Design of practical broadband matching networks with commensurate transmission lines

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ContentslistsavailableatSciVerseScienceDirect

International

Journal

of

Electronics

and

Communications

(AEÜ)

j o u r n al hom ep ag e :w w w . e l s e v i e r . c o m / l o c a t e / a e u e

Design

of

practical

broadband

matching

networks

with

commensurate

transmission

lines

Metin

S¸

engül

∗

KadirHasUniversity,FacultyofEngineeringandNaturalSciences,34083Cibali,Fatih,Istanbul,Turkey

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received15November2012 Accepted4February2013 Keywords: Broadbandmatching Realfrequencytechniques Matchingnetwork Losslessnetworks

a

b

s

t

r

a

c

t

Todesignbroadbandmatchingnetworksformicrowavecommunicationsystems,commercially

avail-ablecomputeraideddesign(CAD)toolsarealwayspreferred.Butthesetoolsneedpropermatching

networktopologyandelementvalues.Therefore,inthispaper,apracticalmethodisproposedto

gener-atedistributed-elementmatchingnetworkswithgoodinitialelementvalues.Then,thegainperformance

ofthedesignedmatchingnetworkcanbeoptimizedemployingthesetools.Theutilizationofthe

pro-posedmethodisillustratedbymeansofthegivenexample.Itisshownthatproposedmethodprovides

verygoodinitialsforCADtools.

1. Introduction

In the design of high frequency communication systems, if thewavelength of theoperationfrequency is comparable with physicalsizeofthelumpedcircuitelements,usageofdistributed elementsisinevitable.Therefore,atRadioFrequencies(RF),design of broadband matching networks withdistributed elements or commensuratetransmissionlineshavebeenconsideredasavital problemforengineers[1].

Although analytic theory of broadband matching may be employedfor simple problems [2,3], it iswell knownthat this theoryisinaccessibleexceptforsimpleproblems.Therefore,for practicalapplications,itisalwayspreferabletoutilizeCADtools, to design matching networks with distributed elements [4–6]. Matchedsystemperformanceisoptimizedbyallthecommercially available CAD tools. At the end of this process, characteris-tic impedancesand thedelay lengths ofthe transmissionlines areobtained. But performanceoptimization ishighly nonlinear withrespecttocharacteristicimpedancesanddelaylengths,and requiresproperinitials[7].Furthermore,selectionofinitialvalues isvitalforsuccessfuloptimization,sincetheconvergenceofthe optimizationdependsontheselectedinitialvalues.

Therefore,inthispaper,awell-establishedprocessisproposed, todesignbroadbandmatchingnetworkswithequallengthor com-mensuratetransmissionlines.Theselinesarealsocalledasunit elements(UEs).

∗ Tel.:+902125336532;fax:+902125335753.

E-mailaddresses:msengul@khas.edu.tr,mtnsngl@gmail.com

2. Broadbandmatchingproblem

The broadbandmatching problem can beconsidered as the design ofalossless two-portnetwork betweenageneratorand complexload,insuchawaythatpowertransferfromthesource totheloadismaximizedoverafrequencyband.Thepower trans-fercapabilityofthelosslessmatchingnetworkisbestmeasuredby meansofthetransducerpowergainwhichcanbedeﬁnedasthe ratioofpowerdeliveredtotheloadtotheavailablepowerfromthe generator.

The matching problems can be grouped basically as single matching and double matching problems. In the single match-ingproblems,thegeneratorimpedanceispurelyresistiveandthe loadimpedanceiscomplex.Ontheotherhand,ifbothterminating impedancesarecomplex,thentheproblemiscalledasthedouble matchingproblem.

Letusconsidertheclassicaldoublematchingproblemdepicted inFig.1.Transducerpowergain(TPG)canbewrittenintermsof therealandimaginarypartsoftheloadimpedanceZL=RL+jXLand thoseoftheback-endimpedanceZ2=R2+jX2,orintermsofthe realandimaginarypartsofthegeneratorimpedanceZG=RG+jXG andthoseofthefront-endimpedanceZ1=R1+jX1ofthematching networkasfollows:

TPG(ω)= 4R˛Rˇ

(R˛+Rˇ)2+(X˛+Xˇ)2

. (1)

Hereif˛=1,ˇ=G,andif˛=2,ˇ=L.

