can be found in numerous classic microwave power terms and trigonometric power terms. This makes the textbooks [1]–[4]. Over the years, many filter structures existing methods, such as Richard, Euler, or digital trans- have also been analyzed and synthesized involving (lossless) form method [1], [9], [10], and coupling matrix eigenvalue lumped elements and transmission-line structures, such as approach [11]–[15] inapplicable for the pole-zero analysis steppedlines,stubs,andcoupledlines [5]–[8].However,most of the filters. For blind modeling methods such as vector analyses have not considered in detail the order and locations fittingtechnique [16],theyalsoinvolvemanyuncertaintiesand of poles and zeros of synthesized filters. In particular, most inconsistencies of poles and zeros depending on the chosen filter analyses of microwave textbooks are subjected only for number of poles and fitting bandwidth. low-pass filter prototype and typically in lumped elements. In this paper, we present the application of Belevitch theo- When replaced by transmission-line structures after various remforthepole-zeroanalysisofmicrowavefilterssynthesized transformations, the transfer functions of these filters may with transmission lines and lumped elements. In Section II, the challenges in solving poles and zeros for such filters Manuscript received October 15, 2017; revised February 28, 2018 and will be exemplified in detail, demonstrating the difficulties June 25, 2018; accepted July 29, 2018. This work was supported in part byresearchprojects throughDSOunderGrantDSOCL12016andinpartby and deficiencies of existing methods, including transform theSingaporeMinistryofEducationTertiaryEducationResearchFundunder method, eigenvalue approach, and vector fitting technique. In Grant2015-1-TR15. (Correspondingauthor:DingYuHeh.) Section III, the Belevitch theorem will be discussed. Using The authors are with the School of Electrical and Electronic Engi- neering, Nanyang Technological University, Singapore 639798 (e-mail: the Belevitch theorem,it will be shownthat the scattering (S) [email protected]; [email protected]). matrix determinant (), called Belevitch determinant, com- Color versions of one or more of the figures in this paper are available prisespolesandzerosthatareseparatedindifferenthalf-plane onlineathttp://ieeexplore.ieee.org. Digital ObjectIdentifier 10.1109/TMTT.2018.2865928 regions. Once the poles and zeros are separated completely, 0018-9480©2018IEEE.Personaluseispermitted, butrepublication/redistribution requires IEEEpermission. Seehttp://www.ieee.org/publications_standards/publications/rights/index.html formoreinformation. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES the contour integration method based on argument principle will be applied, which allows them to be determined with certainty.Several applicationswill be discussed in Section IV to demonstrate the use of the Belevitch theorem and the contour integration method to determine the poles and zeros of various microwave filters on the complex plane. II. POLE-ZERO ANALYSIS OFMICROWAVE FILTERS WITH Fig.1. Schematic ofaparallel transmission-line filter. TRANSMISSION LINES AND LUMPEDELEMENTS roots may be solved directly [9], [10]. Following [9], we set For microwave filters involving transmission lines, it is Z =50, Z = Z =2Z ,andtheelectricallengthsofthetwo often challenging to solve for the poles and zeros of transfer 0 1 2 0 parallel transmission lines are θ = π/2, θ = 5π/2, which functions due to the involvement of transcendental functions 1 2 give a = 5. Substituting this a into (6) and upon changing such as the exponentialand trigonometricfunctions.Consider the variable z =u1/2, we obtainan equationthat is consistent the example from [9], where the filter section consists of two parallel transmission-line sections with characteristics with [9, eq. (3)]by notingu =ejπ(f/fd) and the same ak, bk impedances and electrical lengths Z , Z and θ , θ , respec- in [9, Table I]. 1 2 1 2 Thus far, the transform method has been applied for the tively [see Fig. 1(a)]. The expression of S for such a filter 21 case of integera to solve for polynomialroots. However,it is section is given by no longer able to simplify root solving for noncommensurate S21(ω)= NS21(ω)/DS21(ω) (1) lines or when a is not an integer. To demonstrate this, we let θ =3.6514π instead of 5π/2 in Fig. 1(a). Now, substituting where 2 a = 7.3028 into the denominator of (6) gives an equation, NS21(ω) which consists of fractional powers that can no longer be = −2jZ Z Z (Z sin(θ ω/ω )+Z sin(θ ω/ω )) (2) solved easily as polynomial roots. In fact, when a is not 0 1 2 1 1 0 2 2 0 D (ω) integer, it is also mentioned in [9] that the filter cannot be S21  analyzed within the digital-inspired (z or u) framework (e.g., = Z02Z12+ Z02Z22+Z12Z22 (sin(θ1ω/ω0)sin(θ2ω/ω0)) only approximate when θ =0.497π). 1 +Z02Z1Z2(cos2(θ1ω/ω0)+cos2(θ2ω/ω0)+sin2(θ1ω/ω0) The difficulties in solving poles and zeros of transfer + sin2(θ ω/ω )−2cos(θ ω/ω )cos(θ ω/ω )) functions are further exacerbated when the filters consist of  2 0 1 0 2 0 −2j Z Z2Z cos(θ ω/ω )sin(θ ω/ωa ) both transmission lines and lumped elements. To illustrate 0 1 2  2 0 1 0  this, the same filter section in Fig. 1(a) is added with two + Z Z Z2 cos(θ ω/ω )sin(θ ω/ω ) (3) 0 1 2 1 0 2 0 capacitors of capacitance C1 and C2, as shown in Fig. 1(b). ω0 is the center angular frequency. To convert (1)–(3) into The denominator of S21 can be derived as complex frequency s domain, where s = σ + jω, ω is D (ω) substitutedwith−j·s =−j·(σ+jω).Itisworthpointingout S21  = −105C C ω2 p4−2p2p2− j8p2p q +2p2q2 that the S21 denominator of the filter section is already rather 1 2 1 1 2 1 2 2 1 1 complicated even for one section only. As exemplified from +2p2q2− j8p p2q +24p p q q + j8p q q2+ p4 1 2 1 2 1 1 2 1 2 1 1 2  2 (1)–(3),therootscannotbedirectlysolvedforthesenonlinear +2p2q2+2p2q2+ j8p q2q +q4−2q2q2+q4 2 1 2 2 2 1 2 1 1 2 2 functions. Furthermore, as the trigonometric functions are +200ω(2p p + jp q + jp q )(2C p q − jC p p periodic, there are infinitely many poles on the whole s 1 2 1 2 2 1 1 2 1 2 1 2 − jC p p +2C p q + jC q q + jC q q ) domain. It is therefore, difficult to ascertain and solve for the 1 1 2 2 1 2 1 1 2 2 1 2 poles within certain frequency band of interest. −(2p p + jp q + jp q )2 (7) 1 2 1 2 2 1 For commensurate lines whose electrical lengths are the where same (ormultiples)forallsections,onepossible wayto solve     for the poles is by utilizing the Richard transform or Euler’s θ θ p =cos 1ω/ω , p =cos 2ω/ω (8) identities [1]. In particular, using 1 2 0 2 2 0     z+1/z z−1/z θ θ cos(−jsθ /ω ) = , sin(−jsθ /ω )= (4) q =sin 1ω/ω , q =sin 2ω/ω . (9) 1 0 1 0 1 0 2 0 2 2j 2 2 za +1/za za −1/za cos(−jsθ /ω ) = , sin(−jsθ /ω )= It is evident that due to the combinations of various inte- 2 0 2 0 2 2j ger power terms (ω2, ω) and trigonometric power terms (5) [sin((θ/2)ω/ω ),cos((θ/2)ω/ω )],(7)cannotbetransformed 0 0 where a =θ /θ , S of (1) is transformed into z domain as intoanothersimplerformtosolvefortherootsdirectly.Hence, 2 1 21 thetransformmethodisinapplicableforthepole-zeroanalysis 2z2a +2za+1−2za−1−2 S (z)= . (6) of filters with transmission lines and lumped elements. 