CHAPTER 8 TIIB DESIGN OF WIDEBAND IMPEDANCE-MATCHING NETWORKS 8 . 1 INTRODUCTION .Animpedance-matchingnetwork usually matchesa load to a sourceinside the passband ad may alsobe usedto attenuateunwantedsignalsoutsideit. Whenthe load impedanceandthe soruceimpedancearepurely resistive,inductorcapacitor(LC) networkscanbe designedrelativelyeasilyto fulfill the filter specifications n widebandmatching networks.It is difficult, however,if not impossible,to scale inpedancesover a widebandby using only a limited numberof inductorsand capacitors. '!tis transformationfunction can only be done with transformerswhen the bandwidth,"- .sformationproduct becomeslarge. If the requiredbandwidthis relatively small, bwever, it is possibleto transformresistance overlargedistanceswith LC networks. When the load impedanceor the sourceimpedanceis reactive, part of the hpedance transformationfunction ofthe matchingnetworkis to removethis reactiveness. Tbeextentto which this canbe doneis a functionof the loadimpedanceitself, aswell as t Fansducerpowergain versusthe frequencyresponserequired. The limitationsof a specificloadimpedancefor a specificfrequencyresponsecan bc determinedin at least three ways. Fano'sset of integral equationscan be used to j': ' :rminetheseconstraintsI while Youlaformulatedtheconstraints in termsof Laurent I ], . :s expansions[2]. Carlin advancedan iterativeprocedurefor this purpose[3]. Becauseof its relativesimplicity,only the iterativetechniquedevelopedby Carlin " . bepresentedhere. While theunderlyingtheorywill not beconsidered here,theintegralconstraintson resistor-inductor(RL) and resistor-capacitor (RC) networkslead to simple and -.-:ul upperlimits on the gain [4]. Thesegainlimits will be considered in Section8.3.3, r-og with the Youla gain-bandwidthconstraintsassociated with a parallelRC or a series ; :oad(Chebyshevresponse). With the limitations of a particular load (or source)known, a network that will idetherequiredpowergainversusfrequencyresponse canbedesignedby usingdirect resisor iterativetechniques. Bothoftheseapproaches will bediscussed in thischapter. Networksfor matchinga complexloadto a complexsourceareoftenrequired.A tr,.rctical approachto solving this classof problems was developedby Chen and Satyana-
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Design of RF and Microwave Amplifiers and Oscillators
rayana[5], andmorerecentlyanaltemativeandsimplifiedtheorywasintroducedbyCarlin and Yarman[6]. Carlin and Yarmanalsodevelopediterativetechniquesfor matchinga complexloadto a complexsource[6, 7]. Becauseof its relativesimplicityandits superior results[8], only iterativetechniquesfor matchinga complexloadto a complexsourcewill be considered. It is often possibleto designmatchingnetworksfor complexterminationsby arethenabsorbed initially assumingtheterminationsto bepurelyresistive.Thereactances the effortrequiredto parasiticallyinto thenetworkwhenthedesignis completed.Because designa networkin this way is minimal whenit canbe done,this approachwill alsobe consideredin this chapter. networksare often requiredto providea transducerpower Impedance-matching LC networkscanbe with a positiveslopein thepassband. gainversusfrequencyresponse There is, however,the fulfill this requirement. or directly to interactively designedeither in the frequencies at the lower be mismatched will inevitably thatthe source disadvantage in the to changes gain tend to be sensitive slopesalso passband.LC networkswith RLC impedance-matching componentvalues.Becausethis doesnot necessarilyapplyto networks,the designof thesenetworkswill alsobe examinedin this chapter.
8.2 '
F'ITTING AN IMPEDANCE OR ADMITTANCE FUNCTION TO A SET OF IMPEDANCE VBRSUS FREQUENCY COORDINATES
When impedance-matchingnetworksaredesigned,impedance(or admittance)functions that will approximatea setof discreteimpedanceversusfrequencycoordinatesareoften input impedanceof a transistoror mightbe themeasured required.The setof coordinates (admittance)of a networkto be input impedance output or it could be the &tenna, or dcsigned. It is sometimespossibleto approximatethemeasuredimpedanceof a devicewith rimple RC, RL, or RLC equivalentcircuits.This canusuallybe donewhenthe resistive pt of the impedanceor admittanceis moreor lessconstantoverthe frequencyrangeof interest. The componentsof suchan equivalentcircuit canbe determinedby settingup an ofthe networkchosen,andequatingits real oqgationforthe inputimpedanceor admittance od imaginarypartsto the measuredvalues.Althoughthis techniquecanbe used,more techniquesareoftenrequired. sophisticated A major problem in finding an impedancefunction that will fit a given set of coordinatesis its realizability.The functionobtainedmustbe positive-real. A techniquethat usuallygivesgoodresultsis basedon the fact that the reactance (admitiance) functioncanbedeterminedwhenthe (susceptance) of a minimum-impedance is known [9]. The equivalentcircuit of a minimum-impedance rcsistance(conductance) functionthathasa parallelcapacitoror inductorasthelastelementis shownin Figure8.1. is known,it follo'*'s canbe determinedwhenthe resistance Becausethe reactance
The Designof WidebandImpedance-Matching Networks
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that the impedanceitself is known when its resistivepart is known. In termsof equations, if
= IEc,/(ro -
