CIIAPTER 6 TRANSMISSION-LINE TRANSFORMERS 6.I

INTRODUCTION

The high-frequencyresponseof a magneticallycoupledtransformeris limited by the leakageinductanceandthe parasiticcapacitance ofthe transformer. The leakageinductancecanbe decreased significantlyifa balunor a stackedcore (insteadof a toroidal core)is used.It is moredifficult, however,to decrease the parasitic capacitancebetweenthe two windings andthe tums of eachwinding. Ifthe outerand centerconductorofa coaxialcableareusedas the primary and secondarywindings of a 1: I transformerand connectedas shownin Figure6.1(b),a I :4 impedancetransformationcan be obtained(similar to that obtainedwith the autotransformershown)andtheparasiticcapacitance betweenthewindingscanbe controlled. One would expectthe performanceof this transmissionlinetransformerto be .ptimumwhenthecapacitance betweenthewindingsis low,thatis,whenthecharacteristic rmpedance of the line is high. Fortunately,this is not true; rather,thereis an optimum performance characteristic of the transformer impedancefor the line. Thehigh-frequency :sthereforeimprovedby the transmission-line effect. transformercan be consideredto be a At low frequenciesthe transmission-line :onventionaltransformerwith excellent coupling. Becauseof the transmission-line :apacitance,however,this model is not valid at high frequencies. In fact, the magnetic :ouplingbetweenthewindingscanbe removedcompletely(by not usinga magneticcore rndstraightening performance of thetransformerwill not theline), andthehigh-frequency .e influencedat all (i.e.,if the losseswerenegligible).

F{ure 6.1

(a) A conventionalauto-tansformerand (b) a l:4 transmission-linetansformer.

179

rt0

Desigrrof RF and Microwave Amplifiers and Oscillators

I+Ie-iF I

I e-fr f

It+Ve-F

Figure 6.2

The voltagesacrossandthe currentsin a transmission-linetransformerat high frequencies.

by assumingthecurrentsin thetransmission Thatthis is possiblecanbeappreciated line to be balancedand the line to be short enoughfor the phasedifferencebetweenthe voltagesacrossthe line andthe currentsin the line to be small.This is illustratedin Figure 6.2. It has been assumedin Figure 6.2 that the currentsin the tansmission line are balanced.If the currentswere perfectlybalanced,as shownin Figure6.3(a),the output voltage would havebeenzero, which is not the case.The currentsin the line must be unbalancedfor the output voltageto be non-zero. Because (6.1)

Yo= 2sLrl I ru=2 jaLr(. I tu

In (6.1)L, is the inductanceper the unbalancedcurrentis very smallat high frequencies. line, and 4, the unbalanced the length ofthe unbalanced current, 0 length for the unit two conductors of the line. in each of the componentof the current performance transformeris of the transmission-line Althoughthe high-frequency response is seriously not affectedby the removalof themagneticcore,the low-frequency degradedwhenthis is done.The reasonfor this is the increasein the unbalancedcurrent. This currentincreasesapproximatelyinverselywith frequency.Whenmagneticmaterial currentis reduced.Ifthe is use{ the inductancein (6.1) is increasedandthe unbalanced ofthe transformerwill bevery currentis smallenough,thefrequencyresponse unbalanced goodat both high andlow frequencies. The bandwidthobtainablewith a transmissionJinetransformeris significantly bcfier than that obtainablewith a conventionaltransformer.

- V,,+

: (a)

Egprc 6.3

O) asa functionof (a)thebalanced transformer Theoutputvoltageof thel:4 transmission-line currents in theline. and(b) unbalanced

Transmission-Line Transformers

181

Becauseof the wide bandwidth,transrnission-linetransformersare often usedto transformresistance.When only one transmissionline is used,the only transformation ratiosthat canbe obtainedare l:1 and 1:4.Whenmorethanoneline is used,it is possible to realizetransformerswith othertransformationratios. Apart from impedancematching,transmission-line transformersarealsousedto performvariouscombiningandsplittingfunctions. The mostcommonlyusedconfigurationswill be presentedin the next section. The analysesof the l:4 transmission-line transformerandothertransmission-line transformers will be discussed in detailin Section6.3.It will be shownin this sectionthat the basiccomponentof any transmission-line transformeris an unbalanced transmission line with increasedinductancefor the unbalanced currentsin the line. This simplifiesto a balancedtransmissionline at high frequenciesand a l: I transformerwith magnetizing inductanceat low frequencies. primarilyrequiresthatthetansformer Thedesignof transmission-line transformers may meetsthelow-frequencyspecifications. Compensation at low and/orhighfrequencies alsoberequiredto extendthebandwidth.Thevariousstepsin designingthesetransformers will be consideredin detailin Section6.4. Transmission-line transformers areusedextensivelyin RF poweramplifiers.The designof impedance-matching networksfortheseamplifierswill beconsideredin Section 6.5.