21 4z2a+1−z2a−1−2za −z For certain microwave filter synthesis [11]–[15], its cou- If a is an integer, the numerator and denominator of (6) plingmatrixmayalsobeusedtofindtheS-parameterpolesvia contain only polynomials(with integer powers) for which the eigenvalue approach. Considering, for example, an n-coupled This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. TANANDHEH:APPLICATIONOFBELEVITCHTHEOREMFORPOLE-ZEROANALYSISOFMICROWAVEFILTERS 3 lumpedelementresonatorfilter,its S and S canbederived network can be represented as [17] 21 11 using the coupling matrix as [3]   S S 1 u v S = √ 2 A−1 (10a) S= S1211 S1222 = g v∗ −u∗ (13) 21 q q n1 e1 en  where g is a strictly Hurwitz polynomial and g, u, and v are 2 −1 S = ± 1− A (10b) related by 11 q 11 e1 uu∗+vv∗ = gg∗. (14) where A = q+ pI − jm (11) The notation g∗ (with subscript “∗”) denotes g∗ =[g(−s∗)]∗,   j ω ω wherethesuperscript“*”indicatescomplexconjugation.Note p = − 0 (12) that the effect of g∗ is to reflect all complex roots of g FBW ω ω 0 symmetricallyabouttheimaginaryaxis.Therefore,therootsof qe1 andqen arethescaledexternalqualityfactorsofinputand g and g∗ are symmetricalabout the imaginary axis. Applying outputresonators, FBW is the fractionalbandwidth,ω is the theBelevitchtheorem(13)and(14),itcanbeshownthattheS 0 center angular frequency, I is the n×n identity matrix, q is matrix determinant, aptly called Belevitch determinant, takes an n × n matrix with all entries zeros except q = 1/q the form 11 e1 and qnn = 1/qen, and m is the n ×n coupling matrix. If a = S S −S S = −g∗. (15) narrowband filter can be represented in the coupling matrix 11 22 21 12 g form of (11), the poles can be found by first solving p from Since g is strictly Hurtwiz, (15) shows that all poles of  are the eigenvalues of matrix comprising m in (11). Thereafter, located in the left half-plane (LHP) of complex s plane, with the poles are obtained via (12). However, a broadbandfilter can no longer be representable thesamenumberofzeros(rootsofg∗)locatedexactlyopposite of jω axis in the right half-plane (RHP). Note that there will in the coupling matrix form of (11) should it comprise com- be no zeros in the LHP and no poles in the RHP for . As binations of noncommensuratetransmission lines and lumped such, the poles and zeros have been effectively separated into elements. Indeed, the impedance or admittance matrix or A in (11) for such a filter should have various powers of ω, different half-plane regions through (15). This is crucial to (1/ω), as well as sin and cos terms, in much the same allow them to be determined separately with certainty in the subsequent analysis. Furthermore, the Belevitch determinant manner like (7). In other words, the p term of (12) not only involves (ω/ω ) and (ω /ω), while the m matrix of (11) not in (15) is an irreducible expression due to g being strictly 0 0 Hurtwiz. Consequently, the degree of a lossless 2-port is the only contains constant coupling coefficients, but also various nonlinear functions of ω. This makes it difficult or impos- degree of the polynomial g of the canonic form (13) [17]. Byretrievingthepolesof,wehaveessentiallyretrievedthe sible to extract the poles using the eigenvalue approach same poles for S sharing the common denominator g. directly.Hence,theeigenvalueapproachisinapplicableforthe 21 Oncethe polesandzerosare separatedcompletely,one can pole-zeroanalysisoffilterswithtransmissionlinesandlumped proceed to solve for them using several possible ways. To elements. that end, the contour integration method based on argument The blind system modeling methods such as the vector principleisonepossiblewaythatmaybeappliedconveniently. fitting technique, Loewner matrix, or Cauchy method are The argument principle is given by [18], [19] sometimes used to locate the system poles and zeros through fitting of measurement data points. We shall comment briefly 1 1 f(s) on the vector fitting technique [16] that is one of the most Z − P = 2πCargf(s)= 2πj f(s)ds (16) C popularmodelingmethods.Whileitisablindfitting/modeling method, the number of poles to estimate is often uncertain where Cargf(s) is the change in argument of f(s) along and the poles obtained are very much dependent on the closed contour C in the counterclockwise direction, f(s) is a fittingbandwidth.Forafilter withnocomplexzeros,spurious meromorphicfunction, Z and P are the number of zeros and complex zeros may even appear that are not symmetrical poles of f(s), including their multiplicities within the closed about the jω axis, thus violating the S-parameters’ unitary contour, and f(s) is the derivative of f(s). The contour C conditions [11]. Henceforth, this technique and other blind alsoshouldnotpassthroughanypolesandzeros.Thecomplex fitting/modeling methods are beyond the scope of this paper. function f(s) in (16) is to be replaced by the Belevitch determinant (s) of a microwave filter on complex s plane. Since  has no zeros in the LHP (Z = 0), we are able III. APPLICATION OFBELEVITCH THEOREM to evaluate (16) to find the number of poles in the LHP. It FORPOLE-ZERO ANALYSIS should be emphasized here that without the use of Belevitch To overcome the difficulties and deficiencies mentioned determinant in (15), the argument principle could not be above, we shall present the application of Belevitch theorem applied directly if f(s) contains both poles and zeros in the for the poles and zeros analysis of microwave filters syn- LHP. Most argument principle-based methods, such as [20] thesized with transmission lines and lumped elements. The and [21], are able to solve only for the zeros of f(s) when it Belevitch theoremstates that the S matrix of a lossless 2-port is analytic within C (P =0). This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES The contour integration of f ≡  can be evaluated numerically via 1 (s) 1 (s+δs/2)−(s−δs/2) ds = . 2πj (s) πj (s+δs/2)+(s−δs/2) C s (17) Here, centralfinite differenceis used for the derivative(s), while the averaging scheme is used for (s). This reduces the number of stencils from three to two for every point on the contour. δs is the spatial step size chosen along the path to ensure convergence.It should be emphasized that to evalu- ate(17),theoverallanalyticalexpressionofisnotrequired, while only the numerical values of  along the contour path are needed. In practice, for microwave filter circuits, the net- work parametersfor individualsections (such as transmission lines,lumpedelements,etc.)canbecomputedandmanipulated readily to obtain the  values of the overall circuits. Fig. 2. S-parameters of a coupled-line filter with Chebyshev response Tolocatethepoles,thecontourcanbesuccessivelydivided (N =3). Inset: layout of N+1sections coupled-line filter along witheven andoddcharacteristic impedances oftheindividual section. into smaller sections until they are tightly enclosed. The contour method could even be combined with other efficient iterative rootsearching algorithms, such as Newton’s method,  can then be obtained subsequently by reflecting all the Muller’s method, etc., which often require rather good ini- zeros of f(s) symmetrically along jω axis. This will bypass tial guess. Having divided the contour into small sections, the computation of 1/ inversion for greater simplicity and the contour centroid may serve as good initial guess to robustness.Apartfrom the sole determinantin (15), the Bele- these iterativealgorithms.Afterthe poleshavebeenretrieved, vitch theorem can also be applied for other S-parameter we can further determine (if any) the complex zeros of a expressions or combinations. For instance, one may consider microwave filter. This can be done by repeating (16) for f(s) = S / to solve directly for the transmission zeros 21 f ≡ S21 and dividing the contour into smaller sections (zeros of S21) or consider f(s) = S11/ or f(s) = S22/ successively to find Z with known P. to solve directly for the reflection zeros of a microwave filter. Thepolesandzerosmayalso be solvedusingthe approach All in all, many opportunities arise from the application of in [22], employing Newton’s identities. However, one would the Belevitch theorem, which could be further explored for need to estimate the number of distinct zeros and poles, the pole-zero analysis of microwave filters. which is done through forming successive system matrices and checking if they are singular. This is often difficult in IV. APPLICATIONS FOR MICROWAVE FILTERS practice to numericallyascertain whether a matrix is singular. A. Applications I: Classical Filters Furthermore, a large number of zeros and poles often result We shall demonstrate the applications of the Belevitch in high polynomial order and ill-conditioned problem. On theoremandthe contourintegrationmethodtosolve forpoles the other hand, alternative searching technique, such as [23], and zeros of several microwave filters. First, consider the involvesexhaustive triangulationof 2-D domain.Still, it does classical coupled-line filter in Fig. 2 [1]. The specifications not guarantee that all the roots can be found and the risk of of the bandpass filter are Chebyshev response with N = 3, missingrootishigherduetotheirregulardiscretizationofthe passband ripple L =0.5 dB, center