i,i

...-.

.

:

CHAPTER 4 NARROWBAND IMPEDANCE.MATCHING WITH LC NETWORI$

I.1 INTRODUCTION to certainrequired networksareusedfor transformingimpedances lmpedance-matching When yalues,which may or may not betheconjugateof the sourceor theloadimpedance. power is transferred (i.e., maximum Zr: Z"'), to a load I sourceis conjugatelymatched bctweenthem.This is importantwhenthe powergainof a transistoris low, asis the case sith mosttransistorsat higherfrequencies. networksarealsooftenusedto contol Apart from matching,impedance-matching rhegain,the noisefigure, or the outputpowerof the different stagesin an amplifier. These Etworks aremoreaccuratelyreferredto ascontrolnetworks(gain,noisefigure,or power networks. control),but they arealsogenerallyreferredto asimpedance-matching When a matching network is designed for maximum power transfer, the rminations areusuallyknown. The terminationsto be usedwhena controlnetworkis . cnedaredeterminedby theparameter to becontrolled.This aspectwill be considered --.rhapter10. Independentof how the terminationsareestablished,the designprocedurefor the . urching network remainsthe same.The designof nanowbandimpedance-matching .rorks,mostly for maximumpowertransfer,will be consideredin this chapter. Narrowbandimpedancematching is done with t'wo or more components.Where :rie 'H "ornponentsare usedto bring about an impedancetransformation,the matching .i'ork matchingnetworksare usuallyT- or PI:. is called an L-section.Three-element :ons.The namesaredescriptiveof the configurationformedby the reactiveelements' The designof L-, T-, and Pl-sectionswill be discussedin this chapter.Trans'-:ation of real andreactiveloadswill be considered. When T- and Pl-sectionsare used,it is possibleto bring about the required r-sformation andto control the bandwidthof the network.Although the 3-dB bandwidth du L-sectioncanbe determinedeasily,it is not a designparameter. to know the bandwidthresultingfrom a transforming It is sometimesnecessary . r:onmore accuratelythan is possiblewith the approximationmethodthat is usually . : In thesecases,as well as in instanceswhere a bandwidthother than the 3-dB : . iwidth is of interest,the procedureoutlinedin Section4.9 canbe used.

r25

126

Designof RF and Microwave Amplifiers and Oscillators

It was shownin Chapter3 that losslessreactivecomponentsdo not exist. For this networkswill havesomeinsertionloss.Theselossescan all impedance-matching nsason, quite pronounced whenthe bandwidthof a circuit is very narrow.A simpleprocedure be for calculatingthe insertionloss causedby a cascadedLC networkwill be outlinedin 4.8. Section impedance-matching networks Apart from matchingandtransformingimpedances, aresometimesalsousedto rejectunwantedsignalsoutsidethe passband(this practiceis networks are designed).The not recommendedwhen widebandimpedance-matching networkswith high rejectionrequiredcanoftenbe obtainedby usingimpedance-matching great. too required is not rejection that is, ifthe Q-factors, The rejection obtainableby using parallel and seriesresonantcircuits will be consideredin Section4.2. Whenthe requiredrejectionbecomesvery high, the p of the components,their temperaturestability, and any tuning requiredcanbecomea problem.If the associated insertionloss can be toleratedand the filtering occursat low power levels,the required rejectioncan often be obtainedby using surfaceacousticwave(SAW) devices,ceramic arevery stableandcanprovideextremelysharp filters,or crystalfilters.Thesecomponents presented by thesedevices,some(low Q) impedance rejection.Becauseof theimpedances matchingis usuallyalsorequired.

4.2

PARALLEL RESONANCE

A parallel resonantcircuit is shown in Figure 4.1. Although it is not an impedancematchingnetwork,it is of interestherebecauseof its frequencyresponse. The frequencyresponseof this circuit is determinedby the zeroat the origin, the zeroat infinity, andthe two poles.That is, V,(s)= 71511

= I / U l R r +s C + 1 / ( s Z )

+ V" RL

Figure 4.1

A parallelresonantcircuit.

Narrowband Impedance-Matchingwith LC Networks

sLI s2LC+sL/Rr+I

t27

(4.1)

From Figure4.1 it is obviousthat the highestpossibleoutputvoltagewill occur where coZ= 1/ (coC) that is, when

(4.2)

oo=l1.,1rc

The 3-dB frequenciesof the circuit canbe determinedby using (a.1) and (4.2). Thesefrequenciesoccur where +| / (iaoL)l I t n, + jrlrC+| | (jo,l)l= {Z lt I R, + irl,oC

=Jitn,

(4.3)

After somemanipulation,the solutionsof this equationarefoundto be ,-------03dB= rrro{t+ | / gQ\

tl I (2RC)

(4.4)