6.2 TRANSMISSION-LINETRANSFORMER CONFIGURATIONS Transmission-line transformers areusedto changeresistance levelsin impedance-matching networksandamplifiers,aswell asto performcertainsplittingandcombiningfunctions. The transformationratiosobtainablewith thesetransformersarelimited to those shownin Table6.I , that is, if lessthanfive transmissionlinesareused. Thenumberof linesusedin a practicalapplicationis limitedby theavailablespace. To ensurea good low-frequencyresponse, eachline mustbe woundarounda magnetic @re.

More than oneline cansometimesbe wound aroundthe samecore.Thepolarity of lhe voltageinducedby the flux in the coreandthe relativesizeofthe voltageacrosseach windingmustbe takeninto accountwhenthis is done. The configurationcorresponding to a particulartransformationratio canbe found by usingthetechniqueillustratedin Figure6.4 [ ]. Whenthistechniqueis applied,theratio of the input to output voltagechangesfroln x I y tolx l(X+ y)l.The impedance - ,nsformation ratiois changedfrom (x I y)" to fx I (x + y)]'. In the simplestcase, x=l=! rnd the applicationof this techniqueresultsin the configurationfor the 1:4transmissionIre transformer.

Design of RF and Microwave Amplifiers and Oscillators

182

Table 6.1 Transformation ratios obtainable with transmission-linetransfonners as a function of the numberof lines used Number of lines

l:1.00(l/l)' l:4.00 (li2f Transformation ratios obtainable

2

I

t:2.25(213f l:4.00(r/2f l:9.00 (l/3)'

_

J

l:1.78 (3/4)' l:2.78(315f r:6.25(2/5)' l:16.0(l/4)'

:

4

l:1.56 (4/5)' 1:1.96(5171 l:2.56 (5/8\' 1:3.06(4/7)' r:5.44(3/7)2 l:7.1l (3/8)' r:r2.3 (2/7)2 l:25.0(l/5)2

If the techniqueis applied to the l:4 transformer,the l:9 transformershown in is obtained. 6.5(a) Figure thehigh impedancesideof the l:4 transformerto be the inPut,the considering By in Figure6.5(b)is obtained. shown 1:2.25transformer (3/5)2transformerscanbe (ll4)2 andthe(215)2, The configurationsfor the(314)2, 4:9 and 9:4 transformers, the 9:l and 1:9 and to the found by applying the technique areshownin Figure transformers 4:25 and the I : 16 respectively.Theconfigurationsfor the 6.6.

Figure6.,f

The influence of adding an extra line on the impedance transformation ratio ofa transmission-linetransformer.

183

Transmission-Line Transformers

- 2Y+

3V

lsR (a)

leR o) Fgure 6.5

Derivation of the configurationsfor the (a) l:9 and (b) 4:9 transmission-linetransformers.

The transformersmost often used in power amplifiers are the l:4 and 1:9 ransmission-linetransformers.The high cut-off frequencyof the I :4 transformercanbe increased considerablyif two linesinsteadof only oneareused,asshownin Figwe 6.7 l2l. - 3 V +

l,-

- 3 V +

(a)

- 3 V +

J,,^

- 3 V +

(b) figure 6.6

The configurationsofthe (a) l:16 and (b) 4:25 transmission-linetransformers.

Desigr of RF and Microwave Amplifiers and Oscillators

Zo:3tn N2

fhe configuration of a I :4 transmissionline transformer that has no high cut-off frequency (theoretically).

Figure6.7

Impedancehansformationbetweena balancedsourceanda balancedloadis often transformers required.The configurationsfor the balanced1:4 and 1:9transmissionJine areshownin Figure6.8(a)and (b), respectively.

