Design of RF and Microwave Amplifiers and Oscillators

- I II I SHAPTER -rE\r r -r-r\ 1 r I t I I CHARACTERIZATION AND ANALYSIS OF LINEAR CIRCUITS AT RF AND I MICRowAVE ...

Author: Pieter L. D. Abrie

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I

II I

SHAPTER -rE\r r -r-r\ 1 r

I t I

I CHARACTERIZATION AND ANALYSIS OF LINEAR CIRCUITS AT RF AND I MICRowAVE FREQUENCIES I

I r.t I I | I I f \ I I I I I I I f I I

rNrRoDUcrroN

t o*-frequency circuits areusually analyzedin termsof transferfunctions.This approach -'ldomusedat RF andmicrowavefrequencies.Analysisat thesefrequencies is usually m termsof oneof the manysetsof single-frequency parameters. The parametersmost frequentlyusedare they-, Z-, z-, ands-parameters.The first fue setsof parametersrelatethe terminal voltagesand currentsin different ways, while & $parametersarecloselyrelatedto thepowerincidentto andreflectedfrom a network. Becauseof the relative easewith which S-parameters can be measuredand the :ul informationdirectlyobtainedfrom them,components areusuallycharacterized by asuringtheirS-parameters,andcircuits areanalyzedbycalculatingtheir,g-parameters. fe otherparametersare often usedto simpliff the computationsnecessaryfor circuit - *i sisandsynthesis. Each of thesesetsof parameterswill be consideredin detail in the following 'onsi, of the voltages,currents,or power levelsin a linearN-port networkcan be c{crilated in termsof the extemalsignals(independent variables)whenoneof thesesets d perametersis known at the frequencyof interest.Conversionbetweenthe different tracters is straightforward.

I l, ,

'-'ARAMETERS

l I

|

IL )

l b

, - i -pu*-eters of an.ly'-portnetwork are definedby the expression v .-ne

( l . l )

Desigrr of RF and Microwave Amplifien and Oscillaton

Il I2

(r.2)

I_ IN

vr v2 (1.3)

VVN

Ir" Y_

ltzt

!p

!tN

ln

lzx

1,",

!Nz

(1.4)

!xu

d is the currentflowing into the ith terminal,and ( is the voltageacrossthe fth port of the network. Eachelementofthe /-parametermatrix canbe calculatedor measuredby usingthe relationship

I y,,=+1,, "

h e f t , 2 , 3 , . . . Jh, *t jl

(l.s)

V'Yt="t J

that is, yu is given by the ratio of the currentflowing into the ith terminal (output signal) andthe voltageacrosstheTthport (input signal),with all the othervoltagessetto zero. to any given set of terminal By using (1.1),the terminalcurrentscorresponding is, therefore,completely of the network voltagescanbe determined.The linearresponse known. matrix are characterizedwhen the N2elementsof the )'-parameter can be usedto calculatethe the lz-parameters As with any othersetof parameters, impedancesand gain ratios correspondingto any set of terminations.By using the apply equivalentcircuit in FigureI .1,it canbeeasilyshownthatthefollowing expressions to a two-port network terminatedas shown:

{

characterization and Analysis of Linear circuits at RF and Microwave Frequencies

ilr, ,#,

lrz Vz

] ttrrl

l.l

An equivalent circuit for a two-port in terms of its l-parameters.

- I, Vt

ltzlzt

', c- - 7 7 - . Y- t l., -

Iz Y ' t - 1- 7 - r 2- 2 -. -

lnlzt

V2

(r.7)

hr+Y"

V, - a

(r.6)

-

lzz+YL

Vtr

(1.8)

- -

Vr

ln+YL

t =+=-+=Aryr/y;,, Ir

(1.e)

Ir

/ : -- P L - ly r rY+rY, r l ' G L P" R " (rJ _ P , _ Pn-t

=

rI3

+ Y,)(yzz+ Yr) - lnlzr

(1.10)

4GLG,

(l.ll)

pu,-o

=| ,r, l' c, P,"* lr" *r"I R"(rJ

(r.r2)

I

i,

u

.'i:"*

Desigrr of RF and Microwave Amplifiers and Oscillators

il

li ilii

lJ.* Available PowerGain

[.

t

zh'

lli i

o

I-

I

x

I

OperatingPowerGain

II t II

TransducerPowerGain

Ir il tl

zn'

ti

o

L-

lii ]i

MAG /MSG

ill ]t lti til lil

G^-o*

Figure 1.2

The equivalentcircuits relevantto the different power gain definitions.

I

Characterization and Analysis of Linear Circuits at RF and Microwave Frequencies

In theseequations,)', : G, + jB,is the loadadmittance,Y": G" +78- is the sor.nce ,Jmittance,P, is the powerdissipatedin the load,P,, is the powerenteringthe input port : thenetwork, P"u-,is thepoweravailablefrom the source,P"u-,, is the availablepowerat -.eoutput terminals of the two-port, Iln is the input admittance,md I"* is the output lmittance. The availablepower of a sourceis definedasthe power dissipatedin a load which .cnjugatelymatchesthe source,andis givenby the expression

"*-

- l E l 2- l l , l ' 4R" 4q

( 1 .l 3 )

rere E is the sourcevoltage,{ is the equivalent(l.Iorton)sourcecurrent,andRr and G5 -: definedby r'.=e"+jB,

(r.r4)

-=ft,+jx,

(1.l s)

.tere Z" is the sourceimpedance,and I. is its inverse. Note that the operatingpower gain (G") will be equalto the transducerpower gain ' rheinputis conjugatelymatched(seeFigure1.2).Similarly,theavailablepowergainwill c equalto the transducerpowergainwhenthe outputis conjugatelymatched. The maximum availablegain (MAG) of a two-port is definedas the transducer .r"*er gainwhenboth sidesareconjugatelymatched(ifpossible). If the MAG cannotbe :,lculated(negativeresistance),the maximum stablegain (MSG) is of interest. The -aximum stablegain (MSG) is the MAG associatedwith the deviceafter adding the -jnimum shuntconductance requiredfor the MAG to exist. or seriesresistance

rglre 1.3

r

Two networks connectedin parallel.

Designof RF andMicrowaveAmplificrsandOscill*ors

is the availablepower gain associatedwith an optimum noisematchon the Gon_o* (i.e., Z"is chosento minimizethe noisefigure of the two-port). input side Whena circuit is analyzed,the l-parametersarefrequentlyusedto find a singleset of parameterscharacterizingtwo networks connectedin parallel. This is illustrated in Figure 1.3.Note that the terminalvoltagesfor the two networksarethe same,while the currentsadd. The l-parametersof two networksconnectedin parallel simply equalthe sum of the l-parametersof eachindividualnetwork: (1.16)

Y, =Y^+Y,

EXAMPLE 1.1

Derivationoftheequationforthe inputadmittanceofa twoport.

The input admittanceis definedby (1.6): Y,n= I, /V, To find the input admittanceit is thereforenecessiryto find an expression for Z, in termsof d. Ohmslaw andKirchhoffs currentlaw appliedto the input port vield v, = [11 - tp v2)/ tn

(r.r7)

The output voltage is given by Vz = -Iz/ Yr = -A^ V, + \2 V2)I Y,

( 1 .l 8 )

that is,

I r" ^ = -

l" v, !rr. * Y,

substitution of (1.19)into (1.17)yields(1.6): After somemanipulation,

Y^=ltt-L' ? lzzl t

( 1.1e)


1.2.1 The IndefiniteAdmittanceMatrix ie indefinite admittancematrix is a useful tool by which the l-parametersof a network rn be determinedif they are known for the samenetwork connecteddifferently. For parameters \rmple, if the common-emitter of a bipolartransistorareknown,this matrix .:,nbe usedto determinethe common-base parameters. or common-collector An admiuancematrix is indefinite whennoneof the networkterminalshavebeen :onnectedyet to ground,andthe total currentflowing into it is thereforeequalto the sum i the currentsflowing into eachterminal. It can easily be shownthat the sumof the elementsin eachrow or eachcolumn of r indefinite admittancematrix is equal to zero. Consideringa three-portnetwork, this rplies that if four of the nine parametersareknown, then all the parametersare known. The proof that the sum of the elementsof eachrow must equal zero follows by .roosing the terminal voltagesto be equal.Eachof the currentswill then be zero and rtractionofeach individualequationfrom (1.1)yieldsthe desiredresult. That the sum of the elementsin eachcolumn should also equal zero follows by 'etting two of the voltagesequalto zero and addingthe threecurents, the sum of which .rstbe equalto zero.

EXAMPLE 1.2

Calculation of the common-baseparametersin terms of the corrrmon-emitterparameters.

The common-baseparametersof a transistorwill be determinedin terms of its cornmon-emitterparameters,as an exampleof using the indefinite admittance matrix. The indefinite admittanceparameters,which correspondto the commonemitterparameters, canbeidentifiedby settingZ2in FigureI .4 andin ( I .20)equal to zero.

gure 1,4

I

:

An indefinite three-port.

,

Desigr of RF and Microwave Amplifiers and Oscillators

It'l [r', rn ,','lin'-l It , l = l n l z zt n l l v , I

Lr,,lLy,,rn v")lr,)

commonBecausethe currentin the emitter (1r)is not of interestwhen the (l '20) thenreducesto emitterconfigurationis considered,

[r,l lv,, v,,l[r,l-lhr" t,z,[v,]

(1.21)

lv,* v,,,L4) [r,l=[r,,vuXv,)-

dnd1.*arealso known,!t,!n,!tv parameters With the cornmon-emitter appliedto determine known, andthe rule for the ""ro .ol rlnr, androw cannow be the common-base identify to is step the other parameters.The only remaining this is doneby parameters, farametersin (1.20).Similar to the common-emitter settingVlin(1.20)equaltozeroandeliminatingtheequationgivingthebase "orr"nt (1,)asa functionof the voltages'It follows that

t,,ol-ln,"f ln* lzzt ln J LYxt

J lln

The cornmon-collectorparametersare given by

| v,, ln* vp"1 l=l

1v,,"!zz")

1.3

llzr

Yrzf Yrr)

Z-PARAMETERS

of an N-port network aredefinedby the expression T\e Z-parameters V=ZI

':

I


Z"

I,

I,

n--+-f""'

't--_-- ",,fl-------. :--i *--i f - r re 1.5

':un'\"

An equivalentcrrcuit for a two-port network in termsof its Z-parameters.

s- - I' and 1 aredefinedby (1.3)and(1.2),respectively. Theequivalentcircuitassociated rith the two-portcaseis shownin Figure 1.5. Eachelementin (1.25)canbe computedor measuredby usingthe relationship

=?1,a h e1r,2,3,"',N) h* i

(r.26)

th is, zuis the ratio ofthe voltagesacrosstheTthport (output signal)andthe currentat the * oort (input signal) with all theother ports idle (open-circuited). Equation(1.24) canbe usedto find the terminalvoltagescorresponding to any , r set of terminal ctments. Comparisonof (1.24) and (1.1) revealsthat the Z-paranetersof a network are -:'':ed to its l-parametersin the following way:

Z =Y-|

(r.27)

l,ttVsr

. grre 1.6

t-

VA2+Vn

Two networksconnectedin series.

.

'

Designof RFandMiuowaveAmplifienandOscillaton

l0

Z-panmetersare frequently usedto find an equivalentset of parametersfor two networksconnectedin series,as illustratedin Figure 1.6.Note that when networksare connectedin series,the terminalcurrentsarethe same,while the voltagesaddtogether. of two networksconnectedin seriesare given in terms of the The Z-parameters individual Z-parametersby Zr=Zn+Zo

I.4

TRANSMISSIONPARAMETERS

ofatwo-portaredefined (Z-parameters oTABCDparameters) Thetransmissionparameters by the equation

u1ln,1 ln'1=lnD)l_Ir)

(r.2e)

L/,1 lc

with the voltage andcurrentasdefinedin Figure 1.7.Note that 1, is the output cunent and not the current enteringthe output terminal as in the caseof the I- and Z-parameters.

Figure 1.?

The voltageand currentrelevantto the definition ofthe transmissionparameters.

The expressionsfor the individual elementsof the transmissionmatrix can be obtainedby setting eitherV, or 1, in (1.28)equalto zeroafter extractingthe individual equationsfrom the matrix equation. Z-parameterscan be convertedto l-parameters by using the following set of

equations: yrr=D/B

(1.30)

ln=C-AD/B

(1.31)

{

Characterizationand Analysisof Linear Circuits at RF and Microwave Frequencies

11

(r.32)

\':t=-llB

(1.33)

vz= A/B '| re inverseexpressions are :

'=-!zz/lzr

l

(1.34) (1.35)

B=-lllzr r -- ln - !n!n

(1.36)

I lzr

(1.37)

-)=-ynlyzr ffi

:

Transmission parameters are frequently used to find an equivalent set of parameters : rwo cascadednetworks. The transmission matrix for the equivalent network is given in -:ns of the matrices for the individual networks by

7 =TnT,

( r .38)

' . is illustrated in Figure1.8.

' qur l.t

:5

+

+

v2

V.

,.{

Two cascadedtwo-port networks.

SCATTERING PARAMETERS

'.errse of the easewith which scatteringparameters(S-parameters)canbe measured,as :il rs stabilityconsiderations andthe physicalmeaningsattachedto them,S-parameters . -'d extensivelyto characterizecomponentsand alsoto analyzscircuits. The definitionsrelevantto theseparameters,their physicalmeanings,and their ltircation in analyzingcircuitswill be consideredin the following sections.Both single:qrFrcy S-parameters and thosein the complexfrequencyplane will be considered.

r

.

.

.

Design of RF and Microwave Amplifiers and Oscillators

12

Becauselosslessnetworksareof considerableinterestin this text, the constraintson the ^Smatrix of a losslessnetworkwill alsobe examined.

1.5.1

S-ParameterDefinitions

aredefined Similar to the reflection coefficientsin transmission-linetheory,S-parameters however,anincident InS-parametertheory, intermsof incidentandreflectedcomponents. componentis definedasthat componentwhichwould existif theport underconsideration were conjugately matchedto the normalizing impedanceat that port. The normalizing impedancesarethe equivalentsof the short-circuitandopen-circuitterminationsusedto They canbe definedto have characteiz-ea network in termsof its I-, Z-, or T-parameters. anyarbitraryvalue(aslong astheresistivepartis positiveandnot equalto zero),but 50O impedances areusedin mostcases. In terms of the current and voltage at eachterminal, the incident and reflected componentsaredefinedby the following setof matrix equations: Eo=V + ZoI

(1.3e)

Ii =lZo+ Zil r,

(1.40)

f=fi-f,

(r.41)

Vi = zilt

(r.42)

V =Vi *Y,

(1.43)

r, o=fttzr+ziltl2

(1.44)

n=fttzo+z;ltt2ri

(1.4s)

Zot

0 Zo=

0 0

0 0 Zo, o o zrt ;

;

0 0 0

(1.46)

;,,.

a


0 "frto, 0 0 JRo, o = 0 ^Fo, 1l7o+7i1tt2 0 0

0

0

0 0 0

13

(r.47)

: VRo"

' 'h 4 the normalizing impedance at portj , Z i the matrix with conjugateelementsof ' * of Zo, I1 and V1ithe incidentcurrentand voltage atportj, Irand Zrithe reflected . :=nt andvoltzge,a, thenormalizedincidentcomponent,and6,thenormalizedreflected . lponent atportj. The voltageandcurrentrelationshipsareillustratedin Figure 1.9 for a two-port -trrork. Note that the incident voltage is equal to the product of the conjugateof the r*malizing impedanceand the incident current;that is,

.=z;r, '

: equivalentrelationshipin tansmission-line theory is

, = ZoI , By using(1.40)to eliminateEsin (1.39)andsubstituting (t.al) and(1.43)in the -: :lting equation,it can be showneasilythat, similar to transmissionJine theory,the - .:ionshipbetweenthe reflectedcurrentsandvoltages is 1 = ZsI ,

(1.48)

Therearethreedifferenttypes ofs-parameters,which aredefinedinthe following

:

= SI I,

(r.4e)

,# 1. = SvVl

l=^Sa

(1.s0) ( l.s1)

; :separametersetsarethe current,voltage,andnormalizedS-parameters, respectively. For a two-portnetwork,(1.51)reducesto .l}

; } , 1


F

V2=V2fY2,

za

Zor v Eo,

)

' +

v

+

,s

,

(

v2

Eoz

(c)

I

Zor

+ ,:,

Eo,

: Ftrrc

za

1,,

zo,

+ l/",

Eoz

(d) f .9

'.

(a), (b) The voltage and current relevant to the S-parameterdefinitions; (c) the two-port of (a) and (b) augmentedby the normalizing impedances;(d) the equivalent circuit for calculatingthe incidentcurrentand voltage.

[ql=f*, ",,1[o,'] tz, Lb,) Lrr,

)la, )

(r.s2)

The definitions given aboveare summarizedtogetherwith other useful relationshipsin

-

characteization and Analysis of Linear circuits at RF and Microwave Frequencies

trr

l.l0

15

A diagramof S-, I-, and Z-parameterrelationships.

I ;-* 1.10. ft" impedancematrix Z n inFigure l.l0 is definedby ,

n '; :." :l t, I

;;

,^l

(1.s3)

tc matrix E uby

-*'

En "42

;

(1.54)

I il|r * -

E4 refersto the sourcevoltageat theJthport ofthe.l/-port augmentedby the actual

16


+ vl

Figure l.ll

v2

The two-port augmentedby th" u.tu, load and sourceterminations(84 is usually equal to ze(o).

of interest( Z, ). Thesedefinitionsareillustratedin FigureI . I I sourceandloadimpedances for a two-port network. variablesandemanate Note thatthevectorsin FigureI .10flow into thedependent to eachbranch.If next are shown multipliers The branch variables. from the independent (t4 be used. should matrix the unit no multiplier is shown, It can be shown that Eo (the source voltages of the N-port augmentedby its norralizing impedancesas illustratedin Figure 1.9)is given in termsof Ea (the source andsourcevoltageof interest) by theactualimpedances voltagesof theN-port augmented by the expression

Eo=II x - (Z o - Z )(1, - S' )(Zo+ Z;){rt E A EXAMPLE 1.3

r.

l ,

(1.s5)

Derivation of the relationshipbetweenthe reflectedcurrent andvoltage.

To usethe diagramin Figure I . I 0, considerthe derivationof the equality( 1.4S): V,=ZoI , It follows by inspectionof the diagramthat in orderto find a relationship to relateV to I. Theeasiestpossibleway would betweenV, andI , it is necessary be to usethe expression Eo =V + ZrI Eocanthenbe replacedin termsof 1,, Zin termsof V, andV, Z, in terms of Zs' andl,, and 1 in terms of 1, and1,. After a few manipulationson the equationthusobtained,(1.48)follows.

"-


EXAMPLE 1.4

17

Calculationof the incidentandreflectedcomponentsfor a two-port.

In order to make the definitions given abovemore real, considerfinding the incidentandreflectedcomponents whentheterminalvoltageandcurrentof a twoport aregiven by Zr = l'OV Vz = 0'5Y '

1 r= 0 ' l A Iz = -0'2A andthe normalizingimpedances arechosento be Zot = 5Cl Zoz = looThe first stepis to find the sourcevoltage in the equivalentcircuit shown inFigure 1.9(d)inorderto findthe incidentcurrentandvoltage. Inspectionofthe diagramyields(1.39): Eo=V +ZrI

ol[ o.rI ll I 10.5JL0 rOJL-0.2J

[s - l [t.ol l+l

=I r . s l L-t.rl The incident componentscan now be obtainedby using the equivalent circuitin Figure1.9(d):

ft,,.l=ltrpno; o l["0,]_[o.rsol Lrr,) L

0

U(zPYJ))lqor)f_o.ozs_J

.l lr',,1-ltoir,,l_[ o.zs Irr,,l-Lri,,,)- [_o.zsJ The normalizedincidentcomponentsfollow by applicationof (1.44):

r}

....

.:.j:1.''

Dcsigr of RF and Microwave Amplifters and Oscillators

o::s+ [o'l=| J5r" l[ I Ro,Ir, F

a

f

-

Lo,J l,l

'

1

) ) L-0.2372

Thereflectedcomponents canbe obtainedby applying(1.41),(1.48),and(1.45):

[r,,]_| r,,- r,l _fo.osol Lr,, )- lr,,- r,l- fo.rzsJ lr,,f-l zo,I,,l - [o.zs'l rr,J Lvr, )- lzrr [r.zs-J [6,I - [r/4,r,,I _[o.rrrr I Lt,)-l,[4 4,)- fo:rsrl

F

1.5.2

The Physical Meanings of the Normalized Incident and Reflected Components of an N-Port

Thcnormalizedincidentandreflectedcomponents aredefinedin (l.44) and(1.45)interms of the incident and reflectedcomponentsof the terminal current.It is useful to have ogcssions for thesecomponents in termsof theterminalvoltageandcurrent.Theinverse r=6lisnshipsarealsoof interest. The requiredexpressionfor a, canbe obtainedeasily by using the relationship bam the incident current md En:

o,=,tS Ii,

tu $r

(l.56)

= ,!n, ro,/[Ro,+ Ror]

-w _Y,

* ZoiIi

fr

( 1.s7)

*r$"

Tb qtssion for the normalizedreflectedcornponentcanbe derivedby usingthis result in thc following way:

t, =,{ntt,

(1.58)

G

* .

Cbaracterization andAnalysisof LinearCircuitsat RF andMicrowaveFrequcncies

I

'

=,[[f,r,,-,[$r,

I

19

u

=,[nrg,,r,l

I I

I

- ' -r r =vir*z:ili 2-tR^ J,-ut,

I r-

(r.se)

f

lL

inverserelationshipsfollow easilyby manipulating(1.57)and(1.59):

-l , =JRo,

(l'60)

I'

I-

.:;

zo',o, o,b, = f !z^ J

(1.61)

I

f ." J I f r J I

It follows from (1.60) thatthenormalizedcurrentat anypoint in thecircuit canbe ' nodasthedifferencebetweenthenormalizedincident andreflectedcomponentsat that Notethat, if squared,the unitsof the normalizedcurrentwould be that of power. When

rI

''simplit-resto

I I

x

1

= .[{ta.

(1 A',\

-h.r

--rs case, the normalized voltage at any point can be obtained as the sum of the I ! -AizEd incidentand reflectedcomponents.The units of the normalizedvoltageare I :- - thatof Powerif it is squared. f An expressionfor the power entering any port can be derived in terms of the l" lr?r,dcomponentsbyusing(1.60)and(1.61)inconjunctionwiththeexpressionfor ]-a

*ol),"),'.,,,,,

(,63)

F -

-

.

20

Design of RF and Microwavc Amplifiers and Oscillrtors

',b',

Zoja , *' -zt to i"i

=15-tti'i

a L

6

4i -bi

&;

ziia i + Z o b , "t -b;

6

6

=lo,l'-V,l' The power enteringany port is, therefore,simply equalto the differencebetween lhc squareof the normalizedincident and reflectedcomponentsat that port. The last statementcanbe takena stepfurther.It canbe showneasilythat larl2 is the availablepowerat theTthport of theN-portaugmented by its normalizingimpedances (seeFigures1.9(c)and 1.9(d). From this and from (1.64),it follows that lD,l2is the reflectedpower at the7th port of the augmentedN-port, and, consequently, the power enteringanyport ofa networkis equalto thedifferencebetweentheavailableandreflected powerat theTthport of the l/-port augmented by its referenceimpedances. It is important to realizethat the availablepower in the N-port augmentedby the ryaplizing impedances is not equalto theavailablepowerin theN-portaugmented by the Ehral sourceand load impedances, unlessthe two setsof impedances areidentical. Thesimpleexpressions for thevoltage(1.61),current(1.60),andpower(1.6a)in tms of the normalizedincidentandreflectedcomponents aresummarizedbelow.

Ir=(ai-b1)/l\i Y, =(Zs,a1+Zorb) t ,t\i

= rlnrg, +bj) if zoi=zii

P,=V,l'-Vtf t.53

The Physical Interpretations

of the Scattering Parameters

Considcr the definitions of the elements of a two-port scattering matrix. The input reflectim parameter.r,is definedby

n,=**loz=0 d

the forward transmissionparameters, by

(1.6s) !

Characterization and Analysis of Linear Circuits at RF and Microwave Frequencics

= --2t

2l

(1.66)

arlar=o

The constraintson the cunent andvoltageat the outputterminals,when a2= 0, carr gtermined by using(1.57): Q = at' =V'

+ Zo'I'

2^lRo,

#L pr::ng

[g

J =Zw[-Izl

(r.67)

l '

In order for a, tobe equal to zero, the load impedanceacrossthe output port must -: be equal to the normalizing impedance at that port, and the electromotive force ':rust be equal to zero. This is illustrated in Figure l.l2(a).

af0

zo,

Z02

E02

J

o) I!

illi-

The conditionsunderwhich (a) a2--0 and (b) a, = 0.

it this stage(1.57)and(1.59)canbesubstituted into (1.65)and(1.66)to find an :o for the parameters in termsof the terminalcurrentandvoltage:

-': *'

22

DesignofRFandMicrowaveAmplifiersandOscillators

z ,n -zi , I v , -z; t l t I \ r = y 3 4 1 , l ' , - o= q 3 7 r l ' , - n

J2l

-

mv2-z;2r2 | _ - @zoret)-zi,t, Eo, Vn6J, l"=o

{&

1&,

(1.68)

1 la'=o s*t,

'-I^l

= -2JRo,Roz fr\"='

I]

NT

(1.6e)

a.ls

vhereZin is the input impedanceof the two-portterminated,asshownin Figure l.l2(a). The equivalencebetween(1.68)andthe expression

-r r. n- Z a - Z o r z^+2,

k?.:

(1.70)

for a reflectioncoeffrcientin transmissionlinetheoryis obvious.When Zot = Rot will be identical. asis oftenthe case,the two expressions is equal,the forwardtransmissionparameterstt Whenthe normalizingresistance is simply the voltage gain Vr l(Eotlz) of the two-port augmentedwith its normalizing impedancesand with Es2setequalto zero. Becausethe S-parametersare defined in terms of the normalized incident and reflectedcomponents,and the squareof thesecomponentswas shownto be the incident and reflectedpower at the relevantport of the two-port augmentedwith its normalizing respectively,it follows that impedances,

=l*11",=, b,,[ = Pr, I e*-^1.,=o rd

l't'l'

(1.71)

L

=l#1.,=.

I {


23

lu,l'-lo,l' t

l*=o

P

larl -

P L I

(r.72)

Tl",a 'av-E

tr'berc Pu,-u is the poweravailablefrom the sourcewhenthetwo-port is augmentedby the mrmalizing impedances,and P,, is the power reflectedfrom the input port when it is andEoris setequalto zero. ogmented by the normalizingimpedances areillustratedin Figure1.13. Themeaningsof ls,,| 2andls21l2 Similar expressionsapply to the outputreflectionparameters22andthe reverse :nnsmission parametersrr.