Therefore,the bandwidthof the circuit is -o:os-r =I I (RC\ B = (Dros-z

(rad/s)

(4.5)

It can be seenfrom (4.4) that the circuit responseis not symmetricalaroundthe resonantfrequencyoo. It canbe provedeasily,however,that the resonantfrequencyis the geometricmeanof the two cut-off frequenciesby multiplying the two solutionsgiven by 4.4),that is, oo =

Jo:as_r

.o:ae_z

(4.6)

The p-factor of the circuit is defined as the ratio of the center frequencyto the :andwidth; that is, Q = a o /B = ooCR

(4.7) (4.8)

{

128

Design of RF and Microwave Amplifiers and Oscillators

(4.e)

= R / (cooZ)

A high Q, therefore,implies a very small relative bandwidthand, in the caseof with low impedance comparedto thatof theloadresistance. parallelresonance, reactances The reactances areshownin Figure4.2 for a Q of 10. Whenthe Q of thecircuit is high, the arithmeticandthe geometricmeanof the cutoff@uencies areapproximatelythe same(see(4.4)).

i)

Figure 4.2

|"rroo #-.rroo

A parallel resonantcircuit with p:

100()

10.

Extremely sharprejection can be obtainedby using a parallel or seriesresonant to be ideal,it canbe shownthat the ratio canbe considered circuit. Whenthe components (P,',J andthat at any other frequency load at resonance power to the transmitted of the the following equation: (P"(f )) canbe calculatedby using

+6=

+(roo / r,r)21 1-ze\ +Q'IG't oo)2

(4.10)

of the circuit, the attenuationis As an illustrationof the rejectioncharacteristics givenin Table4.1 asa functionof thenormalizedfrequency(f lfo) for differentvaluesof becausethe response areconsidered, aboveresonance the circuit Q. Otnythe frequencies curvelevelsoff to a singleis closeto symmetricalwhenthe Q is high.Wheretheresponse pole response, no moreentriesweremadeinto the table. therateofrejection,a-30-dBqualityfactory(Q-r) is defined In ordertoappreciate hereas

r I'

I

F i ;

Q+o=.folB-so

(4.1l)

"bandwidth" of the circuit (in Hertz). whereB-rois the -30-dB p-factorsusedin Table4.1me0.315(0 : l0), the three for The 30-dBQ-factors (Q:250), (Q: respectively. 100),and7.90 3.15 It follows by observationof the resultsobtainedthat the -30-dB Q-factorof a resonantcircuit is relatedto the 3-dB Q-factorin a simpleway whenthe 3-dB Q-factoris greaterthan l0:

Q_n=0.0315Q

(4.r2)

129

Narrowband Impedance-Matchingwith LC Networks

The normalized -30-dB bandwidth of the circuit is therefore given to good approximationby the following equation: (4.13)

B-to=31.75/Q

It canbe shownthat the two normalized-30-dB rejectionfrequenciesare given to good approximationby the following equation:

(4.r4) By using (4.13),the Q-factorrequiredfor a specified-30-dB bandwidthcan be calculatedeasily.

Table 4.1 The frequenciesat which the output signalofan ideal parallelor seriesresonantcircuit is attenuatedas listed for somevaluesofthe circuit quality factor

(/fr) frequencies Normalized

Attenuation(dB)

8: r0 0 -3 -10 -20 -30 -40 -50 -60 -'t0

EXAMPLE 4.1

1.0000 1.0512 1.1615 1.615

,:

Q=t00 1.000 1.005 1.015 l.051 l .l 7 l 1.620 3.46 4.24

Q:250 1.000 1.002 1.006 1.020 t.065 1.22 t:,

Establishingthe Q-factorrequiredfor -30-dB rejectionat two specifiedfrequencies.

As an exampleof the applicationof (4.13),the p-factor necessaryto provide -30-dB rejection at 40 and 60 MHz with a parallel resonantcircuit will be determined. The resonantfrequencyofthe circuit is fo = "[40 "60 = 48.99MH2

Design of RF and Microwave Amplifiers and Oscillators

130

' '":! The normalized-30-dB bandwidthis B-so= (60 - 40)I 48.99 = 0.4082 The requiredQ is obtainedby using(4.13): (4.15)

Q = 3 1 . 7 5 1B - t o = 7 7 . 7 6

of the parallelresonantcircuit have Up to this point the lossesin the components been ignored. When the requiredQ of the circuit becomesof the sameorder as the unloadedQs ofthe componentsused,this cannotbe done. When the lossesare taken into account,the effective load resistance(-Rr)at the resonantfrequencyis then given bY (4.16)

I I Rr = | I R, +l I (QLx) +l I (QcXc)

where X. andXrare the reactanceof the inductor and capacitor,respectively, at the resonantfrequency(seeFigurea.3).Qtffid Q, arethe unloadedQ-factorsof the inductor andcapacitor,respectively.