-V Figure 6.E

I

-2V

(a)

-3v

-V

rb)

The configurationsfor the balanced(a) I :4 and (b) 1:9 transmission-linetransformers.

the l:1 transformershownin When eitherthe load or the sourceis unbalanced, transformation. Figure6.9 canbe usedto providethe requiredunbalanced-to-balanced

Figure6.9

1:l transmission-linetransformer. Theunbalanced-to-balanced

Transmission-Linc Transformers

185

I : I balanced-to-unbalanced transformer

Figure6.l0

lllustration of the equivalence betwecn the l:4 balanced and the l:1 balanced-tounbalancedtransmission-linetransformers.

the frequencyresponseof the I : I transformeris exactlythe At high frequencies, szrme asthat of a transmissionline terminatedin the sameload. Becauseof the symmetry,the high-frequencyresponseof the l:4 balanced transformeris identicalto that of the 1:I transformer.The equivalenceis illustratedin Figure6.10. transformation canbeobtainedby combiningthe I : 1 A 4: I unbalanced-to-balanced and 1:4 balancedtransformers,as shownin Figure6.1l(a), or by using the transformer shown in Figure 6.11(b). The latter transformeris less sensitiveto nonoptimum characteristic impedances thantheformer,althoughit hasa lowercut-off frequencywhen theoptimumcharacteristicimpedanceis used. Theoutputcurrentsof thetwo transistorsin a push-pullclassB amplifierareoften

-21/ (b) Figure 6.11

(a) Combination of a I : I transformer and a 4: I transformer to obtain an unbalanced-to. balancedtransformation.(b) A l:4 unbalancedtransmission-linetransformer.

Designof RF and Microwave Amplifiers and Oscillators

186

OA .V (b) Figure 6.12

(a) I :4 and (b) I :9 transformersfor combiningthe currentsof the transistorsin a classB amplifier into a single load.

combined (at lower frequencies where the conduction angle is 180

")

by using either of the

shownin Figure6'12. l:4 or l:9 transformers Although they areusedfor different purposes,it canbe seenthatthe configurations of the I :4 transformershownin Figure6.12 aresimilarto that of the I :4 unbalanced-tobalancedtransformershownin Figure6.11(b)' planeof the I :4transformerin Figure6.12(a)(asshown By redefiningthereference ofboth areidentical. in Figure6.13),it becomesclearthat the frequencyresponses

I

2V Figure 6.13

The configurationofthe l:4 transformerfrorn Figure6.12with a redefinedreferenceplane (ground)'

Transmission-LineTransformers

187

(a) Fgure 6.14

(a) A transformerfor combining two in-phasesignalsinto the sameload; (b) the same transformerusedas an in-phasepower splitter.

The combinershown in Figure 6.la(a) is often usedto combinetwo in-phase sipals at radio frequencies. As indicatedin the figure,the voltagedrop acrossthe l:l transformerusedin the combineris equalto zerowhen the two input signalsareequalin amplitudeand are inthetwo sourceswill be isolatedfrom eachother abase.Whenthe signalsareunbalanced, by the transformer.This is illustratedin Figure 6.15 for the casewhereE2 = 0. If the transformeris assumedto be ideal, and R,' = 2R" = 4Rr, no currentwill flow in the resistance R"r. At low frequencies,the isolation obtainablewith this transformeris a function of tb magnetizinginductanceof the 1:1transformer;that is,

(6.2)

$ = [ 4 r o ! 1/ ( R , / 2 ) ] 2+ l Rr=2R"

Fgure6.15

Illustration of the isolating action of the hybrid nansformer shown in Figure 6.14(a).

188

Design of RF and Microwave Amplifters and Oscillators

where.Sis the ratio of the powerdissipatedin the load (R,,: R.l2) andthe power dissipated (R.), whenEz:0. in the sourceresistance Theisolationat high frequencies is a functionof theelectricallengthof theline and the characteristicimpedance. The combinercan be changedto a splitterby connectingit as shownin Figure 6.14O). As in the caseof the combiner,thevoltagedropacrossthe 1:1transformerwill be zeroas long asthe loadsarebalanced.If not, the transformerwill causethe powerto be distributedmore evenly betweenthe two loadsthan would be the casewith a direct connection.

6.3 ANALYSISOF'TRANSMISSION.LINE TRANSFORMERS transformeris anunbalanced transmission Thebasicbuildingblock of a transmission-line line (seeFigure6.16).The line canbe woundaroundmagneticmaterialor canbe shaped asa solenoidalcoil. The lattercanbe doneby usingsemi-rigidcoaxialcable. transmission-line arenotequal, Thecurrentsin thetwo conductorsof anunbalanced but are relatedby the equation

I

Ir(x\ = /, (x) + /o(x)

(6.3)

Because the effective current entering the line at any point (d6 = d (r) - Ir(x) : - 1o@))must be equal to the current flowing out of the line at any other point further along the line (see Figure 6. I 7), the unbalancedcurrent (10(x))is independentofthe distance(x) along the line. Therefore, (6.3) simplifies to

It

(6.4)

Ir(x)=It(x)+Io

It(x):Irn@)-Io/2

Ir(x)=Iro@)+Io12

Figure 6.16

The balancedand unbalancedcomponentsofthe currentin a transmissionline.