(a)

Zor

az= 0

,s

Eot

zo,

PL: ls2rlz P"-E

(b) r-':rc

l.l3

The physicalmeaningsof the scatteringparameters(srrl szr) illustrated.

When the normalizing impedancesare also the impedancesin the actualnetwork - " rterest,the transducerpower gain andthe ratio ofthe reflectedpower at the input to the r ,:lablepowerfrom the sourcearegivendirectlyby s21ands11,respectively. aredisplayed arepurelyresistive,ands11and,r22 Whenthenormalizingimpedances o a Smith Chart, the input and output impedancesof the network can be readdirectly.

1.5.4

Constraints Imposed on the Normalized Incident and Reflected Components by the Terminations of an N-Port

.: order to derive expressionsfor the gains and impedancesof an l/-port with arbitrary -

_

_

_

.

A

Desigr of RF and Microwwe Amplifien and Osciltators

terminations, it is necessary to derive expressions for the constraints imposed by the terminations on the normalized incident and reflected components. Consider port n of the l/-port terminated in an impedance Znnin series with a voltage source E , as shown in Figure L14.

Figur l.14

The N-port under consideration.

Theterrrination forcesthe following relationshipbetweentheterminalvoltageand current: E,1"=Yn+Znrl,

(r.73) I

By usingthis relationshipin conjunctionwith (1.57)and(l .59),it follows that

r !:

2,t-R*a" = Vn+ Zo,I, = E n, - (Z nn- Z or)I n

|r t:

leadingto

2,t{a,

- E nn= -(Z e, - Zo) I,

(r.74)

rd I

zr[{0, - vo- zoil, = En - (z no- zoi)t,

t: k

r*tich leadsto

zrt 4u, - E^, = -(Zen+ zr)1t^

(r.7s) I

Dividrng(1.7a)by (1.75)yields {-

25

Characterization and Analysis of Linear Circuits at ItF and Microwave Frequencies

2j{â^ - E nn - -l.Zn, - Zrnll n z,[Re,b^- EA, -l,Z,cn+ zo)t^ ntich leadsto

z^*- (z;;)

,l&*

o"=-Tb,*-YEnn Zn, * (z;") zAN+ (z;n)

( r .76)

$'ith

E,-=0 '' : sesond applies: termin (1.76)is equalto zero,andthefollowingrelationship

-

,,$;

(r.77)

=ctn16,=?o-9z n, + 1Zi,)

This expressionclearly has the form of a reflection parameterwith normalizing of to be the interconnection =danceZo,*.Theterminationcanthereforebe considered i ::e-portnetworkwith a port ofthe two-port. Thenormalizingimpedanceofthe one-port - ,-irtrethe conjugateof that at the corresponding port of thetwo-port.This is illustrated - : igure1.15. One would expect that the normalizedcomponentincident on the one-port (a1) i. .rldbe equalto thecomponentreflectedfrom thetwo-port(b) andthatthe component ':cted from the one-port(b.) shouldbe equalto that incidenton the two-port(ar), that

, bt and bL: az

-

Th" proof follows easily from the fact that the voltage acrossthe one-port is the * Dc:fs that at the correspondingport of the two-port U/L: Vr) andthatthe curents are td.n,t"ut except for a difference in sign (It: -Ir). It follows from (1.57) and (1.59) :.:l

Vr+ Zo, I,

t/L- (z;).IL

2 ,$,

z ,[F*

v2- z; 12 _ VL+ (Z;)IL

2 ,lF* r-

-

(1.78)

=bt

(r.7e)

=aL

2 ,[a* '

25


4,

z-'

Two-Port

Ftgurc 1.f5

One-Port

Cascading a one-portwith a two-portnetwork.

The componentincidenton the N-port (a,) is, therefore,reflectedfrom the oneport, andthe componentreflectedfrom theN-port (b") is incidenton the one-port. The normalizingimpedancefor the single-portis the conjugateof that for the Nport.

F h

t

1.5.5

Derivation of Expressions for the Gain Ratios and Reflection Parameters of a Two-Port

Considerthe two-port with terminationsas shown in Figure l.16 and the associatedSparameterexpression:

*]=[;;] [l]=[l

(1.80)

r F

Figure 1.16

The two-port under consideration.

-

characterization and Analysis of Linear circuits at RF and Microwave Frequcncies

27

In (1.80) a, is an independentvariable,the magnitudeand phase of which are rrcnninedby the sourcevoltageE andthe fixed normal-izing impedanceZo,. Accordingto (1.77),D,is constrained to z'' = _ "r, an /

(z;;)

2, a@rj

= a, / S,

(1.81)

with at the independentvariableand b, known in termsof ar, (r.g0) amounts to lro 6qrratisns with two unknownsandvaluesfot ar, byand6, canbedeterminedin terms . - :-rescatteringparameters anda1.Theresultsareasfollows: - = l

(1.82)

r' -, srrsrr'S, - Jtt

"

-

(1.83)

I - sr,s,

JzrSz (1.84)

l- srrs,

.: -ar/5,

(1.8s)

At this stage, the reflection paxametersand the gain ratios of interest can be rnined. The expressions mostfrequentlyusedarerepeatedbelow.

,

-, --=44-vr- zirI, -b,- "t' - . srzrzrsz z^+h- Vn4 i=a= "' *;;t

+, =T4=3 Zour+2, r =2,-(zii) Z,+(Zi,)

-

? Vr+2,I,

r+-bz=-zz s4"= s",4.s,zsrA "', ' a2 l_s,, ^S,

(1.86)

(1.87)

(1.88)

2t


_ =-+

'), '

- ls,l2l l"r,l2tt

-lrrr(t - szzS t)+ s,rsr,,s,12 fl srrS.l'

(1.8e)

ten s'h ten

Z:

-Vrf =Prl' P*_t

l'I

k'i det

t l'r,l'[t- ls,l']tr- ls"l'

(1.e0)

lI slrs,l[1-szzSr] s,rsr,,S"Srl' Grqr=o= Gr,u

-ls"lz-l-lsrl'br,l2 t-z't -

=, I

lt s,,s,l2

(1.er)

ll rrrsrl'

ufrere Gr, is the unilateraltransducerpower gain

G,^ =

Pnn

(r.e2)

Pn-t

_ =

- ls"l'l l"r,l'tr

1

(l.e3)

uficte P,,-o is the maximum availablepower at the output terminalsof the transistor A=,s,,sr-Sr2Jzr

(1.e4)

d

Q="r,-As;

(1.e5) -


,[6 a, + bz sr,[l+.S,] @ ar+b, 1Ro,l*srr - szzSL-s,,sr'S, .,!Ro., +s,rs,S,

29

(r.e6)

In orderfor (1.96)to apply,the normalizingimpedances mustbe purelyresistive. rn (1.86),s,,. isdefinedto be the input reflectionparameter with the two-port r:ninated in the actualload of interest(normalizingimpedanceon the input side:Zo,), " :le,s22u is defined in (1.87) as the outputreflectionpararneter with the two-port - rinated in the sourceimpedanceof interest(normalizingimpedanceon the output side, Similarly,sr,, is definedhereas s1 whentheoutputnormalizingimpedanceis the r":.ral loadimpedanceof interest(Zoz:Zr) and theinputnormalizingimpedanceis taken -e theconjugateofthe input impedance of thetwo-port(Zor:Zn\. It follows from this r::nition that t

2 n | =u.

(r.e7)

Similarly,srrois definedassl whentheinputnormalizingimpedanceis the actual :ce impedanceof interest(Zor: Z,) andthe outputnormalizingimpedanceis takento :neconjugateof the outputimpedanceof the two-pofi (Zoz= Zour,).Itfollows that &

.,;::..

(l.gg)

sl.is defined as s1 when the normalizing impedanceon the load side is the actual ird of interest(Zot: Zt) and that on the input side the actual sourceimpedanceof interest l&.= Z"). This implies that

4 , rl' = G ,

(r.ee)

Thesedefinitions are relevantduring circuit synthesis.

EXAMPLE 1.5

Derivationof the expressionfor thetransducerpowergain.

As an exampleof the applicationof (1.82)to (l .85), considerthe derivationof

(1.e0).

An expressionfor the powerdissipatedin the load follows directly from (1.84)and(1.85):

r, =ltrl2-lorl'

=ldol tr-;s,l'tl',1'

(1.100)

Design of RF and Microwave Amplifien and Oscillators

for Pu"-6, it is necessryto use(l .76). Application ln orderto derivean expression of (1.76)to port I yields

r, o,' = z" (z;i)u,*&Z,+Zot

-, .. t....-..*!

Z"*(Zor)'

fromwhich it follows that

Er=+(2,+zor)

(1.101)

{fior

Substitutionof (lI0l)

in the expressionfor the availablepowergain yields

l"'-$?ll' pn-E =E? tl4R,l=l'::t:'ls"a,l' "r"rl k, l-r ,

t:,,

ot12

4RorR"

(1.102)

t-F,lt

After substitutionof b1 in termsof c, (see(1.83) in this equation,it follows that

l

N32400AA SolutionsI 2 25:1:1999

l

l3:t0:33

F

0 slt + S2l a s22 o st2

t' R01: RM:

ftrrc t.t7

50.00 50.(x,

(50O normalization)ofa fansistor displayedon a polar plot (the constant The S-parameters resistanceandconstantreactancecirclesonly applyto s,, ands22;s,, andsrrwerenormalized as shown). The one set oftraces is usedfor the pafiImetersas supplied by the manufacturer "2" of the small(faces markedwith a ), while the other set is usedfor the S-parameters signal model fitted [2]. Note that the highest frequency point on each curve is not marked.

*.


p - - l[

3l

-s,,S"][ -szSr] -szrsrz,S",Sr,12

t av-E -

(r.l 03)

Combinationof (1.103)and(1.100)yieldsthedesiredexpression. The S-parameters(50O normalization) for a typical microwave transistor are 'playedin Figure 1. 17.Theperformance with differentterminationscanbe obtainedbv :ngthe equationsprovidedin this section.

5.6

Signal Flow Graphs

: 'gparameterequationsshownabovecanalsobe derivedby usingsignalflow graphs .{dditionalinsightinto thedifferentrelationships arealsogainedfrom thenow giaphs. The following rulesapplywhena signalflow graphis created: l.

Eachvariableis designated with a node(in the caseof thetwo-port,nodes will be usedfor e1,ct2,by br. andb.).

2.

A multiplier is associated with eachbranch.

3.

Branches emanatefrom independentvariable nodes and terminate on dependent variablenodes(dependence andindependence areestablished by the associatedequation).The directionof the flow is indicatedwith an arrow on eachbranch.Thebranchmultipliers areappliedto theindependent variablenodes.

4.

The value of eachdependentvariableis determinedby the multipliers and independentvariablesassociated with the branchesenterinsthe relevant node.

Theserulesareillustratedbelowby buildinga signalflowgraphforthe normalized and reflectedcomponentsof a two-port(Figurel.lg). Apart from representing the relationshipsofinterestgraphically,flow graphscan f,E ir- be usedto calculatethe valueof any of the dependent variablesin the graphin terms --eindependent variableofthe graph(0"in thiscase).This is doneby applyingMason's ' : :o the graph.The following terms arerequiredbeforethe rule can be formulated: c.rht

I.

A first-order loop product is defined asthe product ofthe branchmultipliers encounteredin ajourney starting from any specific node and moving back to the same node in the direction of the arrows. The first-order loop products in Figure 1.18are srr f, , s22lr, arrdsr, lr, s12f"

Desip of RF and Microwave Amplifiers and Oscillators

*--r-*_

b,l

bl

+r" L-

ol

-*l',

I

b2

, D

o

"

^-

U .

-

A

l

?2,

* t

|

"

(-

. I

I

IX

.tr

a2

T

*

I

qt

b"

|.

F

bz

szr

stt

bt

b,

at

bt

at

szz srz

a2

szr

b2

t, a2

bz

Ltrr; Flrnc l.lE

A flow gaph for the incident and reflected componentsof a two-port'

{


33

2.

Loops are nontouchingwhen they haveno nodesor branchesin common.

3.

loop productis theproductformedby combiningthe loop A second-order productsof any two non-touchingfirst-orderloops.

.1.

with any threenonA third-orderloop productis the productassociated touchingfirst-orderloops.

5.

An nth-order loop product is the product associatedwith any n nontouchingfirst-orderloops.

6.

A pathis any forwardroute(routein the directionofthe arrows)emanating from the independentvariable of the graph and terminating on the dependentvariableof interest.

\lason's rule canbe formulatedat this point:

p,[t - Ern]r", * E L'*r, - 1 + prll - Ez,i, pz+ ... 1-tZt*fI2-!13"...

(1.104)

dG

' -

! ' is the sum of all the rth orderloop product.,E Inlr.- is the sum of all the nth with the loopsnot touchingpathm, didP. is the productof the a productsassociated 'rrch termsalongthe pathz. Note that the denominatorof (1.104)is only a functionof the graphtopologyand 'r samefor all thedependent variables.It followsthatthistermwill be cancelledif the .fany ofthe dependent variablesis taken.

EXAMPLE 1.6

Calculationof a, in termsof b",andbr, br, andarin termsof cI1.

To demonstrateapplicationof (1.104), ar in Figure 1.18 will be calculatedas a functionof 6". The sum of all the first-orderloop productsis srr

s2lr+s, l" s, l.

loop (loop factor s1 s, f" fr). Thereis only one second-order The only loop that doesnot touchthe pathleadingto a, is the loop on the right-handsideofthe flow graph(loop factorsz lr). This leadsto -


[ 3 4

;

",, ."0

szr

r,

r"

",c.,

Jru

b " l

)"'

br"

Filrrc f.f9

o t =

b

J

F

The frst-order loops and the forward pathsrelevantto calculationofthe ratio DtlD,.

l- srrl, I [s,,1" * szzlr * sztl" s,rflJ * srrszzl"l,

( l.l0s)

In the previous section4r was takento be unity, which leadsto

br=

I - [s,,I" * szzlz * szr\ s,r[1 * Jrrrrrl" Iz l- srrT,

br, b,, ndc, cannow be derivedin termsof atby applyingMason'srule in each case.The results obtainedwill be the sameas those in the previous section. To illustratethis, considerthe derivationfor 6r: t '

b,

s,, (1- srrT) * szrlr s,, (l) I - [",rf" * szzlL * Jzrf" s,rflJ * ",r"zz\ lr

(1.107)

-

i


35

Note that there are no nontouchingloops associatedwith the secondpath termin the numerator of (1.105)(sr,f.s,r(l)). Substitutingb" in this equationproducesthe sameresultas(1.83).

;l

1.5.7

The Indelinite ^S-Matrix

'

:rlarto the indefiniteadmittancematrix.thesumof theelementsin eachrow or column .neindefiniteS-matrixis equalto a constant.In this casethe constantis unity. In orderto provethat the sumof the elementsin eachrow mustequal1, consider : . three-portshownin Figure 1.20. Under the conditions shown,all the incident componentsare equal,and = Sirar* Sizaz't Sildl

rlifies to

!

I

= [sr1+ si2r sit)at

S':xstitutionof 6, anda, in termsof the reflectedandincidentcurrentsyields /. =[s;1 *s;2 r sp)Iri

(b) - : re 1.20

t=

Circuits usedto prove that the sum of the elementsin (a) any row or (b) any column of an indefinite S-marix is eoualto l.

F F

'Design of RF and MicnowaveAmplifiers and Oscillators

36

and becausethe terminal currentsmust equalzerowhen all the sourcevoltagesareequal, 1rimustequald,. It follows that (1.l 08)

F

S;1+s7z*Sr3=l

l

The circuit in Figure 1.20(b)canbe usedto provethatthe sumof the elementsof the first at terminals columnofthe indefinitematrixis equalto 1.Becausetheincidentcomponents condition two andthreeareequalto zero,the necessary Ir+Ir*1r=Q simplifiesto Ir, -- Ir, + Iz, + 13, with Qz=0- clz b, = srra,+ snaz + sl3d3 br=s2rar+szz02+s'a3 br=srrar+s32az+\3a3 simplifiesto b, = sra, b, = s'rra, b, = srta, and,therefore, ( l . 11 0 a )

It" =.srr1r, Ir, = srrlr,

(l.l 10b)

Ir, = s3rl1,

( 1 . 1l 0 c )

Equation(l .109)combinedwith (1.110)yields *Srr=l Jll+.S21

{

(1.111t

By moving the voltage sourcein Figure L20(b) to the othertwo ports andfollow-

I

T -


15 lbe smre procedure,it can also be shownthat the sum of the elementsin eachof the frtwo columnsof the indefiniteS-matrixis equalto 1.

Extension of the Single-FrequencyS-Parameter Definitions to the Complex FrequencyPlane ' csary conditionfor a matrix to betheS-parametermatrix ofa linear,lumped,passive ' normalizedto Nminimum reactance *: functions(i.e.,impedancefirnctionswith no ii :r the real-frequencyaxis) is that noneof its elementsmay haveany polesin the -rght-handside(RHS) of the complex frequencyplane[3]. fhe definitionsgiven for a and 6 in Section1.5.1are adequatefor any single; .'rm] application,aswell as in the complexplanewhenthe normalizingimpedance rfrs (Z.ls)) do nothaveanyfinitepoles(i.e.,purelyresistivenormalizingimpedances, lr .rlrdances of the form Ro,+ sZo;).However,whentheseimpedancefunctionsaremore ,sti-- r\. it is necessary to extendthe definitionsof the normalizedincidentandreflected ,',F :ents.The following definitionsarerelevantto the moregeneralcase:

o l

0 Z*(t)

=f,.fn L;

o l 0 l

0

(l.l l2)

ZorG))

1.(s) is the normalizingimpedanceat port7,

ro,(s)

0

o l

0

rr(s)

0 l

o l _ 0

(1.113)

ror(s)J

- t(s)ft(-s)

( 1 . 1l 4 )

= 0.5lZo,@) + Zo,(s)l

-

j

(1.11s)

Design of RF and Microwde Amplifien nrd Oscillators

' r;

ln$)/n!) o

rt(s)= |

0 l'" o l

0 mr(s)/n (s)

L ;

l,^-A>l

mr(s

wherern,(s) andn, (s) arepolynomialsandthe zerosof 4 (s) (polesof [ (s)) areconstrained to the openleft-handplane(LHP) andthe zerosof m,(s) (zerosof h, (s))areconstrained to the closed right-hand plane (RHP).

a(s) = ft(-s)I,(s)

(1.117)

D(s)= ft(s)I,(s)

( 1 . 1I 8 )

Wherea(s)is the matrix of normalizedincidentcomponents,D(s)the normalizedreflected components,andwith I, andI, asdefinedin SectionI .5.I . Note thatthe elementsof r(s) areevenfunctions(i.e., rs,(s): ro,(-s)) andarethe partsof the corresponding effectiveseriesresistance normalizingimpedances. With thesedefinitionsfor thenormalizedandreflectedcomponents, it followsthat . _ ,_\ - Vt(s\+ Zo,@)I1G) r' 2h,(s)

s . t r ,

I

(s)= D. r' '

v,(s)- zo,Fs)I iG)

Sr(s) =

2h,(-s)

(1.lle)

(1.120)

h,(s) Zi,1(s)- Zot(s)

h,(-s) Zin,G)+ zo,@) (s)

/. s4(s)= -2h,(s)h*(")fr

(r.r22)

"0k

Tbese relationshipsare identical to those derived previously for single-frequency applicationsas long as

=,r@, = h,(-s) h1G)

(1.123)

This relationship will apply in all caseswhere the normalizing impedancesare purely resistiveor of the form Ro,+ szor. {

Characterization and Analysis of Linear Circuis at RF and Microwave Frequencies

39

Independentof the complexity of the normalizing impedances,the incident and respectively. :flccted powerarestill given by la)2 and l6112,

Calculatingft(s) for atwo-port.

EXAMPLE 1.7

As an example,ft(s) will be calculatedfor the normalizing impedancesshown in Fisure1.21. lo -v-i Eor ')

I l

o

s

L"-r lo

Es2

(a)

Es2

(c) ;;rrc

1.21 ,

(a) The normalizing impedancesunder consideration;(b) the equivalentcircuit usedto determines,,(s) and str(s);(c) the equivalentcircuit usedto determinesrr(s)andstr(s).

Because Z o r G )= l + s / [ l + s ] it follows that ror(s)= 0.5[Zot(s)+ Zot(s)] l-2s2 =1-s2 : :

t-J-zst+Jis l+s

l-s

4tf


r, .

h.?.

and, therefore;

::i

4 (s) = (1- J-2s)/ (1+s)

:

Similarly, '' hG\=s/(l+s) 15.9

Constraints on the Scattering Matrix of a Lossless N-Port

Tb averagepowerenteringa passivelosslessdevicemustbe equalto zero.This imposes thc following constraintson the scatteringmatrix: 0= P,* =051V''(ir,l) I(ia) + r''ffo) V(ito)l

=ta''(jo) c(j f;;;l 'l

f-

I;@l =lr,f,>,; v^@ I e B)

(2.2e)

lv)1t1r,1tS I"(t)I;(t)l

Crrrent Representation

(zB) r:2(t)t/ , , =l';)'r','rlrr;,r,r (2.30)

-17âmI,IQ\ I;zQ) -lnGV"J,)

f,r,ut

I,r(t) IirQ)

\

:ageRepresentation

,=W/(z' d

(2.3r\

=lr,,rfraM1t(za ) lr)U>v,r(t) v,r(t)v)r(t)

!"'

By usingdefinitions(2.4),Q.l6), and(2.17)it follows that

55


Co = 2kT R*

Y;,I Gri I R*l

f[L,'

Tbc sccondtern of Q.32) is derivedas follows: I,(t)=IûQ)+Y*,V,(t) impliesthat

+i

T

I,(t) v: dt = l

f I,(t) I/,'(t) * Y"o,v,(t) dt T J

0

0

= o * I Y"o' T

T

I0

v,(t) v^. (t) dt

= 4kTBRn Y"*

(2.33\

It is a simplematterto show that(2.32)is alsoequivalentto

t

l Co = 2 kT R*

I

r

-l 1F.," - r"-o" lL 2z Rv

Fîn-r- y .l -'-"ot

2 R*

II'"-"0, lt

I

(2.3-l

l

It is possibleto transformany of the correlationmatricesdefinedaboveto anr . . tbc otber types. The transformation matrices required for this purpose are summarized

Table 2.I [4]. In this table Y,Z, andT arethe I-parameter,Z-parameter,and parametermatrices,respectively,of thenetworkunderconsideration. The rcquiredis doneby usingthe equation C* x,bc

= XCoriX'' ''

indicatesthe transposed conjugateofX. The equationssummarizedin Table 2.1 can be derived easily by using : rclaionship betweenthe noise voltagesand currentsin the different representatio: Bccauseofthe principleof superposition, the equivalence canbe derivedby assumingi;r ? noisegeneratorsto be the only excitationspresent.

t:

-t

Characterization and Analysis of Active Circuits at RF and Microwave Frequencies

EXAMPLE 2.1

57

Derivationof expressionsfor the equivalentnoisesources 16@ and 12,ft)interms of V,(t) andI"(t).

Considerderiving expressionsfor the equivalentnoise sources/t"(t) and lr,(t) (current representation)in terms of V,(t) and/,@ in the cascaderepresentation. 1,(r) is clearly part of lr,(t) and,therefore,only the equivalentcurrent sourcesfor V,(t) arerequired. Becauseofsuperpositionand becausethereis no theloadcanbeshortedand representation, noisesourceontheouput in thecascade using the l-parameters: the currentsresultingfrom 4(r) canbe calculatedby I, = -yu V,(t)

Iz = -lx Y,(t)

Adding 1,(/)yields Ir^(t): -y, V,(t)+ I,(t) Ir^(t): 'yz, V,(t) leadingto

t] =[ 2 " ol [,'xnl [",n1 L/r"(r)ll-r,, [1,(t)]

(2.36)

Equations(2.37) through (2.39) are generallyused to calculate the equivalent crrelation matrices of two networks connectedin cascade,parallel, and series, :ryectively. The relevantequations[4] are ''- =Cot+T Co2Tt' ': = CyiCyz

C. =Crr+C",

(2.37) (2.38) (2.3e)

T in(2.37) is thetransmissionmatrix of the networkclosestto the generator(i.e., conjugateof networkon the input side).The superscriptusedindicatesthe transposed 'rnsmissionmatrix. for a networkarecalculated,it is usefulto know that Whenthe noiseparameters

2kT*(4

(2.40)

58


cr=2krfi(n

Q.4r)

for any passivenetwork[4].

Table 2.1 The matrix (X) required to tansform any of the noise conelation matrices to another (Cn* : X C.aX,.)

F

T

(::, I

(;l)

Y

(;I t;A

z

z

')

T

2.23

z

Y

\rieinal Neo, \

(:

( t -",,)

fo -,,J

(;I

Calculating the Noise Figure of a Cascade Network

The noise figure of a cascadenetwork (seeFigure 2.5) is often of interest.Given the definitionof the noisefigure in termsof the availablenoisepowerat the input sideof the network,it is a simplematterto provethat

E=R*Fr-l* Gor

F -l

*...

GorGoz

' 2 "

E

F'(Z)

Fr(Zû)

F,(Z^q*tt)

G^(Z)

G*(Zû)

1-(266s)

ZL

L

F

tr|grre 2.5

The circuit usedto calculate the noise figure of a cascadenetwork.

-

characterization and Analysis of Active circuits at RF and Microwave Frequencies

59

*trere Fr is the noise figure of the first stage(input stage)andGo,is its availablepower --:rn.Similarly,F, is thenoisefigureof thenth stagewhenterminatedon its input sidewith -.coutputimpedanceof the previousstage,andG*is its availablepowergain. EquationQ.a\ is known asFriiss' formula. It is clearfrom Friiss' formula thattheproductof the gainofthe stagespreceding my givenstagemustbe high in orderfor it to havea negligiblecontributionto the overall ,-.isefigure of the cascade. It is alsoclearthatanystageaddedwill havea degradingeffecton thenoisefigure. -1r contributionof anystageto theoverallnoisefrgureis a functionofboth its noisefigure -rd its availablepowergain.The noisemeasure(Al) of a networkis a figure of merit for .:rseffect and is defined as

(2.43) ,bcre F_ is the noisefigure of an infinite chainof identicalstageseachwith noisefigure ' andavailablepower gainG". By usingthe identity

' - x=l+X*.X2*...

Q.44)

r canbe shownthat the noisemeasure,M,is givenby F - l | - t/G,

(2.4s)

"nereF is the noisefigure of the stageof interestandG" is its availablepowergain. Theassociated noisefigureis ofgreaterinterestandis givenby substituting(2.43) lrro (2.45): _F-l/G" | - llc"

EXAMPLE 2.2

(2.46)

Calculationofthe effectofthe lossesof a passivecascadeon the noisefisure of a transistor.