RL

Figure 4.3

A parallel resonantcircuit with lossy components.

areequal,and(4.16)canbe the capacitiveandinductivereactances At resonance simplified to X L / R r = X t l R , + [ 1l Q r . + I / Q c )

.

(4.r7)

The lastterm in this equationis definedasthe unloadedQ (Q) of the circuit: I l Q u = 1 /Q r + l l Q "

(4.18)

The effectiveQ of thecircuit (0"") is thereforegivenby llQ"n=IlQt +l/Q, areassumedto be lossless. whereQ, is the p whenthe components

(4.1e)

Narrowband Impedance-Matchingwith LC Networks

131

The highest Q obtainablewith a parallel resonantcircuit is limited by component lossesandthe temperaturestabilityof the components.

I 4.3 }

SERIESRESONANCE

The resultsobtainedfor a parallel resonantcircuit can be applieddirectly to a series :esonantcircuit by usingthe principleof dualism. According to this principle, for every circuit thereis anothercircuit for which ,,vhatever appliesto thecurrentof onecircuit,alsoappliesto thevoltageofthe othercircuit, rnd vice versa. Thisequivalentcanbeobtainedbyfollowingtheprocedureillustratedin Figure4.4. \ nodeis placedin everyloop of the first circuit, aswell asin the spaceoutsideit. These rodesarethenconnectedby passingfrom oneloopto anotherthroughthe componentsof :he different loops. Inductorsare replacedwith capacitors,capacitorswith inductors, :esistorswith conductors,andconductorswith resistors.The valuesassignedto the new :omponents(H, F, O, S) arenumericallyequalto thoseof the originalcomponents. The output voltageof the parallel resonantcircuit in Figure4.4 is given by the rcllowingequation: j',,= I I / R + s C + I / (s L)] = 0,2/ 10.5+ 3s + I / (5s)l [1

ir

llgure 4.4

N2

I-;

(4.20)

N3

The principle ofdualism appliedto a parallelresonantcircuit.

a

The output of the seriesresonantcircuit is obtainedby replacing V,,uirthIo, I sith4 R with G, Cwith L,and Z with C. Thus. 5 3 s +1 / ( 5 s ) ] I , = E / [ 1 l G + s Z + 1 / ( s C ) ]= 0 . 2 1 [ 0 . +

(4.21)

It follows from Figure 4.4 that the output current of the seriesresonantcircuit is eedgivenby this equation. By applyingtheprincipleof dualismto theresultsdeducedin theprevioussection,

132

Design of RF and Microwave Amplifiers and Oscillators

'ti-?l-

':n:..

.t1

I

s

l, ?..f;

Efrt

.,

r

.r,. V,,l

{.5

.

u

The seriesresonantcircuit.

b

the following equationsarefoundto applyto the seriesresonantcircuit of Figure4.5: Q=aoLl R=1/(rooCft)

(4.22)

o il

o 3 @= , o " l t * u

(4.23)

e1

GA)

l-Rl(2L)

r( B= Rl L (radls)

(4.24)

Rr=Rr+XrlQt+XrlQ"

(4.2s)

I I Q"n = Rr / X t + (l I Qt +l I Qc)

(4.26)

L

It follows from (4.26)that similarto the parallelresonantcircuit, the unloadedp for the seriesresonantcircuit is givenby l l Q " = l l Q t + L /Q "

(4.27)

The reactancefor a seriesresonantcircuit with O: l0 are shown in Figure4.6 atthe to theloadresistance. valuesarehighcompared rcsonantfrequency.Notethatthereactance With the sameloadedp, the frequencyresponseof the seriesresonantcircuit is identicalto that ofthe parallelresonantcircuit.

FEcrc 4.6

cfucuitwith O: 10. A seriesr€sonant

a

133

Narrowband,Impedance-Matchingwith LC Networks

4.4 L.SECTIONS An L-section is a two-elementmatching network. The fow possibleconfigurationsare shownin Figure4.7. Dependingon the positionof the first component(asviewed from the load),the oadresistancecanbe transformedupwardsor downwardswith an L-section. When the first reactivecomponentis a seriescomponent,the transformationis upward;andwhenit is a parallelelement,the transformationis downward. caused Thesecondelementin theL-sectionis usedto removetheresidualreactance .v thetransformationelement(i.e.,the first element).This secondelementis thereforethe . ompensating element. The basicprincipleusedin narrowbandimpedancematchingis thatthe resistance .f a complexload is not the samewhenviewedin impedanceor admittanceform. This is .lustrated in Figure4.8. When a reactive element (X,) is added in series with a resistor (R) and the ,'quivalentparallel combination is considered(seriesto shunt transformation),the -csistance increaseswith a factor

D r = r +0 l

(4.28)

.i here

(4.2e\

)1= X1/ R

When a reactive element (Xt) is added in parallel with a resistor (rR) and the :quivalent series combination is considered (parallel to series transformation), the 'esistancedecreaseswith the samefactor (D,). In this case,however,the Q-factor in (4.28) s defined by

ffi

n'-

-T-r*1

jX'

trx,

fn

^'-

(a)

(c) r*glrr.e4.7

The four possibleconfigurationsfor an L-section.

