Broadband impedance matching via lossless unsymmetrical lattice networks

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Int.J.Electron.Commun.(AEÜ)66 (2012) 76–79

ContentslistsavailableatScienceDirect

International

Journal

of

Electronics

and

Communications

(AEÜ)

jo u rn al h om e p a g e :w w w . e l s e v i e r . d e / ae u e

Broadband

impedance

matching

via

lossless

unsymmetrical

lattice

networks

Metin

S¸

engül

∗

KadirHasUniversityEngineeringFaculty34083Cibali,Fatih-Istanbul,Turkey

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received16September2010 Accepted11May2011 Keywords: Broadbandnetworks Losslessnetworks

Latticenetworks,Impedancematching Synthesis

a

b

s

t

r

a

c

t

Inthispaper,abroadbandimpedancematchingnetwork(equalizer)designalgorithmhasbeenproposed.

Intheequalizer,alosslessunsymmetricallatticenetworkhasbeenutilized.Thebranchimpedancesof

thelatticenetworkareconsideredassinglyterminatedlosslessLCnetworks,sinceitisnotdesiredto

dissipatepowerintheequalizer.Aftergivingthealgorithm,itsusagehasbeenillustratedviaanexample.

1. Introduction

Designofbroadbandmatchingnetworksisanessentialproblem formicrowavecommunicationsystemengineers[1].Soanalytic theoryofbroadbandmatching[2],[3]andcomputer-aided-design (CAD)methodsareessentialtoolsforthedesigners[4–6].Butit is wellknownthat analytictheory isdifﬁcult toutilizeeven if thesourceandloadimpedancesaresimple.Therefore,itisalways attractivetouseCADtechniques.AlltheCADtechniquesoptimize thematchedsystemperformance.Butperformanceoptimization ishighlynonlinearwithrespecttoelementvaluesandneedsvery goodinitials [7].Asa result,initialelementvalues arevital for successfuloptimization.

In matching network design problems, ladder networks are used,sincethesestructureshaveverylowsensitivity[8].In addi-tiontheyarepreferablesince theyare unbalanced,thusallthe shuntbranchescanbegrounded.Ifnon-minimum-phasetransfer characteristicsaredesired,thenetworkcomplexityincreases.This addedcomplexityistheresultofright-halfplanezerosofatransfer function,whichcanberealizedonlybyasignal-cancellation pro-cess.Thisrequiresmorethanonepathoftransmissionbetween theinputandoutputports.Inladdernetworks,thiscanbe real-izedbyparallelorbridgedstructures.Withoutthecommonground betweentheinputandoutputports,righthalf-planezeroscanbe realizedbyabridgestructureoftheformshowninFig.1.

Therefore, in this paper, two-port bridge structures are employedinmatchingnetworks.Inthefollowingsections, two-portbridgenetworksandrealfrequencybroadbandmatchingwill

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

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

besummarized.Then,theproposeddesignalgorithmandan exam-plewillbepresented.

2. Bridgenetworks

When a voltage source of frequency ω0 with the

resis-tance RS is applied to the input port of a bridge network

(Fig. 1) and the branch impedances are related such that [Z1(jω0)Z4(jω0)=Z2(jω0)Z3(jω0)],thevoltagedropacrosstheload

impedanceZLiszero,resultinginazerooftransmissionorazero

ofthevoltagetransferratioat(p=jω0).Abridgenetwork

satis-fyingtheseconditionsissaidtobebalanced.Ifoppositearmsof thebridgehavethesameimpedances [ZA(p)=Z1(p)=Z4(p)] and

[ZB(p)=Z2(p)=Z3(p)],thenetworkissaidtobesymmetrical.Ifthe

bridgeleadsaretwisted,theconﬁgurationseeninFig.2isobtained, whichisknownasaunsymmetricallatticenetwork.Althoughany physicallyrealizabletransferfunctioncanbesynthesizedbyusing unsymmetricalRLClatticenetwork,asymmetricallatticerestricts therangeofrealizablefunctions.Especially,certainphysically real-izabletransfer functionshavingpoles onthejω-axiscannot be synthesizedbyusingsymmetricallatticestructures[9].