The effect of the insertion lossof a lossypassivenetworkon the noisefigure of an activestagewill be calculatedby usingFriiss' formula. The noisefigure of a passivenetworkis givenby


Fe""(zJ = Frct(Zouts,t)

llG,#(2")

Go-ô(zou4n)

Go-pu(Z")

[-

Figure 2.6

ZL

The effect of insertionlosson the noisefigure of an amplifier stage.

Fe^(n

E!= PnolPno-r*= -krB G,_p*(f)

= llc"-e*(

-f)

(2.47)

that is, if the passbandis narrow enoughfor the availablepower gain and the mismatchfrom the outputof the networkto its loadto be consideredconstant. Enteringthis into Friiss' formulafor the cascadecombination(seeFigure yields 2.6) fET = = Ff p u *" F " o - 1 -=1 I t // (7t a - p a s. ' F o o - l qr^ O"**

- l + F * t - l Go-r* = F^"rl Go-pu

Expressedin decibels, (2.48) becomes Fr = F*, - Go-r*

(dB)

(2.4e)

It follows from (2.49) that the noise figure of any stageis degraded proportionatelywith any lossesdirectly precedingit (G"-0",in (2.49) will be negativefor any passivenetwork).This is illustratedin Figure2.6.

2.3

THE OUTPUT POWER OF'LINEAR AMPLIFIERS

point) Themaximumoutputpowerobtainablefrom a linearamplifier(l-dB compression

a

Characterization and Analysis of Activc Circuits at RF and Microwave Frequencies

61

will be consideredin this section. The transistorsused in a linear amplifier are usually biasedin class A (360' ' conductionangle),classB ( I 80 conductionangle),or classAB mode.ClassAB is often nsedat microwavefrequenciesinsteadof classB, mostlybecausethe gain obtainablein Thevoltageandcurrentwaveforms classB modeis usuallytoo low at thesefrequencies. in Section2.3.1. with classA and B stageswill beconsidered andtheloadlinesassociated point areusually intercept pointandthethird-ordertwo-tone The 1-dBcompression ofthe linearityof anamplifier.Therelevantdefinitionsandthedefinition usedasmeasures rhedynamicrangeof an amplifierwill be consideredin Section2.3.2. point andthethird-orderinterceptpoint of anamplifierwill The I -dB compression -.' reducedby anydriverstagesadded.This effectwill alsobeconsidered in Section2.3.2. The maximumoutputpowerobtainablefrom a classA amplifiercanbe estimated by usingtheapproachintroducedby Cripps[l]. RF,aswell asatmicrowavefrequencies, in Section2.3.3. will be considered e Crippsapproach The Cripps approachcanbe generalizedandmanyofthe inherentinaccuraciescan h removedby using the powerparameterapproachintroducedin [2]. This approachis -lined in Section2.3.4. The Cripps approach and the power parameterapproach are based on the .sgmptionthat the maximum power obtainablefrom a linear amplifier is determinedby "c powerlevel atwhich the intrinsic outputcurrentand/orvoltageof thetransistor(s)used -rts to clip; that is, the power is limited mainly by the limited swing in the intrinsic :tput current andvoltage. Thepowerparu.*t", upptoachis suffrcientlygeneralto handleanyloadingeffects, :cdback,"h*g", in the transistorconfiguration,cascadenetworks,and/ormultistage _:rplifiers.All of theseaspectswill alsobe consideredin section2'3.4. tone The power p**"1", approachcanalsobe usedto initialize the fundamental ir 1' amplifier' the of simulation nonlinear flrantitiesin a full Larmonicbalance

:3.1

Load-Lineconsiderationsin class A and classB Amplifiers

Wtrena transistoris biasedfor classA operation,the averagevoltageacrossits ouQut mustbe equalto the dc voltage (Vosor Vs6; rminals (drain-sourceor collector-emitter) andtheaveragecurrentmustbeequal power is important), ;ually thesupplyvoltage,K, if ' I rhedc current(Io, or 16) (thedc currentmay changeasthe drive level is increased)'If jr distortion in the *uu"rot-. is negligible, the voltage and current will swing ;rmmetrically aroundthe averagevalues. in practiceby the The maximumpossiblevoltageswing(I/r5 ot v66)is decreased (R,)' The effectof resistance guration voltageof the transistor(4J andany saturation at thetransistor presented ..resaturationresistance canbe lumpedwith the loadresistance ':rminals. at RF The maximum outputpower obtainablefrom a classA or a classB amplifier is givenby [6] G'equencies

62

P.*

Desigp of RF and Microwave Amplifiers and Oscillators

(V, - V,),

RL

(2.s0)

2(R,+ oRrJ R,+cRru,

where Z"is the supply voltage(assumingthat no drain or collectorresistoris used)andR, presented to the outputterminalsof thetransistor.a is equalto 2 is the parallelresistance for classA amplifiersandequalto 1 for classB amplifiers.It follows from this equation by the saturationvoltageandthatthe effective that theeffective supplyvoltageis decreased intrinsic load resistanceis increasedby the saturationresistance. presentat the output In deriving (2.50), it was assumedthat any susceptance terminals of the transistorwas removed.

2.3.1.1

Class A Load Line

The output current and voltage and the associatedload line in a classA stagewill be considered next. In general, if Vr, (t) :

(2.51)

lVr,l ei^

and Yt_i* = -1.r,/V2,: llr_rol eF

(2.s2)

the drain voltage and current (dc and ac components) are given by Io(t) :

IDs - YL,in V2i(t)

(2.s,?

Vo(t):

VDS+ V2iQ)

(2.s4

With I/2,(r) replaced in terms of (2.51), it follows that Ir (t) :

Ior - lVr) ei-

Vo@ :

V o s +l V r , l e 4

l lr_i"nl ei9: Ios lYr_innvr, 1 si@*o)

(2.56

It follows from the last two equations that the dynamic load line is defined by Iz,O: Vr,(t) :

IoG)- Irx:Vo$) - Vos :

/, 5()

lllll'

l Y r _ i n n V zl s, i ( ' ' * o )

(2.s-

lVr, lei*

(2.58

I -

63

Characterization and Analysis of Active Circuis at RF and Microwave Frequencies

Va" *rr-:

:t

?t=

The dynamib load line ofa transistorbiasedfor classA operation(reactiveload line).

:.-

the load is reactive,the loadline will be similarto that shownin Figure2.7. The dc powerdissipatedin a classA stageis constantandis givenby Q.59)

vÎ^ The powerdissipatedin the inhinsic loadis givenby

ffi.';*" ,ri*,r 1ilil

#

1-o-o

1I

l/Rb,*

Ir"

Vo" -

tj.G

}

2.S

vr*

Clipping in a classA amplifier can occuron any of the four line segmentsshown(resistive load lines shown).

64

Design of RF and Microwave Amplifien and Oscillaton

Pu = lv2i 12GL_inn I2

(2.60)

< Pel2 or by P' -- Vr, 12RLi,oI 2

(2.61)

< Pel2 If the voltageis clippedfirst, the maximumoutputpowerwill be givenby (2.60). If thecurrentis clippedfirst, (2.61)will apply.In general,clippingcanoccuron anyof the four line segmentsshownin Figure2.8 (resistiveloadlines shown).

2.3.1.2 ClassB Load Line The conductionanglein a classB amplifier is 180' . A parallel-tunedcircuit or a push-pull theharmonicsin thevoltagewaveform.Whenthi: configurationis usuallyusedto suppress assumed to be sinusoidaland can thereforebe is done, the output voltage can be represented by usingthe sameequationsasin the classA case. The intrinsictransistorcurrent(/r(/)) is a half-sinusoid.Thepeakamplitudeof the theacpower)canbeobtainedfrom theFourierserie. frrndamental tone(whichdetermines expansionfor the half-sinusoid(referto Figure2.9). The Fourierseriesexpansionofa half-sinusoidis givenby hG):(Ir**ln)

F F

Figure 2.9

cos4rot+...] cos2ro/- (2115) [ + (n 12)cosrot+(213)

(2.62')

The relationship between the actual (intrinsic) output current and its fundamentaltonc component.

{


65

Note that the half-sinusoidoutput curr€nt and its fundamentaltone are in phase. This implies that if the amplitudeof the fundamentaltone output current is 16no(/) at any givenmomentin time,thenthe amplitudeof theactualoutputcurrentis 2 {un6(t). This can beusedto translatetheleft-sideboundaryandtheupperboundaryfor the transistorcurrent onthellV-planeto equivalentboundarylines for the fundamentaltonecomponent[2]. It follows from(2.62) that the peakamplitudeof the fundamentaltoneis equalto half of the peakamplitudeof the half-sinusoid(1, *J: I2,l : 17*1"12

(2.63)

The averagevalue (dc component)of the transistorcurrent(1r (r)) is given by (2.64)

In= I, n*l n " 'ollows that the dc dissipation in the transistor is given by P*:

(2.6s)

V a "I r * / n

.le the output power is given by P :

lVr,l'Gr-rnnl2:

l I 2l,Y r , i n n l ' G L - i i1l ,2

= l/rp"* I (2 YL_i")12GL i"nI 2 = llr_p"rr/ Yr_,nnl' Gr,_inI I

(2.66)

:

(2.67)

l1r-p*t 12'Rr-iot/ I

fb efficiency is calculatedas the ratio of the output power (P,) or the effective output --'.r'er(P, - Pj to the dc power(P6"): ' = P. /P6

(2.68)

'

(2.6e)

(P,- Pin)/ Pd" 1.68) is used, the effrciency is given by - (Zp*r-n'ol V*) @I\

(2.70)

Defficiency(q) ofaclassBamplifierincreaseslinearly with increasingoutputvoltage ! to a maximumof 78.5%. If theintrinsicloadterminationis reactive,theefficiencywill ' .ower. Whenthe outputpoweris lower thanthe maximumpossible,the efficiencyof a & B stagewill be observedto vary with the angularposition aroundthe constantoutput lb

I Design of RF and Microwave Amplifiers and Oscillators

t 6 6

power contours. The efficiency of a classA amplifier is constaniarounda constantoutput powercontour. The dynamicload line for a classB amplifieris shownin Figure2.10.Whenthe effective load line is purely resistive,the output current of the transistor and the voltage acrossit are constrainedas shown in Figure 2.10(a).When the effective load line is asshownin Figure2.10(b).Note that the reactive,the currentandvoltageareconstrained currentis zeroduringhalfofthe cycle. Tlte l|V-constraints of a class B stage apply to the total current through the transistor(half sinusoid)and the voltage acrossthe transistor.The constraintson the fundamentaltone quantities are, however, of greaterinterest. Becausethe voltage ofthe fixedrelationshipbetween waveformwasassumedtobeapuresinusoidandbecause the total current and its fundamental tone (see Figure 2.9), the constraints on the fundamentaltonequantitiescanbe takento be asillustratedin Figure2.11. Note that the new origin (V^' , Io,')shouldbe moveddown far enoughto allow the firndamentaltone currentto swing symmetricallywithout clipping whenthe instantaneousvoltageis higher thanV&. Underthe transformationillustratedin Figure2.11, aclassB stagecanbe treated asa classA stagewhenits outputpoweris calculated.This canalsobe donewhena setof load-pullcontoursis generatedfor the transistor. Thedc l|V-contraintsfor a powertransistorareoften suppliedby the manufacturer Theseconstraintscan be takento be the RF constraintsof the intrinsic devicetoo, if the currentsourceand currentis interpretedasthe sumofthe currentofthe voltage-controlled circuit. the intrinsic output resistancein the equivalent

1-o-o

Io"

Va,*-o

t t

I/d"

il

*

(a)

o) Figure 2.10

The dynamicload line ofa transistorbiasedfor classB operation:(a) resistiveload line an: (b) reactiveload line.

I ? ' r * {

Characterization and Analysis of Activo Circuits at RF and Microwave Frequencies

67

2Y 2X

ltool

.o

&

X

lfirndmoal--u

Id"'

(Y*,1*\ ),0)

Vd"

V^' tlure

2.11

23.2

vt*-o

lllustration ofthe conversionofthe 1/Zconstraintson the total output current and the output voltageof a classB amplifier to thoseapplyingto the fundamentaltone quantities.

Distortion in Linear Amplifiers

The l-dB compressionpoint (single tone) andthe third-orderinterceptpoint for two-tone poducts areusuallyusedasmeasures of the linearityof an amplifier. point is definedasthe level (usuallyexpressed The l-dB compression in termsof tb ouput power) at which the operatingpower gain (G,) is I dB down from its smallrisnal level. The third-ordertwo-toneinterceptpoint (TOI) is definedasthe powerlevel .. .rhicheachextrapolated third orderproduct(2f, - f,and2f, -f components) is equal - magnitudeto the extrapolatedfundamentaltonecomponent. At low signallevelsthe slopeofthe fundamentaltonecomponent(P"* in decibels -susPinin decibels) is I : 1, andthat for the third orderproductsis 3: 1. The definitionsareillustratedin Figure2.12. The third-order interceptpoint of a linear amplifier is usually about l0 dB higher point [7]. the l-dB compression h Thedynamicrangeof an amplifieris usuallydefinedasthedifferencebetweenthe : B compressionlevel andthat ofthe minimum detectablesignal,referencedto theouput -

Desip of RF and Microwave Amplifien md Oscillators

Pout (dBm)

MDS.*=MDS;'+G1

MDSin= l?4dBm+ 60dB+ 3dB + NF (dB)

Pin

(dBm)

Pout (dBm)

Pin

(dBm)

t t l Io Figure 2.12

The dynamic range (DR) and the spurious free dynamic range (DR;) of an amplifier.

ofthe amplifier[5]: DR=Pras-MDSo,n

The minimum detectablesignalcould be definedas3 dB abovethe noisefloor of the amplifier,that is, = kT B+ F +Gr+3 (dB) MDSout

lfril

(2.7r)

(2'72\

,

{

whereF is the noisefigure of the amplifierandG7is its transducerpowergain.

I

{

characterization and Analysis of Active circuits at RF and Micttrvave Frequencies

69

The spuriousfree dynamicrange@R) is often also of interest.The definition is illustratedin the lowerpanelof Figure2.72.

2.3.2.1 The Third-Order Intercept Point of a Cascade Gaincompressionandany additionalfrequencycomponentsgeneratedarethe resultofthe ,,i'eak)nonlineartransferfunctionof the amplifier [7]. At a givenbiaspoint (V,, Vo),the utputsignal(v,,= 6Vu; v, = 6V) canbe calculatedby usingTaylor's theorem: =

'

av o

AV, ' '

v -t

*

&vo v? dv v?l t + o t 2 AzV. t

dV l

6

"'

(2.73\

This canbe simplifiedto

r, = otri + azv? * atvl + ...

(2.74)

The coefficientsin (2.74)areusuallytakento be real,but they could be complex general.Ifthe coefficientsarereal,any distortionproductsgenerated will havea fixed :traserelationshipwith the input signal. If

':l

=acos(|)t

(2.7s)

. :hstitutedin (2.74\,it canbe shownthat: l.

Oddorderharmoniccomponents (31 5f, ...) aregenerated by the oddorder terms.In addition,eachodd orderterm will alsogeneratea componentat the fundamentalfrequency(/). Thesefrrndamental tone componentsare responsiblefor the gaincompression observedin amplifiers.

2.

Evenordercomponents will generateevenorderharmonics(2f, af, ,,.).ln addition,eachevenorderterm will also generatea dc component.These componentscausethe shift in bias point observedwhen an amplifier is driven strongly.

P ^

u@

The distortioncreatedby thethird-orderterm in(2.75) is usuallyof mostinterest: ar(acosto/)3 = a:a3cos3r,lt = ara3cosco/0.5(l + cos2or) = a3a3 0.5cosc,rt+ 0.5(cosr,ltcos2o/) = a3a3O.5cosz i v )

.

(2.103)

1

"!1..*(N-MSry t ' Z i

J ' Z i

'-

R

requiredexpressionfor I'

follows afterrearrangingthis equation:

$ r , - f r -f f w - f f i t - x

Q.r04\

'tcreXis givenby = l M R n R O

(2.105)

r - "lion (2.104)canbe usedto find the intrinsicload associated with any externalload, L-

;

Desigr of RF and Microwave Amplifien and Oscillators

80

atwhichpointthemaximumoutputpowercanbecalculatedbyfindingthepowerlevelat which (hard) cliPPingwill occur' of I''' With '4' known' the Equation(Z.f Oj) can alsobe used* f-*-dl" in terms extemalload (f) follows directlyfrom (2'100)' genelatecontorrrs similar to the cnpps approach,ttreseequationscanbe usedto is of interestwhen latter of constantoutputpower or "onrt*t "ff"ctive outputpower. The an oscillatoris designed. AnimportantdifferencebetweentheCrippsapproachandthepowerparameter approachisthattheassumptionthatthegytnutpowerwillbeamaximumwhenthe while no such intrinsic power generatedis amaximum is inherentto the cripps approach, may lead to *r,r-pti,on is madewhen the power parametersare used.This assumption for the lossless "rrors if th" output circuit is loadedwith the optimum power termination generatedwill still be a case.If the externalload impedanceis a short-circuit, the power maximum but no power will be deliveredto the load' FET) by Thepowercontoursgeneratedfor a transistor(TexasInstrumentsFoundry line load optimum The 2'19' Figure in shown [2] are using the power p*u-"t"i, correspond closely GHz l0 at predicted contours (maiimurnpo*"rj i, also shown.The (the location and with the measuredload-pull contoursprovided by the manufacturer rounder).The Sare contours measured orientation of the contoursarethe same,but the with the compared are parameters parametersof the model used to calculatethe power in Figure2.18. measuredpararneters (referencedto Note that srr, in nig*" 2.19 is the input reflectioncoefficient srr,'istheconjugateoftheoutput %r:50O)associatedwithth-optimumpowerload(S;); matched(if possible)' is conjugately side input the when reilectioncoefficient

FS14120C I2 Solulions 26:'l:1999 l6:8:30

o slt + s21 a s22 o s12

50.00 50.00

Figure 2.18

Comparisonofthemeasured.s.parametersofthe-transistorusedinFigure2.l9withth: parametersassociatedwith the small-signalmodel'

I

Characterization and Anolysis of Active Circuits at RF and Microwave Frequencies

Rol: R02:

50.00 50.m

O r.mcHr +

o.smcHz

A ro.soocHz

Grrre2.19

o

tLmrek

E

25.08028,98027.880dh

+

25.s1026.01027.010d8m

A

25.s0 26.s0 27.o6odBm

O

2o.o2o2z.o2o28.uodgm

Rot: R02:

50.00 50.00

The Ioad-pull contours (-l dB; -2 dB) and the optimum load termination (Sr) fora transistoras predictedby using the power parameters[2].

g2

''


If ,Szands22,' were on top of eachother, the optimum power andoptimum output match(VSW\* = l) pointswould havebeenthe same' Voltage-shuntfeedbackcanbe addedto this transistorto improvethe ouput match with maximumpowerwithout losingtoo muchpower(aroundI dBm). associated

2.3.4.2

Modifrcation of the Power Parameters of a Two-Port by Adding a CascadeNetwork on Its OutPut Side

gasgade on theoutputsideof an activetwoMten a passive netw1rk(two-port) is addedin aremodified.The derivationfor the port, €rs.t o*o in Figure2.20, ilspower parameters new parametersis shownbelow. voltages The intrinsic voltagesof the original network aremappedto the external by Vr=MVt,+NY,

(2.106)

Vz=OVr,+PYr,

(2.r07)

is Z3instead of z, is alsothe input voltage of the combination,but the new outputvoltage I/3' of a function to find V2as V2. ltistherefore necessary terms of the Theinputcurrentandvoltageofthecascadenetworkaregivenin output quantitiesbY

l;,;lfl Y;l

(2.108)

that is,

+

JJ

Y2

I

Y :'

Cascade network

M N o P

Figure2.20Addingacascadenetworktotherightofanactivenetwork'

{


:':'

Vz = Az V, + B, I,

E3 (2.10e)

Iz=CzVr+DrI,

(2.1 l0)

Eliminating 13from the last two equationsgives t I" V z= A zV z* B r l ; \"2

C"r

;lr,

B^ B. C^ = A ^ V . +" I ^ - ' " V z

' '

)

D

"

D

J

,Jingto B^' C^ ' \ V . + -B^ -'1^ ' D 2 ' D r '

=U"-

(2.111)

|i}M*

'

;an be eliminatedfrom this equationin termsof V, andZ2by using the l-parametersof irr original network:

(2.r12)

:=lztVr+YrrV, ading

to

f - u2t - t?'rrr, . ! rru y, - r, v2) Dr. u2 ing this equationyields

B^ B^ =Vr**yrrVr*i!"rV, -

D2

Dr-"

z

=BN v,+ ( r+ ? r u r r ,2 D2 fitz,

"

B.

ir,' " 2

= .4- 2

-

B. C,

B" t**yn v' l

12D2 - B2C2

"

D2

vr

2

B, c,

A,^ -D

" D2

lzr Bz

t*

*

v''

,

Dr+YuB, A2D2 - Bz C2

v2

(2.r13)

.:

84


Aftersetting ctl

-

lzrBz AzD2-B2C?

(2.rr4)

and g- =

"

D"+ 'v."8" " ' '

(2.1 rs)

V-

ArDr- BrC,

'

it follows that Vt=drVr+arV, = a1(Ml/ti * NTz) + e2(Ovti + PV2) = (atM + d,z0)Yr,+ (arN + urP)Vr,

/

(2.1l6)

The new power parametersof the trvo-port are therefore given in terms of the original powerparameters by (2.106): Vt=MVr,+NV, and (2.111)

V, = (arM + azo)Vr, + (urN + arP)V,

I

{

2.3.4.3

Modificationof the PowerParametersof a Two-Portby Adding a CascadeNetwork on Its Input Side

When a cascadenetwork is addedto the input sideof a two-port, the input voltage for th. ari combinationis different from that of the originalnetwork,andthe powerparameters thereforealsochanged.The effectof the cascadenetworkis derivedbelow.

Cascade Network

+

vl At Bt Cr Dt

{

Y; X,,

I

a t

Characterization and Analysis of Active Circuits at RF and Microwave Frequcncies

85

The new input voltage and current (% and.I) are given in terms of the previous input voltage and current:

ln,]=ln,u,llnl

l'.11.,",1lrl

( 2.1r 8)

Therefore, Vo=ArVr+8,I,

1,in this equationcan be replacedin terms of Zt and Z, in this equationby using the I)arametersof the original two-port:

(2.rre)

Ir=lnVr*lnVz Therefore, l'o = ArVr * B jrrV, * lrrV')

Q.r20)

= (A + B yn)Vr * B lnVz q':.h ::, z

= A * B l u

(2.r2r)

= Blp

(2.r22)

d 3

Ilowsthat '

= d, v, + urrv, = ar, (M Vt, * N Vr,) + ar, (O Vr, + P Vr) = (o, M * azr O) Vr, + (a, N * o, P) Vz,

Fe modifiedpowerparumeters are,therefore,givenby (2.123)and(2.107): .

= (arrM

,*

arro)Vr, + (urrN + arrP)V,

= OVr, * PV, b

(2.r23)

+

,? {

+ Yzt

t

I T x

The new input and output voltagesare given by (2.12(

I'

I-

vr=vro+v*

f

I

'i i

' 1-'i':

y, = v, * vr.o

(2'121

f

[

rJ "

F

.

o

.

'

{

'

t

l

F

l

}

l


vro = ztttlt * zrztlz

(2.r28)

vro = zzttlt * zzztlz

(2.rze)

and It=ltrVro*lrrVro

(2.r30)

Iz=lztVro*!rr.V*

(2.r3r)

it follows that Vrb = zttb jrrVro -- (zrn ln

* lnVz)

* znt OztVro * yrrVr)

* znt !zr) Yv + (ztt ln

:b= zzlb 9rt

Vro + lp

= (zzrblt

* zzzr !z)

* znr lzz) Vu

Yvr\ * zzzt (!zt Vro * !r, Vn + (zzn ln

(2.r32)

Vro)

+ zzzt !zz) Vzo

(2.133)

. .., = zltb ln

* znt lzt

Q.r34)

2 y = Zttb ln

* znt lzz

(2.13s)

= Z2tbln

* zzzt lZt

(2.136)

.:L = Zzrbln

* zzzolzz

(2.r37)

;,.,

?.132)and(2.133)reduceto )'.t=&ttrVto*&lNVzo

(2.r38)

lzt=&ztrVlo*dzxVzo

(2.r3e)

:]th Zra and Vroknown in terms of the original power parameters,the modified power -,lrameters canbe calculated:


vl = Vlo = Vro * &tl" Vto * &tx Vzo = ( 1 * 4'")

Vro * orr, Vro

= ( 1 * drrr) (M Tt, + N Vzi) * urr, (O Vr, + P V2,) = [(l + crr") M * &n, Of Vr, + [(1 + dll") N * drz" P]

V,,

(2.r40)

= vzo * vzt = V2o * &21, Vto * dzx Vzo

= [dzr"M + (l*urr") O) Vr, + farr, N * (l +azz) Pf Vzi

(2.r41)

2.3.4.5 The Effect of Changing the Configuration on the Power Parameters As was the casewith the two-port parameters,the power parameterschangewhen the below. is established con{igurationis changed.The changein the parameters to Common-GateCase Common-Source configuration(seeFigure2.23) aregiven for the common-source If thepowerparameters by Yrr=MrVr,+N"V,

(2.r42)

Vx=O"Vr,+PrV,

Q.r43)

the parameters for the common-gate configuration can be calculated from the voltage

relationships:

vrr = -vr"

(2.144t

vujv^-v"

(2.r45'

vrr=v^+vr,

(2.146\

f

t {


89

s.ure2.23 The effect of changing the configuration from common-source to common-gate on the power parameters.

re first two parametersfollow easily from (2.144) and (2.142): ,=MrVr,+N"Vr, 'ecomes

-Vrr=MrVr,+NrVr,

(2.r47)

-hich implies ,=-M"Vri-N"Vzi

(2.148)

substitutingthis result in(2.146),it follows that Vrr=Vr.r=O"Vr,*PrVr,

OrVr,*P"Vr,+Vr, = O, Vt, * P, Vr, * f-MrVri - NrVr) :irerefore,that = (O, - M)Vr, + (P - N)Vr, . (O, * Mc)\i

+ (P" + Ns)V2i

(2.r49)

tlr:owerparameters forthe common-gate (2-la$arird(2.14\. configurationaregivenby b,

Desigr of RF andMicrowaveAmplifiersandOscillators

90

Common-Gateto Common-Drain Case

'rr 5: *1, Flgure2.24Theeffectofchangingtheconfigurationfromcommon-gatetocommon-drainonthepower pafiImeters.