Theobjectiveinbroadbandmatchingproblemsistodesignthe losslessmatchingnetworkinsuchawaythatTPGgivenby(1)is maximizedinsideafrequencyband.Sothematchingproblemin thisformalismcanberegardedasthedeterminationofarealizable 1434-8411/$–seefrontmatter © 2013 Elsevier GmbH. All rights reserved.

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G Z

Distributed

Element

Equalizer

L Z 2 2, S Z

E

1 1, S Z

Fig.1.Doublematchingarrangement.

impedancefunctionZ1orZ2.OnceZ1orZ2isobtainedproperly,the losslessmatchingnetworkcanbesynthesizedeasily.

RealfrequencylinesegmenttechniqueproposedbyCarlin (RF-LST)isoneofthebesttechniquestodeterminearealizabledataset forZ2[8,9].Inthismethod,Z2isrealizedasaminimumreactance functionanditsrealpartR2(ω)isresembledbylinesegmentsin suchawaythatR2(ω)=

m

k=1ak(ω)Rk,passingthroughm-selected pairsdesignatedby

Rk,ωk;k=1,2,...,m

.Here,breakpoints (orbreakresistances)Rk areconsideredastheunknownsofthe problem.Then,thesepointsareobtainedvianonlinear optimiza-tionofTPG.

TheimaginarypartX2(ω)=

m

k=1bk(ω)RkofZ2isalsoexpressed bymeansofthesamebreakpointsRk.Itisimportanttonotethat thecoefﬁcientsak(ω)areknownquantitiesandtheyarecalculated intermsofthepre-selectedbreakfrequenciesωk.Thecoefﬁcients bk(ω)are obtainedbymeansof Hilberttransformation relation given for minimum reactance functions. If H{◦} represents the Hilberttransformationoperator,thenbk(ω)=H

ak(ω)

.

InRF-LST,twoindependentapproximationstepsseemtobe dis-advantagesofthemethod.Althoughitispossibletoextendthe methodtosolvedouble matching problems,thecomputational efﬁciencyappliesonlyforsinglematchingproblems.

Thebasicprincipleofthedirectcomputationaltechnique(DCT) issimilartothatoftherealfrequencylinesegmenttechnique[10]. Inthismethod,therealpartoftheunknownmatchingnetwork impedanceR2iswrittenasarealevenrationalfunction.Thenthe unknowncoefﬁcientsofthisfunctionareoptimizedtogetthebest gainperformance.

InDCT, theunknowncoefﬁcients ofR2 must bedetermined sothatR2 isanonnegativeevenrationalfunction,whichinturn ensurestherealizabilityoftheresultingimpedancefunctionZ2. Soinordertoguaranteetherealizability,anauxiliarypolynomial isutilizedforconstructinganintrinsicallynonnegativerealpart functionR2.Bytheintroductionofthispolynomial,althoughthe realizabilityissimplyensured,thecomputationaleffortandthe nonlinearityofthetransducerpowergainwithrespecttothe opti-mizationparametersareincreased.

InFettweis’smethod,parametricrepresentationofthepositive realback-enddrivingpointimpedanceZ2isutilized[11].Namely, thepositive realimpedanceZ2 isexpressedin apartialfraction expansion,andthenthepolesofZ2areoptimizedtogetthebest gainperformanceofthesysteminthefrequencyband.

Theparametricmethodconstitutes anefﬁcientapproach for solvingsinglematchingproblems.Theonlyproblemisthe initial-izationofthelocationofpoles,whichmaybecritical.

Inallthemethodsexplainedbrieﬂyabove,thelossless match-ingnetworkisdescribedintermsofasetoffreeparametersby meansofback-enddrivingpointimpedanceZ2.But,thematching problemcanalsobedescribedbymeansofanyothersetof param-eters.Intherealfrequencyscatteringapproachwhichisreferred toastheSimpliﬁedRealFrequencyTechnique(SRFT),thecanonic polynomialrepresentationofthescatteringmatrixisemployedto describethelosslessmatchingnetwork[12,13].