\rt

I (b)

(d)

"

Design of RF and Microwave Amplifiers and Oscillators

134 js0o

F13lrc 4.t

A complex impedancedisplayed in impedanceand admittanceform.

Qr=-Rl Xr=

(4.30)

g r1 6

The ratiosdefinedin (4.29)and(a.30)aresimilar in form to the Q-factorsof the series or parallel resonant circuits, respectively.These ratios are referred to as transformationps. is caniedoverto the transformationQ. The sign of the reactanceor susceptance It follows that the transformationQis positive when the effectiveseriesreactanceis is capacitive inductive (impedanceformat) or when the effective shunt susceptance (admittanceformat). The reactancechangesby a factor

\=r+U Q?

t

.'

I

(4.31)

in the tansformation step. As is the casewith the resistance,the reactanceincreasesafter a seriesto shunt whena shuntto seriestransformationis considered. transformationanddecreases The reactanceof the first elementused in an L-sectionis determinedby the (R) to the valuerequired(R')' tusformation Q requiredto transformthe loadresistance Tb Q valuecanbe calculatedby usingthe relationship

R ,= D r R = ( l + Q I R

(4.32)

A positive or a negativesign can be assignedto the transformationQ. level. The secondelementin the L-sectionis usedto achievethe desiredreactance of this elementis givenby If a purelyresistiveinput impedanceis required,the reactance

Xt=-Xll+tlQI=R'lQr

(4.33)

if the first element is a shunt element, and by

X z = - X t / ( 1 + ll Q I = - R ' Q r

t I

(4.34)

if the first elementis a shuntelement. Equations(a.33) and (4.34) can be verified easily by using the relationships Z=l I YandY:1 /trespectively.

Narrowband Impedance-Matching

t35

with LC Nefworks

The formulasrelevantto the designof an L-sectionaresummxizedinTable 4.2. of anL-sectionnearthe Whenthetransformationp is high,thefrequencyresponse resonantfrequencywill be similarto that of a simpleseriesor parallelresonantcircuit. The circuit Q of the L-sectionis approximatelyequalto half the transformationQ. If a more accuratevalue for the Q of the circuit is required,the procedureoutlined in Section4.9 canbe followed.

EXAMPLE 4.2

propertiesof an L-section. Illustrationofthe transformation

propertiesof anL-sectionwill be illustrated Thetransformationandcompensation hereby usingtheL-sectionshownin Figure4.7(a)asanexample.In this example,

=," jlo -.-T--r\-/-l Addition of the transforming elernent

*'n

The parallel equivalentofthe seriescombination(Y:1 | 4

Cancellationof the residualreactance

The transformed resistance

flure

4.9

lllustration of the transformation and compensationproperties of an L-section.

Desigr of RF and Microwave Amplifiers and Oscillators ' Table 4.2 FormulasRelevantto the Designof L-Sections Downward transformations

I,,'-rl,,n,

R ' = Rt ( l + Q I

(4.3s)

x'=xt/Q+llQh

gs6)

Xz= Xin+Q,R'

(4.37)

Upward transformations

l/R'+jBin 1

pF

9X

--r-

I

l,*,,l-_l_

I vtP,2

sX

trR

n,=R(r+Q?)

(4.38)

x'=x(l+r/Q?) g.3s) Bz= Bin+ Qt I R'

(4.40)

the resistanceand reactanceat the transformationfrequencyaretakento be R:lfl olZ: lQ llroC:2Q The first elementin the matchingnetwork(seeFigure4.9) is a seriesinductorof is alsoequalto 1Q,thetransformationp is equal 71Q. Becausethe loadresistance

Narrowband Impedance-Matchingwith LC Networks

137

is transformedupwardswith a factor I + 12: 2 to to +1, andthe I Q loadresistance a valueof 20. ThetransformationQhasthe samemagnitudebeforeandafterthe transformation(the sign of the Q changesin the process),and the reactancein parallelwith the transformedresistanceis therefore +72O(still inductive).This reactanceis removedby resonatingit off with a capacitor(-72O reactance),after which the original 10 resistorhas beentransformedto 2Q at the frequencyof interest. element,whilethecapacitor Notethattheseriesinductoris atransformation element. is a compensation

EXAMPLE 4.3

Designingan L-section.