Thecommon-drainpowelparameters(seeFigure2.24)canbecalculatedfromthe asfollows' common-gateparameters Starting with Vrr=Mrvri*NrVx Vrr=Orvrt*PrYr,

(2.150) (2.151)

andthe voltage relationshiPs Vv = -Vzs

V-=vrr-Vr,

(2.1s2) (2.rs3)

it follows that -Vu = Vz, = OrVr, * Prvzi and,therefore,that Vv=-OrV,,-Prvzi and Vu=vrr-Vr, .

= MrVti* NrVr,- OrYrr- PuYx

Q.rs4)


= (Ms - Os) Vti + (N

Comnon-I)rain

ffuure 2.25

- Ps) I/2i

9l

(2.15s)

to Common-Source Case

The effect of changing the configuration from common-drain to common-sourceon the power parameters.

The common-sourcepower parameterscanbe calculatedfrom the common-drain Funeters (seeFigure2.25)asfollows. Startingwith Yro

MoVr, + NoV,

(2.rs6)

yu = OaVr, * PaVz,

(2.1s7)

ad the voltage relationships Vt = -Vu

(2.l 58)

l'u=Vro-V.

(2.rse)

: tollowsthat I u = Yro-V= MaVr, * NaVz,- OoVr, - PoVr, = (Ma - o)\i

+ (N, - P)Y2i

(2.160)

92


Vx = -Vza= -OoVr, - PoV, Q.161)

= -Oa Vri - Pd Y2i

REFERENCES "GaAs PowerAmplifier Design,"TechnicalNotes3.2,PaloAlto, cA: l. cripps, s. c., MatcomInc. 2.MultiMatch RF and Microwave Imp,e4sr..-rttching, AmpliJier and Oscillator synthesissoftware,somersetwest: Ampsa(Pty) Ltd.; http://www.ampsa.com. 3. Haus,H. A., and R. B. Adler, circuit Theoryof Linear NoisyNetworfrs,New York: Wiley, 1959. "An Effrcient Method for ComputerAided Noise 4. Hillbrand, H., and P. H. Russer, Analysisof LinearAmplifier Networks,"IEEE Trans.Circuitsand Systems,Yol. CAS-23,No. 4, APril 1976. 5.Vendelin,G. D., A. M. Pavio,andu. L. Rohde,Mi crowavecircuit DesignUsingLinear New York: JohnWiley, 1990. and Nonlinear Techniques, 6. Kraus,H. L., C. W. Bostian,andF. H. Raab,SolidStqteRadioEngineering,NewYork: JohnWiley, 1980. "Harmonic and IntermodulationDistortion in GaAsFETAmplifiers," 7.Cripps, S. C., TechnicalNotes2.1,PaloAlto, CA: MatcomInc.

SELECTBDBIBLIOGRAPHY Reston,VA: RestonPublishing Roddy,D., and J. Coolen,ElectronicCommunicatiorzs, 1981,pp. 103-136. Company,Inc.,

CHAPTER 3 RADIO-FREQUENCY COMPONENTS 3.1 INTRODUCTION !n orderto designrealizableradio-frequencyandmicrowavecircuits, someknowledgeof 'jrelimitationsof andthe parasiticsassociated is essential.The with practicalcomponents :haracteristicsof practical capacitors,inductors,magneticmaterials,and microstrip :ransmissionlineswill be consideredin this chapter. networks,filters,coupling Thecapacitorsusedin anRF circuit(impedance-matching one of the many manufacturers obtained from andde-couplingnetworks) canusually be apply to inductors.The design :f thesecomponents.Unfortunately,this doesnot always c: .nductorswill, therefore,also be consideredin this chapter.Single-layerair-cored aductorsandinductorswith magneticcoreswill be considered. In orderto get the circuit manufacturedto perform asexpected,careshouldbe taken }. ensurethatthe circuit realizedis the sameasthe onedesigned.Apart from theparasitic .:-Jcts of the componentsused,care shouldalso be takenwith any connectionsmade liween components.The effect of all the connectionsmadeshouldbe includedin the dnrulation. Connectionsto the ground plane shouldalso be made with care. Ground loops ground connections) should be avoided and connections cannot be made

thatall pointson the groundplane r . ;rarily to thegroundplaneon the(false)assumption !c at the samepotential (as would be the caseon the circuit diagram).When any Dertainty arisesasto exactly wherea connectionshouldbe madeto the groundplane,it -,;efulto realizethat the electric signal is traveling as a wave through the circuit and ;' -rndat any point is wherethe waveis. When an active circuit is manufactured,RF and microwave decouplingof the dc . -.rit is essential(introducing an RF ground). Parasiticresonancescan easily be roduced inadvertentlywhenthis is done.It is oftenpossibleto eliminatesuchresonances - -singsmallresistorsin thedecouplingcircuit(thevoltageacrosstheseresistorscanalso " -sedto checkthe dc current).A numberofcapacitorscanalsobe usedin parallel. The rcitanceofthe differentcapacitorsis usuallychosento differ by a factorof I 0 whenthis : Jne.

ofthe areusedin parallel,theseriesresonatingfrequencies Whendifferentcapacitors ':rent capacitorsshouldbe takeninto accountwhenthe valuesarechosen(thesmaller

93

CHAPTER 3 RADIO.FRE QUENCY COMPONENTS INTRODUCTION -'ier to designrealizableradio-frequencyandmicrowavecircuits, someknowledgeof is essential.The with practicalcomponents nitationsof andthe parasiticsassociated -.:teristicsof practical capacitors,inductors,magneticmaterials,and microstrip usmission-lineswill be consideredin this chapter. networks,filters,coupling Thecapacitorsusedin anRF circuit(impedance-matching n:d de-couplingnetworks)can usually be obtainedfrom one of the many manufacturers -- :lresecomponents. Unfortunately,this doesnot alwaysapplyto inductors.The design luctors will, therefore,also be consideredin this chapter.SingleJayerair-cored J -crrctorsandinductorswith masneticcoreswill be considered. In orderto get the circuit manufacturedto perform asexpected,careshouldbe taken .:-.urethatthe circuit realizedis the sameasthe onedesigned.Apart from theparasitic -.s of the componentsused,careshouldalso be takenwith any connectionsmade It.

.1rrrc€ncomponents. The effect of all the connections made should be included in the .rtion.

Connectionsto the groundplane shouldalso be madewith care. Ground loops ground connections) should be avoided and connections cannot be made

thatall pointson thegroundplane frtrarily to the groundplaneon the(false)assumption .: the samepotential (as would be the caseon the circuit diagram).When any -certaintyarisesasto exactlywherea connectionshouldbemadeto the groundplane,it rseful to realizethat the electricsignalis travelingas a wavethroughthe circuit and r-..'undat any point is wherethe wave is. When an active circuit is manufactured,RF and microwave decouplingof the dc it is essential(introducing an RF ground). Parasiticresonancescan easily be coduced inadvertently when this is done. It is often possibleto eliminate such resonances

rsing smallresistorsin thedecouplingcircuit(thevoltageacrosstheseresistorscanalso usedto checkthe dc cunent).A numberofcapacitorscanalsobe usedin parallel. The itanceofthe differentcapacitorsis usuallychosento differ by a factorof I 0 whenthis done. When different capacitorsareusedin parallel, the seriesresonatingfrequenciesofthe

capacitorsshouldbe takeninto accountwhenthe valuesarechosen(the smaller

93

94

Design of RF and Microwavo Amplifien and Oscillators

frequencywillbe)andcareshouldbetaken thehighertheresonating thecapacitancevalue, used. betweenthe components to avoidparallelresonances usedatmicrowaw Thethin-filmresistorsandparallelplate(singleJayer)capacitors fr,equencies cannotbe accuratelysimulatedaslumpedcomponents.Thedistributednature will bc of thesecomponentsmustbe takeninto accountin the design.Thesecomponents consideredin Chapter7. Additional complicationsare introducedby the steps,T-junctions,and crossc withplanartransmissionlines. Theidealconnectionisapointjunction,butthesc associated junctionsarenot pointjunctions.Theseeffectswill be consideredin Chapter9.

3.2 CAPACITORS Capacitorsdiffer in capacitance,resonantfrequency,losses,temperature stabilr tolerances,packaging,and size. Most of thesecharacteristicsare determined by thc dielectric material used.The parasiticinductanceis, however,also a function of tlr packagingandthe leadlengthsofthe capacitor. The equivalentcircuit for a practicalcapacitoris shownin Figure3.I . The parasiticinductancecausesthe impedanceof the capacitorto be lower tl expected.The impedanceat the seriesresonantfrequencyis equalto the seriesresistar of the capacitor.Above this frequencythe impedancebecomesinductive. belowthe resonantfrequencyis givenby The effectivecapacitance

(l

ca=coltl-U l.f,)'f and at low frequencies whereCois the capacitance

f,=

I 2rr!LCo

log lZl

&

c o L

-,1-tt-r (a)

t|rrc

3. f

(b)

(a) An equivalent circuit for a capacitor; (b) the effect of the parasitic inductance resistanceon the impedanceofa capacitor.

tl. rA, 3.

95

Radio-FrequencyComponents

Table 3.1 The resonantfrequenciesfor somecapacitors[1- 4] Capacitance

Mica: Disk Ceramic Porcelainchip capacitors Parallel-plate capacitors

I pF

7-10 GHz 20GHz

l0 pF

100pF

z-tis,

170MHz I GHz 2GHz

7 GHz

I nF

l0 nF

60 MHz 230MHz 600MHz

20MHz

.vhere/] is the resonantfrequencyofthe capacitor. The resonantfrequenciesfor somecapacitors(with very shortleadlengthsor no leads)areshownin Table3.1 [-4]. As can be seen from Table 3.1, even chip capacitorshave some parasitic Therearetwo reasonsfor this: First,the finite dimensions(andthereforethe :nductance. ofthe capacitorplates,andsecond,the finite distanceacrossthe plates. .nductance) with the finite separationof the That theremust be someinductanceassociated :apacitorplatesis obviousif Maxwell's law 9xH=i+OD/Ot magnetic currentgenerates :s inspected.Accordingto this equation,evena displacement with it. Theinductancecanbeminimizedby :lux and,therefore,hasinductanceassociated :hoosingthe smallestcapacitoravailable(with voltage and power ratingstaken into x.count). The lossesin a capacitorareusually specifiedby the quality factor (Q), where

(3.2)

!=Xr/R,

ofthe capacitor. ofthe capacitor,andX"is theeffectivereactance r?,is theseriesresistance It is, The quality factor (Q-factor) is frequency-and temperature-dependent. which the tcrefore, importantto speciff the measuringfrequencyandthe powerlevel at : ,.:surgrn€Dt wasmade. While the lossesof thecomponentarespecifiedin termsofthe p-factor,thelosses ;: Jielectricmaterialsare specifiedinterms of the dissipationfactor (DF) or the loss zrgent (tan 6).

i

t fl,.*

r^Lr^ t r Table 3.2 (e) factors for some commonly used materials and dissipation constants The dielectric

DF (low frequencies)

DF (@l00MHz)

0.03 0.002 0.00007

96


to thepowerstored Thedissipationfactorspecifiestheratioofthe powerdissipated in the material:

(3.3)

DF=Poi.r/P**o

F

; f

i

r $

I

:. :

The relative power dissipationof dielectricmaterialsis directly proportionalto the with high dielectricconstants. dissipationfactor.High lossesareassociated usedmaterialsaregiven in Table3.2 for three commonly The dissipationfactors dielectric constantdrops,as well as the [2]. Note the decreasein lossesas the relative increasein dissipationat higherfrequencies. It canbe easilyshownthatifthe parasiticinductanceofa capacitorcanbe ignored, the dissipationfactorandthe Q-factorarerelatedin the following way:

(3.4)

DF =ll Q

specifiedin termsofthe aresometimes Thelossesofthe dielectricmaterialsandcapacitors losstangent(tan5). Thedefinitionof the losstangentis the sameasthat of the dissipation factor. butincreasewith temperature Dissipationfactorsarenotonlyfrequencydependent, and,therefore,with powerlevel.Thepowerdissipationinsidea typicalchip capacitoronly to that of commonlyused needsto be on the orderof 40 mW to increasethe temperature solderingirons [2]. At high temperaturesthe dissipationfactor can be an order of magnitudehigher than at room temperature.As the temperatureinside a capacitor which causesa furtherincreasein temperature the dissipationfactorincreases, increases, with more losses.This thermalmnawayphenomenonis particularlyimportantat low impedanceandhigh powerlevel pointsin a circuit. The series resistanceand Q-factorsof two high-quality capacitorsat room temperatureare given at two different frequenciesin Table 3.3 l2l. Even for good capacitors,the p-factor is surprisinglylow at high frequencies.

I

t Table 3.3 The quality factor and resistanceoftwo capacitorsat high frequencies Frequency

l0 pF 100pF

'

100MHz 2200(0.0s50) 7oo(0.0180)

500MHz r80(0.l6eo) 60(0.055cl)

Not only the dissipation factor, but also the capacitanceofa capacitor, are affected by a changein temperature.The changein capacitancecan be very small (NPO) and linear (class I ceramics), or large and nonlinear (class2 ceramics).Class I ceramicswith positive (up to 150 ppm/'c) and negative (up to -5500 ppm/"c) temperature coefficients are available [5].

97


As afinal remarkoncapacitors, it shouldbe notedthatthe capacitanceofcapacitors rth high dielectric constantsis usually also voltage-sensitive.The capacitanceofClass 2 ,'ramics can change by more than2}%o if the voltage is varied from 0% to 150% of the .tedvalue [5].

\ummarv \e

following points are important when choosing a capacitor for a particular purpose:

l.

The parasiticinductance;

2.

The toleranceof the capacitor;

3.

The p-factor at the desiredfrequencyand power level;

4.

changes,aswell as The influenceofvoltage on the capacitor(capacitance the breakdownvoltage);

5.

The influenceoftemperatureon the capacitor(ambientaswell as increases dueto the powerdissipationin the capacitor);

6.

The sizeandpackagingofthe capacitor.

:

INDUCTORS -i

performance ofpractical inductors are degradedby parasitic capacitanceand resistive

(seeFigure3.2) causesthe resistance ofthe inductorto The parasiticcapacitance gherthan expected.This effectis very pronouncednearthe resonantfrequency(/).

(a)

r

los.f

(b) (a) The equivalentcircuit ofa practicalinductor;(b) the effect ofparasiiic capacitanceand losseson its impedance.

I Design of RF and Microwave Amplifiers and Oscillators

l e 8 Inductor losses consist of copper losses(R) and, if magneticmaterial is used, The hysteresisand eddycurrentlosses(R). All oftheselossesarefrequencydependent. copperlossesincreaseaboveits dc valuebecauseof the skin andproximity effects. By usingmagneticmaterial,the sizeof theinductorcanbe reduceddrasticallyand will, therefore,alsobe considerablylower' Unfortunately,there the parasiticcapacitance in the material.Theselossesaremainly hysteresislossesin the losses be some will also caseof ferrite materials. The effect ofparasitic capacitanceon the Q-factorandthe inductanceofinductors, the skin and proximity effects,the designofair-cored solenoidalcoils,the propertiesof magneticmaierials,and the designof inductorswith fenite coreswill be discussedin the following sections.

3.3.1

The Influence of Parasitic capacitance on an Inductor

By using the equivalentcircuit shown in Figure 3.2,it can be easily shown that the effectiveinductance(I"6) of an inductoris givenby L"n=L,lU-(flf,)'j

(3.s)

wherc.f,is the parallel resonantfrequencyof the inductor. This equationappliesonly if the approximation

l+l/fi =-1

(3.6)

where Qr=aLrl R, canbe made. As can be seen from (3.5), the inductanceincreasesrapidly as the resonant frequency(,f ) is approached. Under the sameconditions,the effectiveresistance(Roignored)is given by

R"n= R" ttt- (f I f,)'l

(3'n

ofthe parasiticcapacitance because hasincreased Becausethe effectiveresistance present,the lossesin the coil arehigherifthe input currentto the inductoris consideredto thecurrentin theparasiticcapacitoris outof phasewith Le thesame.This happensbecause inductor. part the of that in the inductive The effective Q-factorof the coil will thereforebe lower than without parasitic The effectiveQ-factotis givenby capacitance.

Q"n=Q,U-U/f)'1

(38


99

When /= 0.707f,the effective Q-fa6or will be half that of the inductive part of re inductor. Theseeffectscan be minimizedby keepingthe parasiticcapacitance as low as ossible. The capacitance of an air-coredsolenoidalcoil is givenin Figure3.3 asa function f the length-to-diameter ratio andthe meanradiusof the coil [6]. The capacitanceof the coil is not a frmction of the number of turns as might be .rspected; it is a strongfunctionof the coil size(radius)and a weakfunctionof the coil ,aape(length-to-diameter ntio,l/D). The capacitancecan thereforebe minimizedby :uking the coil as small aspossible.An initial valueof 2 canbe usedfor the length-to:'zneter ratio.

,-/D :Flcn)

alD

l3

:l tt"

Theself-capacitance ofa single-layer solenoidal coil (Source:[6]).

For high inductance,the tums of a coil shouldbe spacedascloselyaspossible.It shownlaterthat this distanceis determinedby the desiredQ-factorof the coil. Whenthe coil capacitanceis known,the resonantfrequencycanbe found by using :'dation

I

(3.e)

-r n/2"c" vpical resonantfrequenciesfor someinductancevaluesaregiven hereasa guide -rn be achievedeasily[l]: lfl)nH: '! uH: ) pH:

400-800MHz 100-200MHz 25-60MHz

100


Table 3.4 The wire diameterand resistancefor wire gatges 12-32 (20'C; coppermaterial) Gauge

Bare diameter (mm) AwG (SWG)

t2 l4 t6 l8 20 22 24 26 28 30 32

2.052 (2.64) 1.628 (2.03) l.2el (1.63) r.024 (r.22) 0.812 (0.914) 0.644 (0.71l) 0.511 (0.5s9) 0.405 (0.457) 0.321 (0.376) 0.255 (0.3r5) 0.202 (0.274)

Doubleenamel(mm) coateddiameter AWG (SWG) 2.r3 (2.73) r.1t (2.r2) r.37 (r.7r) l.l0 (1.29) 0.879(0.984) 0.70t (0.774) 0.564(0.617) 0.4s2(0.512) 0.366(0.424) 0.295(0.361) 0.241(0.316)

Resistance (A/km) AWG (SWG)

5.5 (3.1) 8.6 (s.2) (8.2) ts.2 22.0 (14.5) 34.3 (25.8) 61.0 (42.6) 87.8 (6e.1) 133.9 (103.2) 212.9 (ts2.6) 338.s (217.4\ 538.5 (286.6)

rangingfrom frequencies Miniaturechip coils (0305,1008,...) with self-resonant nH are commercially n}{to 2.2 250 MHz to above6 GHz for valuesrangingfrom 1500 frequencyclaimedfor a 100nll(22 nH) miniaturechip inductor available.Theresonance is 1.5GHz (3.2 GHz)for a chip sizeof 0805(8mils x 5mils) and I GHz (2.4 GHz) for a l50MHz(25}MHz) and100MHz chipsizeofl00S [7].TheminimumQ-valuesquotedat are40 and 50, respectively[7].

3.3.2

Low-Frequency Losses in Inductors

The resistivelossesin a conductorare approximatelyconstantat low frequencies.The resistanceis a functionof the materialusedandthe wire diameter.The diametersandthe resistanceof copperwire with wire gaugesrangingfrom 12 to 32 aregiven in Table3'4. The American wire gauge(AWG) valuesare listed with the correspondingstandardwire gauge(SWG) values.Note that the wire diameterdoubleswheneverthe wire gauge by a factorof6. decreases It canbeseenfrom thetablethatthediameterof AWG No.I 2 wire is approximately of No. 12wire is 5.5 O/km and 2 mm andthat of AWG No. 22 is 0.2 mm. Theresistance correlates thatofNo. 32 wire is 538O/km.Theincreaseof approximately100in resistance well with the decreasein the diameterby a factorof 10 (R* l/A , whercI is the crosssectionareaof the wire).

3.3.3

The Skin Effect

A conductorcan be viewed as a guide for the electrical andmagneticfields aroundit, as


101

is shown in Figure 3.4. The c.trrent flowing in the conductor is caused by the changing magnetic flux that penetratesinto the conductor. This current opposesthe magnetic field that causesit. The result is that the magnetic field decreasesin strength (exponentially) as it penetrates the conductor.

Ftgure3.4

Theelectric,magnetic,

andinsidea circularconductor(after [9]).

The inducedelectricalfield within the conductoris siven as a function of the oenetrationdepthx by E, = Eroe-r'

(3.10)

whereE, is theelectricfield strengthatthesurfaceof theconductor(in thedirectionof the conductor). The propagationconstantof the electricalfield in the wire is

f = .//ro pT t'-'--;-(r + /) lTcJ ILy

= Cf+,tF

(3.1l)

wherey is the resistivityof the conductor. The inverseofthe attenuationconstantc is definedasthe skin depth6:

6= l / a = t t J ; f w

(3.r2)

Therefore,the amplitudeof the electricalfield at a distancex insidethe conductor

toz


tha is tur

Table 3.5 The skin depth of somematerials as a function of frequency Material Brass Aluminum Gold Copper Silver Mu-metal

Skin Depth (cm) 12.7/fn 83/fn 7.7tfn 6.6lf tn 6.2/f n 0.4/f n

E(x) = E(0)e-'16

(3.13)

Becauseof the decreasein the field strength,the current density will be higher closerto the surfaceof the conductor.Whenthe conductoris at leastsix skin-depths(or dcpthsof penetration)in diameter,all the currentcanbe consideredto flow uniformly in e layer one skin-depthdeepalongthe surfaceofthe conductor. The resistanceof the conductorcanthen be calculatedwithin l0% by using the following equations[9]: R- = {nr2 /lnrz -r(r - 5)2]}Ro"

(3.14)

= {nr2 llnrz - n(rz -26r + 62)11R0" = lnr2 /l2n6r - n62llRd"

&e

(3.1 s)

rrfue2ris the outsidediameterof the conductor. where6 < 2r, this equationsimplifiesto At high frequencies, :

F

& =[t/(26)]Rdc

(3.16)

Becausethe skin depthis inverselyproportionaltothe squareroot ofthe frequency, tb rcsistanceR""will increaseproportionallyto theroot of the frequency,that is, if 6 ( d (r*tere d is the diameterof the conductor). The skin depthsfor somematerialsare given in Table 3.5 as a function of the frequency. As an illustration of the changein skin depth with frequency,considerthe skin dcpthfor copperat variousfrequencies: 6 = 0.66mm at 10kHz 6:66 pm at 1 MHz :6.6 pm at 100MHz i

it is importantto ensure Becausethe skin depthis very smallat high frequencies,

t!\ E: t::


103

fiat conductorsurfacesaresmoothifthe lowestpossibleresistancewith a specificmaterial s required.When materialswith low conductivitiesareused(usuallyto ensuretempera:urestability),it becomesworthwhileto platethe conductorswith silverabove100MHz. To get anideaofthe increasein resistancewith frequencycausedby the skin eflect eonsiderthe resistanceof 1 m of AWG No. 22 wire asa functionof frequency: R:0.06 O at dc .R=0.60QatlMHz R:5.95 O at 100MHz Note that the resistanceat 100MHz is approximatelyl00tn times that at I MHz. causedby the skin It is obviousfrom thesenumbersthatthe increasein resistance -'frectcannotbe ignoredat high frequencies.

,13.4

The Proximitv Effect

A conductorcarryingaltematingcurrenthasa changingmagneticfield aroundit. If another conductoris broughtcloseto it (seeFigure3.5),the changingmagneticfield throughor round it will causeeddycurrentlossesin it (whend>56, thepenetrationdepthof thefield -. .mall comparedto the diameter).Theselossesare reflectedin the first conductoras ,:easedresistance. is proportionalto theroot of the Similarto the skineffect,theincreasein resistance rency at high frequencies(d>56). When only two conductorsare in closeproximity, the influence of the proximity eftct is relativelysmallcomparedto thatofthe skineffect,but whenmoreconductorsare cd it shouldbetakeninto account.Becausea solenoidalcoil consistsof manyconductors gb6e to one another,the proximity effect can significantly affect its resistanceat high , uencies.As an exampleof this, the resistanceof a single-layersolenoidalcoil with ratio of 0.7 is almostsix timesthat of the same s touchingand a length-to-diameter - : whenstraightened out (thatis, if morethan 10 turnsareused). Whenthe tums of a coil arespacedwell apart,theproximity effectcanbe ignored.

r '- rrc 3.5

The proximity effect.

104 3.3.5

AmplifiersandOscillators Designof RFandMicrowave

Magnetic Materials

The inductanceof an air-cored coil can be increasedsignificantly by using a magnetic material as the core. The reasonfor this is that the magnetic flux density increases substantiallywhenthe relativepermeabilityof the materialis high. Typical values for the relative permeability (p) of ferrite materials at radio with cut-off frequencieson the frequenciesare 10-150. The highervalue is associated order of 20 MHz, while lower value is associatedwith cut-off frequenciesof around sharply. I GHz. Above the cut-offfrequency,the relativepermeabilitydecreases lossesin Apart from the relative permeabilityand its frequencydependence, points. at high voltage especially be taken into account, must also magneticmaterials When ferrite materialsareused,theselossesare mainly hysteresislosses.When materialswith higherconductivitiesareused,the eddy-cunentlossesin the materialalso becomesignificant. Lossesin a fenite coreareproportionalto the energystoredin it. Theenergystored is proportional to the energy density and the volume of the core. The volume is areaandthe meanpath length. approximatelyequalto the productof the cross-sectional Therefore,lossesin a ferrite corearegivenby an equationofthe form

4o.,= k(pr,.f ,B^^)B"^^*AI

(3.17)

whereI is the averagecross-sectionalareaof the core,/ the meanpath length of the core, B.* the maximum root mean square(rms) flux density in the core, and k a constant dependenton the frequency,relativepermeability,flux density,andmaterialused. Thepowerlossesin a fenite corearebestspecifiedin termsofthe ratio lt,RrlL and in parallelwith the inductance(Z) of the magnetic(3.17). R, is the lossresistance not by coredinductor. This ratio is independentof the core dimensionsand is only a function of the shouldbe independent materialusedandthe maximumflux density.Thattheratio 1t"Rn/L asfollows. of the coresizecanbe established the lossesin thecore,thepowerlossin the coreis givenby BecauseRorepresents 4o,,=V;/Rp

(3.l 8)

whereVnisthe rms voltageacrossthe inductor. This voltage is relatedto the maximum flux densityB.o bY Vo= ja(N@)= jaNABô where-ly'is the numberof turns. Rnisfoundto be By usingthesetwo equations,the resistance Ro=v] / Pr",,

(3.1e)

Components Radio-Frequency

a 2 N 2 A 2B 2 ^ * 4or.