3. Realfrequencybroadbandmatching

Letusconsidertheclassicalsinglematchingproblem(resistive sourceimpedanceandcomplexloadimpedance)showninFig.2.

ThematchingconditionsofthecomplexloadZLtotheresistive

generatorRScanbeformulatedintermsofthenormalized

reflec-tioncoefficientsatports1and2.Theinputreflectioncoefficients fNormcanbedefinedby

1=Zin−RS

Zin+RS

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M.S¸engül/Int.J.Electron.Commun.(AEÜ)66 (2012) 76–79 77 ) ( 1 p Z Z 3 ( p ) ) ( 2 p Z Z ₄( p ) ) ( p Z L E + S R

Fig.1.Generaltwo-portbridgenetwork.

whereZin istheinputimpedanceseenatport1whenport2is

terminatedbytheloadZL.Similarlythereﬂectioncoefﬁcientatport

2canbedeﬁnedby 2= Zout−ZL∗

Zout+ZL

(2) whereZoutistheimpedanceseenatport2whenport1is

termi-natedbythesourceresistanceRS,andtheupperasteriskdenotes

complexconjugation.Here,2isthenormalizedreﬂection

coefﬁ-cientatport2.Sincethetwo-portisconsideredaslossless,wehave ontheimaginaryaxisofthecomplexfrequencyplane

1

2

=

2

. (3)

Then,thetransducerpowergain(TPG)atrealfrequenciescan beexpressedas TPG(ω)=1−

1

2 =1−

2

2 . (4)

Thegoalinbroadbandmatchingistodesignthelossless net-workN,whichconsistsofthearmimpedancesZ1(p),Z2(p),Z3(p)

andZ4(p),suchthatTPGgivenbyEq.(4)ismaximizedinside a

desiredfrequencyband.Obviously,maximizingTPG(ω)meansto minimizingthemodulusofthereﬂectioncoefﬁcients

1

or

2

.In

thiscontext,thematchingproblemisreducedtothedetermination ofarealizableimpedancefunctionZinorZout.

LettheequalizerinputimpedanceZinbeexpressedintermsof

itsrealandimaginarypartsontherealfrequencyaxisas

Zin(jω)=Rin(ω)+jXin(ω). (5)

ByusingEq.(5),Eq.(4)andEq.(1)weobtaintransducerpower gain(TPG)intermsoftherealandimaginarypartsoftheinput impedanceZinoftheequalizerNandthesourceresistanceRSas

follows:

TPG(ω)= 4RSRin(ω)

(RS+Rin(ω))2+(Xin(ω))2

. (6)

NamelythematchingproblemconsistsofﬁndingZin(jω)such

thatTPG(ω)ismaximizedinsideadesiredfrequencyband.OnceZin

isdeterminedproperly,theequalizernetworkNcanbesynthesized directlybyusingthe obtainedimpedance orthecorresponding reﬂectioncoefﬁcient. ) ( 1 p Z ) ( 4 p Z ) ( p Z _L S R E + Z 3 ( p ) Z 2 ( p ) in Z , 1 ρ ρ 2 , Z out N

Fig.2.Unsymmetricallatticenetwork.