frequency f =2 GHz, domain.Overall,thereare stilluncertaintiesanddifficultiesin Ar 0 andfractionalbandwidthFBW= 0.1.Thefilterisrealizedby solving for poles and zeros using these methods. Unlike the four (N +1 = 4) coupled-line sections. The S-parameters of approachabove,by using the Belevitch determinant (15), one thefilterareshowninFig.2.FromtheS plot,threereflection will have the poles and zeros separated completely, with only 11 zeros can be observed, which may lead one to deduce that poles and no zeros in the LHP. There is no need to estimate there are three poles within its passband in line with the the number of poles or zeros, thus obviating the forming of third-orderspecification.Usingthecontourintegrationmethod successive system matrices and checking if they are singular. fortheBelevitchdeterminantwithcontourpathsenclosingthe The poles can then be solved as the roots of polynomial. passband around f = 2 GHz, the poles are determined as Hence, there is no need to divide the contour excessively 0 (normalized by 1 GHz) until the poles are tightly enclosed. In practice, one would still consider dividing the contour sufficiently to reduce the s = 2π(−0.03124+ j1.89849) p1 number of enclosed poles in each contour. This should lower s = 2π(−0.06259+ j2.00000) p2 the polynomial order to avoid ill-conditioned problem. s = 2π(−0.03124+ j2.10151) The Belevitch determinant in (15) further allows us to p3 evaluate (16) on the contours in the RHP with f(s) =  sp4 = 2π(−1.88540+ j2.00000) [instead of in the LHP with f(s) = 1/]. The poles of s = 2π(−1.88549+ j2.00000). (18) p5 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. TANANDHEH:APPLICATIONOFBELEVITCHTHEOREMFORPOLE-ZEROANALYSISOFMICROWAVEFILTERS 5 passband ripple L = 0.01 dB. f is still maintained at Ar 0 2 GHz. The S-parameters are also shown in Fig. 2. Three reflection zerosare still visible fromthe S plot. Meanwhile, 11 the numberof polesdeterminedfromthe determinantcontour integrationmethodisstillfivearoundthepassband.Thepoles are subsequently solved as s = 2π(−0.15601+ j1.67367) p1 s = 2π(−0.71330+ j2.00000) p2 s = 2π(−0.58349+ j2.00000) p3 s = 2π(−0.44182+ j2.00000) p4 s = 2π(−0.15601+ j2.32633). (21) p5 For verification, the transform method is again utilized to obtain the roots in the z domain as z = 0.22427+ j0.85578 p1 z = j0.57108 p2 aFnigd.(3d.)im(aa)gRineaarlyapnadrt(sbo)fimS2a1g.inary parts of S21.Absolute errorsin(c)real zp3 = j0.63237 z = j0.70680 p4 It can be seen that there are five poles within the contour.For z = −0.22427+ j0.85578. (22) p5 verification,we also solvethe polesbyutilizingthe transform methodandfindthepolynomialroots(withpositiveimaginary Using(20),thepoles(ins domain)areinagreementwith(21). part only) in the z domain as It can be seen that all five poles are now near to each other and to jω axis. In addition, note that the two extra poles z = 0.07771+ j0.97266 p1 in (18) and (21) do not lie on/near the Chebyshev ellipse zp2 = j0.95203 on the complex s plane. While the other three poles that z = −0.07771+ j0.97266 are around the Chebyshev ellipse may contribute most to the p3 z = j0.22746 third-order Chebyshev response, the presence of two extra p4 poles would still affect the filter response to a certain extent z = j0.22744. (19) p5 (see Fig. 3). From these two examples, one should appreciate the importance of poles analysis, whereby the number and Thepolesins domaincanthenbededucedfromtheroots(19) locations of the poles of a synthesized filter may be different through the inverse of Richard transform or Euler’s identities from those originally specified. s =ω /θ ·lnz. (20) 0 0 It is found that they agree well with (18) obtained via B. Applications II: Advanced Filters our Belevitch determinant contour integration method, thus Complex zeros may sometimes be utilized to optimize validating our proposed method. the filter phase response for better group delay equalization. Interestingly, from (18), it can be seen that the last two We now apply the Belevitch theoremand the contourintegra- poles sp4 and sp5 are located further away from the jω tionmethodtoanalyzealinearphasefilterwithcomplexzeros axis. To investigate further the consequence of retaining from [3].Thefour-polelinearphasefilterhasapassbandfrom all five poles or only first three poles sp1, sp2, and sp3, 920to975MHzandcanbesynthesizedbycouplingmatrixas ⎡ ⎤ Fig. 3(a) and (b) shows the real and imaginary parts of S 21 0 0.9371 0 0.1953 cthoenystraurectecdomuspianrgedaltlofivSe21poobletsaianneddodnirleycfitlrystfrthormeespimoluelsa,tiaonnd. m =⎢⎢⎣0.90371 0.60196 0.60196 0.90371⎥⎥⎦. (23) Therealzeroslocatedonthe jωaxishavealsobeenincluded, 0.1953 0 0.9371 0 consideringaswellthenegativefrequenciesandperiodicityof this coupled-line filter. Fig. 3(c) and (d) shows the absolute The poles and zeros are solved using the contour integration errors in the real and imaginary parts of S for the results method with Belevitch determinant. The poles and zeros are 21 constructed using five and three poles compared to the simu- plotted on the complex s plane in Fig. 4, marked by “x” and lation.Itcanbeobservedthatifoneexcludesthelasttwopoles “o,”respectively.Itcanbeobservedthatfourpolesarevisible in (18), the errors are larger in both real and imaginary parts withinthepassband.Moreover,acomplexzeroisalsopresent. of S . These errors may at times be tolerable, which explain This further shows the effectiveness of our proposed method 21 whymostliteraturestill regardthe (N =3)coupled-linefilter in retrieving both poles and zeros on the complex plane. For as “third” order effectively with three poles only. verification,the polesare also calculated using the eigenvalue Let us now modify slightly the specifications of the approach.Tothatend, pcanbesolvedfromtheeigenvaluesof coupled-line filter with increased FBW = 0.2 and reduced matrix comprising (23) in (11). Then, the poles are obtained This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 6 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES Fig.4. Polesandzeros plotofalinearphasefilter. Fig. 5. Poles and zeros plot of a linear phase filter cascaded with parallel transmissionlines andlumpedelements. via (12) and it is found that they are consistent with those TABLEI shown in Fig. 4. ROOTSOBTAINEDUSINGMULLER’SMETHODWITHDIFFERENT Consider nextthe linear phase filter of (23)being cascaded INITIALGUESSES(NORMALIZEDTO1GHz) with the parallel transmission lines and lumped elements in Fig. 1(b). The capacitances are set as C = 1 pF and 1 C = 2 pF, respectively, while θ = π/2 and θ = 3.6514π. 2 1 2 This cascaded filter couldno longer be analyzedby the trans- formmethodortheeigenvalueapproach,asitnowconsistsof transmission lines and lumped elements. Using the Belevitch theorem and the contour integration method, we are still able to determine the poles and zeros of the cascaded filter, as shown in Fig. 5. It is observed that there are additional poles and complex zeros introduced due to the cascaded parallel transmissionlinesandcapacitors.Thisexamplehasshownthe capabilities of the proposed method in solving the poles and zeros of filters with transmission lines and lumped elements. The contour integration method is useful to identify the regionthatiscertaintocontainoneormorezeros/poles.Once responses that meet different specifications of each band. In identified, one may use various efficient methods, such as our case, it is a Butterworth response in the first passband Muller’smethod,to locate the rootsforzeros/poles.Although (lowerband)andaChebyshevresponseinthesecondpassband being more efficient, the roots may at times go out of (higher band). Fig. 6 (inset) shows the photograph of the range or may not even converge as exemplified for some fabricated filter, realized by noncommensurate transmission cases in Table I. To demonstrate this, the zeros of filters lines and stubs. The simulation and measurement of the in Figs. 4 and 5aresearchedwithinthecomplexsplaneusing S-parameters have been performed and found to be in good Muller’smethodwith differentinitialguesses.Fromthetable, agreement [24, Fig. 7].Actually,ourmainconcernhereisnot we can see that if the initial guesses are not sufficiently close only the S-parameters but also the