_ @ 2N 2 A 2 B 2 ^ * k At 82^

105

': '' r

-lr'lklNzAll

(3.20)

re resistanceR, is, therefore,proportionalto the Squareof the numberof turns and the ,ss-sectional areaof the coil. It is inverselyptoportionalto the meanpathlength. This is alsotrue for the inductance,which is givenby

-

No = l t o P r-{24 fv

A

t=7=

I

(3.2r)

T

of the coredimensions. Theratio p81L is, therefore,independent By using (3.20)and(3.21),it follows that

(3.22)

..RolL=a2 /(ktto)

Because,t is a function of the flux density and the frequency,the ratio 1t,R,/L is soa functionof the flux densityandthe frequency. curves for this ratio asa functionof frequencyareshownin Figure3.6 [8]. These fltnresapply at small-signalconditions(thatis, whenB.*is small).

lo"

Y,RolL l0rI (s")

l0ro

t0

100

f (MIlz) rgure3.6

Curvesof the ratio p.RolL (ro1t,/tan6) plotted againstfrequencyfor two fenite materials (4,*- 0) (Source:[8]).

106

Desigl of RF and Microwave Amplifien and Oscillators

By using thesecurvesand a value of 120for the relativepermeability,it can be showneasilythat the highestunloadedQ (8, = Rn/ @tL))that canbe expectedar 6 MHz by using 4C6 materialis approximately125. Whenthe flux densityincreases, the lossesin thecoreincreaseaswell. Curvesfor the ratio 1t,Ro/Lasa function of the productBôf areshown for 4C4materialat different frequenciesin Figure3.7. The product B^^f is used becauseit is independentof the frequencyif the maximumvoltageacrossthe inductor(2,) is assumed to be constant.

l0r2

15MHz 1t,Rr/L

(s'')

10.

lot

B*f (THz)

Figure3.7

(op,/tan6)plotted Curves of yt,R"o/L (8,;f) for4C4material against theproduct atvariou. (Source: frequencies [8]).

By usingthe curvefor 1.6MHz, it followsthatthelossesdoublefrom their smallsignalvaluewhenthe flux densityis approximately14mT (140 Gauss). As a final remark on magneticmaterials,it should be noted that the relative permeability of magneticmaterialsis temperature-dependent. Materials with higher permeabilitiesareinfluencedmoreby temperature changes. Becausethe temperatureof the materialchangeswhenheatis dissipatedin it, the rclativepermeabilitywill alsochangewhenmorepoweris dissipatedin it.

Summary The following points shouldbetakeninto accountwhena magneticmaterialis selectedfor a particularpurpose:

-


t.

Thehighestfrequencyofoperation;

2.

The maximumallowableamountof losses;

a

The sizeof the inductorand,therefore,the relativepermeability;

4.

The temperaturedependenceof the magneticmaterial.

107

3.3.6 The Design of Single-Layer Solenoidal Coils i inglelayer solenoidal coils are often used at radio frequencies.Their use is limited by the :rductancevalues and unloaded Q-factors obtainable,as well asby the associatedparasitic ,:pacitance. The inductance of a single-layer solenoidal coil is given approximately by

L -- Nzrll22.9l lr +25.41 (pH)

(3.23)

/ thelengthof thecoil (in centimeter), *tere r is themeanradiusofthe coil (in centimeter), md Nthe numberof turns. of thesecoils is givenin Figure3.3 asa functionof the The parasiticcapacitance ingth-to-diameter ratio (//D) andthe radiusof the coil. The capacitanceis small whenthe :oil radiusis small. The unloadedQ of air-coredcoils is a functionof the frequency,inductance,dc ofthe coil. :esistance, skin effect,proximity effect,andself-capacitance theunloadedQ is given neglected, can be wheretheself-capacitance At frequencies t! t6l

Q.= lrrJ7

(3.24)

.- :re the radiusmustbe specifiedin centimetersandthe frequencyin Hertz. ratio of the coil and the relative The factor k dependson the length-to-diameter for variouscoil shapesand wire plotted 3.8 in Figure is Its value tums. of the facing g'.ing ratios(dlc),wherec is the distancebetweenthe centersof two adjacentturnsand ; :hediameterof the wire used. The following factscanbe deducedfrom the curvesin Figure3.8 andQ.2\:

ffi

l.

Higher unloadedQ-factorscan be obtainedby using coils with larger ratios(//D). diametersandlength-to-diameter

2.

Theturnsofan air-coredsolenoidalcoil shouldbe spacedcloseenoughto ensnrethat the dlc ratio is largerthan 0'4 d, andin shortercoils (//D =1) they shouldbe spacedfar enoughapartto ensurethatthedlc ratiois smaller than0.8 d

108


When larger coils are used the turns can touch without any significantreductionin the unloadedQ (lessrhan25%). By usingthe curvesin Figure3.8 andthe equationsgiven,solenoidalcoils canbe designedto have a specified inductance and unloaded Q.The parasitic capacitancecan be determined by using the curve in Figure 3.3. The design can be done as summarizedbelow.

0.16

F

0.14

F

0.12 0.10

tlD 0.08 0.06 0.04

r

0.02

0.2

F

r

dlc

I

t

Figure 3.8

F I

A Dcsign Procedure for Controlling the Inductance and Quality Factor of an AirCored SolenoidalCoil

F

Curvesfor calculatingthe unloadedp of single-layersolenoidalcoils at high frequencie. (Source:[6]).

l.

Choosethe length-to-diameter rario(llD) equalto l.

2.

Calculatethe radius (r) of the coil (in centimeter)by using the equation

r=Qu/GJ7)

(3.25

whercQ, is theunloadedQ required,andk:0.1 for //D:1.0 (seeFigure

r i tF ;

F

3.8). 3.

Findtheparasitic capacitance of thecoilby usingFigure3.3.Calculate th, resonantfrequencyby usingthe equation

.f,=rlJrc lQn)

(3.26'

109


whereClD = 0.45pF/cmfor llD: l. 4.

cannotbe reached Ifthe resonantfrequencyis too low, the specifications and it will haveto be changed.

5.

Calculatethe required.numberof tums by using the equation N =lL(22.9(l I r)+25.4)I rltt2

6.

7.

(3.27)

Calculatethe requiredwire thicknessby using the dlc ratio usedin step2:

d = (d I c\ ll / (N - 1)l= (l / D) (d / c) [2r / (N -r)]

(3.28)

whered is the wire diameterto be used,andd/c = 0.55 for l/D: Figure3.8).

I (see

If the requiredwire thicknessis small,a coil formerwill be needed.If the it canbe redesigned. coil is to be self-supporting, .: In order to increasethe wire diameter,it will be necessaryto increasethe size of the coil. Whenthe resonantfrequencyis a potential problem, the llD ratio can be increased.The resonantfrequencywill decreaseif the radius is increased. Wheretheresonantfrequencyis notaproblem,theradiusofthe coil canbeincreasedin orderto increasethewire diameter.Themaximumvalue of the radiusis tô = c^ / (2c)

(3.29)

where C. is the maximum self-capacitanceallowable, and C is the per centimeterasgivenby Figure3.3. capacitance withuD: l,C = 0.45pF/cm.

EXAMPLE 3.1

Designinga single-layerair-coredsolenoidalcoil to havea specifiedp andresonantfrequency.

As an exampleof the applicationof the procedureoutlined, a I pH coil was designedto havea minimumunloadedQ of 300at 50 MHz andresonantfrequency above250 MHz. The resultsof the differentstepsareasfollows: - !

l.

l/D: I

2.

r=0.42cm

110


3.

1d";,.,

f,=256MHz

4. 5.

N: 13

6.

d:0.36 mm

7.

Becausethewire diameteris small.it will benecessaw to usea coil former.

It is not possibleto increasethe wire diameterby increasingthe coil radius in this case(/: 250 MHz). It is possible,however,to increaseit by increasingthel/D ratio ofthe coil. Unfortunately,it is not possibleto increasesufficientlythe wire thickness to makethe coil self-supporting. Theresultsfor differentl/D ratiosarecomparedin Table3.6.Notethatthe wire diametercanbe doubledif the length-to-diameter ratio is chosento be equal to 4. Although the wire thicknessis a strongfunction of the length-to-diameter ratio, the resonantfrequencyof coils with length-to-diameter ratiosfrom 0.6 to 4 doesnot vary significantlyif theyaredesignedto havethesameunloaded,Q-factor. The volumesof the coils in Table 3.6 increasewith increasingllD ratio. Whena smallcoil is required,the length-to-diameter ratio canthereforebe chosen to be equalto 0.6.

Table 3.6 The dimensions, unloaded Q, and resonantfrequency for a I pH coil as a function of the I/D ra6o

UD

r

N

(cm)

0.6 1.0 2.0 4.0

0.48 0.42 0.37 0.32

d/c (mm)

l0 13 18 26

0.31 0.36 0.52 0.63

Q" (MHz)

0.55 0.55 0.63 0.63

)G | L

(3.46) J

f w/h)*f,*f?!)'l'".1 ?_=oonl \t4/)

(3.47)

p =270{t - *t[t.t e2+0.706(t +H2/ h)tz

(3.48)

L

lwh

I I

#rJ

- tanh-r{t0.012 w I h+0J77(wI D2 -0.027(wI D3l J = 1.0109 ll+ Hz l hlzl

(3.4e)

2..,=Zoo--PQ

(3.s0)

z

r 0.053

e' - o'9 D= -0.564[ | \ e, + 3.0/

(3.51)

c = I + (r/49) ln{(w/h)2[(w/D2 +r/52\/l(w /h)4+0.432]\ + (r I 18.7)ln {r + UVI (18.1r)13} -=ob

.

ltl +rTh I wli -21(tn2)I nl (t I h)/ (Ir I n1tt2y +0.121(H,I h)-1.164| (H2| h)) tanh[l.043

(3.s2) (3.53) (3.54)

Designof RF and Microwave Amplifien and Oscillators

e.-l

r-+l eetr=--*q

z

(3.ss)

zo=zoo/JG

(3.s6)

vo=c/r[4o

(3.s7)

ufurc vois the phasevelocity in the microstrip,

fo=Zr/l2stohl

(3.s8)

G = (n2 / I2)l(e, -l) I e"ul(Zo | 60)tt2

(3.5e)

t,-"r(.f)=",-ffifu

(3.60)

c

s=

"2

4f2[e,-"u(f)-l]

(3.61)

! = sl3-(wlr2

(3.62)

YaQ)= r2onhtlzoJil)

(3.63)

'P = ( W / 3 ) 3+ ( s / 2 ) [ W " u Q-)I r | 3 ]

(3.64)

F ,=(p'+y3)'t2

(3.6s)

VrnU\ = W / 3 +[r + plvi -f, - pfttt

(3.66)

Zo(f)=

I20nh

W'-G,

(3.67)

Radio-FrequencyComPonents

t2l

The frequencydependence(dispersion)of the chatacteristicimpedanceand the effective dielectric constantof a microstripline result from the non-TEM nature(inhomogeneity)of the modeof propagationalongthe microstrip. As anexampleof theapplicationof (3.45)to (3.67),thewidth-to-heightratiosand the effectivedielectric constantsof a 50O line on an alumina(e': 10.2)and a Teflon Theresults,respectively, (e,:2.5) substrateat2 GHzwrthH2lh: 20.0werecalculated. areasfollows: Wlh= 0.85with €,41= 6.6945

Wlh: 2.75 with e,"6: 2.0775 to takeinto accountthelosses necessary it alsobecomes At microwavefrequencies of theselossesis usually main source The lines. ;onductorand dielectric)in microstrip given by thefollowing setof is a. constant ,rnductorloss.The conductorlossattenuation quations [4, l5]:

g*-!-*-!, = 8'68R"a ' 2nZoh

W.o

ynyL++)l w th)> 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:


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


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


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


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


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


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> 2(w + f) mustbe satisfied. Bond Wire Inductors Bonding wire inductorshavethe advantageover strip inductorsthat higher Q-factorscan be expectedbecauseofthe largersurfacearea.Furthermore,touch-uptuning is possible with bondingwire inductors,while the inductanceis fixed for strip inductors.The fixed inductance,however,is an advantage in a first-time-rightdesign. The inductanceassociatedwith a long (lld >100) free-spacebonding wire of diameterd andlength/ canbe calculatedby usingthe equation[4]

L(r*I I mm) = 0.20[1n( I / d) +0.386]

(e.17)

The effectofa groundplanecanbe incorporatedby usingthe equation[4, 6] l-;--

z(nH / mm)=Q.2 U!

d

*rnl

+ tl l' -+d' /-4 r+JP +qh2

.wffi,1.*, An approximateexpressionfor the Q of a round wire inductor is [5]

(e.18)

Microwave Lumped Elements,Distributed Equivalents, and Microstrip Parasitics

"' (cu')\"' (f 4 = [eg'l) 3.38x r03 r(nH) o Y/ i(-p J \ z )

327

(e.le)

Equations(9.17)and(9.18)areonly accuratewhenlld >lA0 [9]. Whenshortbond for the free-space wiresareused,the following equationis recommended case[9]:

z(H)=[p0 / Gn)] r ulrztray*,[-* q r afl+ d/ (2t)

(e.20)

I+(d /(2/))2+ p,6) Whenthe wire is manufacturedwith nonmagneticmaterial,as is usually the case,F, = L the intemalinductanceof the wire. The skin depthterm (6) in (9.20)represents The effect of the ground plane is similar to a currentimage reflectionof the whena qround inductor.Becauseofthis effecttheinductanceofthe bondwire is decreased planeis present.The effectiveinductanceis this caseis givenby [9]

(H)= r -[Fot (2n)].1. z"m trn[lt Qh)+s*

r rzn>f]

I + ( 2 h I t ) 2+ 2 h I I \

(e.2r)

where 2h is the center-to-centerseparationbetweenthe wire and its image,and ft is the distancefrom the groundplane. in [9] that hin(9.21) shouldbe replacedby It is recommended

h ' = h+ 4 . 6 6 to accountfor the nonperfectground(finite conductance).

(e.22) ,.

SquareSpiral Inductors For square spirals the inductance(in the absenceof any ground plane) is given approximatelyby [0]

r(nH) =o.8sJiNsp

(9.23)

whereI is the areain squaremillimeters andN the numberof tums. line length(in squaremillimeters)is approximately The associated

l, = N [ 8 a+ d ( a N -3 )] in this equationaredefinedin Figure9.2. The parameters

(9.24)

328


SquarespiralsareoftenusedasRF chokesin MICs. Circular Spiral fnductors The inductanceof a circular spiral inductor can be calculatedby using the following equations: z(nH) =3.930a2N2/10.8a+1.lc1 a(mm)=(do*dt)/4.0

(e.26)

c(mm) = (do - dt) 12.0

(e.27)

where d, andd. are the inner and outer diameterof the spiral, respectively,s the spacing betweentwo adjacentconductors,andN the numberofturns. For minimum losses,the outer diameterof a spiral inductorshouldbe approximately five times the inner diameter [l]. Under this constraint,the Q is given approximately(+20%) by [5]

w O_1.3x102 K'

(e.28)

where K' is a function of the width of the conductor(w) and the spacingbetweenthe conductorsand is given by [4] K,=1.009 + 0.g594"-@+w)/w +0.6376"-2(s+w)tw *1.g43 e3('*n)t*

e.2g)

In orderfor (9.28)to apply,d. shouldbe greaterIhanl.2d,,iy'greaterthan l, and thethickness(t) greaterthanfive skin depths[5].

- t Fd

J'F Figure 9.2

A squarespiral inductor.

329

Microwave Lumped Elements,DistributedEquivalents,and Microstrip Parasitics

Typical valuesfor the conductingstrip width of a spiral inductor are 50-250 pm. ratio of unity is recommended For closeto optimumresults,a width-to-spacing [5]. Single-LayerSolenoidalAir-Cored fnductors At microwave frequencies,solenoidalinductors are often used as RF chokesin hybrid circuits.Whenthe sizeis not prohibitivelysmall,they canalsobe usedasinductors. The inductanceofa solenoidalcoil is givenby

(e.30)

I + 2.54rf r(nH) = lo.orzN2 | [2.29

wherer is the radius(in millimeters),/ is the length(in millimeters),andNis the number of turnsof the coil. In order to remain essentiallylumped,an inductormust be electrically short. Reasonable resultscan be expectedwith shuntinductorswhen the associatedelectrical will lengthis shorterthan30' (thedeviationfrom theexpectedlinearincreasein reactance are more severe restrictions inductor, the thenbe lessthan l0%). In the caseof a series becausethe resistancein serieswith the inductorwill be transformedbecauseof the effect. transmission-line In orderto provideanideaofthe boundson realizableseriesinductances,the inducwith a line of 38'(Q:0 ande,: l) werecalculatedandaretabulatedin tancesassociated Table9.2 at differentfrequenciesfor eachof the inductorsdiscussedabove.Becausethe inductiveand capacitivecouplingwereignored,the boundson the inductanceof square spiraland solenoidalcoil inductorsareonly approximate. Theinductancevaluesin Table9.2 areoptimisticin thesensethatthe Q of the load wasassumedto be zero,the relativedielectricconstantwasassumedto be unity, andthe with the lumped influenceof the finite incrementalcharacteristicimpedanceassociated inductorswas ignored.The influence of the effectiverelativedielectricconstantis to increasethe electricallengthof the inductorby a factore,tt2,andtheQ andZsinfluences

Table 9.2 Upper bounds on the seriesinductancerealizable (e, = l; 0 = 38') with different inductors as a function of frequency Inductance(nH) Frequency (GHz)

I 2 4 6 8 l0 12

Bonding wire (d:25 pm)

48.0 22.0 9.7 6.1 4.3 J.J

2.7

Strip inductor (w:50 pm)

48.0 22.0 9.9 6.2 4.4 3.4 2.7

Squarespiral (r,:20 pm (25 pm))

Solenoidalcoil (c:25 pm)

(4 = lo pm(sopm)) 10e.0 (65.0) 41.0 (25.0) l5.o (9.1) 8.2 (s.0) 5.3 (3.2) 3.8 (2.3) 2.e (1.7)

r 44.0 50.0 17.0 9.4 6.1 4.3 3.1

330


Tabte 9.3 The inductanceofdifferent inductorsas a function ofthe lenethofthe conductor Length (mm)

Inductance(nH) Sfrip inductor w: 50 trrm

1.0 1.5 2.0 t
.t 9.1 13.0 20.0 28.0 36.0

Squarespiral r,=25 pm (20 pm) d,: 50 pm (10 pm)

Solenoidal coil c=25 pm

0.3(0.6) 0.7(r.2) l.l (l.e) t.6 (2.6\ 2.1(3.5) 3.3(5.4) 4.6(7.6) 8.4(14.0) 13.0(21.0) 23.0(38.0) 34.0(s7.0) 47.0(78.0)

0.7 1.3 2.1 2.9 3.9 6.t 8.6 16.0 25.0 46.0 71.0 100.0

are tabulatedin Table 9.1. An idea of the lowering in the inductanceboundscausedby thesefactors can be obtainedby using Table 9.3 in conjunctionwith rable 9.1. The inductanceof the different inductorsis tabulatedin Table 9.3 as a function of the conductorlength. The inductanceofthe solenoidalcoil inTables9.2and9.3wascalculatedbyusing (9.30)andthe following setof equations: = 0.3788,tl".c roo'lop,=0.4202 r[t]"

( e.31) +"

N =0.4202 W

(e.32) (e.33)

where c is the wire thickness(in millimeters), ro* the optimum radius, /, the conductor length, and /oo,the optimum coil length. Thewire thicknessof the solenoidalcoil shouldbechosento optimiz,etheQ $efer to Section3.3.6). Equations(9.31)to (9.33)werederivedby settingthederivativeof theinductance, as given by (9.30),equalto zeroin orderto find the highestinductancecorresponding to a specifiedconductorlength. EXAMPLE 9.I

Calculation of the inductancebounds for a matching network.

The matchingnetwork in Figure 9.3 wasdesignedto matchthe output impedance


331

of a GaAs FET to a 50Q load over the passband2-6 GHz As an exampleof the applicationof the material derived in the previoussections,the feasibility of realizingthe inductorsin the network in lumpedform will be investigated. Inspectionof Table 9.2 yields that the maximum realizableinductance is (e,: l; Zs- *; Qr:0; ls2,l:0.25 dB) at 6 GHz (solenoidal coilsexcluded) approximately8.2 nH, which is higherthanthe inductancevaluesin Figure9.3.

4.57nH

Figure 9.3

4.2lnH

2.l5nH

The matchingnetwork consideredin Example9.1.

It follows from Table9.3 that a conductorapproximately4 mm long will be required to rcalize the 4.58 nH inductor. Assumingthe effective relative dielectricconstantto be2.l7 (stripinductor),it follows thattherequiredelectrical lengthis approximately

0 = 1 2 0 x 1 0 - rt rJ + t = l 2 0 x l 0 - r r x 4 ' , l r n x 6 x l } e = 4 2 o Table 9.1 showsthat even with an infinite value for the characteristic significantlydegrading impedance, the4.58-nHinductorcannotberealizedwithout the match.The 4.21-nHinductorpresentsan evenbiggerproblem becauseit is locatedat a higherQ point (2.01comparedto 1.37). Theelectricallengthofthe 2.15-nHinductoris approximately22.8o,andthe load Q atthatpointis equalto zero.lt follows from Table9.1thatthis inductorcan impedanceas be realizedin lumpedform evenwith an incrementalcharacteristic valueof - 0.07dB for the low as 100O.Applicationof (9.11)yieldsanapproximate errorin gainwith Zot*enas 1000.

9.5

LUMPED MICROWAVE CAPACITORS

Lumped microwave chip capacitorscan be usedup to very high frequencies.The selfvaluesasspecifiedby onemanufacturer for somecapacitance resonantfrequencies [2] are 0.154 small as are as tabulatedin Table 9.4. The dimensionsof these capacitors pF and 0.1 and 5.6 valuesbetween by 0.508mm and2.032by 2.540mm for capacitance 0.254 mm. The 3.0 and 62pF, respectively.The thicknessesvary between0.076 and

332

Designof RF and Microwave Amplifien and Oscillators

approximateseriesinductanceis 0.05 nH. It should be notedthat the power that can be dissipatedin capacitorswith suchsmalldimensionsis limited. Insteadofusing discretecapacitors,capacitorscanbe integratedinto a microstrip, thin film, or MIC design.Thesecapacitorscanbe smallplatecapacitors,microskip gap capacitors,or interdigitalcapacitors.Microstrip gapcapacitorsp3] areonly usedat the highermicrowavefrequencies.

Table 9.4 The self-resonantfiequencies for somehigh quality microwave chip capacitors Capacitance(pF)

Self-resonantfrequency(GHz)

0.t

50

I

2

t0

8

i

9 3

t00 1000

I

In6rdigital capacitorswith capacitorsrangingfrom 0.1-15 pF canbe realizedon MICs andthin film. The approximatecapacitance of an interdigitalcapacitoris given by the equation

C(F)= [(e, + l\ /Wlt'[(N -3)A, + Arj

(e.34)

whereNis the numberof fingers,l, and.,{,areweightingfactorsassociatedwith the inside andoutsidefingers,respectively,and / is thelengthof overlap,asillustratedin Figure9.4. pF/mm and When the substrateis thick enough,these constantsare 8.85826x10-3 pF/mm,respectively. 9.92125x10-3 Formaximumcapacitance,the linewidthsandspacings shouldbe equal[14]. Spacingof l0-25 pm betweenthe fingersis typical [5]. The parasiticsassociated with interdigitalcapacitorscanbe ignoredaslong asthe productis smallerthan2.0x10-3[14]. capacitance-frequency

W

I (a)

Figure 9.4

T

T

cr

cl

Ct

Cr

(b)

(a) An example of the layout of an interdigital capacitor; (b) a low-frequency equivalent circuit for a seriesinterdigitalcapacitor.


Interdigitalcapacitorsareconsideredin detailin [

333

].

9.6 DISTRIBUTEDEQUIVALENTSFOR SIIUNT INDUCTORSAND CAPACITORS If the required inductance is low enough, a shunt inductor can be replacedto good transmissionline. Similarly, impedance, approximationby a shorted,high characteristic a shunt capacitorcan be replacedwith an open-endedstub having low characteristic is small enough.The accuracywith which these impedanceif the requiredcapacitance on the linearityof thetangentfunction.To give an canbemadeis dependent replacements indicationof the frequencyrangeover which this functioncanbe consideredlinear,the valueof (tanO- 0 ) / 0 is summarizedfor severalvaluesof 0 (radians)in Table9.5. If a the maximumelectricallengthfor an equivalentline is 30". l0% deviationis acceptable, canbereducedto lessthan5% with the same Themaximumdeviationacrossthepassband line lengthby averagingthe deviationacrossthe passband. The equationsapplyingto replacingthe lumpedcomponentexactly at a frequency fn arc Zrttan(Pl\= X rt

(inductive)

(e.3s)

(capactive)

(e.36)

and Zo" ltan(Pl.)= X ,c

whereXsl and Xo6arelhercactancesto be replacedat frequency/1, andZsl (short-circuited stub) and Zor(open-ended stub) are the characteristic impedancesofthe stubs.