In anothermethodproposed in[7,14],the back-enddriving pointimpedanceofthematchingnetworkZ2ismodeledasa min-imumreactancefunction,then,ifnecessary,aFosterimpedanceis connectedinseries.

Astheresultoftheexplanationabove,itisdesiredtoexpress theback-endimpedanceZ2ofthematchingnetworkintermsof anysetoffreeparameters.Thengainperformanceofthematching networkisoptimizedvia(1).Butthedeterminationofthe back-endimpedanceexpressioniscomplicated.Thereisaverysimple andobviouswaytodeterminetheback-endimpedanceZ2or front-endimpedanceZ1ofthematchingnetwork.Thisisthecruxofthe proposedmethod.

Intheproposedmethod,thesedrivingpointimpedances(Z2 orZ1)aredeterminedutilizing thescatteringparametersofthe losslessmatchingnetwork,sourceandloadreflectioncoefficients. Soin thenext section,canonic polynomial representation of a distributed-elementtwo-portnetworkisbrieflysummarized,and thenrationaleoftheproposedmethodisgiven.

3. Canonicpolynomialrepresentationofadistributed elementtwo-portnetwork

Mostof thedesign methods for microwavenetworks incor-porateﬁnite homogenoustransmissionlines ofcommensurable lengthsasidealUEs[15].Bycommensurate,itmustbeunderstood thatalllinelengthsinanetworkaremultiplesoftheUElength. Richardshasshownthatthedistributed-elementnetworks com-posedofcommensuratetransmissionlines(UEs)canbeproceeded inanalysisorsynthesisaslumpedelementnetworksunderthe transformation

=tanhp,

whereisthecommensuratedelayofthetransmissionlines,pis theusualcomplexfrequencyvariable(p=+jω)andistheso calledRichardsvariable,=+j.Speciﬁcally,ontheimaginary axis,thetransformationtakestheform=j=jtanω.

ReferringtothedoublematchingconﬁgurationshowninFig.1, thescatteringparametersofthelosslessmatchingnetworkcanbe writtenintermsofthreerealpolynomialsbyusingthewellknown Belevitchrepresentationasfollows:

S11()= h() g(), S12()= f(−) g() , S21()= f() g(), S22()=−h(−)_g() , (2)

wheregisastrictlyHurwitzpolynomial,fisarealpolynomialwhich isconstructedonthetransmissionzerosofthematchingnetwork andisaunimodularconstant(=±1).Ifthetwo-portis recipro-cal,thenthepolynomialfiseitherevenoroddand=f(−)/f().

Thepolynomials

f,g,h

arerelatedbytheFeldtkellerequation

g()g(−)=h()h(−)+f()f(−). (3)

Itcanbeconcludedfrom(3)thattheHurwitzpolynomialg() isafunctionofh()andf().Ifthepolynomialsf()andh()are known,thenthescatteringparametersofthetwo-portnetwork, andthenthenetworkitselfcancompletelybedeﬁned.

Inalmost allpractical applications,thedesignerhasanidea abouttransmissionzerolocationsofthematchingnetwork.Hence thepolynomialf()isusuallyconstructedbythedesigner.For prac-ticalproblems,thedesignermayusethefollowingformoff()

f()=f0()(1−2)n/2 (4)

wherenspeciﬁesthenumberofequal-lengthtransmissionlines incascade,andf0()isanarbitraryrealpolynomial.Apowerful classofnetworkscontainsseriesorshuntstubsandequal-length

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transmissionlinesonly.Series-shortstubsandshunt-openstubs producetransmissionzerosat=∞.Series-openstubsand shunt-shortstubsproducetransmissionzerosat=0.Forsuchnetworks, thepolynomialf()takesthemorepracticalform

f()=k₍₁₋2₎n/2 ₍₅₎

wherekisthetotalnumberofseries-openandshunt-shortstubs, andthedifferencen−(n+k)givesthenumberofseries-shortand shunt-openstubs.Here,nisthedegreeofthematchingnetwork, whichisalsothedegreeofthepolynomialg()orh().

