IEEE MTT-V066-I11 (2018-11)

478 Pages·2018·157.128 MB·English

by  FANTOMASPING

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. Mirabbasi 5129
IEEE MICROWAVETHEORYAND TECHNIQUES SOCIETY
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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,
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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
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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).
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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
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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
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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