Table 9.5 The value of(tanO - 0)/0 (in radians)as a function ofthe angle0 (in degrees)

(tanO- 0.)/0

(tan0-0)/0

(") 5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0

(o/o)

0.3 0.6 1.0 1.6 z.J

3.2 4.3 ).1

6.9 8.5 10.3

(") 35.0 37.5 40.0 42.5 45.0 47.5 50.0 52.5 55.0 57.5 60.0

(%)

14.6 t7.2 20.2 23.5 27.3 3l.6 36.0 42.2 48.8 s6.4 65.4

334

F


Table 9.6 Approximate values for the minimum capacitive and maximum inductive shunt reactance that can be replacedwith shunt stubs

I

2.17

ef

X".-i"

(Q)

F

34t2 85

r6t2 43

(+2070deviation)

2412 ll8

t2/2 60

(+4.47o deviation : 2-6 GHz)

38/2 78

r8/2 40

27/2 109

13/2 56

X"r-* (O) Xo"-.* (o) Xz,*-(o) X""-^r (Q) X"r* (o)

[ ' '

gr

(+10% deviation)

X"r.* (O)

X""*'" (o)

lot2 1)

z^*(a) ]

10.3

cl re

21t2 t4l

4*n(O)

(+8.3% deviation: 2-6 GHz\

br

A

To give an ideaof the rangeof reactance valuesthat canbe replacedin this way, the minimum capacitivereactanceandthe maximuminductivereactancecorrespondingto a perfectmatchat low frequencies, anda llYo and2lYodeviationat thehighestfrequency in the passband aretabulatedin Table9.6.This is donefor €": 2.17 ande,: 10.3.In derivingthis table,the minimum andmaximumwidth-to-heightratiosweretakenas 0.3 and 10.0,respectively.Theminimumwidth is determined by theamountof (unpredictable) under-etching andtheacceptable resistivelosses.Themaximumratiois determinedby the electricalwidth of the stub. In calculatingthe minimum capacitivereactanceenteredinto Table 9.6, the capacitorwasreplacedwith two parallelstubs(cross-junction). As an exampleof the improvementpossibleby averagingthe deviationacrossthe passband, thereactance corresponding to a passband of 2-6 GHzandmaximumdeviations (0 :29') and+8.3%0 = 39.5") arealsogivenin Table9.6.Theequations of *.4.406 used to calculatethesereactancesare Z o t = 1 . 8 0 8X x 2

(e.37)

Zoc=Xncll'808

(e.38)

and )

F

Zot =1.28 X nL

(e.3e)

Zoc=Xnc lI'209

(e.40)

respectively.

ce I

335

Microwave Lumped Elemenb, Distributed Equivalents, and Microstrip Parasitics

Becausea significantreductionin the deviationin reactanceis possiblein wideband designs by averagingit across the passband,an equation for the optimum characteristicimpedance(admittance)as a function of the inductance(capacitance)to be replacedandthe line lengthwill be derivedhere. When an inductor is replacedwith a short-circuitedstub,the srror in reactanceis givenby û _Zotan0-oZ aL tane-rrr'LlZo

(e.4r)

aLlZ, Under the equality h Zg =:a u.*

(e.42)

^^*L

(9.41) can be changed to *_tan9-Q

(e.43)

lb

gtb

The optimum value for D, and thereforethe characteristicimpedance.can be calculatedby settingthe error at 0.o in the passbandequalto the negativeofthe errorat 06n : 0.* /2, wherez is the relativebandwidth.The resultis

tun}^* tan(O.*/ z) 6= 2 1l * I 0** lu J L 0.*

(e.44)

The optimum value for the characteristicimpedancecanbe obtainedasa function ofthe phaseshift at the highestfrequencyin the passband(0,*) and the reactanceto be replacedby substitutingtheresultof (9.44)into (9.42).Theseimpedances €retabulatedin Table9.7 togetherwith the corresponding errorsin reactance.The error in reactanceis smallwhenthe bandwidthis relativelynarrowandthe electricalline lengthat thehighest frequencyin the passbandis short. The characteristicimpedancerequiredis clearly a weak function of the relative bandwidthand a strongfunctionof the stublengthandreactance requiredat the highest hequencyin the passband. EXAMPLE 9.2

Replacinglumpedcapacitorswith open-ended stubs.

Considerthe matchingnetwork in Figure 9.5 (passband2-4 GHz). Assumingthat theinductorscanberealizedin lumpedformwith negligibleerror,equivalentopen-

336

k

Design of RF and Microrvave Amplifiers and Oscillators

Table 9.7 . The optimum normalizedcharacteristicimpedance(admittance)and the correspondingerror in reactance (susceptance) for a short-circuited(open-ended)stub as a function ofthe line lengthat the highest frequencyin the passbandand the relativebandwidth(u:fr/-fr)

0* (") 10.0 I1.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.O 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 31 . 0 32.0 33.0 34.0 35.0 36.0 3?.0 38.0 39.0 40.0 41.0 42.0 43.0

u.0 45.0 46.0 47.0 48.0 49.0 50.0 51.0 52.O 53.0 54.0 55.0

Zo-"orllarLl; reactance enor (o/o) Yo."rJl.orQ; susceptance error (%) u= 1.5

5.687 5.162 4.724 4.353 4.033 3.756 3.513 3.298 3.107 2.935 2.780 2.639 2.511 2.393 2.285 2.185 2.092 2.006 1.926 1.851 t.780 l;l 14 1.652 t.593 1.537 t.485 1.434 1.387 1.341 1.298 1.256 1.216 1.178 l.l4l t.106 1.072 1.039 t.007 0.971 0.947 0.918 0.890 0.863 0.837 0.81I 0.786

+0.3 +0.3 rO.4 +0.5 +0.6 +0.6 +O.7 +0.8 +0.9 *l.l +1.2 +l.3 +1.4 *l.6 +1.7 +1.9 +2.0 x2.2 +2.4 *2.6 +2.8 +3.0 +3.2 +3.5 +3.7 +3.9 *4.2 +4.5 +4.8 +5.1 +5.4 +5;l +6.1 +6.4 +6.8 +7.2 +7.6 +8.0 +8.5 +8.9 +9.4 +9.9 *10.5 +l 1.0 +l1.6 +12.2

u=2.0 5.693 5.169 4.731 4.360 4.041 3.765 3.522 3.308 3.117 2.945 2.791 2.651 2.523 2.406 2.298 2.199 2.106 2.02t 1.941 1.866 1.797 1.731 1.669 l.6l I 1.555 1.503 1.466 t.406 1.36t 1.318 1.276 1.237 1.199 1.163 t.t28 1.094 1.06t 1.030 0.999 0.970 0.941 0.9t4 0.887 0.861 0.835 0.810

+0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +l.l +1.3 +1.4 *l.6 +1.7 +1.9 +2.1 +2.3 +25 +2.7 +3.0 *t.2 +3.5 +3.7 +4.0 +4.3 +4.6 +4.9 +5.2 +5.6 +5.9 *6.3 +6.7 +7.1 +7.5 +8.0 +8.4 +8.9 +9.4 +9.9 *10.4 +l1.0 *t 1.6 +t2.2 +12.8 +13.5 +14.2 +14.9 +t5.7

u= 3 . 0 5.697 5.173 4.736 4.365 4.401 3.771 3.529 3.315 3.124 2.953 2.799 2.659 2.531 2.415 2.310 2.208 2.t16 2.031 1.952 1.8't7 1.808 1.742 1.681 1.623 1.568 1.5t6 1.471 1.419 1.374 1.331 1.290 1.251 1.213 l.l'7'l 1.142 1.109 t.076 1.045 1.015 0.985 0.957 0.929 0.902 0.8'16 0.851 0.826

u=4.0 +0.5 +0.6 +0.7 +0.8 +0.8 +1.0 +t.2 rl.3 *1.5 +.1;l *1.9 +2.1 +2.3 +2.5 +2.7 +3.0 +3.2 r3.5 +3.8 +4.1 +4.4 +4.7 *5.0 +5.4 +5.7 +5.1 +6.5 *6.9 +7.4 +7.8 +8.3 +8.7 +9.2 *9.8 +10.3 +10.9 +t 1.5 *.12.1 112.7 +13.4 +14.0 +14.8 +16.2 *16.3 +17.1 +17.9

5.698 5.174 4.737 4.367 4.049 3.7'13 3.531 3.317 3.t26 2.956 2.801 2.662 2.534 2.410 2.310 2.211 2.120 2.035 1.955 1.881 1.812 1.746 1.685 1.627 1.572 1.520 1.47 | 1.423 1.379 1.336 t.295 1.256 t.219 1.182 1.147 l.l13 1.081 1.050 1.020 0.991 0.962 0.935 0.908 0.881 0.856 0.83I

+0.5 +0.6 i0.7 +0.8 +0.9 +l.l +1.2 +1.4 +1.6 +1.8 +2.0 +22 +2.4 +2.6 +2.9 +3.1 +3.4 +3.7 +4.0 +4.3 +4.6 +4.9 +5.2 +5.6 +6.0 +6.4 +6.8 +7.3 +7.7 +8.2 +8.7 +9.2 +9.7 +10.2 +10.8 +l L4 +12.0 +12.6 *.13.2 *14.0 +t4.7 +15.4 +16.2 +17.0 +17.8 +18.7

ti:


337

endedstubswill be determinedfor the capacitors(e,: 2.17). It follows from Table9.6thatthelowestpracticalcharacteristic impedance on a substratewith e, : 2.17 is approximately25Q. The susceptance of the 0.485pF capacitoris 12.189mS at 4 GHz,whichleadsto a valueof 3.28for the Yo/(aoQ ratio in Table9.7.Inspection of this tablefor u:2 (4 GHz/2 GHz), yieldsthattherequiredline lengthwill be around17" (at4 GHz) if theerrorvalues arethe sameat thepassband edges.Theerrorin thereactance valueswill bearound l%o.T\e expectederrorfor the0.47'l pF capacitoris moreor lessthe same. If the error is not averagedover the passbandand the capacitorsare transformed exactly at the highest frequency in the passbandinstead,the line lengthsrequiredfor the two capacitors(at 4 GHz) are,respectively, B/=tan-r

7 ":ic

xnc

=tan-r(25 /1000 / (2nx4 x 0.485)=tan-t125/82.041=16.9"

and

pl=16.7 " (0.477pF). The expectederrorsat 2 GHzare 125I tAngL - LI (a rC)l I (l / a rC) = 2.6Yo and2.lYo,respectively. While the error in the reactanceis larger in this case,the performance obtainedin a wide-bandnetworkby replacingthe shuntcapacitorsexactlyat the highestfrequencyin thepassband is oftenbetterthanthat obtainedwhenthe error valuesat the passbandedgesarechosento be the same.The main reasonfor this is that the effect of a shunt capacitoris significantly greaterat the higher frequenciesin the passband whenthe passband is wide. It follows from theabove,thatifthe erroris not averaged,seriescapacitors and shunt inductorsshouldbe replacedexactly at the lowest frequencyin the passband, while seriesinductors(andshuntcapacitors)shouldbe replacedexactly at the highestfrequencyin the passband.

Figure 9.5

The matchingnetwork consideredin Example9.2.

338


9.7 A TRANSMISSIONLINE EQUIVALENT F'ORA SYMMETRIC LOW.PASST.SECTIONOR PI.SECTION Seriesinductorsin lumpeddesignsareoften replacedwith high characteristicimpedance tansmissionlines.It was shownin Section9.3 that the rangeof inductances that canbe replacedin this way is limited. Wherean inductorforms part of a low-passPl-section, significantly better results can be obtainedby replacingthe inductanceand someof the capacitance with a seriesline. Similarly,shuntcapacitorsforming part of a low-passTsectioncan also be replacedwith serieslines. Thesetwo possibilitiesare illustratedin Figure9.6. An exacttransmissionline equivalentfor anysymmetriclow-passT- or Pl-section can be obtainedat any particularfrequencyby equatingthe transmissionmatrix of the sectionto be replacedto that of a transmissionline. The transmissionmatrix of the T-sectionshownin Figure9.7(a)is

lvr' tc L ,r.

jaL(2-@2 LC)t 1t-az tq] j l-azLc

(e.4s)

By equatingthis to

cos(P/) 7zosin(Bf'l I [rrssin(FD cos(P/)I

(e.46)

L

L2

I

t

z

o

h

nVr_---:-__'--!V'-

L

C

0:Bo

L

(a)

Figure 9.6

The partial replacement of (a) a low-pass T-section and (b) a low-pass Pl-section with a seriesline.

339

Microwave Lumped Elements,Distributed Equivalents, aad Microstrip Parasitics

it follows that a transmissionline with the following parametenwill be exactlyequivalent to the T-sectionat the frequencyofinterest(ro):

L.::fz-lJ.ztc1 L,=: l-a'LCA

L

'

(9.47)

C

(e.48)

=-----------;-

l- a'LC

(e.4e) gt=tanl(@Jn)

(e.50)

Excellent results can be expectedwhen a T-section is replacedwith a transmission line and the difference betweenthe characteristicimpedancesand line lengths required for exact equivalents at the low and the high endsofthe passbandis negligible. Altematively, the capacitanceand inductance associatedwith a chosen line section at the lowest and at the highest frequency in the passbandcan be compared. The equations required for this purpose are

(DL = Zn "

sin(B/) I +cos(p/)

(e.sl) (e.s2)

aC = Yosin(p/)

where Io is the inverseof Zo. The equationsassociated with the Pl-sectionequivalentof Figure9.7(b) are

nT ' (a) Figure 9.7

T' (b)

(a) A symmetrical low-pass T-section and (b) a symmetical low-pass Pl-section.

340

L'=


L= l-a'LC

(e.s3)

c,=L1z- o, l-co" LC-

tcl

(e.54)

and

(e.ss) B/= an-t(ro,[t: Cl

(e.56)

are Theinverserelationships aL = Zosin(F/)

(e.s7)

and

ac = Y^-gQ2"

(e.58)

I + cos(pi)

It follows from the equationsgiven abovethat the lengthof the equivalentline for aL/Zoandthenormalized TPl-section is only a functionof thenormalizedreactance a or can aC/Yo,respectively.The following equations be usedto calculatethe susceptance aC/Ys andtheline lengthcorresponding to a specified requirednormalizedsusceptance normalizedvaluefor the reactanceof the inductorin a Pl-section:

{=4ltYo .rL

W

(e.5e)

aL/Zo

(e.60)

and

p/ = tan-r

[-@;d the samesetof equationsappliesto a T-section. With coC,roZ,and Yo,Zointerchanged, corresponding to differentline lengths The normalizedreactanceandsusceptance aretabulatedin Table 9.8. The deviationsin the equivalentinductanceand capacitance

Microwave Lumped Elemens, Distributed Equivalents, and Microstrip Parasitics

341

Table 9.E The normalized reactance/susceptance and susceptance/reactance ofthe componentsofthe lumped Pl-section/T-sectionequivalentof a seriestransmissionline as a function of the line length and the percentdeviationbetweentheselumpedcomponentsand thoseassociatedwith a line length of 10"

p/ (.) 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0 32.5 35.0 3' 7.5 40.0 42.5 45.0 47.5 50.0 52.5 55.0

) /.) 60.0

allZol(oClYo) (-;Y"\

0.1736 0.2164 0.2588 0.3007 0.3420 0.3827 0.4226 0.4617 0.5000 0.s373 0.s736 0.6088 0.6428 0.6756 0.7071 0.7373 0.7660 0.7934 0.8192 0.8434 0.8660

(0.0) (-0.3) (-0.6) (-1.0) (- 1.5) (-2.0) (-2.6' ) (-3.3) (-4.0) (- 4.8) (-5.6) (-6.5) (-7.4) (-8.4) (-9.5) (- 10.6) (- 11.8) (-t2.9) (-r4.2) (- 15.5) (-16.9)

aClYol @LlZo) (-; o/o)

0.0875 (0.0) 0.r0e5 (0.r) 0.r317 (0.3) 0.1539 (0.s) 0.1763 (0.7) 0.1989 (1.0) 0.2217 (1.3) 0.2447 (1.7) 0.267e (2.1) 0.2915 (2.s) 0.3153 (3.0) 0.3395 (3.5) 0.3640 (4.0) 0.3889 (4.6) 0.4142 (5.2) 0.4400 (s.e) 0.4663 (6.6) (7.3) 0.493l 0.5206 (8.2) 0.5486 (9.0) 0.s'774 (10.0)

comparedto thevaluesassociated with a 10" line (samecharacteristic impedance)arealso listedin the table.With the necessary changes,Table9.8 alsoappliesto T-sections. Table 9.8 servesto provide an idea of how much the componentvaluesin the equivalentcircuit changeasthe line length(andthereforethe frequency)is increased.If thepassband stretches from I 0 " up to 20o(octavebandwidth),thechangein theequivalent inductanceis lessthan 1.5%,while the capacitance changesby lessthan0.7Yo. Table9.8canalsobe usedasa designaid whenan inductor(or a capacitor)is to be replacedwith an equivalentline. The changethat canbe toleratedin the inductanceover thepassband would determinethemaximumelectricalline lengthat thehighestfrequency in the passband.The reactanceofthe inductorat the highestfrequencyin the passband should be calculatednext, after which the characteristicimpedancerequired can be calculatedby usingthe normalizedreactance listedin the table.Theparasiticcapacitance is obtainedsimilarly. As an exampleof this, if the inductancevariationshouldbe lessthan 10%,the line length can be 45' at the highest passbandfrequency.It follows from this that the characteristic impedancerequiredis 70.7Q.Theparasiticcapacitivesusceptance required is 5.86mS (0.4142/70.7).

Design of RF and Microwave Amplifiers andOscillators

342

EXAMPLE 9.3

Replacinga lumpedinductor with a line.

a transmission As an exampleof the applicationof the Pl-sectiontransformation, the passband over determined will be nH inductor 2 a series for equivalent line 2-8GHz. with % = 150o, applicationof (9.59)and (9.60)yields that the required capacitanceandthe line length correspondingto an exactequivalentat 8 GHz are C : 0 . 0 5 1p F and

Bt:42.08 ThePl-sectionequivalentfor thisline at 2 GHz(pI = 42.08|4 : I 0'52' ) can be foundby using(9.57)and (9.58).The resultsare I:2.18 nH and C: 0.049pF -7.3% respectively). which arecloseto the originalvalues(within +g.\yo and be obtained by (nanowband cases) Better results can sometimes by lowering the be done can passband. This minimizing the error acrossthe By selecting this iteratively. frequencyat which the transformationis exact (at GHz) and the 5.8 frequencyas 5.8 GHz, the line length becomes29.07" -3.9Yo 8 GHz. The at difference in inductancebecomes3.9Yoat 2 GHz and -1.9o respectively. and2.0Yo, reducesto differencein the parasiticcapacitance

EXAMPLE 9.4

network. Distributedequivalentsfor a lumped-element

Considerthe matchingnetwork shownin Figure9.8. A distributedequivalentover thepassband2-6 GHzwill bedeterminedfor it. This will be doneby replacingthe with two seriestransmission two seriesinductorsandsomeof theshuntcapacitance : will be replacedwith capacitance remaining which the after (Zq 1500), lines is takento be the material of constant dielectric relative The open-endedstubs. 2.17. By applying(9.61)through (9.68) and changingthe frequencyof transformation iteratively, the optimumtransformationfrequencyfor both inductorsis found to be approximately5.74 GHz.Therequiredline lengthsandcapacitanceare


343

2.05nH

l50Q

t50Q

93.2Q 20"

Figure 9.8

(a) The matchingnetwork consideredin Example9.4, (b) a dishibutedequivalentobtained by minimizing the reactanceerors, and (c) an altemativedistributedequivalent(electrical lengthsspecifiedat 6 GHz).

42" and0.03pF for the 3.26 nllinductor and22.2"and0'047pF for the 2.05nH inductor.Themaximumerrorsin theinductanceoverthe passbandare+7.8Yoand +2.lvo, respectively. After subtractingthe capacitancerequired for the series lines, the new valuesfor the shuntcapacitanceare found to be 0.102pF (previously0'194pF)' 0.402pF (previously0.542pF),and0.097pF (previously0.144pF),respectively. ofthe first and last capacitorsarevery high and the error The reactances resultingfrom transformingthemto equivalentstubswill be very small.It follows will be lessthan | 9% if by inspectionof Table 9.7 thatthe errorin susceptance value for the characteristic With this is, Zo:93.2Q. 2.799;that XHC/ Zois equalto for the 0.107-pF 20" are approximately line lengths impedance,the required capacitorand 19" for the 0.097-pFcapacitor. For minimum error,the 0.402-pFcapacitorshouldbe replacedwith a low characteristicimpedanceline. A 25O line will be usedin this case.The correspondingXs. / Zs ratiois then2.647.Inspectionof Table9.7 yieldsthat the error will be approximatelyl.g%.Therequiredline lengthis approximately21". The transformedcircuit is shown in Figure9'8(b). Theou@utvoltage

g


Table 9.9 comparison ofthe input reflection coefficients(s,1)ofthe threenetworksshown in Figure 9.8 Frequency

str (a)

s" (b)

st ' (c)

(GHz)

(dB,")

(dB,")

(dB,")

-9.58 43.0 -8.91 37.7 -8.38 32.7 -7.97 27.9 -7.67 23.3 -7.48 18.9 -7.38 r4.8 -7.3' 1 tr.r -7.46 7.8 - 7. 6 5 5 . 1 -7.92 3.3 -8.28 2.7 -8.65 3.7 - 8.93 6.8 -8.9t r1.8 -8.42 17.5 -1.44 21.9

2.00 )Ja

2.50 ', 1
t

(10.30)

lt,'l=t

( 10.3r )

and

l'rl=t

(10.32)

that is, the stability circles must lie outside the Smith Chart and the input and output with 50Oterminationsmustbe passive. reflectioncoefficientsassociated Insteadof formulatingtheinherentstabilityconditionsin termsof stabilitycircles, the Rollette stability factor (,t) canbe used.Inherentstability is then establishedby the followine conditions:

*_

t - 1 " , , 1 ' - l s+rl l 2l ', , 2ls',llsr' I

(10.33)

1",r"r,1.11",,1'

(10.34)

l",rr,l. 1-ltrlt

(10.35)

Theseconditionscanbe derivedby establishingthe conditionsunderwhich stt, or will be passivefor passive& or S",respectively[3]. This canbe doneby considering s22o respectively' terminations), 1"f,,1or lsr,l wnenlSrl< t o.lS"l< I (passive Aniiample i,f? S-itfr'Ch'artstabilitycircle is providedin Figure 10.5.

The Design of Radio-Frequencyand Microwave Amplifiers and Oscillators

Figure 10.5

10.2.3

371

An example of a stability circle displayedwith someconstantgain circles(G"circles) on a Smith Chart(sourceplane).Note that a stabilitycircle canbeconsideredasthe gain circle with infinite eain.

The Reflection Gain Approach

Steady-state oscillationswill occurin a circuit when [4]

I lf,n.F,o.l=

(10.36)

and

ANGLE [F,rl,] = -ANGLE [fh, ]

(10.37)

where l,n. is the reflection coefficient to the right of the point of interest, and I,n, is the reflection coefficient to the left. These conditions follow easily from Figure l0.6.lf a, is the signal incident on the load (RHS), it will be reflected as Dr: l,r,, dr..Considering the conditions at steady-state, this reflected signal will be the incident signal on the LHS and, assumingno external signal to be present, will be reflected as b" = I,n" f.n, ar..This signal is in tum the incident signal on the load, which implies that 6" : a, (steady-state).This can only be the casefor nonzero a. if f'n, 1,r,,: l. This condition is equivalent to the zero admittance oscillation condition at the common node. This result can also be obtained from the expression for the transducer power gain


372

b":lu, a" =lrr," 1,r," a,

t|3rrt

10.6

aL

Illustration of calculating the reflection gain at a given point in a circuit.

of a one-port.By usingthe equationderivedfor thepoweravailablefrom a sourceandthe constaintimposedby the load(1.101) and(1.79)),it followsthat

V t - -

][l - lr,n,l'] Il - lr,h.12

(10.38)

ll - f,n.r,n,lt

infinity, which will be the case Oscillationwill occurwhenthe gain approaches l, ils was shownabove, when1161116r: At start-up,the magnitudeof f16,f.6 rlust be higherthanunity. Assumingtheonesideto bepassiveandthe otherto be active,(10.36)and(10.37) canbe modifiedto I ' l l rn a s s t vle->; l ' I l^

(10.3e)

'a I l' activeI

and AI.IGLE Ifp*ri,"l

= ANGLE

[1/ fu"tiu.]

(10.40)

Note that (10.39) and (10.40) are essentiallysteady-stateconditions.If the magnitudeof l,n. f,n, is significantly largerthan unity, start-upcannotbe guaranteed,but the two-portwill certainlybe potentiallyunstable. Conditions(10.39)and (10.40)can be detectedeasilyif the invertedreflection coefficientof the active side is comparedto the reflectioncoefficientof the other side (usually the resonatorside) on a rectangularplot as a function of the frequency.The magnitudeof the invertedreflectioncoefficientof the activesideshouldbe smallerthan point for the that ofthe passivesideat the point wherethe phasetracescross(resonance reflectioncoefficients). Becausenoexplicitfrequencyinformationisavailablewhenit is done,thecommon The practiceof consideringonly thesequantitieson a Smith Chartis not recommended. wrong absenceof explicit frequencyinformation can be misleadingand can lead to conclusions.


373

SEROSC 7:1:'1999 1O:32i23

tr

1/f_rtrs

A

f_ht

50.00 50.00

(a)

SEROSC OE

7:1:1999 '10:8:10

e -oR'1 A . OL2

X t/lsllRl I + lz22Ll2

310.0

- 2.1.83E-3 - 0.356 - 0.688

\

\

219.0

\-...-------

- 1.019

214.5 188.0

- r.350

o

lrfi l - 2.013

127.O

- 2.U1

96.50

- 2,676

:>+.-

- 3.m7 - 3.3tt9 2.000

3.000

3.500

,4.000

65.00 35.50 5.000

FREO(GHz) Fosc=3.8'l3sGHz Grefl=2.550d8

6e=46.9025'/GHz

(b) Flgure 10.7

The start-up reflection performanceofan oscillator displayed on (a) a Smith Chart and (b) plot [1]. a rectangular

qF

b

374


The reflectionperformanceof an oscillatoris displayedin Figure 10.7(b)on a rectangularplot. Thereflectiongain( lf,n,l,n,1) andtherateat whichthephasechanges(in degreesper gigahertz)at theoscillatidnfrequbncywerecalculatedandarealsodisplayed. Note that thephasetracescrossat 3.8I 35 GHz andthat the hacefor the invertedmagnitude of the reflectioncoefficientof the active side (RHS) is below that of the passiveside (LHS).Thereflectiongainis 2.55dB at 3.8135GHzandtherateofthe changeinthephase (A (ANGLE Fp"",i,"l-ANGLE [l / f""d""]) I 0f) is -66.9" lGHz. Thereflectioncoefficientsaredisplayedonthe SmithChartin Figure10.7(a).Note that themagnitudeof the invertedreflectioncoeffrcientof the activesideis aeainsmaller rhenthat ofthe passiveside.

10.2.4

The Loop Gain Approach

(a)

l (b)

Ffurt l0.t

(a) calculation ofthe loop gain ofa feedbackamplifier.(b) The circuir usedto calculatethe loop gain (-pl) ofa seriesfeedbackoscillator Il]. (The actualground ofthe circuit is at the point marked"G".)

't


375

The loop gain of an oscillatoror an amplifier stagecan be calculatedby using (seeFigure10.8).Theclosedloop gain(l"t) is givenby feedbacktheory

a",=#

(r0.41)

where16 is the openJoopgain and -plo, is the loop gain (that is, the gain aroundthe loop). The loadingeffectofthe feedbacknetworkon the relevantloops(at the relevant nodes) should be taken into account when the open-loop gain is calculated.The with thesamplingnetworkshouldalsobetakeninto account. feedforwardeffectassociated theeffectiveimpedancein the Whenanoscillatorwith seriesfeedbackis designed, of the open-loopimpedance(Zin-o) to the sum (Zin_.n: is equal Ri,_.r*"dn_"n) input loop (^Z;"_6). Thefeedbackimpedanceis applied from the feedback andtheimpedanceresulting in the input loop is resistance gain, that the negative which implies a functionof the loop -FAo) resistance negative (G6o= and the gain a functionof the loop gaintoo. The loop arelinked by the simpleequationshownbelow: Zin-b

= -GtoopZin-ot

(r0.42)

The negativeresistancein the input loop,therefore,is proportionalto the loop gain. When an oscillatorwith shuntfeedbackis designed,the admittancesat the input in (10.42)mustbereplacedwith therelevant nodeareof interest,andtheloop impedances admittances. If the open-loopresistancein the input loop (seriesfeedbackcase;open-loop at theinputnodein the shuntfeedbackcase)is positive,oscillationswill start conductance at any frequencyat which: l.