189

Transmission-Line Transformers

10ft) =/r(x) -

Figure 6.17

Io@)= Ir(xr) - I,(xr)= Iogr)

The unbalancedcurrenton atransmissionline as a function ofthe distancealong the line.

The effect of the magrreticmaterial(or solenoidalshape)is to increasethe impedance associated with the unbalancedcurrentsin the line. with the balancedcurrents Becausethereis no extemalmasneticfield associated , { H dl = I ro- I n =0), thesecurrentsie not influencedby themagneticmaterialusedor :y the form in which the line is wound.

Ir(x)

IzQ) Flure 6.lE

The equivalentcircuit of an unbalancedtransmission-line.

If the influence of the magneticcoreor the coil form on the unbalancedcurrentsis ignoredinitially, the equivalentcircuit shownin Figure6.18appliesandequationsfor the ; oltageon and the currentin the unbalancedtransmissionline can be derivedin a way ;imilar to the balancedtransmissionline case(referto AppendixA). The resultsof the Jerivationare shownbelow: l,{.r; = -Io / 2+ 1"-rx * '"+rx

(6.s)

/1tx) = Io / 2+ 1"-rx * B"+fx

(6.6)

llrx) = rr(0) - z0 / 2-(A- B) + zo I 2.lAe-'* - B.t"l 1..j?1!t

+sLuxIo/2

.

(6.7)

f 90

Design of RF and Microwave Amplifiers and Oscillators

Y 2 ( x=) V r ( 0 )+ Z o/ 2 ' ( A - B ) - z o/ 2 . l - A e - "- B " t " l +sLuxIo/2

(6.8)

VrzG\=4@)-Vr(x)=Zoltre-r"- Ber'1

(6.e)

where

f =,tie s=ia Jztc

(6.I 0)

Zo="|ffi

(6.1l)

L andC arethe inductanceand capacitanceofthe line per unit length,respectively,andr is the position of the point of intereston the line (relativeto the LHS). Note that the inductancefor the balancedcurrents(I) and that for the unbalancedcurrents(2,) are not the samebecauseof the magneticcouplingbetweenthe two conductorsof the line. When magneticmaterialis used or the line is shapedas a coil, the reactance associated with the unbalancedcurrents(IoI 2) mustbe changedfrom sZ,/ to Xu = 2sL,

(6.r2)

where I11 is the inductanceassociatedwith each conductorof the line when the other conductoris open-circuitedand/ is the lengthof the transmission-line. The inductanceassociatedwith the unbalancedcurrentin eachconductoris twice the expectedvalue(Zrt) becauseof the excellentcouplingbetweenthetwo conductorsof the transmissionline. When currentis flowing in only oneconductor,the voltageacross the lengthof the otherconductorwill be equalto that of the first, providedthat thereare no resistivelossesin the conductors.Thecouplingfactor,therefore,is very closeto unity. when magneticmaterialis usedor the line is woundasa coil, (6.7)and(6.8)musr be changedto

Yr(x)=vr7g)-(zo /2)(A- B) +(Zo lZ)lAe-r' -B"t'l + s ( 2 L r r ) ( x/ l ) I o / 2

F

(6.7b,

and

I/r(x)=vr1g)+(zo /2)(A- B) +(zo lz)IAe-r'*B.t'l + s(2Lrr)(x / l) Io/ 2

(6.8b

' Before(6.5)through(6.9)canbeusedto determinethevoltageson andtheculr€nl, inanyparticulartransmission-linetransformer,theconstantsl,B,l, Z,(0),andZr(0)mus. be determined. These constantscan be determinedby using the boundary conditions for the transformerunderconsideration.