poles and zeros, as well to the zeros, the roots may go out of range, e.g., converging as whether they have been conforming to our specifications to the RHP zeros or other harmonic band zeros, or may even (e.g., different passband Butterworth/Chebyshev responses). failtoconverge.Forsufficientlycloseinitialguesses,theroots Withthepresenceofnoncommensuratetransmissionlinesand eventually converge within range and they agree well to the stubs, it is difficult to analyze such a filter for its poles using zeros in Figs. 4 and 5. Hence, this underlines the importance existingmethods.UsingtheBelevitchtheoremandthecontour of identifying the correct zeros/poles region for sufficiently integration method, the poles can be solved and plotted in closeinitialguess,whichcouldbeprovidedforbythecontour Fig. 6, showing three poles present around each passband integration method. for the third-order responses. To further verify each passband We next proceed to analyze a realized dual-band filter that response, the reference Butterworth circle (first passband) we have designed, fabricated,and measured as in [24]. In the and Chebyshev ellipse (second passband) are also drawn on design therein, the poles of each passband are properly dis- the complex s plane based on the specification. It can be tributed on the complex s plane to provide differentpassband seen that the poles of each passband lie quite closely on

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478 Pages·2018·157.128 MB·English
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES A PUBLICATION OF THE IEEE MICROWAVE THEORY AND TECHNIQUES SOCIETY JR MTT-S NOVEMBER 2018 VOLUME 66 NUMBER 11 IETMAB (ISSN 0018-9480) THIS ISSUE INCLUDES THE JOURNAL WITHIN A JOURNAL ON MICROWAVE SYSTEMS AND APPLICATIONS REGULAR PAPERS OF THE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES EM Theory and Analysis Techniques Application of Belevitch Theorem for Pole-Zero Analysis of Microwave Filters With Transmission Lines and Lumped Elements .............................................................................................. E. L. Tan and D. Y. Heh 4669 Closed-Form Solution of Rough Conductor Surface Impedance ............................................... D. N. Grujic 4677 Devices and Modeling A GaN HEMT Global Large-Signal Model Including Charge Trapping for Multibias Operation ....................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. P. Gibiino, A. Santarelli, and F Filicori 4684 15-Gb/s 50-cm Wireless Link Using a High-Power Compact III-V 84-GHz Transmitter ................................ . . ... . . .. ... .. . . .. . .. .. .. ... .... .. ... .. . .. . . ... ... . J. Wang, A. Al-Khalidi, L. Wang, R. Morariu, A. Ofiare, and E. Wasige 4698 Nonreciprocal Components Based on Switched Transmission Lines ......................................................... . ... .. .. ....... ..... .. ..... .. A. Nagulu, T. Dine, Z. Xiao, M. Tymchenko, D. L. Saunas, A. Alu, and H. Krishnaswamy 4706 Millimeter-Wave Double-Ridge Waveguide and Components ............. S. Manafi, M. Al-Tarifi, and D. S. Filipovic 4726 Design of Frequency Selective Surface-Based Hybrid Nanocomposite Absorber for Stealth Applications ............. . ..... .. .. .. ..... .... ... . . .. ... ... ... ... .... ... . V. K. Chakradhary, H. B. Baskey, R. Roshan, A. Pathik, and M. J. Akhtar 4737 Conversion Rules Between X-Parameters and Linearized Two-Port Network Parameters for Large-Signal Operating Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Essaadali, A. Jarndal, A. B. Kouki, and F M. Ghannouchi 4 7 45 Passive Circuits Theoretical Analysis of RF Pulse Termination in Nonlinear Transmission Lines .......................................... . . .. .. .. . .. .... ... .. . . ... .. ... .... .. ....... .... .. ... . . ... .......... M. Samizadeh Nikoo, S. M.-A. Hashemi, and F Farzaneh 4757 Integrated Full-Hemisphere Space-to-Frequency Mapping Antenna With CRLH Stripline Feed Network .............. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. D. Enders, J. H. Choi, and J. K. Lee 4765 Multiport ln-Phase/Antiphase Power Dividing Network With Bandpass Response Based on Dielectric Resonator .... ... .. ...... ... .. ........ ...... .. ..... .. .. .. ... .. ... ..... ........... .. ...... ... .......... .... .. .. ..... ...... ... W. Yu and J.-X. Chen 4773 Generalized Synthesized Technique for the Design of Thickness Customizable High-Order Bandpass Frequency-Selective Surface ............................................................. K. Payne, K. Xu, and J. H. Choi 4783 (Contents Continued on Page 4667) +.IEEE (Contents Continued from Front Cover) Asymmetrical Impedance Inverter for Quasi-Optical Bandpass Filters With Transmission Lines of Fixed Length .... ...................................................................................................... P. K. Loo and G. Goussetis 4794 A New Balanced Bandpass Filter With Improved Performance on Right-Angled Isosceles Triangular Patch Resonator ..................................................... Q. Liu, J. Wang, L. Zhu, G. Zhang, F. Huang, and W. Wu 4803 Systematic Evaluation of Spikes Due to Interference Between Cascaded Filters ............................................ ................... A. Morini, G. Venanzoni, P. M. Iglesias, C. Ernst, N. Sidiropoulos, A. Di Donato, and M. Farina 4814 Dual-ModeCharacteristicsofHalf-ModeSIWRectangularCavityandApplicationstoDual-BandFiltersWithWidely Separated Passbands ...................................................................... K. Zhou, C.-X. Zhou, and W. Wu 4820 Hybrid and Monolithic RF Integrated Circuits Analysis and Design of N-Path RF Bandstop Filters Using Walsh-Function-Based Sequence Mixing .................. .................................................................................................... A. Agrawal and A. Natarajan 4830 Compact Series Power Combining Using Subquarter-Wavelength Baluns in Silicon Germanium at 120 GHz ......... ............................................................................................. S. Daneshgar and J. F. Buckwalter 4844 0.3–14 and 16–28 GHz Wide-Bandwidth Cryogenic MMIC Low-Noise Amplifiers ........................................ .............. E. Cha, N. Wadefalk, P.-Å. Nilsson, J. Schleeh, G. Moschetti, A. Pourkabirian, S. Tuzi, and J. Grahn 4860 A 1.8–3.8-GHz Power Amplifier With 40% Efficiency at 8-dB Power Back-Off ........................................... ............................................................................... P. Saad, R. Hou, R. Hellberg, and B. Berglund 4870 Instrumentation and Measurement Techniques Nonuniformly Distributed Electronic Impedance Synthesizer ................... Y. Zhao, S. Hemour, T. Liu, and K. Wu 4883 Jitter Sensitivity Analysis of the Superconducting Josephson Arbitrary Waveform Synthesizer ........................... .......................................... C. A. Donnelly, J. A. Brevik, P. D. Dresselhaus, P. F. Hopkins, and S. P. Benz 4898 JOURNAL WITHIN A JOURNAL ON MICROWAVE SYSTEMS AND APPLICATIONS JOURNALWITHINAJOURNALPAPERS Wireless Communication Systems Effect of Out-of-Band Blockers on the Required Linearity, Phase Noise, and Harmonic Rejection of SDR Receivers Without Input SAW Filter .............................................................. A. Rasekh and M. Sharif Bakhtiar 4913 An Ultralow-Power RF Wireless Receiver With RF Blocker Energy Recycling for IoT Applications ................... ....................................... O. Elsayed, M. Abouzied, V. Vaidya, K. Ravichandran,and E. Sánchez-Sinencio 4927 AFullyIntegrated300-MHzChannelBandwidth256QAMTransceiverWithSelf-InterferenceSuppressioninClosely Spaced Channels at 6.5-GHz Band ............................... Y. Zhang, N. Jiang, F. Huang, X. Tang, and X. You 4943 A Real-Time Architecture for Agile and FPGA-Based Concurrent Triple-Band All-Digital RF Transmission ......... ................ D. C. Dinis, R. Ma, S. Shinjo, K. Yamanaka, K. H. Teo, P. V. Orlik, A. S. R. Oliveira, and J. Vieira 4955 A 0.4-to-4-GHz All-Digital RF Transmitter Package With a Band-Selecting Interposer Combining Three Wideband CMOS Transmitters ............................................................................................................... ................ N.-C. Kuo, B. Yang, A. Wang, L. Kong, C. Wu, V. P. Srini, E. Alon, B. Nikolic´, and A. M. Niknejad 4967 Extraction of the Third-Order 3×3 MIMO Volterra Kernel Outputs Using Multitone Signals ........................... ......................................................................... Z. A. Khan, E. Zenteno, P. Händel, and M. Isaksson 4985 InstantaneousSampleIndexedMagnitude-SelectiveAffine Function-BasedBehavioralModelfor DigitalPredistortion of RF Power Amplifiers .......................................................................... Y. Li, W. Cao, and A. Zhu 5000 CompositeNeuralNetworkDigitalPredistortionModelforJointMitigationofCrosstalk, I/Q Imbalance,Nonlinearity in MIMO Transmitters ...................................................... P. Jaraut, M. Rawat, and F. M. Ghannouchi 5011 Wireless Power Transfer and RFID Systems Increasing the Range of Wireless Power Transmission to Stretchable Electronics .......................................... ................................................................. E. Siman-Tov, V. F.-G. Tseng, S. S. Bedair, and N. Lazarus 5021 Bootstrapped Rectifier–Antenna Co-Integration for Increased Sensitivity in Wirelessly-Powered Sensors ............... .......................................................................................... J. Kang, P. Chiang, and A. Natarajan 5031 Microwave Imaging and Radar Applications A Linear Synthetic Focusing Method for Microwave Imaging of 2-D Objects .............................................. ................................................................................ T. Gholipur, M. Nakhkash, and M. Zoofaghari 5042 W-BandMIMOFMCWRadarSystemWithSimultaneousTransmissionofOrthogonalWaveformsforHigh-Resolution Imaging .... S.-Y.Jeon, M.-H. Ka, S. Shin, M. Kim, S. Kim, S. Kim, J. Kim, A. Dewantari, J. Kim, and H.Chung 5051 (Contents Continued on Page 4668) (Contents Continued from Page 4667) An Efficient Algorithm for MIMO Cylindrical Millimeter-Wave Holographic 3-D Imaging .............................. ................................................................................. J. Gao, B. Deng, Y. Qin, H. Wang, and X. Li 5065 A Portable K-Band 3-D MIMO Radar With Nonuniformly Spaced Array for Short-Range Localization ............... .................................................................................................................. Z. Peng and C. Li 5075 Integration of SPDT Antenna Switch With CMOS Power Amplifier and LNA for FMICW Radar Front End ......... ...................................................................................... B. Kim, J. Jang, C.-Y. Kim, and S. Hong 5087 A C-Band FMCW SAR Transmitter With 2-GHz Bandwidth Using Injection-Locking and Synthetic Bandwidth Techniques .......................................................... S. Balon, K. Mouthaan, C.-H. Heng, and Z. N. Chen 5095 A Fundamental-and-Harmonic Dual-Frequency Doppler Radar System for Vital Signs Detection Enabling Radar Movement Self-Cancellation .................................................................. F. Zhu, K. Wang, and K. Wu 5106 Microwave Sensors and Biomedical Applications Novel Microwave Tomography System Using a Phased-Array Antenna ...................................................... ........................... Y. Abo Rahama, O. Al Aryani, U. Ahmed Din, M. Al Awar, A. Zakaria, and N. Qaddoumi 5119 An RF-Powered Crystal-Less Double-Mixing Receiver for Miniaturized Biomedical Implants ........................... ................................................................................... M. Cai, Z. Wang, Y. Luo, and S. 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IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES 1 Application of Belevitch Theorem for Pole-Zero Analysis of Microwave Filters With Transmission Lines and Lumped Elements Eng Leong Tan , Senior Member, IEEE, and Ding Yu Heh , Member, IEEE Abstract—This paper presents the application of Belevitch change considerably.The numbers of poles and zeros as well theorem for pole-zero analysis of microwave filters synthesized as their locations for a synthesized filter may be different with transmission linesand lumpedelements. Thescattering (S) from the originally specified or designed ones. For instance, matrix determinant ((cid:2)) based on the Belevitch theorem, aptly the poles of a synthesized Chebyshev filter may be deviated called Belevitch determinant,comprises polesandzeros that are separated in different half-plane regions. Using the Belevitch from the reference Chebyshev ellipse after realization. It is determinant, the poles and zeros of filter transfer functions to be shown later that the number of poles of a synthesized can be determined separately with certainty, e.g., by applying classical coupled-line filter [1] turns out to be more. Hence, the contour integration method based on argument principle. the number of poles and zeros should be ascertained after Note that the contour integration can be evaluated numerically synthesis,whiletheirlocationsshouldbeverifiedandchecked without requiring complicated overall analytical expressions. The proposed method is able to solve the poles and zeros for for further tuning wherever needed, especially for filters with filterssynthesizedwithnoncommensuratetransmissionlinesand transmission lines transformed from lumped element proto- lumpedelements,wherethetransformmethodandtheeigenvalue types. Since poles and zeros control the amplitude response, approach are inapplicable. Several applications are discussed phase, and group delay of a filter directly, they need to be to demonstrate the use of Belevitch theorem and the contour analyzedforbetterunderstandingandmanipulationofthefilter integration method to determine the poles and zeros of various microwave filters on the complex plane. response. Unfortunately, for microwave filters synthesized from IndexTerms—Argumentprinciple,Belevitchtheorem,contour transmission-line structures, it is often difficult to determine integration,microwavefilters,numericalmethod,polesandzeros, S matrix determinant. the poles and zeros since the overall analytical expressions maybe verycomplicateddueto the involvementof(nonpoly- nomial) transcendental functions. The difficulties in solving I. INTRODUCTION poles and zeros are further exacerbated when the filters con- MICROWAVE filters have been the subject of much sist of both transmission lines and lumped elements. Their research due to their wide applications in wireless resultant scattering (S) parameters’ expressions, if one could communication.Variousfilter synthesismethods,designs,and ever derive, would comprise combinations of various integer transformations can be found in numerous classic microwave power terms and trigonometric power terms. This makes the textbooks [1]–[4]. Over the years, many filter structures existing methods, such as Richard, Euler, or digital trans- have also been analyzed and synthesized involving (lossless) form method [1], [9], [10], and coupling matrix eigenvalue lumped elements and transmission-line structures, such as approach [11]–[15] inapplicable for the pole-zero analysis steppedlines,stubs,andcoupledlines [5]–[8].However,most of the filters. For blind modeling methods such as vector analyses have not considered in detail the order and locations fittingtechnique [16],theyalsoinvolvemanyuncertaintiesand of poles and zeros of synthesized filters. In particular, most inconsistencies of poles and zeros depending on the chosen filter analyses of microwave textbooks are subjected only for number of poles and fitting bandwidth. low-pass filter prototype and typically in lumped elements. In this paper, we present the application of Belevitch theo- When replaced by transmission-line structures after various remforthepole-zeroanalysisofmicrowavefilterssynthesized transformations, the transfer functions of these filters may with transmission lines and lumped elements. In Section II, the challenges in solving poles and zeros for such filters Manuscript received October 15, 2017; revised February 28, 2018 and will be exemplified in detail, demonstrating the difficulties June 25, 2018; accepted July 29, 2018. This work was supported in part byresearchprojects throughDSOunderGrantDSOCL12016andinpartby and deficiencies of existing methods, including transform theSingaporeMinistryofEducationTertiaryEducationResearchFundunder method, eigenvalue approach, and vector fitting technique. In Grant2015-1-TR15. (Correspondingauthor:DingYuHeh.) Section III, the Belevitch theorem will be discussed. Using The authors are with the School of Electrical and Electronic Engi- neering, Nanyang Technological University, Singapore 639798 (e-mail: the Belevitch theorem,it will be shownthat the scattering (S) [email protected]; [email protected]). matrix determinant (), called Belevitch determinant, com- Color versions of one or more of the figures in this paper are available prisespolesandzerosthatareseparatedindifferenthalf-plane onlineathttp://ieeexplore.ieee.org. Digital ObjectIdentifier 10.1109/TMTT.2018.2865928 regions. Once the poles and zeros are separated completely, 0018-9480©2018IEEE.Personaluseispermitted, butrepublication/redistribution requires IEEEpermission. Seehttp://www.ieee.org/publications_standards/publications/rights/index.html formoreinformation. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES the contour integration method based on argument principle will be applied, which allows them to be determined with certainty.Several applicationswill be discussed in Section IV to demonstrate the use of the Belevitch theorem and the contour integration method to determine the poles and zeros of various microwave filters on the complex plane. II. POLE-ZERO ANALYSIS OFMICROWAVE FILTERS WITH Fig.1. Schematic ofaparallel transmission-line filter. TRANSMISSION LINES AND LUMPEDELEMENTS roots may be solved directly [9], [10]. Following [9], we set For microwave filters involving transmission lines, it is Z =50, Z = Z =2Z ,andtheelectricallengthsofthetwo often challenging to solve for the poles and zeros of transfer 0 1 2 0 parallel transmission lines are θ = π/2, θ = 5π/2, which functions due to the involvement of transcendental functions 1 2 give a = 5. Substituting this a into (6) and upon changing such as the exponentialand trigonometricfunctions.Consider the variable z =u1/2, we obtainan equationthat is consistent the example from [9], where the filter section consists of two parallel transmission-line sections with characteristics with [9, eq. (3)]by notingu =ejπ(f/fd) and the same ak, bk impedances and electrical lengths Z , Z and θ , θ , respec- in [9, Table I]. 1 2 1 2 Thus far, the transform method has been applied for the tively [see Fig. 1(a)]. The expression of S for such a filter 21 case of integera to solve for polynomialroots. However,it is section is given by no longer able to simplify root solving for noncommensurate S21(ω)= NS21(ω)/DS21(ω) (1) lines or when a is not an integer. To demonstrate this, we let θ =3.6514π instead of 5π/2 in Fig. 1(a). Now, substituting where 2 a = 7.3028 into the denominator of (6) gives an equation, NS21(ω) which consists of fractional powers that can no longer be = −2jZ Z Z (Z sin(θ ω/ω )+Z sin(θ ω/ω )) (2) solved easily as polynomial roots. In fact, when a is not 0 1 2 1 1 0 2 2 0 D (ω) integer, it is also mentioned in [9] that the filter cannot be S21  analyzed within the digital-inspired (z or u) framework (e.g., = Z02Z12+ Z02Z22+Z12Z22 (sin(θ1ω/ω0)sin(θ2ω/ω0)) only approximate when θ =0.497π). 1 +Z02Z1Z2(cos2(θ1ω/ω0)+cos2(θ2ω/ω0)+sin2(θ1ω/ω0) The difficulties in solving poles and zeros of transfer + sin2(θ ω/ω )−2cos(θ ω/ω )cos(θ ω/ω )) functions are further exacerbated when the filters consist of  2 0 1 0 2 0 −2j Z Z2Z cos(θ ω/ω )sin(θ ω/ωa ) both transmission lines and lumped elements. To illustrate 0 1 2  2 0 1 0  this, the same filter section in Fig. 1(a) is added with two + Z Z Z2 cos(θ ω/ω )sin(θ ω/ω ) (3) 0 1 2 1 0 2 0 capacitors of capacitance C1 and C2, as shown in Fig. 1(b). ω0 is the center angular frequency. To convert (1)–(3) into The denominator of S21 can be derived as complex frequency s domain, where s = σ + jω, ω is D (ω) substitutedwith−j·s =−j·(σ+jω).Itisworthpointingout S21  = −105C C ω2 p4−2p2p2− j8p2p q +2p2q2 that the S21 denominator of the filter section is already rather 1 2 1 1 2 1 2 2 1 1 complicated even for one section only. As exemplified from +2p2q2− j8p p2q +24p p q q + j8p q q2+ p4 1 2 1 2 1 1 2 1 2 1 1 2  2 (1)–(3),therootscannotbedirectlysolvedforthesenonlinear +2p2q2+2p2q2+ j8p q2q +q4−2q2q2+q4 2 1 2 2 2 1 2 1 1 2 2 functions. Furthermore, as the trigonometric functions are +200ω(2p p + jp q + jp q )(2C p q − jC p p periodic, there are infinitely many poles on the whole s 1 2 1 2 2 1 1 2 1 2 1 2 − jC p p +2C p q + jC q q + jC q q ) domain. It is therefore, difficult to ascertain and solve for the 1 1 2 2 1 2 1 1 2 2 1 2 poles within certain frequency band of interest. −(2p p + jp q + jp q )2 (7) 1 2 1 2 2 1 For commensurate lines whose electrical lengths are the where same (ormultiples)forallsections,onepossible wayto solve     for the poles is by utilizing the Richard transform or Euler’s θ θ p =cos 1ω/ω , p =cos 2ω/ω (8) identities [1]. In particular, using 1 2 0 2 2 0     z+1/z z−1/z θ θ cos(−jsθ /ω ) = , sin(−jsθ /ω )= (4) q =sin 1ω/ω , q =sin 2ω/ω . (9) 1 0 1 0 1 0 2 0 2 2j 2 2 za +1/za za −1/za cos(−jsθ /ω ) = , sin(−jsθ /ω )= It is evident that due to the combinations of various inte- 2 0 2 0 2 2j ger power terms (ω2, ω) and trigonometric power terms (5) [sin((θ/2)ω/ω ),cos((θ/2)ω/ω )],(7)cannotbetransformed 0 0 where a =θ /θ , S of (1) is transformed into z domain as intoanothersimplerformtosolvefortherootsdirectly.Hence, 2 1 21 thetransformmethodisinapplicableforthepole-zeroanalysis 2z2a +2za+1−2za−1−2 S (z)= . (6) of filters with transmission lines and lumped elements. 21 4z2a+1−z2a−1−2za −z For certain microwave filter synthesis [11]–[15], its cou- If a is an integer, the numerator and denominator of (6) plingmatrixmayalsobeusedtofindtheS-parameterpolesvia contain only polynomials(with integer powers) for which the eigenvalue approach. Considering, for example, an n-coupled This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. TANANDHEH:APPLICATIONOFBELEVITCHTHEOREMFORPOLE-ZEROANALYSISOFMICROWAVEFILTERS 3 lumpedelementresonatorfilter,its S and S canbederived network can be represented as [17] 21 11 using the coupling matrix as [3]   S S 1 u v S = √ 2 A−1 (10a) S= S1211 S1222 = g v∗ −u∗ (13) 21 q q n1 e1 en  where g is a strictly Hurwitz polynomial and g, u, and v are 2 −1 S = ± 1− A (10b) related by 11 q 11 e1 uu∗+vv∗ = gg∗. (14) where A = q+ pI − jm (11) The notation g∗ (with subscript “∗”) denotes g∗ =[g(−s∗)]∗,   j ω ω wherethesuperscript“*”indicatescomplexconjugation.Note p = − 0 (12) that the effect of g∗ is to reflect all complex roots of g FBW ω ω 0 symmetricallyabouttheimaginaryaxis.Therefore,therootsof qe1 andqen arethescaledexternalqualityfactorsofinputand g and g∗ are symmetricalabout the imaginary axis. Applying outputresonators, FBW is the fractionalbandwidth,ω is the theBelevitchtheorem(13)and(14),itcanbeshownthattheS 0 center angular frequency, I is the n×n identity matrix, q is matrix determinant, aptly called Belevitch determinant, takes an n × n matrix with all entries zeros except q = 1/q the form 11 e1 and qnn = 1/qen, and m is the n ×n coupling matrix. If a = S S −S S = −g∗. (15) narrowband filter can be represented in the coupling matrix 11 22 21 12 g form of (11), the poles can be found by first solving p from Since g is strictly Hurtwiz, (15) shows that all poles of  are the eigenvalues of matrix comprising m in (11). Thereafter, located in the left half-plane (LHP) of complex s plane, with the poles are obtained via (12). However, a broadbandfilter can no longer be representable thesamenumberofzeros(rootsofg∗)locatedexactlyopposite of jω axis in the right half-plane (RHP). Note that there will in the coupling matrix form of (11) should it comprise com- be no zeros in the LHP and no poles in the RHP for . As binations of noncommensuratetransmission lines and lumped such, the poles and zeros have been effectively separated into elements. Indeed, the impedance or admittance matrix or A in (11) for such a filter should have various powers of ω, different half-plane