An L-sectionwill be designedto transforma loadof 50Oto 250Oat 50 MHz asan above. exampleof the applicationof the theorydiscussed Becausethe transformationis upwards,the first elementof the L-section mustbe a serieselement.The diagramin Figurea.l0(a) applies. It follows from the diagramthat the transformationQ must be equalto 2; equalto it follows that a seriesinductoror capacitorwith reactance X=QR=2x50=1000 is required. If the first elementis chosento be an inductor,the requiredinductanceis I = 1 0 0/ ( 2 n x 5 0 x 1 0 6 )= 0 . 3 1 8p H Theparallelequivalentofthe transformingsectionis therequired250Qin parallel with a reactance ,t-:

X'=R'lQr=25012=125{l

p200O

0 . 3l 8 p H

2500 -t s50Q

(b)

(a) rgure4.10

(a) The transformation diagram and (b) the L-section relevant to Example 4.3.

138

Designof RF and MicrowaveAmplifiers and Oscillators

Note that the Q-factorsof the seriescombinationandits parallel equivalent mustbe equalin magnitude(thesignof theQ changeswhenthe transformationis done). Thecapacitance requiredto removethereactivepartofthe inputadmittance of the resistorandinductorcombinationis C = l / | 2 5 ( 2 n x 5 0x 1 0 6 )=l 1 2 5p F

4.5

o

a tl

't

The designed networkis shownin Figure4.10(b).

ir

PI-SECTIONSAND T.SECTIONS

4

Pl-sectionsand T-sectionsare three-element matchingnetworks.A Pl-sectionhas two parallelelements,andthe T-sectionhastwo serieselements,asshownin Figure4.11.

T F

ut

ri

1:

r

("'

i

{

I

t I

ftrrc

4.1f

Topology for (a) a Pl-sectionand (b) a T-section.

a I

3 The first two elementsin thesesectionsaretransformingelements.One of these elementscausesthe resistance to increase.while the othercausesit to decrease. The reactancelevel is set by the last elementin the section(the compensating elmt). Becausethe resistanceis transformedtwice, there are two transformationQs in bb sections.ThehighesttransformationQ canbechosento haveanyvaluehigherthan that requiredin an equivalentL-section. As in the caseof L-sections,the bandwidthof Pl-sectionsandT-sectionsarealso &mined by the transformationQs. Wherethe two Q-faclorsaredifferent, the Q of the R'

R'

{

F

G F

t

t

R'I

(a) ftrrc

{f2

r

{

R'

O)

process An alternative viewof thetransformation in (a)a PI- or (b) a T-section.

rf GI

Narrowband lmpedance-Matchingwith LC Networks

139

network will be approximatelyequalto half of the highesttransformationp. the bandwidthof a Pl-section Becausethe highesttransformationp is adjustable, or a T-sectioncanbe controlled. Thetransformationpropertiesof a Pl-sectionor a T-sectioncanalsobe considered, as illustratedin Figwe 4.12. The fact that the sourcetermination(R") and the load termination(R) mustbe transformedto the sameintermediatevalue(R') is consideredin this case.Both terminationsaretransformeddownwardsin a Pl-sectionandupwardsin a element T-section.Thesecondelement(ascountedfromtheloadside)is thecompensation in this case.The bandwidthis determinedby the side with the highesttransformationQ.

4.5.1

The Pl-Section

approach)areilllstrated in causedby a Pl-section(cascade transformations Theresistance Figure4.13. The resistanceis first transformeddownwardsby a factor I + U and then upwardswith a factor | + Qi . Q, is the first transformationQ and is associatedwith the load resistanceandthe first elementof the network.Q2is the secondtransformationQ. in series Thesecondtransformationp is equalto theratioofthe effectivereactance with the transformedresistance(R') andthe transformedresistanceitself.

r+Qr'

Figure 4.13

(a) Upward transformationof the load resistancewith a Pl-section and (b) downwmd transformationofthe load resistancewith a PI-section.

The transformedresistancewill be lower than the load resistancewhen the first :ransformationQ is higherthanthe second.An upwardtransformationrequiresthe second :ansformationQ to be higherthanthe first. by therequiredbandwidth Thevalueof thehighesttransformationQ is determined ,f the network.The Q of the networkis approximatelyequalto one-halfof the highest :ansformationp when the transformationQ factorsare sufficiently different. Thetransformationof a l0Q loadto 50Oby usinga Pl-sectionis illustratedin detail r Figure4.14.

Design of RF and Microwavc Amplifiers and Oscillators

r.r0

-jl0o

A designedPl-section

]J,* +

Shunt to series transformation

-jl0o

-j'a

€ Seriesto shunttransformation

=,," rcactancc of theresidual Canccllation

Ittrc

4ff

The transformationof a l0O load to 50Q with a Pl-section.