Transmission-LineTransformers

19l

Whenthevoltagesandcurrentsareknown,the powergainandthe input andouQut of the transfofinercanbe determinedeasily. impedances it is sufficientthat lossless, line canusuallybeconsidered Becausethetransmission rheinput impedanceof a transformeris known(in thelosslesssnse,the magnitudesof the input and output reflection coefficients are equal, and the magnitudeof the transducer power gain is only a function of the reflectioncoefficient).The input impedanceof a transformeris a functionof the frequency,the load impedance,andthe transmission-line lengthandcharacteristicimpedanceof thetransmissionline used.The expressionfor the rnputimpedanceis thereforeusuallyquitecomplex. Although the complexityis not a problemwhen a computerprogramis usedto transformer,it is possibleto simpliff theequationfor theinput atnlyznatransmission-line rmpedanceat low and high frequenciesby making appropriateassumptions.At high tiequencies,the reactanceassociatedwith each conductoris high comparedto the impedanceof the line, andthe approximation characteristic sLul>> Zs

(6.13)

rn be made. Under this approximation, the input impedanceof the transformer is only a ..nction of the balanced currents in the line. As far as the impedance is concerned, the -rnsmission line can then be consideredbalanced. At low frequencies,the line is electrically very short and the approximation

(6.14)

:f I _1 - I

rn be madeand the input impedanceof the transformeris independentof the length and -.echaracteristic can transformer line.Thetransmission-line ofthe transmission impedance :enbe consideredto be a conventionaltransformer. transformerreduces It follows that thebasicbuildingblock of a transmissionJine '' a balancedtransmissionline and a conventional1:1 transformerwith magnetizing rductanceZllathighandlowfrequencies,respectively.ThisisillustratedinFigure6.l9.

EXAMPLE 6.1

Theinput impedanceofa I :4 transmissionJinetransformer.

transformer(seeFigure transmission-line of a I :4unbalanced Theinput impedance of (6.5)to (6.11). asan exampleof theapplication 6.20)will be determined The boundarvconditionsfor the transformerareasfollows:

vr(0) = g

(6.15)

Vr(l)=Yt1g1

(6.16)

Vr(l)=Z t11171

(6.r7)

F

t92

Designof RF and Microwave Amplifiers and Oscillators

Iu@)-1, Ib@)+1, Unbalancedtransmissionline (a)

I{r)

Balancedtransmission

I

(b)

Ltr l:l Ideal l: I transformer with magnetizing inductance (c) Ftrrl

*,

(a) The basicbuilding block of a transmission-linetransformersimptified at (b) high and (c) low frequencies.

6.19

Theseconditionswill be usedto find two independentequationsfor the unbalancedcurrent1oin termsof A andB. In this way, the relationshipbetweenI and B can be establishedandthe input impedancecanbe found: ,r,(0)= VnQ)= ZofAe-r' - Ber*1=7oU- B)

?.4.1

.1t

Figure 6.20

I

a..

v{0)

V'(l)

1'(o)

Ir(D

V'(o)

Vz(D

Iz(o)

Ir(D

The l:4 unbalancedfansmission-linetransformer.

.:

Transmission-Line Transformers

193

v2(l) = 0 + 0.5z rfA - B) + s L, t I 0 I 2 - 0.5Z ofAe-il- BerI f ((0) andY2(l)areequalthesetwoequations canbeusedto obtain Because thefollowing an equationfor 1oin termsof I andB. After somemanipulation, equation is obtained:

sLlIo = z0/ (sL,t)'LA-B)+Zo/ GL,t)'fAe-''- B"''l

(6.r8)

The second equation is establishedby using the constraint imposed by the load: Vt(l) =Vr(o\ -0'5 Z0[A - B]+ sl,l Io l2 +0'5ZolAe-r-t- Bertl and

zLI{t) = Zrl-Io /2+ Ae-rt+BeF/l it followsthat By equating thesetwo equations, fZ, + sLulll' =2A[Zre-n -0.5 Zo@-rt+l)] + 2BfZrert+ 0.520(er/ + l)l

(6.1e)

The relationshipbetweenI and B can now be determinedby using (6.18) and (6.1e):

B _ Z0E2U+ZL / G L, t)l-2[Z Le-rt- (ZoI 2) E2] A Z,Eil+ ZL / @L,l)l+2lZrert+ (Z0l2)EJ

(6.20)

where Er=l+en and Ez=l+e-rt The input impedanceof the transformeris givenby the equation Z,^ =V1(0)/[1r(0) + I2Q)] "

I_B/A Ez+@lA).El

(6.21)

194

Design of RF and Mioowave Amplifiers and Oscillators

If the approximation etr/ = I is used,the equationsfor the ratioBlAand the input impedanceof the transformer simplifiesto

B _ 2ZosLrl + Z rZo - Z rs Lul A 2ZosLul + Z LZo+ Z,s Lul

(6.22)

and 1-B /A _ (ZL/4).sL,l12 -

7 -_ / 7 t o' zi' \Lo', , rfrEn

(z/ 4)+ sL"u,

(6.23)

If magaeticmaterialis used,the reactancesZ,/ in theseequationsmustbe replaced with (22,,)s.The input impedanceis then

'v' " -_ 7

-

(Zrl4).sL, \-L

't

(6.24)

--ll

17r14y+sL, At high frequencies,the approximation s Lul >> Zo can be made,and the expressionfor the ratio B/A simplifies to B _ZoU+e-ftl-ZLe-ft A Zol+ "*t'l + Z r"*''

(6.2s)

The impedanceis still givenby (6.21). The transducerpower gain for the transformercanbe determinedby using the equation (6.26)

where Z" is the impedance of the source driving the transformer.