regions through (15). This is crucial to (1/ω), as well as sin and cos terms, in much the same allow them to be determined separately with certainty in the subsequent analysis. Furthermore, the Belevitch determinant manner like (7). In other words, the p term of (12) not only involves (ω/ω ) and (ω /ω), while the m matrix of (11) not in (15) is an irreducible expression due to g being strictly 0 0 Hurtwiz. Consequently, the degree of a lossless 2-port is the only contains constant coupling coefficients, but also various nonlinear functions of ω. This makes it difficult or impos- degree of the polynomial g of the canonic form (13) [17]. Byretrievingthepolesof,wehaveessentiallyretrievedthe sible to extract the poles using the eigenvalue approach same poles for S sharing the common denominator g. directly.Hence,theeigenvalueapproachisinapplicableforthe 21 Oncethe polesandzerosare separatedcompletely,one can pole-zeroanalysisoffilterswithtransmissionlinesandlumped proceed to solve for them using several possible ways. To elements. that end, the contour integration method based on argument The blind system modeling methods such as the vector principleisonepossiblewaythatmaybeappliedconveniently. fitting technique, Loewner matrix, or Cauchy method are The argument principle is given by [18], [19] sometimes used to locate the system poles and zeros through fitting of measurement data points. We shall comment briefly 1 1 f(s) on the vector fitting technique [16] that is one of the most Z − P = 2πCargf(s)= 2πj f(s)ds (16) C popularmodelingmethods.Whileitisablindfitting/modeling method, the number of poles to estimate is often uncertain where Cargf(s) is the change in argument of f(s) along and the poles obtained are very much dependent on the closed contour C in the counterclockwise direction, f(s) is a fittingbandwidth.Forafilter withnocomplexzeros,spurious meromorphicfunction, Z and P are the number of zeros and complex zeros may even appear that are not symmetrical poles of f(s), including their multiplicities within the closed about the jω axis, thus violating the S-parameters’ unitary contour, and f(s) is the derivative of f(s). The contour C conditions [11]. Henceforth, this technique and other blind alsoshouldnotpassthroughanypolesandzeros.Thecomplex fitting/modeling methods are beyond the scope of this paper. function f(s) in (16) is to be replaced by the Belevitch determinant (s) of a microwave filter on complex s plane. Since  has no zeros in the LHP (Z = 0), we are able III. APPLICATION OFBELEVITCH THEOREM to evaluate (16) to find the number of poles in the LHP. It FORPOLE-ZERO ANALYSIS should be emphasized here that without the use of Belevitch To overcome the difficulties and deficiencies mentioned determinant in (15), the argument principle could not be above, we shall present the application of Belevitch theorem applied directly if f(s) contains both poles and zeros in the for the poles and zeros analysis of microwave filters syn- LHP. Most argument principle-based methods, such as [20] thesized with transmission lines and lumped elements. The and [21], are able to solve only for the zeros of f(s) when it Belevitch theoremstates that the S matrix of a lossless 2-port is analytic within C (P =0). This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES The contour integration of f ≡  can be evaluated numerically via 1 (s) 1 (s+δs/2)−(s−δs/2) ds = . 2πj (s) πj (s+δs/2)+(s−δs/2) C s (17) Here, centralfinite differenceis used for the derivative(s), while the averaging scheme is used for (s). This reduces the number of stencils from three to two for every point on the contour. δs is the spatial step size chosen along the path to ensure convergence.It should be emphasized that to evalu- ate(17),theoverallanalyticalexpressionofisnotrequired, while only the numerical values of  along the contour path are needed. In practice, for microwave filter circuits, the net- work parametersfor individualsections (such as transmission lines,lumpedelements,etc.)canbecomputedandmanipulated readily to obtain the  values of the overall circuits. Fig. 2. S-parameters of a coupled-line filter with Chebyshev response Tolocatethepoles,thecontourcanbesuccessivelydivided (N =3). Inset: layout of N+1sections coupled-line filter along witheven andoddcharacteristic impedances oftheindividual section. into smaller sections until they are tightly enclosed. The contour method could even be combined with other efficient iterative rootsearching algorithms, such as Newton’s method,  can then be obtained subsequently by reflecting all the Muller’s method, etc., which often require rather good ini- zeros of f(s) symmetrically along jω axis. This will bypass tial guess. Having divided the contour into small sections, the computation of 1/ inversion for greater simplicity and the contour centroid may serve as good initial guess to robustness.Apartfrom the sole determinantin (15), the Bele- these iterativealgorithms.Afterthe poleshavebeenretrieved, vitch theorem can also be applied for other S-parameter we can further determine (if any) the complex zeros of a expressions or combinations. For instance, one may consider microwave filter. This can be done by repeating (16) for f(s) = S / to solve directly for the transmission zeros 21 f ≡ S21 and dividing the contour into smaller sections (zeros of S21) or consider f(s) = S11/ or f(s) = S22/ successively to find Z with known P. to solve directly for the reflection zeros of a microwave filter. Thepolesandzerosmayalso be solvedusingthe approach All in all, many opportunities arise from the application of in [22], employing Newton’s identities. However, one would the Belevitch theorem, which could be further explored for need to estimate the number of distinct zeros and poles, the pole-zero analysis of microwave filters. which is done through forming successive system matrices and checking if they are singular. This is often difficult in IV. APPLICATIONS FOR MICROWAVE FILTERS practice to numericallyascertain whether a matrix is singular. A. Applications I: Classical Filters Furthermore, a large number of zeros and poles often result We shall demonstrate the applications of the Belevitch in high polynomial order and ill-conditioned problem. On theoremandthe contourintegrationmethodtosolve forpoles the other hand, alternative searching technique, such as [23], and zeros of several microwave filters. First, consider the involvesexhaustive triangulationof 2-D domain.Still, it does classical coupled-line filter in Fig. 2 [1]. The specifications not guarantee that all the roots can be found and the risk of of the bandpass filter are Chebyshev response with N = 3, missingrootishigherduetotheirregulardiscretizationofthe passband ripple L =0.5 dB, center frequency f =2 GHz, domain.Overall,thereare stilluncertaintiesanddifficultiesin Ar 0 andfractionalbandwidthFBW= 0.1.Thefilterisrealizedby solving for poles and zeros using these methods. Unlike the four (N +1 = 4) coupled-line sections. The S-parameters of approachabove,by using the Belevitch determinant (15), one thefilterareshowninFig.2.FromtheS plot,threereflection will have the poles and zeros separated completely, with only 11 zeros can be observed, which may lead one to deduce that poles and no zeros in the LHP. There is no need to estimate there are three poles within its passband in line with the the number of poles or zeros, thus obviating the forming of third-orderspecification.Usingthecontourintegrationmethod successive system matrices and checking if they are singular. fortheBelevitchdeterminantwithcontourpathsenclosingthe The poles can then be solved as the roots of polynomial. passband around f = 2 GHz, the poles are determined as Hence, there is no need to divide the contour excessively 0 (normalized by 1 GHz) until the poles are tightly enclosed. In practice, one would still consider dividing the contour sufficiently to reduce the s = 2π(−0.03124+ j1.89849) p1 number of enclosed poles in each contour. This should lower s = 2π(−0.06259+ j2.00000) p2 the polynomial order to avoid ill-conditioned problem. s = 2π(−0.03124+ j2.10151) The Belevitch determinant in (15) further allows us to p3 evaluate (16) on the contours in the RHP with f(s) =  sp4 = 2π(−1.88540+ j2.00000) [instead of in the LHP with f(s) = 1/]. The poles of s = 2π(−1.88549+ j2.00000). (18) p5 This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. TANANDHEH:APPLICATIONOFBELEVITCHTHEOREMFORPOLE-ZEROANALYSISOFMICROWAVEFILTERS 5 passband ripple L = 0.01 dB. f is still maintained at Ar 0 2 GHz. The S-parameters are also shown in Fig. 2. Three reflection zerosare still visible fromthe S plot. Meanwhile, 11 the numberof polesdeterminedfromthe determinantcontour integrationmethodisstillfivearoundthepassband.Thepoles are subsequently solved as s = 2π(−0.15601+ j1.67367) p1 s = 2π(−0.71330+ j2.00000) p2 s = 2π(−0.58349+ j2.00000) p3 s = 2π(−0.44182+ j2.00000) p4 s = 2π(−0.15601+ j2.32633). (21) p5 For verification, the transform method is again utilized to obtain the roots in the z domain as z = 0.22427+ j0.85578 p1 z = j0.57108 p2 aFnigd.