The formulasrelevantto the designof a Pl-sectionaresummarizedin Table4.3.

EXAMPLE 4.4

A Pl-sectionexamPle.

A matchingnetworkfor transforming50Oto I 2.5O will bedesigned.Themaximum transformationQ will be takento be 5. Becausethe transformationis downwards,the first tansformation p will highest. the be The next stepis to choosethe networktopology to be used.The network is arbitrarily assumedto have an inductor asthe first element,while the other componentsarechosento be capacitors.(It is not possibleto chooseboth ofthe first two componentsto be inductors or capacitors.If this is done, the second transformationQ will be the highest.)

Narrowband Impedance-Matchingwith LC Networks

t4l

Table 4.3 Formulasfor designinga Pl-section

Qt=Q^o=2Q

(4.4r)

R ' = R/ ( t + Q ? )

(4.42)

I+fi=R"lR'

(4.43)

Y1=Q1R l

(4.44)

Xz=R'(Qr+Qz)

(4.4s)

Y t = Q 2R I "

(4.46)

Increasing the load resistance

R

Qz=Q^*=2Q

(4.47)

R , = R , l,( t * d )

(4.48)

l + Q l = P 1P '

(4.4e)

Yt=Q1/R

(4.44)

Xz=R'(Qr+Qz)

(4.4s)

Yr=Qt I R"

(4.46)

Becausethe first transformationp is Qr=

-5

the reactanceof the inductor must be xr:50/5: l0o The secondcomponentmustchangethe transformationQto 2.35.Tlre Q must be positive (inductive) if the last componentis to be a capacitor:

r42

Design of RF and Microwave Amplifiers and Oscillators

+ (-5)) = -5.1o Xz = R'(Qr+ Q) = 1.92(2.35 is of thelastcomponent Thereactance X3= -R" I Qz= -12.512.35= -5.3C1 networkis shownin Figure4.150). Thedesigned

-j5.lo t+Q22:12.51t.92 Q2:+2.35

12.5il

st.92

o)

(a)

(a) The transformation diagram corresponding to Example 4.4; (b) a Pl-sectionfor matching50Q to 12.5Q.

Figure 4.15

4.5.2

The T-Section

Thedualof a Pl-sectionis a T-section.Therefore,the formulasfor designinga Pl-section can also be usedto designa T-section.In orderto do this, it is necessaryto replacethe respectively. resistanceandreactancein theseformulaswith conductanceandsusceptance, The terminationsusedmust also be inverted(i.e., if the actualterminationsfor the Tsectionare 50Q terminations,the terminationsfor the equivalentPl-sectionshouldbe

l/s0o). The reactanceresultsofthe Pl-sectionapply directly to the T-sectionifthese are requiredfor a Pl-section To illustratethis,ifthe components interpretedto besusceptances. requiredin the T-sectionare7l0S,-i5S, andi3S. are7100,-i5O and73O,the components

12.55

Figure 4.16

-r -15.33

An exampleof finding the dual of a Pl-section.

NarrowbandImpedance-Matching with LC Networks

143

Table 4.4 Formulasfor designingT-sections Decreasingthe load resistance

Qz = Q.* =2Q

(4.s0)

n =n " Q +Q ? )

(4.s1)

I* 4 = R'/R

(4.s2)

Xr = QtR

(4.s3)

Yz= (Qr+ Q) I R'

(4.s4)

X, = QrR"

(4.s5)

lncreasingthe load resistance

t*Qr'

Qr=Q^^=2Q

(4.s6)

n'=R(t+01)

(4.s7)

1 + Q l = P ' 1P "

(4.s8)

Xr=QrR

(4.s3)

Y z = ( Q t + Q )I R '

(4.s4)

Xt=Q, R"

(4.ss)

This approachis usefulwhena programto designPI- andT-sectionsis developed. Theprogramcanbe written to designPl-sectionsonly, andby enteringthe specifications correctly it can also be used to designT-sections.When the designis not done by computer,it is betterto follow the procedureoutlinedin Table4.4.

4.6 THE DESIGNOF PI.SECTIONSAND T-SECTIONS WITH COMPLEX TERMINATIONS Theprocedures outlinedin theprevioussectionscanbe extendedeasilyto thegeneralcase *'herethe loadandsourceimpedances arecomplex.The approachis illustratedin Figure 4.t7.

t44

!