4. Fundamentalsoftheproposedmethod

Consider thedoublematching arrangement shownin Fig.1. Inputreﬂectioncoefﬁcientofthematchingnetworkwhenits out-putportisterminatedinZLcanbeexpressedintermsofscattering parametersofthematchingnetworkas

S1=S11+

S12S21SL

1−S22SL (6)

whereSListheloadreﬂectioncoefﬁcientandexpressedas SL=ZL−1

ZL+1. (7)

Similarly,outputreﬂectioncoefﬁcientofthematchingnetwork whenitsinputportisterminatedinZGcanbewrittenintermsof scatteringparametersofthematchingnetworkas

S2=S22+

S12S21SG

1−S22SG (8)

whereSGisthesourcereﬂectioncoefﬁcientandexpressedas SG= ZG−1

ZG+1. (9)

Sothefront-endandback-enddrivingpointimpedancesofthe matchingnetworkcanbecalculatedviathefollowingequations, respectively; Z1= 1+S1 1−S1 , (10) Z2= 1+S2 1−S2 . (11)

Astheresult,thefollowingalgorithmcanbeproposedtosolve bothsingleanddoublebroadbandmatchingproblemswith dis-tributedelements.

5. Proposedalgorithm

Inputs:

• ZL(measured)=RL(measured)+jXL(measured), ZG(measured)=RG(measured)+ jXG(measured): Measured load and generator impedance data, respectively.

• ωi(measured):Measurementfrequencies,ωi(measured)=2 fi(measured). • fnorm:Normalizationfrequency.

• Rnorm:Impedancenormalizationnumberinohms.

• h0,h1,h2,...,hn:Initialrealcoefﬁcientsofthepolynomialh(). Heren is thedegreeofthepolynomialwhich is equaltothe numberofdistributedelementsinthematchingnetwork.The coefﬁcientscanbeinitializedas±1inanadhocmanner,orthe approachexplainedin[16]canbefollowed.

• f():Apolynomialconstructedonthetransmissionzerosofthe matchingnetwork.Apracticalformisgivenin(5).

• ıc: The stopping criteria of the sum of the square errors.

Outputs:

• Analyticformoftheinputreflectioncoefficientofthelossless matchingnetwork,S11()=h()/g().Itisnotedthatthis algo-rithmdeterminesthecoefficientsofthepolynomialsh()and g(),whichinturnoptimizesthegainperformanceofthesystem. • Circuittopologyofthelosslessmatchingnetworkwithelement values:Thecircuittopologyandelementvaluesareobtainedas theresultofthesynthesisofS11().Synthesisisaccomplishedby extractingpolesat0and∞,correspondingtostubs,while equal-lengthtransmissionlinesareextractedbyemployingRichards extractionmethod[17] orthemethodgiven in[18]. Alterna-tively,thesynthesiscanbecarriedoutinamoregeneralfashion usingthecascadedecompositiontechniquebyFettweis,whichis basedonthefactorizationofscatteringtransfermatrices[19].As aresult,S11()issynthesizedasalosslesstwo-portwhichisthe desiredmatchingnetwork.

Computationalsteps:

Step1:Normalize themeasuredfrequencieswithrespectto fnormandsetallthenormalizedangularfrequencies

ωi=fi(measured)/fnorm.

Normalizethemeasuredloadandgeneratorimpedanceswith respecttoimpedancenormalizationnumberRnorm;

RL=RL(measured)/Rnorm, XL=XL(measured)/Rnorm, RG=RG(measured)/Rnorm, XG=XG(measured)/Rnorm over the entire frequencyband.

Step2:CalculatecorrespondingvaluesofRichardsvariablevia i=ji.=jtanωi.Thedelaycanbeobtainedasusualfromthe lengthlofthedistributed-elementandthephasevelocityc:=l/c. If l is chosen a fraction 1/K of thewavelength =c/fm (where fmisthemaximumnormalizedfrequencyinthefrequencyband, ωm=2 fm),itfollowsthat=2 /Kωm.Toprovideasafelimitfor theendofthestopband,ascaledfrequencyωm,>1,maybe advantageous[17].Then=2 /Kωm.

Step3:Obtain thestrictlyHurwitzpolynomialg() from(3). Thencalculatescatteringparametersvia(2).

Step4:CalculateloadandsourcereﬂectioncoefﬁcientsSLand SGvia(7)and(9),respectively.