4. Rationaleofthematchingprocedure

Foralosslesstwo-portliketheonedepictedinFig.2,thecanonic formofthescatteringmatrixisgivenby[10],[11]

S(p)₌

S11(p) S12(p) S21(p) S22(p)

= _g(p)1

h(p) f(−p) f(p) _−h(−p)

(7) wherep=+jωisthecomplexfrequencyvariable,and=±1 isaunimodular constant.Ifthetwo-portisreciprocal,thenthe polynomialf(p)iseithereven orodd.In thiscase, =+1iff(p) iseven,and =−1iff(p)isodd.Thus, foralossless,reciprocal two-port

=f(−p)

f(p) =±1. (8)

Foralosslesstwo-portwithresistivetermination,energy con-versationrequiresthat

S(p)ST₍_−p)₌_I, ₍₉₎

whereIistheidentitymatrix.TheexplicitformofEq. (9)is knownastheFeldtkellerequationandgivenas

g(p)g(−p)=h(p)h(−p)+f(p)f(−p). (10) In Eqs.(7) and(10),g(p) is a strictlyHurwitz polynomialof nthdegreewithrealcoefﬁcients,andh(p)isapolynomialofnth degreewithrealcoefﬁcients.Thepolynomialfunctionf(p)includes alltransmissionzerosofthetwo-port.

ConsiderthebridgenetworkseeninFig.1orFig.2.Sinceitis notdesiredtodissipateanypowerintheimpedancesZ1(p),Z2(p),

Z3(p)andZ4(p),theymustcontainonlyinductorsandcapacitors.

Alsotheirterminationsmustbeeithershortoropen,notresistive terminations.Sotheseimpedancesaresinglyterminatedlossless LCnetworks[12].

In[12],ithasbeenshownthatforalosslesssinglyterminated network,theinputreﬂectioncoefﬁcientcanbeexpressedas S11(p)=±

g(−p) g(p) =˛

g(−p)

g(p) , (11)

where ˛=+1 and ˛=− 1 corresponds to an openand short termination,respectively.Sotheprocedureproposedin[12]can beused todesign theimpedancesZ1(p),Z2(p),Z3(p) andZ4(p).

Thenthefollowingalgorithmisproposedtodesignthebroadband impedancematchingnetworkbyusingunsymmetricallattice net-works.

4.1. Algorithm 4.1.1. Inputs

• ωi(actual)=2fi(actual);i=1,2,...,Nω:measurementorcalculation

frequenciesselectedarbitrarily.

• Nω:Totalnumberofmeasurementorcalculationfrequencies.

• ZL(actual)(jωi)=RL(actual)(ωi)+jXL(actual)(ωi);i=1, 2,...,Nω:

mea-suredorcalculatedloadimpedancedataatNωfrequencypoints.

• RS:Givensourceresistance.

• fNorm:Normalizationfrequency.

• R0:Normalizationresistance,usually50.

• nk;k=1,2,3,4:Desirednumberofelementsinthearmsofthe

bridgenetwork.

• ˛k=±1;k=1,2,3,4:Desiredterminationtypeofthearmsofthe

bridgenetwork.

• gk(p);k=1,2,3,4:Initializedpolynomialg(p)describingthearm

impedancesofthebridgenetwork.

• T0:Desiredﬂattransducerpowergainlevel.

• ı:Thestoppingcriteriaforthesumofthesquareerrors.Formany practicalproblems,itissufﬁcienttochooseı=10−3.

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78 M.S¸engül/Int.J.Electron.Commun.(AEÜ)66 (2012) 76–79

4.2. Computationalsteps

Step1:Ifthegivenloadimpedanceandfrequenciesareactual values, not normalized, then normalize the frequencies with respecttofNormandsetallthenormalizedangularfrequencies

ωi=

fi(actual)

fNorm

Normalizetheloadimpedancewithrespecttonormalization resistanceR0overtheentirefrequencybandas

RL= RL(actual) R0 ,XL= XL(actual) R0 .

Itshouldbenotedthat iftheloadis speciﬁedasadmittance data, then the normalization resistance R0 multiplies the real

andimaginarypartsoftheadmittancedata(i.e.,GL=GL(actual)R0,

BL=BL(actual)R0).

Butifthegiven loadimpedance andfrequencies arealready normalized,thengotothenextstepdirectlywithoutany normal-izationprocess.