Design of RF and Microwave Amplifiers and Oscillators

The reactivepartsof the load andsourceimpedance(T-section)or admittance(PIsection)are ignoredinitially, andthe networkis designedto matchthe load and source resistanceto each other. The first and last componentsare then changedto take the imaginary parts of the load and sourceimpedanceor admittanceinto account. to a new seriesvalue,the Becausea T-sectiontransformsa seriesloadresistance load and the required input impedancemust be specifiedin seriesform, that is, as for a Pl-sectionmustbe of parallelform. The first stepin The specifications impedances. desiping a matchingnetworkwhenthe terminationsarecomplex,therefore,is to getthe terminationsin the right form. The following equationsapplyto Figure4.17:

xi=Q,R"

(4.se)

Xt=QrRt-Xt

(4.60)

Bi=Qr/R'

(4.6r)

Br=QtlRL-BL

(4,62)

R'+jxb

llR'+jBio

Figure 4.17

The design of (a) a T-section and (b) a Pl-section when the terminations are complex.

EXAMPLE 4.5

Designinga T-sectionwith complexterminations.

A T-sectionfor matchinga l0 +710Qloadto 50 +740O(seeFigure4.19)with a maximumtransformationQ equalto 5 will be designed.Thesespecificationsare in impedanceform, asrequiredfor a T-section.

Narrowband Impedance-Matchingwith LC Networks

145

p260 *Q22:260150 Q2:2.0s

Figure 4.18

The transformationdiagramfor the T-sectionof Example4.5.

The transformationdiagramfor this problem is shown in Figure 4.18. Becausethe transformationis upwards,the first transformationp must be the highestin this case.The secondtransformationQ mustbe equalto 2.05. With the p-factors known, the next stepis to choosea topology.If the network shown in Figure 4.19 is chosen,the bandwidthof the circuit can be calculatedas was done before. Since the Q-factorsof the load and source impedancesarelow comparedto the maximumtransformationQ, predictableresultscanalsobe obtainedwith othertopologies.Theonly majordifferencewill be in the rateat which the slopeoutsidethepassbandlevelsoff becauseof thehigher numberof poles. The componentvalues of the chosennetwork can now be determinedby usingthe valuescalculatedfor the transformationQ's. In orderto havea transformationQ of 5 at the load,a reactanceof+/40O must be addedto the existing+/0O; that is, Xr=(5Q)-10=40O After the first transformation,the transformationQ is still equalto 5. In orderto changeit to 2.05,a capacitorwith susceptance d,

Y z = ( Q r + Q ) / R ' = ( 5 + 2 . 0 5 )/ 2 6 0 = 2 7 . l m S

740O

jl03o

j40a

(Dr740o.+

ftrrc 4.19

A T-sectionfor transforminga l0 +jl00 load to 50 + j40Q.

jl0o

Design of RF and Microwave Amplifiers and Oscillators

must be used;that is, both Qr andp2 must be positive. It is not possiblein this caseto usea capacitorwith susceptance

Y= (Qr-lQrDt n' sincethe last componentof the network was chosento be an inductor. and Thelastcomponentofthe T-sectionmustremovetheresidualreactance changeit to the requiredlevel of +740O.In orderto do this, a 14.3Oinductoris required. The designednetworkis shownin Figure4.19.

4.7 FOUR.ELEMENTMATCHING NETWORKS Whenfour elementsareused,the controloverthe frequencyresponseof the impedancematchingnetwork increases.The bandwidthcan be lower or higherthan that of an Lsection. One possibleapproachto designinga network to have a very high Q is shown in Figure4.20. The two low-Q sectionstransformthe load impedanceandthe sourceimpedance (R < Rr,R < Rt with thenetwork,asshownin Figure4.20),and to havethesameresistance the high-p sectionsetsthe reactancelevel andprovidesthe requiredrejection. Strictly speaking,only one downwardtransformingsectionis requiredin this network.Whentwo downwardtransformingsectionsareused,however,it is oftenpossible it is oftenpossibleto use to decrease the insertionlossof the circuit.This follows because p when this is done. in the circuit components higher

High-p section

;

i(RQ-4-x) R+j4 Low-Q section ftrre

4.20

R+iXz Low-p section

network. A high-Q, easilytunable,four-elementimpedance-matching

When the approachillustratedin Figure4.21 is followed,the bandwidthcan be to be smallerthanR'). In this wider thanthat obtainablewith an L-section(jR,is assumed aretransformedto their geometricmeanby using casethe sourceandthe load resistance two L-sections.

It pn un

147

Narrowband Impedance-Matchingwith LC Networla

Low-p section

Figure 4.21

4.8

Low-Qsection

A widebandfour-elementimpedance-matching network.

CALCULATION OF THE INSERTION LOSS OF AN LC IMPEDANCE.MATCHING NETWORK

was shown in Chapter3 that the ideal componentdoesnot exist. For this reason,all -acticalcircuitswill havesomeinsertionloss.If the insertionlossis to be kept low, the r loadedQ-factorsofthe components mustbesignificantlyhigherthantheQ ofthe circuit. The insertionlossof anycascaded LC networkcanbe computedby following the :ocedureoutlinedbelow: l.