EXAMPLE 6.2

The input impedance of a l:l balanced-to-unbalanced transformer.

transmissionlinetransformer Theinput impedanceof a I : I balanced-to-unbalanced (seeFigure6.21)canbe determinedby usingthe following boundaryconditions:

Transmission-Line Transformers

195

a

Ziot

:F

Znz -

Figure 6.21

The l: I balanced-to-unbalancedtransmissionline transformer.

t / t ( t ) = z L Il t)

(6.27)

vr(l)= g

(6.28)

V{0) = -Y210)

(6.2e)

By using (6.27), the unbalancedcurrentis found to be Is / 2 = Ae-ft1l- Z0 I Z Ll+ Ber/[ + Z0 I Z L]

(6.30)

for determiningthe When (5.28)and (6.29)used,the secondequationnecessary ratio BlA is found to be

s L uI I o / 2 = ( Z o/ 2 ) ' f A e - r t - B " ' ' ]

:

(6.3 r)

TheratioB/A cannowbeobtainedby usingthesetwo equations: B

e-t'[l - zo I z L]-lzo I (2sL,l)le-rt

7=-

(6.32)

When B/A is known, the input impedances Z,n and Zin can be determined. These impedancesare given by the equations.

7.= -'nr

Zin2 =

zo[1.-BtA] -lz,r (sL,t)l [e-tt- @ I A) e''1+z1r+B I A]

z o [ r -B I A 1

(6.33) (6.34)

+ l z 0 l ( sL , t ) l [ e - r r- ( B I A ) e r t l + 2 0 + B I A l of (6.33)and(6.34) It is clearfromthedifferentsignsin thedenominators

that the two input impedancesare not equal at low frequencies.

196

Desigrr of RF and Microwave Amplifiers and Oscillators

When sI,/ 27 Zo, the two impedances are approximately equal, independentlyof the characteristic impedancevalueof the line. Furthermore,the input impedanceof the transformeris identicalto that of a balancedtransmissionline terminatedin the sameload impedance(Zr). By using this equivalence, it follows that the input impedanceof the 1:1 balanced-to-unbalanced transmission-line transformerwill be purely resistiveat high frequenciesif Zo: Rr. Because of the symmetry, the same applies to the l:4 balanced transmission-line transformer.

;

6.4

DESIGN OF TRANSMISSION LINE TRANSFORMERS

The designof transmissionlinetransformersconsistsof the following:

,

l.

Determining the characteristicimpedanceand the diameterof the transmissionline to be used;

2.

Determiningthe minimum value of the magnetizinginductanceof the transformerat the lowestpassband frequency;

3

.

Selectinga suitablemagneticmaterial(if needed);

4.

Determiningthe type andsizeof the coreto be used;

5.

Calculatingthe line lengthandthe correspondinghigh cut-offfrequency of the transformer;

6.

Compensating the transformerfor nonoptimumcharacteristic impedance,

, ,,)- 7.

Extendingthe bandwidthby using LC impedance-matching networks,if necessary.

Eachof thesepointswill be discussedin detailin the following sections.

6.4.1

Determining the Optimum Characteristic Impedance and Diameter of the Transmission Line to Be Used

At high frequencies, the input impedanceof a transmission-line transformeris a function of the characteristic impedanceof the transmissionline. Theoptimumcharacteristic impedancecanbeestablished by takingtheratio of the

197

Transmission-Line Transformers

voltageacrossoneendofthe transmissionline andthecurrentpassingthroughit. Thebasic buildingblock of the transformeris thenconsideredto be an ideal l:l transformer. transformeris The applicationof this rule to a l:4 unbalancedtransmissionJine illustratedin Figure6.22.

R=2V/l

r igurc 6.22

Determiningthe optimumcharacteristicimpedanceof an I :4 unbalancedtransmissionline transformer.