Step2:Calculatetheinputimpedancevaluesofthearmsofthe bridgenetworkas Z(k)(jωi)= 1+S(k),11(jωi) 1−S(k),11(jωi) , (12) k=1,2,3,4 and i=1,2,...,Nω

whereS(k),11(jωi)=˛kggkk(−jω(jωi)i) istheinputreﬂectioncoefﬁcient

ofthearmsofthebridgenetwork.

Step3:Calculatetheinputimpedanceofthebridgenetwork whenport2isterminatedbytheloadimpedanceZLviathe

follow-ingequation(seeFig.2) Zin= N D where N=Z1(Z4ZL+Z3ZL+Z2Z3+Z3Z4+Z2Z4)+Z2Z4ZL+Z2Z3ZL+Z2Z3Z4 and D=Z1(ZL+Z2+Z4)+Z2Z3+Z2ZL+Z4ZL+Z3Z4+Z3ZL.

Theinputimpedanceexpressionhasbeenobtainedbyusing −to−Ytransformationequations[13].

Step4:CalculatetransducerpowergainviaEq.(6)asfollows

TPG(ωi)=

4RSRin(ωi)

(RS+Rin(ωi))2+(Xin(ωi))2

whereRin(ωi)=Real (Zin(jωi)) andXin(ωi)=Imag (Zin(jωi)).

Step5:Calculatethesumofthesquarederrorvia

ıc= Nω

i=i

ε(jωi)

2 whereε(jωi)=T0−TPG(ωi). Step6:Ifıc≤ı,synthesizeS(k),11(p)=˛kg_gk(−p)

k(p) andobtainthe

arm networks of the bridge, then stop. Otherwise, change ˛k

(terminationtypes)andgk(p)(initializedpolynomials)viaany

opti-mizationroutineandgotoStep2.

5. Example

Inthissection,anexamplewillbegiventoillustratethe pro-posedalgorithm.Hereallthecalculationswillbemadebyusing

) ( 1 p Z ) ( 4 p Z L R S R E + ₍ ₎ 2 p Z ) ( 3 p Z L C L L

Fig.3. Thesourceandloadterminations,RS=1,LL=1,CL=3,RL=1(normalized).

Table1

Calculatedloadimpedancedata(normalized).

ω RL XL 0.1 0.9174 −0.1752 0.2 0.7353 −0.2412 0.3 0.5525 −0.1972 0.4 0.4098 −0.0918 0.5 0.3077 0.0385 0.6 0.2358 0.1755 0.7 0.1848 0.3118 0.8 0.1474 0.4450 0.9 0.1206 0.5743 1.0 0.1000 0.7000

normalizedvalues.Afterdesigningthematchingnetwork,all

com-ponentscanbede-normalizedbyusingthegivennormalization

frequency(fNorm)andresistance(R0).

Thesourceresistanceandtheloadimpedancewhichisselected

asaseriesinductorandaparallelconnectionofacapacitoranda

resistorinnormalizedvaluescanbeseeninFig.3.

Sincethegivensourceandloadterminationshavenormalized elementvalues,thereisnoneedanormalizationprocess.InTable1, thecalculatedloadimpedancevaluesaregiven.

Theselectedinitialcoefﬁcientsofthepolynomials(gk(p)),the

alphaconstants(˛k)andthedesiredﬂattransducerpowergain

level (T0) are as follows, g1=[ 4 2 3 ], g2=[ 2 4 3 ], g3=

[ 3 5 2 ],g4=[ 1 2 4 ],˛1=+1,˛2=−1,˛3=−1,˛4=−1,and

T0=0.7,respectively.

After running the proposed algorithm, the fol-lowing polynomial coefﬁcients and alpha constants are obtained, g1=[ 6.0437 23.1923 3.1920 ], g2=

[ 6.3061 7.7312 0.2542 ], g3=[ 13.1356 6.4255 0.0907 ],

g4=[ 1.3511 13.3529 12.2343 ], ˛1=+1, ˛2=−1, ˛3=−1,

˛4=−1,respectively.