The loop gain (gainaroundthe loop) is greateror equalto 0 dB;

2.

The phaseshift aroundthe loop is 0o or a multiple of 360o.

in the input Oscillationsarealwayspossibleif the sumof theopen-loopresistance (series case). feedback negative is the feedback from loop and the resistanceresulting (conductance) may resistance parallel this negative with Suitablereactancein seriesor unity. than bigger Rollette factor inhibit oscillationsandmay evenresultin a of an oscillatorare displayedin Table The open-loopand feedbackimpedances = Zin_or t Zin-n. l0.l . The eflectiveimpedancein the input loop is givenby Zin_"rr It is importantto realizethat the reactancein the input loop will not necessarily resonatewhenthe loop gainis in-phase(oscillationswill startup aroundI 1.5GHz in the in the oscillator consideredin Table 10.1).Resonanceofthe reactance(susceptance) (loop gain conditionat steady-state relevantloop (at therelevantnode)is only a necessary to 0 dB). compressed It shouldalsobe realizedthat the gain will be different arounddifferent loops.This

:'

t76


Table l0.l The open-loopimpedanceof an oscillatoris displayedwith the impedanceresulting from the feedbackand the associatedloop gain [1]

F t F

Frequency

Zn-a

zn-r

Loop gain

(GHz)

(o)

(0)

(dB;')

9.01 -j37.7s 8.30 - D 7 1 n 7.62 -jrr.16 7 . 3 r -j5.27 7.28 -j4.72 7.26 - j 4 . 1 7

-22.80 j10.69 - l8.29 jl0.o9 - 14.85 j9.72 -13.42 j9.55 -13.29 j9.s2 - 13.l5 j9.53 - 13.03 j9.51 -12.88 j9.50 -12.75 j9.49 -12.63 t9.47 - 12.51 j9.45 -12.39 j9.43 -12.27 i9.41 -t2.r4 j9.40 - 10.05 j8.94 - 8.3l j8.59

9.0000 I 0.0000 I 1.0000 I 1.5000 I L5500 r l.6000 I 1.6500 l l.7000 I 1.7500 l 1.8000 I 1.8500 I t.9000 l 1.9500 12.0000 13.0000 l 4.0000

1 )'t

-i't

1)O

-j3.02 -12.42 -jr.88 -jr.34 -j0.80 -j0.26

1.17 7.14 7.tl 7.09 7.06 7.03 6.6r 6.29

6')

70.33 jil.15 jzt.05

-3.76 51.45 - 1.60 41.82 22.45 2.36 0.36 5.23 5.50 35'l.31 5.76 353.94 6.00 350.46 6.24 346.36 6.45 342.02 6.60 337.88 6.72 333.58 6.78 329.14 6.80 324.6r 6.77 319.55 0.32 258.99 -5.29 240.68

canbe appreciatedeasily by consideringatransistorwith both cunent-seriesandvoltageshunt feedbackloops (considerthe casewith significantvoltage-shuntfeedbackand feedback). negligiblecurrent-series With with the gaincompression. Theloop of interestis usuallytheloop associated ofthe will be compression gain compression reason for the main a well-behavedload-line, causedby thevoltageswingacrosstheinputjunction(nonlineartransfer transconductance function). The loop gain for an amplifrer stageis shown in Figure 10.9. Note that the at this frequencyis listedwith theloop gainandthe slopein thephaseresponse resonance amplifier gain of the margin frequencyis alsothe frequency.Theloopgainattheresonance stage.The gainmarginis 20.5dB. In this case,thereis clearlyno chanceof oscillationsat dl. The loop gain for two oscillatorsare shown in Figure 10.10.The oscillation at this frequency frequencyis listedwith the loop gainandthe slopein thephaseresponse beloweachplot. The effectiveloop resistance(sum ofthe open-loopresistanceand the feedback resistance)for theseoscillatorsis negativewhen the 0-dB loop gain level is marked (horizontalline segments)on the plots. Oscillationis not possiblewhenthis level is not marked. The first oscillator is a dielectricresonatoroscillator(DRO) (seriesfeedback oscillator;puckon thegateside).Oscillationswill startup at 15.6435GHzwith a loop gain is - 895"/GHz at thispoint.Theloopphase of 6.961dB. The slopein thephaseresponse

,.*


POWEX2

a - F'lAl

3Tl

7: l:lm t0:S:2

dB

t.m

s.0

- 5.3C1

ru.o

' it.7t

2{.0

-'ll.l7

2m.o

21.$

!60.0

-

PHNE

le_C - 37.S

$.m

- 13.74

40.00

- s.t3

0.m

- s.52

- 40.00

- c.gl

-s.00 25.fl

0.5m

5.m

9.0m

t3.m

l7.m

FREO(GE) FE=t6.S93B2 tup=-20.gldA

Figure 10.9

6O--11.2nl'reHz

and voltage-shunt The loop gain calculatedfor an amplifier stagewith somecurrent-series feedback[l].

againapproacheszero at a higherfrequencybut the loop gain is too low for oscillation whenthis happens. The secondoscillatorwill startup at 4.4819GHz with a loop gainof 4.474d8.11rc slopein the phaseresponseis -399'lGHz at this point. The ioop gain performanceof the oscillatorconsideredpreviouslyin Figure 10.7 plot aswell ason a SmithChart.Notethatthe is displayedin Figure10.11 on a rectangular gain approachis 3.7873 GHz insteadof loop the start-upfrequencypredictedwith gain approach.The loop gain at start-upis 3.813t GHias predictedwith the reflection I 14'lGHz' is 5.225dB andthe slopein thephaseresponse of this oscillatoris well-behavedandthat oscillations Note thatthephaseresponse arenot possibleat the higherfrequencies. Theloop gainof theoscillatoris displayedon a SmithChartin Figurel0' I I (b).The gainwas scaledsothat its maximumwould fall on the edgeof the SmithChart(theSmith Chartshouldbe viewedasa polarplot whenthe gainis considered)' Note that the unity loop gain circle is alsodisplayedon the Smith chart (Figure 10.11(b)).Start-upwill occurif the loop gaintraceis outsidethis circleat the point where thephasepassesthroughzero.Multiple crossingsshouldnot be allowedon the horizontal axis on the right-handsideof the unity gaincircle. to the transistor(T,p) atstart-upis alsodisplayed The loadterminationpresented in Figure10.11(b). Note that the terminationusedon the left-handside (unconnectedside) of the oscillator(Ro,)waschosento be l0k0 (choosinga highervaluemay b-esafer)andthatthe to theoscillatoris 50O(Ror).st,in Figure Theloadpresented s,, displayedis meaningless. two-portterminatedin \r (10kOin this of the side input tO.t t(ty wascalculatedwith the case).

37r

DROOSC

7:1:1999 l0:,{7:,19

da 25.(x)

110.0

21.51

6E.50

18.23

67.00

14.U

45.50

I 1.46

24.0O

I

Loop_G

PHASE

IV-\ , [ \

4.687

- 19.00

r\

1.302

- 40.50 - 62.00

- 2.06i4 - 5.469

Vr

- E.E55 13.000

14.250

15.400

- 83.50

+ - 105.0 20.000

16.750

FREO(GHZ)

Fosc='15.6435GHz Gloop'6.961 dB

6€*495.2755"rGH2

(a)

OSCEXA

e - PltAl

7: l:1s99 10:57:15

dB 65.00

15.00

36.00

A

0.710

7.000

- 6.435

- 22.00

- t3.58

- 51.00 PHASE

L@p_G - 27.87

- 109.0

- 35.01

- r38.0

- 42.16

- 167.0 - 196.0

- 49.30

\,V,2

- 56.45 2.000 Fosc=4.48 1gcl-lz Gloop=4.747d8

3.500

5.000

6.500

8.Un

9.500

- 225.0 12.000

FREO(GHZ) 60=-399.9925"/GHz

o) Flgure 10.10

The loop gain calculated for (a) a dielectric resonatoroscillator and (b) a varactor-tuned oscillator [].

The Desigr of Radio-Frequencyand Microwave Amplifiers and Oscillators

7:'l:1099 1 1 :i l : 1 0

SEROSC 't9.00

75.00

13.92

'O.0O

E.84E

5.000

3.773

- 30.00

\

- 1.303 Lmp_G

- 65.m PHASE

\

- t1.45

- !35.0

- 16.53

- 1t0.0

- 21.61

- 205.0

- 26.68

- 2,O.0

- 3r.76 2.m0

379

4.000

E.000

- 275.0 4 r4.000

FREQ(GHz) FosF3.7873GHz Gl@p=5.225d8

66=-114.2677'rGHz

(a)

SEROSC 7: l:l9gg l l : 5 :E

o

sll

+

Loop_G

L

s22

o

TI-LL

L6_tlu:6.99d8

10.0083 50.00

FREQUENCYMNGE 3.5000- 4.5000GH2

(b) Figure l0.l I

The loop gain andphaseofthe oscillatorin Figure 10.7displayed(a) on a rectangularplot and (b) on a Smith Chart Ul'

380

F


The rateat which the phaseis changingat the oscillationfrequencyis an indication ofthe loadedQ ofthe circuit. In the specialcasewhen a single-tunedresponsecan be assumed andwhentheoscillationfrequencyis alsotheresonance frequencyor a frequency closeto it, the Q canbe estimatedby usingthe following equationsfor a parallelresonant circuit[3]: /.=

G+ jaC +I/(jaL) I G + 7(coo+ Aco)C+ 1/ (7(oro+ Aco)Z)

=

x

I G + ja oC+ jLroC+ I / (7(roo(l+ AorI a ))L I G + j a o C + j L r o C+ ( 1 - A r o/ r c : l l ( , r r o o I )

(10.43)

^R l+ j2Q(La /ror) where the approximation applies at frequenciesclose to the resonantfrequency. (fhis equationexplainswhy the phaseof a resonantcircuit is linear close to the resonant frequency.) Note that at start-upthe oscillation frequencymay not also be the resonance frequency.However,this will alwaysbe the caseat steady-state. It follows from (10.43)that the Q of an oscillator(single-tunedresponse)canbe estimatedas

A=#(^s t 4nfo

p

where the phaseslope (L1/Lf) is specifiedin degreeper gigahertzand the resonant frequencyin gigahertz. By using this equation,theQ for the oscillatorconsidered in Figure 10.10(b) (F* = 4.4819GHz"399 "lGHz) is estimatedto be 15.6at start-up. Note that the loadedQ will decrease asthe transistoris driveninto compression.

10.2.5

l

(10.44)

Stabilization of a Two-Port with Shunt or Series Resistance

Any transistor(two-port) can be stabilizedby addingshuntor seriesresistanceto it. It is sometimesnecessary to addresistance on bothsidesof thetransistor.This may bethe case whentherealpart of!*!zz,z1,ot z22isnegative. The shuntconductance requiredcanbe calculatedby usingthe equationsderived

The Designof Radio-Frequencyand MicrowaveAmplifien and Oscillators

381

for the stabilitycirclesin termsof the f-parameters(Section10.2.1):

(r0.4s)

Grrr*"=Gi+ni and

Gzz,o=Cj + nj

j

(10.46)

requiredon the input side,Grr"otheconductance where G1,,.ois the shuntconductance requiredon the outputside,Gr.'therealpartof the centerof the stabilitycircle on the load plane(admittanceplane),Rr'the radiusof this circle,G"'the realpartof the centeron the sourceplane,andR""the radiusof this circle. canbederivedby derivingequationsfor the Theequationsfor the seriesresistance stabilitycirclesin the impedanceplanefirst. The equationsareidenticalin form to those shouldbereplaced Theonly differencesarethateach)'-parameter for theshuntadmittance. for the Thereasonfor this is clearif the expressions with the correspondingZ-parameter. input/outputadmittanceand impedanceare compared: vt i n _ - ..

Yll

lnlzt

(r0.47)

!zz* rL

and 7 --

ztzzzr

Lin-Lll-'-

(10.48)

2 2 2 +z L

Theform ofthesetwo equationsis identicaland,becausetheinput conductanceand respectively, arecalculatedby takingtherealpartofthe two equations, theinputresistance the equationsfor the stabilitycircleswill alsohavethe sameform. An exampleof the seriesand shunt resistancerequiredto stabilizea Fujitsu thetransistor FFD(35LGtransistoris providedin TableI 0.2.If seriesloadingis considered, in serieswith the input or the outputside. canbe stabilizedby addingresistance l77Q is requiredon theinputsideto stabilizethetransistorat 100MHz, while only 0.82Ois requiredat l l GHz A parallelcombinationof a 200Q resistorand a 1.07-pF capacitorin serieswith the input terminalwill providethe requiredseriesresistanceat may be requiredfor inherent 0.1 GHz and I I GHz. Someadjustmentin the capacitance stability over the completefrequencyrange. requiredis well-behavedandonly a smallamountof Note thatthe seriesresistance loadingis requiredat the higherfrequencies. In contrastwith the seriescase,stabilizationby usingshuntloadingis simply not with increasingfrequency(greaterloadingis an option.The (shunt)resistancedecreases requiredat the higher frequencies),and loadingis alsorequiredon the othersideofthe withytror yrr. Thevalue associated transistorin orderto removethenegativeconductance

F

382


Table 10.2 The series(top) and the shunt(bottom)resistancerequiredto stabilizea transistor[] Frequency (cHz)

Source loading

F r

F I

Frequency (GHz)

t77.0 I18.0 58.4 28.4 17.6 tt.7 7.95 5.69 4.1I 3.03 2.52 t.92 0.82

(R)

1.07pF

(o)

(&)

(&)

9.240 + 4.E9pF

1il.0 7.38

t860 il 0.87pF

Load loading

(c))

12.7k 1.46k 770.0 399.0 265.0 191.0 t45.0 104.0 74.O 44.5 il.8 0.8 0.48

& 166.0 262.0 145.0 70.8 44.2 28.7 18.8 12.8 8.84 6.l8 4.87 3.43 1.32

2000

Sourceloading

& 0.10 0.50 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.0 l t.0

(o)

(R)

& 0.t0 0.50 t.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 t0.0 l1.0

Load loading

(0)

6.03k 279 66.6 6.8E

Ro

64.5 3.50 7.31 4.86 3 .l 6 2.79 3.90 3.88 2.14 0.10 1.09 1.38 0.51

0.46 + 0.3lnF

requiredon theothersideis listedundertheheadings(R,,)for theinput sideand(R) for the outputside. Note againthat ingeneral,theintentionis not necessarilyto actuallystabilizethe transistorin this way, but ratherto evaluatethe degreeof instabilityby gettingan ideaof the resistancerequiredfor inherentstability.Furthermore,even if the goal is inherent stability,betterresultscanusuallybe obtainedby usingtwo modificationsectionsinstead ofone.


383

10.3 TUNABILITY Whena designedamplifier is realized,it may be necessary to tune it to obtainthe exact resultspredicted.Whenthe influenceof the reversetransfergainof a transistor(s,r)is not negligible,its input impedancewill be a functionof the loadtermination,and,if the load changes,whetherbecauseof tuning, temperaturedrift, or a changein load, the input impedance will alsochange.Theconsequent dependence of theinputmatchonthechanges in the outputcircuit (andvice versa)is usuallyundesirable. The tunability factor [5]

u =p Yl aL /rY,Lrl r r l

(r0.4e)

lYrrYrrYrl

-

ltr,+rrl'lr,^l l.,l =lfrY,sec0, /seco,n

(r0.s0)

where Oin= tan-1[B io I Gnf

( 10.s1)

and

or=tan-tfBr/GL)

(10.s2)

is a measureofthe relativedependence of the inputmatchon changesin theoutputcircuit. It is obvious from (10.49)that the tunability factor is a strongfunction of the operatingpowergain.If the gainis decreased enough,the outputcircuit will usuallyhave very little influenceon the input circuit andvice versa" A tunability factorof lessthan0.3 is usuallyadvisable[5]. Whenthetwo sidesof a transistorarecompletelyisolated(s,, : 0), the tunabilityfactorwill be equalto zero. Therelativechangein theoutputadmittanceasa functionof therelativechangein the sourceadmittanceis given by

/ r"",I 6,=lar"", aY,/y, I

I

(10.s3)


384

=b;r"m

(10.s4)

lYtYrr\l

The order of magnitudeof the two tunability factorsareusually the same. in termsof thereflectioncoefficientsis An expressionfoi thetunabilityexpressed given by [2]

ls,rs,frl

(10.55)

ll-srrfrlls,,-Arrl be Becauseoftunability diffrculties, the MAG or MSG of a transistoroften cannot the of idea realistic a more provides (MTG) usually gain realized.The maximumtunable the iterativelyby decreasing gainobtainablewith a transistor.TheMTG canbeestablished acceptable. is factor tunability the associated iain from its MSG or MAG value until It shouldbe notedthatpoortunabilityis not alwaysundesirable'Whena low noise of the input admittanceon the load admittancecanbe stageis designed,the dependence with anoptimumnoisematchbychangingthe ,rr"ldtoi-prove theinpufVSWR associated VSWR loadadmittanceappropriately.This effectcanalsobe usedto improvethe output with an optimumpowermatch. associated

I0.4CONTROLLINGTHEGAINoFANAMPLIFIER gain(G') Thebestway to controlthe gainof anamplifieris to controltheoperatingpower and/ortheavailablepowergain(G,) ofeach stage(referto Figure 10.12). Ifthe noisefigureis critical,thedesignshouldbestartedat theinputside,andifthe The powerperformanceis moreimportant,the designshouldbe startedat the load side' done be would point. This some at up linked and a"rign "* alsobe donefrom both sides whenthe dynamicrangerequiredis high. the If the operatin! power gain oithe availablepower gain of the last stagein transistor (modified) the of side other thatthe is contr;iled,iiis impticitty assumed cascade VSWR will be conjugatelymatched(inpractice,a goodmatchwill suffice:if therelevant with associated mismatch is below2.b,ihe gainwill bewithin 0.5dB of thattargeted).The 10.12' as shownin Figure the lastmatchingnetworkis incorporatedin G7.n",3, of The requirementof a conjugatematchmay be too restrictivewhen the last stage to be could option a high dynamicrangeamplifier is designed,and in this casea better termination source controlthe transducerpowergainof this stagewith the currentloador a singlefor the stagein place(referto Figure10.13). If this approachis followedto design the which stageamplifi"r, th" loadterminationcanbe designedfor optimumpower,after

385


G"r(f)+

G'z(f)-

Grn"t3+

Gr(f)

(a)

G,{.f)

-

-

cnz(f )

.----*

-Gr,.a(.f)

-

GrU)

o)

Gr(f)

(c)

Figure 10.12

Calculationof the powergain of a multistageamplifierwhenthe designis doneby starting (a) at the load, (b) at the source,and (c) from both sides(high dynamicrangecase).

: 386


input network canbe designedto control the transducerpower gain of the amplifier. In this case,the sourceterminationsassociatedwith the best noise performancecan then be selectedon the differentgain circles. If the operatingpowergain,the availablepowergain,orthetransducerpowergain is controlled,the gain is controlledexactlyandno approximations aremadeasis the case when the transistorsusedareassumedto be unilateral.It hasbeenshownin [6] that the errorsmadeby,assuminga transistorto be unilateralareseldomnegligible. Beforecontrollingthe gainof a transistorwith a losslessnetwork,it is a goodidea to first levelits gainby usingresistivemodificationsections(feedbackor loadingsections). The gainto be leveledis usuallythe MAG. Levelingthe availablepowergainassociated with the best noise figure (Gon_oo.) is usually a betterchoicewhen a low-noisestageis designed.Similarly, levelingthe operatingpowergainassociated with the highestoutput poweris usuallya goodidea. Insteadof using lossy modification sections,the gain can also be leveled by designingthe impedance-matching networks(gaincontrolnetworks)to havepositivegain slopes,but this routeusuallyleadsto sensitivedesignsandshouldbe avoidedifthe goal is to designa first-time-rightamplifier.

Gr leveled withNF minimized

Modified transistor I

J

trlgure 10.13

Optimum power match 2

lllustration ofa designprocedurefor a high dynamicrangesinglestageamplifier.

In orderto controltheoperatingor theavailablepowergainof a stage,itis necessary to establishwhatthe sourceterminationsor the loadterminationsshouldbe to providethe requiredgain.Theactualsourceor loadimpedance canthenbetransformed to thatrequired by designingan impedance-matching networkfor this purpose. It will beshownherethatthecontoursof interestarecircleson theadmiffanceplane or on the Smith Chart.The centerandradiusof the constantgain circleswill be derived here. Whenatransistorisinherentlystable,thegaincircleswillbeinsidetheSmithChart (passiveterminations).Itwill be shownthat in this case,it is alwayspossibleto transform theactivegainproblemexactlyto anequivalentpassiveimpedance-matching problem.The equivalent passive problem can be solved by using standardimpedance-matching techniques. As long astheseproblemsaresolvedaccurately, thesolutionssynthesized will alsosolvethe original activeproblem. If a widebandproblem is solved by transformingthe active problem to the equivalentpassiveproblem,thedeviationbetweenthe gaintargetedandthatobtainedmay


387

not be insignificant. In this casean extra stepshouldbe introducedin which the solutions obtainedfrom the equivalentpassiveproblemare optimizedfor the bestactiveperformance. At thosefrequenciesat which the transistoris potentiallyunstable,the bestpoint on eachconstantgain circle can be selectedas targets.It shouldbe notedthat while the relevantgainwill remainconstanton the circumference of a constantgaincircle,the other parameters parameters of interestmay changesignificantly.These may includethe noise figure,thepowergain,the stability,theassociated VSWRs,andvarioussensitivityfactors.

10.4.1

Circles of Constant Mismatch for a Passive Problem

It wasshownin Section8.4.3.1thatthelocusof loadadmittances for whichthetransducer powergainGrof apassivesourcewithinternaladmittance Y":G" +TB"terminatedina passiveload Yr: GL + jBLwill remainconstantis a circle in the linearadmittanceplane with center

Go+ jBo = 12I Gr - LlG,- jB,

(10.s6)

andradius (10.s7)

Rys=2 G,

Similarly,the locusof constanttransducerpowergain is alsoa circle on the Smith Chart

O) Figure 10.14

(c)

(a) The equivalentcircuit relevantto the derivationofthe constantmismatchcirclesand an example of these loci on (b) the admittanceplane and (c) a Smith Chart.

3t8

Design of RF and Microwave Amplifiers md Oscillators

(seeFigure 10.14).Thc paramaersof this circle aregivenby

(10.s8)

*"=ffi

(10.5e)

whereCois the centerof the circle andRoits radius. (10.58)and(10.59)canbederivedfromtheexpressionforthetransducer Equations power gain of a one-port:

^ 1r-lrrl' 1r-lr"l'l

(10.60)

vr =---------:;-

l'

- f"rrl-

where l, is the reflection coefficientof the load termination,and |" is the reflection coeffrcientof the sourcetermination.The derivationis repeatedherefor convenience. It followsfrom (10.60)that

crll- p"rrl'= (r- lrrl'Xt- lr"l') from which it follows that

,

,, r-lr-12,

lr-r"r,l'*fflr,l'= that is,

It-r"rrlt+crltrlt=61 whichcanbewrittenas (l - f"frXl - f"'fil + ofrfj

= ct

canbewrittenas Thisexpression

The Desigp of Radio-Frequencyand Microwave Amplifiers and Oscillators

389

(10.61)

which is the equationfor a circle on the Smith Chart. The centerandradiusofthe circle canbe obtainedfrom this equation,and,after givenhereareobtained. somesimplification,the expressions The importantpoint to graspat this point is that while the problemof a conjugate matchimpliestransforminga givenload(source)terminationinto a specificinput (output) theproblemof gettinga specifiedamountof mismatchis a circleproblem.The impedance, of the load(source)terminationcanthenbetransformedto anypoint on thecircumference relevantgain circle,andthe gainof the passivenetworkwill be asspecified.

10.4.2

Constant Operating Power Gain Circles

It will be shownherethat the contoursof constantoperatingpowergainarecircleson the admittanceplane,as well as on the Smith Chart.The equationsfor both caseswill be derivedhere.The admittanceplanecasewill be consideredfirst. The operatingpowergainof a transistor(seeFigure10.I 5) is givenin termsof its by )z-parameters G^=Pt/Pin

'

(10.62)

whereP, is the power dissipatedin the load,andP. is the powerenteringthe input terminalsof the amplifier. with

lnlzr=P+iQ as defined before, and

l n i Y L = ( G t + g z ) + j ( B i L + b r r ) = G L+ i B L (10.62) becomes.

-

v----

r fl^ llul vL

grrGL'i BtrB't"- PGL -QBL

(10.63)


3ql

Lossless impedancematching network

Figure 10.15

The circuit relevant to calculating the op$ating power gain of an amplifier stage'

By multiplying bothsidesof this equationwith the denominatorof the right-hand sideandai"iai"g themby911G', the following equationis obtained:

c,: -

'GL + 8,: -QBL -lY"l'(GL--s") Ttt

gn

(10.64)

SttG^

This equationcanbemanipulatedinto theexplicitform of theequationfor a circle. The centerof this circle is found to be

G^ + jB^

=[*s-)*q.,(*-u,,)

(10.6s)

and its radius (R^) can be obtainedfrom the equation

-{t* -s,,)-l+*l'.} =c?^ R?^

(10.66)

power Whenthe transistoris inherentlystable[0 < C < l;8r r > 0; grr> 0], the operating transistor the gain circles lie entirely in the right-handsideof the admittanceplane' When is is potentially unstable,thesecirclescrossover into the left-handsideofthe plane,as same at the illustratedin Figure 10.16.Note thatthe gaincirclescrossthe imaginaryaxis to aninfinitevaluefor thegainis alsothe two points,andihatthegaincircleconesponding stabilitycircle on the admittanceplane,asderivedin Section10.2. when a transistoris inherintly stable,an expressionfor the maximumrealizable to the power gain can be derivedby calculatingthe operatingpowergain corresponding maximum equalto zeroin (10.66)'the iuin "i.-"t" with radiusequalio zero.With Rr. set

The Designof Radio-Frequencyand Microwave Amplifiers and Oscillators

391

realizablepower gain is found to be

G.-ro

- l Y r , lt'( 2 s , , ) Grr-or,- Gi

(10.67)

where Gr*o* is the real part of the load termination corresponding to the maximum realizable gain, and Gr' is defined by (10.12). Equation (10.67) can be simplified to

l.,lr =lfll G,_.* vc-

(10.68)

The load termination correspondingto the maximumrealizablegain is given by

r----; -R YL_op,=rlC i "i ' +i n i

(10.6e)

with Br'as definedin (10.12). When a transistoris potentiallyunstable,the maximum operatingpower gain obtainableis, theoretically,equalto infinity. The parametersof the constantoperating powergain circlesdisplayedon a SmithChartcanbe derivedby usingthe expressionfor the operatingpowergain in termsof the loadreflectioncoefficient(1.89):

G." -o -

- lsrl'] l"r,l'It -1",,(1-szzsr)+s,rsr,,srl2 lt-srsrl2

(10.70)

The centerof eachconstantoperatingpowergaincircle is givenby 11 -o

-

S'(iz-

----;---

A's")

I+B.(sr,l'-14'l

(10.71)

and its radius by

(l - 2 klsrrsrrlg, + lsrrsrrl2gl)t/2 Rr=

(r0.72)

lt*",f1",,1'-l{'{ The normalized gain, g., is given by

g. = G. lltrrl'

(10.73)

392


(a)

Figure 10.16

10.4.3

o)

The position ofthe constantoperatingpower gain circlesrelativeto the imaginaryaxis of the admittanceplane,illustratedfor (a) an inherentlystabletransistorand (b) a transistor forwhich C> 1.