Step5:CalculateinputandoutputreﬂectioncoefﬁcientsS1and S2via(6)and(8),respectively.

Step6:CalculateinputandoutputimpedancesZ1andZ2via(10) and(11),respectively.

Step7:Calculatetransducerpowergainvia(1).

Step 8: Calculate the error via

(ω)=1−TPG(ω), then ı=

(ω)

2.

Step9:Ifıisacceptable(ı≤ıc),stopthealgorithmand syn-thesizeS11().Otherwise,changetheinitializedcoefﬁcientsofthe polynomialh()viaanyoptimizationroutineandreturntostep3.

6. Example

In this section, a double-matching example is presented to designapracticalbroadbandmatchingnetwork.Thenormalized sourceandloadimpedancedataaregiveninTable1.Itshouldbe notedthatthegivensourcedatacaneasilymodeledasacapacitor CG=4inserieswitharesistorRG=1(i.e.R+Ctypeofimpedance), andtheloaddataasacapacitorCL=4inparallelwitharesistance RL=1(i.e.R//Ctypeofimpedance).Sincethegivenimpedancedata arenormalized,thereisnoneedanormalizationstep.Thesame exampleissolvedhereviaSRFT.

Inthedesign,K=8,=1.3,andωm=1arechosen,eventually leadingtoanormalizedvalueof=0.6042.

In the matching network, it is not desired to have a trans-former,sotheleastdegreecoefﬁcientofthepolynomialh()must

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Z

L

Z

G

E

Z1

τ

Z

τ

2 Z

τ

3 Z

τ

4 Z

τ

5 Z

τ

6

Fig.2. Designeddistributed-elementdoublematchingnetwork;proposed:Z1=1.3114,Z2=1.5137,Z3=0.33667,Z4=1.8733,Z5=1.18295,Z6=1.2459,=0.6042;SRFT: Z1=1.3117,Z2=1.5067,Z3=0.33697,Z4=1.8708,Z5=1.18295,Z6=1.2453,=0.6042(normalized).

berestricted,i.e.h0=0[20].Thenthepolynomialh()is initial-izedash()=−6₊5₋4₊3₋2₊_in_an_ad_hoc_manner._Also thepolynomialf()isselectedasf()=(1−2₎3_._So_in_the match-ingnetworktherewillbesixcascadedunitelementsonly.Inthe example,˛andˇareselectedas˛=1,ˇ=G.Sofront-end driv-ingpointimpedanceZ1andsourceimpedanceZGareusedinthe TPGexpressioninStep7.Thenafterrunningtheproposed algo-rithm,thefollowingscatteringparameterofthematchingnetwork isobtained S11()= h() g() where h()=−21.85646₊_39.70705₊_16.28164₊_0.35353 +9.47042₋_2.3660, g()=21.87926+80.59735+96.01754+67.20693 +33.18402₊_8.8298₊_1.

After synthesizing the obtained scattering parameter, the matchingnetworkseeninFig.2isobtained.

Asitis seenfromFig.3,initialperformance ofthematched system looks very good. However, it can be furtherimproved viaoptimizationutilizingthecommerciallyavailabledesigntool calledMicrowaveOfﬁceofAppliedWaveResearchInc.(AWR)[4]. Thus,theﬁnalnormalizedelementsvaluesaregivenasZ1=0.7517, Z2=1.709,Z3=0.2315,Z4=1.512,Z5=0.1459,Z6=1.145.For com-parisonpurpose,bothinitialandtheoptimizedperformancesof thematchedsystemandtheperformanceobtainedviaSRFTare depictedinFig.3.

InFig.4,transducerpowergaincurvesarezoomed.Thecurves obtainedviatheproposedmethodandSRFTareveryclosetoeach other,nearlythesame.ThealgorithmisimplementedviaMatlab. Theelapsedtimeforthisexampleis84.1322s.Itis84.6948svia SRFT.Alsototalsquarederror(ı=

|

(ω)|2₎_which_is_calculated atStep 8is 1.7189.It is1.7194 forSRFT. Thereis avery small

Table1

Givennormalizedloadandsourceimpedancedata.