(3d.)im(aa)gRineaarlyapnadrt(sbo)fimS2a1g.inary parts of S21.Absolute errorsin(c)real zp3 = j0.63237 z = j0.70680 p4 It can be seen that there are five poles within the contour.For z = −0.22427+ j0.85578. (22) p5 verification,we also solvethe polesbyutilizingthe transform methodandfindthepolynomialroots(withpositiveimaginary Using(20),thepoles(ins domain)areinagreementwith(21). part only) in the z domain as It can be seen that all five poles are now near to each other and to jω axis. In addition, note that the two extra poles z = 0.07771+ j0.97266 p1 in (18) and (21) do not lie on/near the Chebyshev ellipse zp2 = j0.95203 on the complex s plane. While the other three poles that z = −0.07771+ j0.97266 are around the Chebyshev ellipse may contribute most to the p3 z = j0.22746 third-order Chebyshev response, the presence of two extra p4 poles would still affect the filter response to a certain extent z = j0.22744. (19) p5 (see Fig. 3). From these two examples, one should appreciate the importance of poles analysis, whereby the number and Thepolesins domaincanthenbededucedfromtheroots(19) locations of the poles of a synthesized filter may be different through the inverse of Richard transform or Euler’s identities from those originally specified. s =ω /θ ·lnz. (20) 0 0 It is found that they agree well with (18) obtained via B. Applications II: Advanced Filters our Belevitch determinant contour integration method, thus Complex zeros may sometimes be utilized to optimize validating our proposed method. the filter phase response for better group delay equalization. Interestingly, from (18), it can be seen that the last two We now apply the Belevitch theoremand the contourintegra- poles sp4 and sp5 are located further away from the jω tionmethodtoanalyzealinearphasefilterwithcomplexzeros axis. To investigate further the consequence of retaining from [3].Thefour-polelinearphasefilterhasapassbandfrom all five poles or only first three poles sp1, sp2, and sp3, 920to975MHzandcanbesynthesizedbycouplingmatrixas ⎡ ⎤ Fig. 3(a) and (b) shows the real and imaginary parts of S 21 0 0.9371 0 0.1953 cthoenystraurectecdomuspianrgedaltlofivSe21poobletsaianneddodnirleycfitlrystfrthormeespimoluelsa,tiaonnd. m =⎢⎢⎣0.90371 0.60196 0.60196 0.90371⎥⎥⎦. (23) Therealzeroslocatedonthe jωaxishavealsobeenincluded, 0.1953 0 0.9371 0 consideringaswellthenegativefrequenciesandperiodicityof this coupled-line filter. Fig. 3(c) and (d) shows the absolute The poles and zeros are solved using the contour integration errors in the real and imaginary parts of S for the results method with Belevitch determinant. The poles and zeros are 21 constructed using five and three poles compared to the simu- plotted on the complex s plane in Fig. 4, marked by “x” and lation.Itcanbeobservedthatifoneexcludesthelasttwopoles “o,”respectively.Itcanbeobservedthatfourpolesarevisible in (18), the errors are larger in both real and imaginary parts withinthepassband.Moreover,acomplexzeroisalsopresent. of S . These errors may at times be tolerable, which explain This further shows the effectiveness of our proposed method 21 whymostliteraturestill regardthe (N =3)coupled-linefilter in retrieving both poles and zeros on the complex plane. For as “third” order effectively with three poles only. verification,the polesare also calculated using the eigenvalue Let us now modify slightly the specifications of the approach.Tothatend, pcanbesolvedfromtheeigenvaluesof coupled-line filter with increased FBW = 0.2 and reduced matrix comprising (23) in (11). Then, the poles are obtained This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 6 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES Fig.4. Polesandzeros plotofalinearphasefilter. Fig. 5. Poles and zeros plot of a linear phase filter cascaded with parallel transmissionlines andlumpedelements. via (12) and it is found that they are consistent with those TABLEI shown in Fig. 4. ROOTSOBTAINEDUSINGMULLER’SMETHODWITHDIFFERENT Consider nextthe linear phase filter of (23)being cascaded INITIALGUESSES(NORMALIZEDTO1GHz) with the parallel transmission lines and lumped elements in Fig. 1(b). The capacitances are set as C = 1 pF and 1 C = 2 pF, respectively, while θ = π/2 and θ = 3.6514π. 2 1 2 This cascaded filter couldno longer be analyzedby the trans- formmethodortheeigenvalueapproach,asitnowconsistsof transmission lines and lumped elements. Using the Belevitch theorem and the contour integration method, we are still able to determine the poles and zeros of the cascaded filter, as shown in Fig. 5. It is observed that there are additional poles and complex zeros introduced due to the cascaded parallel transmissionlinesandcapacitors.Thisexamplehasshownthe capabilities of the proposed method in solving the poles and zeros of filters with transmission lines and lumped elements. The contour integration method is useful to identify the regionthatiscertaintocontainoneormorezeros/poles.Once responses that meet different specifications of each band. In identified, one may use various efficient methods, such as our case, it is a Butterworth response in the first passband Muller’smethod,to locate the rootsforzeros/poles.Although (lowerband)andaChebyshevresponseinthesecondpassband being more efficient, the roots may at times go out of (higher band). Fig. 6 (inset) shows the photograph of the range or may not even converge as exemplified for some fabricated filter, realized by noncommensurate transmission cases in Table I. To demonstrate this, the zeros of filters lines and stubs. The simulation and measurement of the in Figs. 4 and 5aresearchedwithinthecomplexsplaneusing S-parameters have been performed and found to be in good Muller’smethodwith differentinitialguesses.Fromthetable, agreement [24, Fig. 7].Actually,ourmainconcernhereisnot we can see that if the initial guesses are not sufficiently close only the S-parameters but also the poles and zeros, as well to the zeros, the roots may go out of range, e.g., converging as whether they have been conforming to our specifications to the RHP zeros or other harmonic band zeros, or may even (e.g., different passband Butterworth/Chebyshev responses). failtoconverge.Forsufficientlycloseinitialguesses,theroots Withthepresenceofnoncommensuratetransmissionlinesand eventually converge within range and they agree well to the stubs, it is difficult to analyze such a filter for its poles using zeros in Figs. 4 and 5. Hence, this underlines the importance existingmethods.UsingtheBelevitchtheoremandthecontour of identifying the correct zeros/poles region for sufficiently integration method, the poles can be solved and plotted in closeinitialguess,whichcouldbeprovidedforbythecontour Fig. 6, showing three poles present around each passband integration method. for the third-order responses. To further verify each passband We next proceed to analyze a realized dual-band filter that response, the reference Butterworth circle (first passband) we have designed, fabricated,and measured as in [24]. In the and Chebyshev ellipse (second passband) are also drawn on design therein, the poles of each passband are properly dis- the complex s plane based on the specification. It can be tributed on the complex s plane to provide differentpassband seen that the poles of each passband lie quite closely on

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