If a line with any other characteristicimpedanceis used, the input reflection -oefficientof the transformerwill be affectedadversely. The effect of the characteristicimpedanceon the cut-off frequency of the -rnsformerwill be discussedlater. When the optimum characteristicimpedanceis known, the type of line to be used rnustbe chosen. impedancearefreelyavailable.A Coaxialcableswith 25Qand50Ocharacteristic newith a 12.5Qcharacteristicimpedancecanbe obtainedby connectingtwo 25O lines : parallel,while l00O canbe obtainedby connectingtwo 50Olines in series. can be obtainedby twisting together A wide rangeof characteristicimpedances arerequired(less very low impedances When with various diameters. conductors nirs of together. can be twisted with smaller diameters ran l0O), manyconductors ofthesetwistedlinesareinfluencedby thediameter impedances Thecharacteristic : thewire used,aswell asthe numberof twistsper unit length. Apart from the characteristicimpedance,it is also necessaryto decideon the jiameterof thecableto beusedwhereapplicable.This is determined by thelossesthatcan -e toleratedandthe powerto be transmittedthroughthe line. The attenuationof bifilar or multifilar transmissionlinescanbecomea problemat -,:ghfrequencies, asmentionedin Chapter3.

6.4.2

Determining the Minimum Value of the Magnetizing Inductance of the Transformer

\t low frequenciesthe transmission-linetransformercan be consideredto be a ..rnventional 1:I transformerconnectedin a specialway.

198

Ii I

Design of RF and Microwave Amplifiers and Oscillators

When this model is used,the input impedanceand power gain versusfrequency responseat low frequenciescanbe determinedby usingKirchhoffls voltageand current laws on the simplifiedequivalentcircuit. If the loadconsistsof a singleresistor,only theinputimpedanceofthe transformer Thepowerdissipatedin theload(andthereforethepowergain)can needsto bedetermined. be foundby usingthe equation

(6.35)

PL=v]"G.tr /2

where Vo is the maximum (peak) voltage acrossthe effective parallel input resistance (R"6= l/G"6) of the transformer. When the input impedanceand the transfer function are known, the minimum canbe determined' inductance(2,1)requiredto meetthe low-frequencyspecifications l:4 unbalancedand 1:l the of inductance The minimumvalueof the magnetizing as examples. transformerswill be established unbalanced-to-balanced

& l-l lrrwT"l

4&

f-ff(a)

I

&

4&

LI

(b)

&

LI

&

(c)

Flgure6.23

Simplification of the equivalent circuit of the l:4 unbalancedtransfornier atlow frequencies.

t99

Transformers Transmission-Line

EXAMPLE 6.3

The magnetizinginductancerequiredin a l:4 transmissionline transformer.

With the transmissionline replacedby a l:l transformerwith magnetizing inductance,the equivalentcircuit of the 1:4transmissionlinetransformercanbe simplified asshownin Figure6.23. Ifthe cut-offfrequencyis to be the 3-dB cut-offfrequency,it is obvious Z' mustbe suchthat from Figure6.23(c)thattherequiredmagnetizinginductance (6.36)

tDLr,= ft, /2

If this transformeris to be usedin a power amplifier, the magnetizing inductancemustbehigh enoughfor thespecifiedminimumallowableripple in the passband to be achieved. Becausethe power dissipatedin the load is given by (6.35),the output poweris directlyproportionalto the effectiveparallelinput resistance' if the effectiveload is reactive The efficiencyof the amplifieris decreased of thetransistoris assumed (referto Section2.3.3),thatis, if theoutputimpedance by a factor is it decreased purely Specifically, resistive. to be T't,= | /[ + (R.u I X"u)']U'

(6.37)

whereX"6 is the effective parallel input reactanceof the transformer. BecauseR.6is equalto the optimumvalue(R")in this particularproblem, the power transmittedthroughthe 1:4 transformeris also equalto the optimum value,that is, at low frequencies. The relativeefficiencyis givenby I, = 1/[[l + (R" I (aLrr)121v2 the magnetizinginductancemustbe suchthat If rl, : 0.95is acceptable, = 3R" roZ11

(6.38)

(o211is often chosento be equalto 4X).

R L +

RL

(b) (a) Fryure 6.24

The l: I unbalanced-to-balanced transmission-line transformer at low frequencies.

200

Design of RF and Microwave Amplifiers and Oscillators

EXAMPLE 6.4

The magnetizinginductancerequiredin a l:l transmission-line transformer.