Model eachreactivecomponentin thenetworkasan idealcomponentwith a resistorin seriesor in parallelwith it, dependingon whetherit is a series or shuntelement,respectively. The valueof this resistance canbe determinedfrom the unloaded estimated for the component. Q-factor Unloadedp-factorsfor capacitorsandmagnetic-core inductorscan usually be found by using the datagiven by the manufacturer,while those of air-coredsolenoidalcoilscanbe determinedby following theprocedure outlinedin Section3.3.6. The unloadedQ of a componentmay be a strongfunction of the frequency.

2.

Assumethat the powerdissipatedin the loadis equalto lW.

J.

If the first componentof the network(asviewedfrom the load)is a series element,calculatethe powerdissipatedin it by usingthe equation Pn=(Ro/R)PL

(4.63)

whereRo is the seriesresistance associated with the elementandR, is the (effective)loadresistance. P, is thepowerdissipatedin theload( I W in this case). If the first componentis a parallel element,calculate the power dissipatedin it by usingthe equation

Desigr of RF and Microwave Amplifiers and Oscillators

14t

(4.64)

Po=(Go/G)PL

where Gq is the parallel conductanceassociatedwith the component,and of the load.P, is the powerdissipatedin G1is the (effective)conductance the load. 4.

Add the power dissipatedin the first componentto that dissipatedin the load:

(4.6s)

P, = Pt+ Po

consider the first componentto be part of the load and calculatethe new (effective) load admittanceor impedance. 6.

Repeatsteps3 to 5 until the power enteringthe matchingnetwork (Pp)and the effective input impedanceof the network (Z) ne known.

7.

Calculate the transducerpower gain of the network (Gr) by using the equation

G,=(r-lt"l')+

(4.66)

- 'jl'l,,,,, -l':" =1, 1z^*z,l

(4.67)

)

|

4RinR"

PLI Pr

( R , n+ R " ) 2 + ( X , n + X , ) z

(4.68)

whereS"is the input reflectionparameterwith Z"asnormalizingimpedance, Zrn= Rir+ iXin

(4.6e)

and

(4.70)

Zr= Rr+ iX,

whereZ"is the internal impedanceof the sourcedriving the network.

EXAMPLE 4.6

Calculatingthe insertionlossof a Pl-section.

As an exampleof the applicationof this procedgre,the insertionloss of the PI-

149

Narrowband Impedance-Matchingwith LC Networks

section designedin Example4.4 will be calculatedat the centerfrequency.The unloadedp-factors of the capacitorsare assumedto be 500, while that of the inductoris takenas 100. associated with the inductoris lmS, the seriesresistance Theconductance with the associated andtheconductance capacitor is 0.010, with the first associated is 0.38 mS. secondcapacitor Thepowerdissipatedin theinductoris 50 mW. Thepowerenteringthe last sectionof the networkis therefore1.05W.The input impedanceof this sectionis

Z =2.0+ i9.6O Thepowerdissipatedin thefirst capacitoris 5 mW. Thepowerenteringthe networkat this point is therefore1.055W.The input admittanceat this point is .

Y = (84 - 7186)mS

:'

The power dissipatedin the last capacitoris also 5 mW. The total power enteringthe networkis therefore1.06W.The input impedanceis

zi n = r 1 . 9 - j o.4 5 o powergainofthe networkwascalculatedto be 0.94,thatis, Thetransducer an insertionlossof 0.3dB. :

{.9

: ! .

CALCULATION OF'THE BANDWIDTH OF CASCADED LC NETWORKS

ne bandwidth of a network can be found iteratively if its transducerpower gain is :erminedas a function of frequency.The transducerpower gain of any cascadedLC :rvorkcanbe foundby following the procedureoutlinedin the previoussection. Becausethe cut-off frequencies(3-dB) of L-, PI-, and T-sectionsare known to ,ffi 'tood apptoximation(f-rau: fo+fo I Q^), the exactbandwidthof thesecircuits can be :rerminedquickly by following this procedure.

EXAMPLE 4.7

The 3-dB bandwidthof a matchingnetwork.

By following the proceduredescribed,the 3-dB cut-off frequenciesofthe network in Example4.6 arefoundto be 83 and 130MHz, thatis, if thecenterfrequencyis selectedas 100 MHz. The exactQ of the circuit is therefore2.3 insteadof the 2.5. estimated If a bandwidthother than the 3-dB bandwidthis required,it can be found easilyby following the sameprocedure.

I

150

Desiga of RF and Microwave Amplifiers and Oscillators

. ]: SELECTEDBIBLIOGRAPHY RF PowerTransistorManual,Somerville,NJ: RCA Corporation(Solid StateDivision), 197t.