ω RL XL RG XG 0.1 0.86 −0.34 1.00 −2.2500 0.2 0.60 −0.49 1.00 −1.2500 0.3 0.41 −0.49 1.00 −0.8333 0.4 0.28 −0.45 1.00 −0.6250 0.5 0.20 −0.40 1.00 −0.5000 0.6 0.14 −0.35 1.00 −0.4167 0.7 0.11 −0.32 1.00 −0.3571 0.8 0.09 −0.28 1.00 −0.3125 0.9 0.07 −0.26 1.00 −0.2778 1.0 0.06 −0.23 1.00 −0.2500

Fig.3. Performanceofthematchedsystemdesignedwithdistributedelements.

Fig.4.Closerexaminationofthegainperformances.

differencewhichcanbenegligible.Consequently,itcanbesaidthat

theproposedmethodandSRFTnearlyhavethesameperformance.

7. Conclusion

Designofpracticalbroadbandmatchingnetworksisoneofthe

importantproblemsofmicrowaveengineers.Inthisregard,

com-merciallyavailablecomputer-aideddesigntoolsareutilized.Once

thematchingnetworktopologyandproperinitialelementvalues

areobtained, these toolsare excellentto optimizesystem

per-formancebyworkingontheinitializedelementvalues.Soinitial

elementvaluesbecomeveryvital,sincethesystemperformance

ishighlynonlinearintermsoftheelementvaluesofthe

match-ingnetwork.Therefore,inthispaper,aninitializationmethodis

proposedtoconstructlosslessbroadbandmatchingnetworkswith

distributedelements.

Intheproposedmethod,theback-endorfront-enddrivingpoint

impedanceofthematchingnetworkisdeterminedintermsofthe

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reﬂectioncoefﬁcients.Thenthisimpedanceandoneofthe

termi-nationimpedances(ZGorZL)areusedtocalculatethetransducer

powergainofthesystem.Scatteringparametersofthematching

networkareoptimizedtogetthebestgainperformance.

Finally,it is synthesized asa lossless two-portresultingthe

desiredmatchingnetwork topologywithinitialelementvalues.

Eventually, the actual performance of the matched system is

improvedbymeansofacommerciallyavailableCADtool.

Basicadvantagesoftheproposedmethodcanbesummarized

asfollows:Thetransmissionzerosofthematchingnetworkcan

becontrolleddirectlybythechoiceofthepolynomialf().Since

thetransducerpowergainisquadraticallydependentonthe

opti-mizationparameters,theproblemreducestothatofaquadratic

optimization.Sothenumericalconvergenceofthemethodis

excel-lent.Alsotheproposedmethodis applicabletobothsingleand

doublematchingproblems.

An example is presented to construct broadband matching

network with distributed elements. It is shown that the

pro-posedmethodgeneratesverygoodinitialstofurtherimprovethe

matchedsystemperformancebyworkingontheelementvalues.

Therefore,itisexpectedthattheproposedalgorithmisusedasa

front-endforthecommerciallyavailableCADtoolstodesign

prac-ticalbroadbandmatchingnetworksformicrowavecommunication

systems.

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MetinS¸engülreceivedB.Sc.andM.Sc.degreesin Electron-icsEngineeringfrom ˙IstanbulUniversity,Turkeyin1996 and1999,respectively.HecompletedhisPh.D.in2006 atIs¸ıkUniversity, ˙Istanbul,Turkey.Heworkedasa tech-nicianat ˙IstanbulUniversityfrom1990to1997.Hewas acircuitdesignengineeratR&DLabsofthePrime Min-istryOfﬁceofTurkeybetween1997and2000.Between 2000and2008,hewasalectureratKadirHasUniversity, ˙Istanbul,Turkey.Dr.S¸engülwasavisitingresearcherat InstituteforInformationTechnology,Technische Univer-sitätIlmenau,Ilmenau,Germanyin2006forsixmonths. HeworkedasanassistantprofessoratKadirHas Univer-sitybetween2008and2010.Currentlyheisservingasan associateprofessoratKadirHasUniveristy.Dr.S¸engülisworkingonmicrowave matchingnetworks/ampliﬁers,datamodelingandcircuitdesignviamodeling.