The equivalentcircuit for the 1:l unbalanced-to-balanced hansformeris shownin Figure6.2a@). By transformingthe load on the secondaryside of the transformerto the primaryside,the equivalentcircuit of the l:l unbalanced-to-balanced kansformer canbe simplifiedto that shownin Figure6.24(b). By using this equivalentcircuit, the input admittanceis foundto be rin =

R r + s L r r R r l [ R+, s l r , ] R, +2sL' R?,+ZsLrrR,

_

1 .l+R./(s2,,) 2R, 1+ R, / (2sLrr)

(6.3e)

It is clear from this equationthat the input resistancewill be equalto 2Rrif the magnetizingreactance is relativelyhigh. Therelativepowerdissipationin thetwo loadresistances canbedetermined by transformingthe parallelcombinationof oZ,, andR, in Figure6.24(b)to the equivalentseriesimpedanceshownin Figure6.25. Becausethe samecurrent flows through the two resistors,the ratio of the power dissipatedin each load is equal to the ratio of the resistanceof these resistors.If altt = 4.4Rr

(6.40)

the powerdissipatedin the two loadresistorswill differ by 5%. The input power to the transformerwill thenbe 1% higherthanthe designvalue,andthe relative efficiencvwill be 0.99.

Figure 6.25

The seriesequivalentof the impedanceof the circuit from Figure 6.24(b).

201

Transmission-Line Transformers

6.4.3

DeterminingtheTypeandSizeoftheMagneticCoretoBe Used

transformers.The sizeof the toroidal Toroidalcoresare often usedin transmission-line coreis determinedby the inductancerequired,the maximumflux densityin the core(and thereforethe allowablelosses),andthe line lengthrequiredto meetthesespecifications. It was shownin Chapter3 that if the inductanceandflux densityspecificationsare to be met simultaneously.a corewith

Folr, V3* ,, -- ---------:--

(6.41)

nt

aB'^* aLrt shouldbe used(see(3.33)). It canbe shownthattheline lengthwill alwaysincreaseif a corewith anll-product argerthanthat given by this equationis used. it is possiblethat the line lengthmight be shorter,at If the coresizeis decreased, .ea* initially. Whetherit will be shorteris a function of the extentto which the inductancemust .e increasedto meetthe lossspecification(theflux densityin the corewill be too high if :heinductanceis not increased), aswell asthe dimensionsof the core. the If the lossesin thematerialincreasesharplywhentheflux densityis increased, optimumcoresizewill alwaysbe that givenby (3.39). to providetherequired It is sometimespossibleto reducetheline lengthnecessary ragnetizing inductanceby usinga numberof smallertoroidalcoresinsteadof only one 3rgercore. The ratio ofthe line lengthfor a singlecoreto that of//" stackedcoresis given rpproximatelyby the equation 2wr+2ht+4t

l r ,_ 1,,

(6.42)

(At I A).lU + (1, I lt).(4w, + 4t)

l-

7-F T hl

--rl rurlr-

(a) f4rtre 6.26

;

TI

--1rI

l w 2

O)

of(a) a singletoroidalcoreand(b) a numberofstackedtoroidalcores. Thecross-section

;

202

Design of RF and Microwave Amplifien and Oscillaton

wheret is the outer diameterof the transmissionline used,/r the meanpath length of the largercore,/, the meanpath lengthof eachof the smallercores,11 the effectivecrossareaofeach ofthe sectionalareaofthe largercore, andA, the effectivecross-sectional smallercor€s.lolew2,hr, andft2aredefinedin Figure6.26. Equation(6.42)wasderivedby assumingthe inductanceandthe flux densitiesof the two inductorsto be equal. In order to havethe sameflux density,it is necessarythat (6.43)

Nt/\=N2/12

whereN1is the numberof tums usedwith the singlecoreand N, the numberof turns used with the stackedcore. The inductanceof the two inductorswill be the sameif

N?At/\=N"NlAz/lz

(6.44)

inductor. whereAf is the numberof coresusedin the stacked-core By using (6.43),(6.44) canbechangedto

.

Arlt= N,- A2l2

(6.4s)

andstackedIt follows from this equationthattheeffectivel/-productsof thesingle-cored coredinductorsmust be the same. Equations(6.45)and(6.43)canbe usedto determinethe numberof coresandthe numberof tums required,if usinga transformerwith stackedcoresis worthwhile(i.e., if the coredimensionsareknown). If a core with suitabledimensions(comparableto thoseof the stackedcore) is the line lengthofthe transformer. available,a baluncorecanalsobe usedto decrease

EXAMPLE 6.5

Comparisonof the line lengthsassociatedwith a stacked coreanda singlecoretransmissionlinetransformer.

As an o