After synthesizing the corresponding reﬂection coefﬁcients

S(k),11(p)=˛kg_gk_k(−p)_(p)

thebridgenetworkseeninFig.4isreached. TheobtainedtransducerpowergaincurveisgiveninFig.5.

Actualelementvaluescanbeobtainedbyde-normalization.In thiscase,actualelementvaluesaregivenby

ActualCapacitor= (NormalizedCapacitor/2fNorm)

R0 , 1 L L R S R E + L C L 1 C 2 L 3 L 4 L 2 C 3 C 4 C

Fig. 4.Designed matching network, L1=0.2606, C1=7.2658, L2=30.4138,

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M.S¸engül/Int.J.Electron.Commun.(AEÜ)66 (2012) 76–79 79

Fig.5.Transducerpowergain.

ActualInductor₌

_Normalized_Inductor 2fNorm

R0,

ActualResistor₌(NormalizedResistor)R0.

Sincethematchingnetwork isdesignedbyusingnormalized values,thecutofffrequency ofthenetworkis ω=1(seeFig.5). Afterde-normalizationprocess,itshiftstothegivennormalization frequency,sincefi(actual)=ωifNorm.

AscanbeseeninFig.5,anearlyﬂattransducerpowergaincurve isobtainedwithintherequiredfrequencybandatthedesiredﬂat gainlevel(T0=0.7).

6. Resultsandconclusion

An algorithm has been proposed to design broadband impedancematchingnetworksvialosslessunsymmetricallattice networks.Sinceitisnotdesiredtodissipatepowerintheequalizer, thearmimpedancesofthelatticenetworkareselectedassingly terminatedlosslessLCsections.Inthepaper,singlematching prob-lem(resistivesourceimpedanceandcomplexloadimpedance)has beenconsidered.Butthesameprocedurecanbeusedeasilyfor dou-blematchingproblems(complexsourceimpedanceandcomplex loadimpedance).

Intheexample,thedesiredﬂattransducerpowergainlevelis selectedas0.7.Ascanbeseenfromthetransducerpowergain graph,anearlyﬂatgaincurvearoundthislevelhasbeenobtained.

Itisshownthattheproposedmethodgeneratesverygood ini-tialstoimprovethematchedsystemperformancebyoptimizing theelementvalues.Therefore,itisexpectedthattheproposed algo-rithmcanbeusedasafront-endforthecommerciallyavailableCAD toolstodesignbroadbandmatchingnetworksforcommunication systems.

AppendixA. Supplementarydata

Supplementarydataassociatedwiththisarticlecanbefound,in theonlineversion,atdoi:10.1016/j.aeue.2011.05.005.

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[12] S¸engül M. Reﬂectance-based foster impedance data modeling. Frequenz Journal of RF Engineering and Telecommunications 2007;61(7–8): 194–6.

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MetinS¸engülreceivedB.Sc.andM.Sc.degreesinElectronicsEngineeringfrom

˙IstanbulUniversity,Turkeyin1996and1999,respectively.HecompletedhisPh.D. in2006atIs¸ıkUniversity, ˙Istanbul,Turkey.Heworkedasatechnicianat ˙Istanbul Universityfrom1990to1997.HewasacircuitdesignengineeratR&DLabsof thePrimeMinistryOfﬁceofTurkeybetween1997and2000.Between2000and 2008,hewasalectureratKadirHasUniversity, ˙Istanbul,Turkey.Dr.S¸engülwas avisitingresearcheratInstituteforInformationTechnology,Technische Univer-sitätIlmenau,Ilmenau,Germanyin2006forsixmonths.Heworkedasanassistant professoratKadirHasUniversitybetween2008and2010.Currentlyheis serv-ingasanassociateprofessoratKadirHasUniveristy.Dr.S¸engülisworkingon microwavematchingnetworks/ampliﬁers,datamodelingandcircuitdesignvia modeling.