Constant Available Power Gain Circles

planeor Thecontoursof constantavailablepowergainarealsocirclesontheadmittance presented ontheSmithChart.Theequations for bothcases will be here.Theadmittance planecasewill be consideredfirst. The availablepowergainof an amplifier(seeFigure10.17)is givenby 6 ^.

_

'D av-O

P*-t

n(r,) E(r,)

(r0.74')

wherePuu_o is the maximum power availableat the output of the amplifier, andPuu_" is the maximumpower availablefrom the soruce. Comparisonof (10.7a) and the expressionfor the operatingpower gain of an amplifier yields that if y,, is replacedwithy22, y" *ith Yr, and I"*with {n, the two expressionsare identical.Becauselzv !n, andlzz are constants,and the relationship betweenI. and I"", is identicalto that betweenY,.andI,", it is possibleto determinethe locusof sourceadmittances for whichtheavailablepowergainof a transistorwill beequal to a specifiedvalueby usingtheresultsobtainedfor theoperatingpowergain.By following this approach,the centerofa constantavailablepowergaincircleis foundto be locatedat


393

Y",

Two-

+

* Y "

)

Port Y..

Figure 10.17

The equivalent circuit relevant to determining the available po*cr gain of an amplificr.

I G* + jBro =

ty,,l'f

.+1.'(*-,") l(+-s,,) L I

(10.7s)

and its radius (R.,) can be obtainedfrom the equation '

R?"=G?" s,,)'-14'l Ir*

(r0.76)

Whendisplayedon a SmithChart,thecenterof eachconstantavailablepowergain circle is given by

g,(sir-dtrr) "'-t*;4115 /1 -

(r0.77)

andthe radius by

-Zklsrrsrrlg" g2")tt2 +lsrrsrrl2 o - (l ,

(r0.78)

The normalizedgain,g", is givenby

go = Go/ltrrl'

(10.7e)

Theseequationsarealsoidenticalin form to the operatingpower gain equations. This follows from the fact that the expressionsfor G, and G, areidentical in form too. An expressionfor G, is shownbelow:

394

Designof RF and Microwave Amplifiers and Oscillaton

n u o" =--

l"r,l'tl-ls"l'l

- l"r(t - srrS") + ",r"r,,s"1' lt- ",,S"1'

10.4.4

(10.80)

Constant Transducer Power Gain Circles

The setof sourceadmittancesor reflectioncoefficientsassociated with a specifiedvalue of the transducerpower gain is againa circle on the admittanceplaneor on the Smith Chart.The sameappliesif the load admittanceor reflectioncoefficientis considered. Thederivationof therelevantequationsfor theadmittanceplaneis basedon the Iparameterexpressionfor the transducerpowergain (referto (l.l l)):

Gr=

+ltr,l'GrG, l0zz+ Yr)(yn + r") - yrryr,l'

(10.81)

If the admittanceI. is considered to be fixed, this equationcanbe usedto find the constraintson )t, to ensurethat G. will remainconstant. It follows, aftersomemanipulation,that the centerof the circle is givenby

jBtr=-rou,*l*l'+ GLrt

(10.82)

and the radius by

Rlr=G?,-G'.,,

(10.83)

where Io*= Gou,*-/3ou,is the output admittance of the (modified) transistor terminated in the source admittance I". Ifthe source admittance is taken to be the independent variable (fixed Ir), the center of the circle is given by

Grr+jB,r=-f -l*l'+

(10.84)

and the radius (R"r) by

R?r=G?r-G?^

( 10.8s)


395

whereZ,n=G^* jB,nis the input admittanceof the (modified)transistorterminatedin the loadadmittanceIr. The parametersfor the circleson the Smith Chartcanbe derivedby using the Spamrneterexpressionfor G, (referto (1.90)):

- lsrl'Xr - ls,l') l"r,l'(t

,1 -I v 7 -

(10.86)

- szzSr)-s,rsr,,S",Srl2 K|(l-sr{Xl

Thecenterof therelevantcircleontheloadplane(S.fixed)is givenby VLT

o-[,.###

_

(10.87)

and its radius by

Rrr =

- r-i;J x.[r*",

(r0.88)

l+ X,

where

Xr=

- ls"l'l l"r,l'{t

(ro.8e)

crll-",,S,1'lS"",l' The equations for the circles on the sourceplane (S, fixed) are

vsT

and

-

(10.e0)

396


R"r =

o['.o-#') (l0.el)

L +X ,

where

xr=

(r0.92)

crlr- "rrs,ltls,,'It

The transducerpower gain circles can be usedto control the transducerpower gain and the noise figure of an amplifier when the load network has already been designed to optimize the power performance, and vice versa.

10.5 CONTROLLING THE NOISE FIGURE OF AN AMPLIFIER

i .1,

It was shownin Chapter2 (Section2.2) thatthenoisefigure of a (modified)transistoris determinedby the sourceimpedancepresentedat its input terminalsby the circuit. It was alsoshownthat the contoursofconstantnoisefigure arecirclesin the sourceplane. It follows from

F = 4nn.

- 4_oo,)t+ (4 - 4-"0,)'l

*nO

(10.e3)

(referto (l J7 a{) that the centerof eachconstantnoisefigure circle is givenby

(

p-r, )

Gr+ jBr=l G" "0,*{}l!l-l*,4-*, \

-

Z

K

*

,

(10'94)

/

while its radius(Rn)is given by

RI = G7- G?_op,

(lo.es)

An expression for the noise figure in terms of the reflection coefficient presented at the input terminals of the transistor can be derived by first modifing (10.93) to

F=FL-.?E - r"_"0,1' G" in (10.96)canbe replacedby usingthe following result:

(10.e6)

397


c"_ 1-lr"l'

(10.e7)

Yo lt+r"12

- t"_"0,1canbereplaced by usingtheequality

lt"

2(1, - f"_oo,)

Y, - Y, oo, -=Yo

(l+f")(l+f"_oo,)

(10.e8)

Substitutionin (10.96)yieldsthe following expressionfor the noisefigure:

r,nllr- a"_"*l'

,

------:--------min ,

-,

(t - lf,l-)l1*f"_"*l

(10.ee)

where

,- * -- R * 4

(10.100)

manipulationof ( I 0.99)yieldsthatthecenterof eachconstantnoise Straightforward given the Smith Chart by is on figure circle /1

\-F

'

-

f"-oo, -

(l0.r0l)

l+cr

where

o=*lt*."-*,1' 4r,,

(10.102)

The radius (,Ro)of eachcircle is given by -o-f"-oo,f"-oor) o, - -o(1 ' (l+a)'

(10.103)

with the optimumnoisefigure is too low or If the availablepowergainassociated performance mustbesacrificedto some noise the VSWRsareunacceptable, theassociated extent. The point with the highestgain on a constantnoise figure circle is usually of interest.This point canbedeterminedgraphicallyby findingthenoisefigurecirclethatjust touchesthe gain circle of interest.A better altemativeis to tabulatethe noise figure with

E

398

,#

Designof RFandMicrowaveAmplifiersandOscillators

ATF35076 7: t:10I ll'.2A:17

O a . + gllA A S22A o SllA'

R0t: R02:

Figure 10.18

50.m

s.m

The optimum noisesourceterminationsfor a modified transistor(S) andthe noise circles associatedwith a degradationof 0' I dB in the noiseftgure.

of interestatdifferentpositionsaroundtheconstantgaincircle(or vice theotherparameters versa). Constantnoisefigure circlesfor a modified (seriesandshuntloadingwereusedon the outputside)AvantekATF35076transistoraredisplayedgraphicallyin Figure 10.18, as an example.The optimum points (S") and the contourscorrespondingto a O.ldB degradationin the noise figure aredisplayedfor a numberin frequenciesin the passband (3.54.5 GHz). The output reflection coeffrcientsassociatedwith the optimum source with a conjugatematchon the terminations(srr), andthereflectioncoefficientsassociated output side (s,,r') arealsodisplayed.Becausethe s,,r' andthe S"tracesarecloseto each with theoptimumnoisematchwill be good(around2'5 other,the input VSWR associated in this case).The squareof the magnitudeof sr,, is alsothe availablepower gain of the of interest. modifiedtransistor;notethat the gain is constantoverthe passband

10.6 CONTROLLING THE OUTPUTPOWEROR TIIE EFFECTIVE OUTPUTPOWEROF A TRANSISTOR It was shownin Chapter2 (Section2.2) thatthe powerperformance(l-dB compression point) of a linear two-port (classA and classB) can be controlledby using the power parameterapproach.An accuratesmall-signalmodelandtheboundarylinesto be usedto constrainthe load line on the llV-plane(intrinsic)arerequiredfor this purpose. The power parameterapproachcan be usedto generatepower contoursfor any transistor,with or withoutmodificationnetworks.Whenanamplifierstageis designed,the actualoutputpower(Pou)is usuallyof interest,while theeffectiveoutputpower(P*,-P'J is of interestwhen an oscillatoris designed.


Figure 10.19

399

Typical small-signalmodelsfor (a) FETs and (b) bipolar transistors..

S-parameters A small-signalmodel(seeFigureI 0.19)canbefittedto themeasured by optimizing initial valuesestimatedfor the componentsin the modelto be used.Any informationavailableon thephysicaltransistoror its model(packageparasitics,lines,etc.) the actualdevice.The shouldbe usedto ensurethat the modelfitted accuratelyrepresents with the operatingcurrentandvoltageat the parameters usedshouldbe thoseassociated powerlevel of interest(thebiaspoint usuallyshiftswhenthe amplifieris drivenhard). If a model is fitted to a packagedtransistorandno informationis availableon the packageparasitics,theprocessis usuallysimplifiedby first fitting an inkinsic modelonly (no parasiticsused)to the parametersat the lower end of the frequencyrangeover which dataareavailable.The packageparasiticscanthenbe introducedduring the S-parameter secondphase. It is usuallya goodideato optimizethe fit to the l-parametersof the devicefirst. canbe targeted. fit is obtained,the S-parameters Whena reasonable deviationfrom the actualparametersis usually a good choice. The least-square Duringthe final stagesof theoptimizationprocess,theZ, error(sumof theabsolutevalues is usually at the differentfrequencies) of the relativedeviationfrom the targetparameters a goodchoice. If accuratenonlinearmodels are availablefor the transistorsused,the results obtainedwith the powerparameterapproachcanbe refinedwith a nonlinearsimulator. Constantoutputpowercontoursfor themodifiedtransistorusedin FigureI 0.18 are displayedin Figure10.20,asanexample.Theloadterminationsat whichtheoutputpower with a l-dBm decrease will be a maximum(Sr)aredisplayed,with thecontoursassociated in the output power, for a numberof frequencieiin the passband.The input reflection with theoptimumpowerterminations(s,,,) arealsodisplayed.Note coefficientsassociated is also the operatingpower gain of the modified transistor.The maximum that lsr," l2 power terminationarelistedwith the gain in Table 10.3. and the associated output

10.7

THE EQUIVALENT PASSM IMPEDANCEMATCHING PROBLEM

for which thetransducer It wasshownin Section10.4.I thatthe locusof loadimpedances powergainof a voltageor currentsourceterminatedin a passiveloadwill remainconstant

400


4TF35076 7:i:lte l2:51: E

o sltw + s2tw A 8 L

Rot: R02:

Figure 10.20

50.00 50.m

The constant output power contours generated for a transistor (padsadded; Y".r: 0.4Y; R.",= 0.0; R./"_,o: l00k0; R7".;o : l00ke) by using the power parameter approach.(s,,, is the inpuireflection paramEterassociatedwith the optimum power load .Sr,and lsr,.l2is the operatingpower gain associatedwith this load.)

is a circle in the admittanceplaneor on the Smith Chart.Theseconstanthansducerpower gain circlesalwayslie in the right-handsideof the admittanceplaneor insidethe Smith Chart.Similarly,it wasshownthatthe constantoperating,available,or transducerpower gaincontoursfor anactivetwo-portarealsocircles,and,ifthe two-portis inherentlystable, thesecircleswill alsobe locatedinsidethe SmithChartor in the RHS of the admittance plane.The constantnoisefigure circlesarealwayslocatedinsidethe SmithChart. Tabte10.3 The maximum outPutpower and the associatedload impedanceand operatingpower gain for the transistorusedin Figure10.20(biaspoint: L5V, l0 mA) Frequency

Load termination

Output power

(GHz)

(0)

(dBm)

(dB)

7.2 7.2 7.2 7.2 7.2

21.74 21.49 21.28 21.07 20.87 20.67 20.46 20.26 20.03 t9.82 19.62

3.5 . 3.6 5.t

3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5

86.0+736.5 85.4+j37.1 84.3+j37.5 83.5+/38.2 82.5+/38.4 81.8+739.0 80.6+7j9.3 79.4+tj9.4 78.8+j39.9 77.6+j40.0 76.9+ j40.6

1)

7.2 7.2 7.2 7)

7.2

Powergain

The Desigr of Radio-Frequencyand Microwave Amplifiers and Oscillators

401

Ir out

IL

G.+

Gr

Passiveoroblem

Activeoroblem Illustration ofthe equivalencebetweena constantoperatingpower gain circle andthe circle correspondingto mismatchinga voltagesourceto a passiveload.

Figure 10.21

By consideringtheactiveconstantpowergainor constantnoisefigurecirclesto be the gaincirclesof a passivesourceterminatedin a passiveload,the problemof finding a of the relevant networkto transforma given loador sourceto fall on the circumferences a complexload to source a complex matching that of to transformed be can circles active of frequencies the of at each done be gain. can power This transducer a specified with passband' the inside stable inherently used is the transistor whenever interest The equivalencebetweena constantoperatingpower gain circle and a passive constanttransducerpower gaincircle is illustratedin Figure 10.21.

10.7,1

Constant Operating Power Gain Case

The equationsrelevantto finding the outputadmittanceand transducerpower gain powergaincirclecanbederivedby setting to a givenoperating equivaient G', + jBm = Go+ iBo

(10'104)

Rr. = Rro

(10.105)

where G.,, Br., andn^, and Gs, Bo,and R"o are the parametersof the constant operating power gain and the constant transducerpower gain circles, respectively.

4OZ


It followsfrom (8.91)and(8.92)that ( ' t

\

\ur

/

(10.106)

G , l + - I l =G o and ^ G"*Jt-

Gr = Ryo

(10.107)

\-TT

G"is elirninatedif (10.106)is dividedby (10.107):

2 - t Gr '1

- - . ^ll - G.

- G, =o-

8

0 ltro

(10.r08)

Gr'

After simplificationof (10.108).it followsthat

++Gl -DGr -qgi - l) = o G2,

(r0.r0e)

It followsfrom (10.109)that

-btJb2 - 4(rX-6)

2(r)

=_i(,_,f,.3

(10.110)

[--.-t I

_+G3-D , -( - - l

2 t . 1l qG6-r)) t f

-Dxz(g'.=-2(g20 l)1p = -2G3-r)+2golffi, = z - z s*t z g o Jas i

(10.1 l 1)


403

Becausegois biggerthanone,only the positivesignin (10.1Il) will yield a value of G, that is biggerthanzero. Equation(10.1Il) canalsobe wriffenas

Gr=t-[".JAl'

(10.r 12)

from which it follows that

G"_out= Rlrcr_oot /(2

l-G,

Br-oot=-BL,

(10.1 l3)

(10.14)

(10.1 r5)

While G*ou,appearsto be a function of Gr-u, it can be shown that (10.113) reducesto the output conductance associatedwith the highest value of the operating power gain and a conjugate match at the input (l/,-ou, = Ii-op, ) . If the Smith Chart circles are considered,the equivalent passive problem is given

by F"-oo, = fi-oot /

-

( 1 0 .r16 )

G r = t G , l l + 4 1 A ,- l )

( r 0 . 1l 7 )

''=^;ffi['-F"-*'l']'

(10.1 l8)

with thehighestoperatingpou,ergain. wherefr_o*is theloadterminationassociated 10.7.2 Constant Available Power Gain Case anavailablepowergaincircleto anequivalent for transforming necessary Theequations load admittance (equivalent input admittance ofthe transistor) and transducer power gain

G' Oscillators Design of RF and Microwave Amplifiers and

404

are

(10.1 1e)

Gt_n= Rrocr-inI (2

(10.120) B ,-in= -Bro (10.121)

IfthesmithChartcirclesareconsidered'theequivalentpassiveproblemisgivenby

(r0.r22) (10.123)

Jr*r* -t> -1

Aû=

lc",lt ^ r

t 2

R""Fr-tl

Lt-lt'-'"1'l

(10.124)

with the highest available power galn' where It" o, is the source termination associated

10.7.3

Constant Noise Figure Case

a constantnoise figure to an equivalentload The equationsnecessaryfor transforming the transistor)andtransducerpower gain are admittance(equivalentinf* ua*io*"" Jf

G 7 = R v r G r n t (z',lc^1 t

(10.12s) (10.126)

Bt=-Brr r2

G,n=t[*

gn.l -l R", )

(10.127)


405

and lr-,n = Fj-*p,

(r0.128)

t

(10.rze)

-

G' , = l G 2 lt+4tAo -l)

4=-H

^ [ , - F-,',Il ' l '

(ro.r3o)

Rilr,-'l-t

where f.,n oo,is the source termination associated with the optimum noise figurc'

DEVICE.MODIFICATION

10.8

The main problem during amplifier synthesisis often not the impedance-matching networksto be designed,but ratherthefeedbackandloadingsectionsthatshouldbe added to the transistorbefore the matchingis done, that is, device-modification[1,7]. The resistivesectionsusedusuallystronglymodifu thetransistorat the lower frequenciesand wherethe gainis low andthe noisefigure is havelittle influenceat the higherfrequencies hieh. Device-modificationhasthe following advantages: 1.

The stability of the transistorcanbe improved.Inherentstabilityover the completeworking frequencyrangeof the transistorcanoften be obtained without degradingthe potentialperformancesignificantly.

2.

Theinherentgainslopeof thetransistorcanbereducedor, ideally,removed ofinterest(frequencyselectivefeedbackand/orloading). overthepassband

3.

The gain-bandwidthconstraintsassociatedwith the impedance-matching problemsto be solvedcanbe reduced.

4.

The optimumgain point canbe forcedto be closerto the optimum noise point. This is usuallyessentialif low noisefigures with low VSWRs are requiredwithout usinghybrid couplersor isolators.

5.

The optimumgainpoint canbe forcedto be closerto the optimumpower point.

With referenceto point 3 above,the differencebetweenthe actualimpedancein

406


?:1:lm l3l6:2t

o stt + 9 1 A g o 8i2

REOUENCY

$.m $.m

NOE

o.tm-1.50mH

Figure 10.22

The S-parametersof the MAR8 die beforemodification' 483(l

Ftgure l0.2il

l9.lnH

A lumped-elementmodification network for the transistorin Figure 10.22lll.

MARS-DIE 7 :r l m 13:21:€

o slr + & t A & o s12

S2lW:18.10d4 SI2MU:19.023d8 FREAUENCY NGE O.lm-

Figure 10.24

l-smoHz

of the modified transistor. The S-parameters


407

place and that required to get the specified performancefrom the transistor can be canbeusedas asa reflectioncoefficientor a VSWR.Eitherof theseparameters expressed a first-orderindicationof the severityof the matchingproblemto be solved. In thefirst exampleatransistor Thesepointswill beillustratedwith threeexamples. will be modified for improvedVSWRs, level gain, and inherentstability.In the second examplea low-noisetransistorwill be modifiedto get the optimumgain matchcloserto flat gain and inherentstability.In the third the optimumnoisematchwith simultaneous examplea transistorwill bemodifiedto gettheoptimumpowerloadcloserto theoptimum gain match,againwith simultaneous flat gainandinherentstability.

EXAMPLE 10.1

Modiffing a transistorfor flat gainandlow inputandoutput VSWRs.

for atransistor(MAR8 die)areshownin Figure10.22.Theinput TheS-parameters VSWRs arepoor andthe gainis slopingdownwardoverthe passband andoutput (0.1-1.5GHz).Thetransistoris alsopotentiallyunstable(f > 0.53). modificationnetworkshownin with thelumped-element Theperformance resistorshownwasusedto in Figure 10.24.The 3900 Figure10.23is displayed gain g,, the modified transistoris The of removethe negative of the transistor. + l.l9 andthe outputVSWR is VSWR is lower than 17.96 0.14dB. The input just stable. is inherently lower than 1.14.The modifiedtransistor EXAMPLE 10.2

Modifying a transistorto get the optimum noisematch closerto the optimumgainmatch.

Thetransistorto bemodified(ATF35076)wasusedasthefirst stagein a low-noise amplifier. The goal was to level the availablepower gain associatedwith the

s21rui12.20d8 s12W: -22.69&a FREOUEilCY MW lffi-4.ffiH2

Figure 10.25

ROt:

m2:

$.m $.@

The S-parameters and the optimum noiseimpedanceof the AtF35076 transistorbefor€ modifi cation (passband3.5-4.5GHz).

408


Table t0.4 The characteristicsof an ATF35076 transistorbeforemodification over the passband3.5-4.5 GHz Frequency (cHz) 2.00 3.00 3.50 3.60 3.70 3.80 3.90 4.00 4.l0 4.20 4.30 4.40 4.50 5.00 6.00 7.00 8.00

Frequency

0.17 0.27 0.30 0.30 0.31 0.31 0.32 0.33 0.33 0.34 0.35 0.36 0.37 0.42 0.51 0.5'l 0.62

Itroer

(GHz)

(dB)

2.00 3.00 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50 5.00 6.00 7.00 8.00

0.130 0 . 19 0 0.220 0.230 0.230 0.240 0.240 0.250 0,260 0.260 0.270 0.280 0.280 0320 0.380 0.440 0.500

MAG

MSG

Go

G@

(dB)

(dB)

(dB)

(dB)

(dB)

(dB)

infinity infinity infinity infiniry infinity infinity infinity infinity infinity infinity infinity infinity infinity infinity infinity infinity infinity

20.69 18.99 18.33 18.21 18.09 17.99 17.07 17.77 17.67 17.58 l7.48 17.39 17.30 16.89 16.16 15.61 15.14

13.45 13.21 l3.l4 t3.12 t3.l l 13.09 13.07 t3.05 13.01 t2.98 12.94 12.90 t2.86 12.64 12.18 11.79 11.42

26.28 22.24 21.14 20.95 20.76 20.57 20.39 20.20 20.01 t9.81 19.61 19.42 19.24 18.3r 16.5? 15.47 14.70

12.26 t2.t3 12.10 t2.10 12.05 12.08 12.07 t2.06 12.05 t2.03 r2.0r I 1.99 1t.97 I 1.84 11.50 11.22 10.96

0.88 0.88 0.87 0.87 0.86 0.86 0.85 0.85 0.85 0.85 0.85 0.85 0.84 0.83 0.79 0.82 0.86

G"(2,,-*)

M(Z'n_Q

NF

6(2*-"n)

(dB)

dB)

21.0'l t9.22 18.17 17.95 17.78 17.58 17.43 r7.25 17.13 17.01 16.87 16.75 16.64 1 6t.0 14.95 13.89 13.28

Fn

Gr

0.03 0.05 0.05 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.0? 0.07 0.07 0.08 0.09 0.1l 0.13

0.13 0.19 0.22 0.23 0.23 0.24 0.24 0.25 0.26 0.27 0.28 0.29 0.29 0.33 0.39 0.46 0.52

l.5E 1.70 1.70

' r.70 1.69 r.69 r.69 1.68 1.68 1.69 1.67 1.68 r.67 1.63 1.39 1.26 1.22

optimumnoisefigure andto getthe optimumnoisematchconditioncloserto that for optimumgain.Inherentstabilitywasalsorequired. Theperformance beforemodificationis listedin Table 10.4.Note that,tis lessthan I andG"(2.,_,) variesfrom 17.95to 16.64dB over the passband. Also notethe largetunabilityfactorbeforemodification(around1.70in thepassband). TheS-parameters andtheoptimumnoiseimpedancebeforemodification are


409

0.23pF

Figure 10.26

The lumped-elementmodification network usedIl].

shownin Figure 10.25.It is clearfrom this figurethat the input reflectionwith a 50O load is severeand that the tracesfor the optimum noise match and the input reflection coeflicient arefar apart. It is important to realizethat the terminationsfor the transistoraretakento be 50O in Figure 10.25andthat the actualterminationswill be different. modificationnetwork Theperformanceassociatedwith the lumped-element displayedin Figure 10.26is listedin Table 10.5.Note that G"(2,,_"1hasbeen leveled(in this casethe gainis level overa very wide band).Thenoisefigure has beendegradedslightly (betterthan0.5dB; previouslybetterthan0.28dB), andthe modified transistor is inherently stable at all frequencies.Also note the improvementin the tunability factor(downto 0.35). The S-parametersand the optimum noise impedancefor the modified traces(50O load transistorareshownin Figure 10.27.Note from the s,, andSnooi termination)that the optimumnoisematchis now much closerto the optimum gain match. Padsandconnectinglinesarerequiredin a realmodificationnetwork.The with a more realisticnetwork(Figure 10.28)is listed in performanceassociated Table 10.6andFigure10.28.Notethatthe gainis now around12dB andthe noise figure around0.44dB. The stabilityhasalsoimproved.

ATF35076 It[ |.a

E d l

+ E 6 q

lM al&

'lEgffire lro-a.@

Figure 10.27

ffi

U: E

& &o

The S-parametersand the optimum noise impedanceof the ATF35076 transistorafter modification with lumpedelements(passband3.5-4.5 GHz)'

410


Table 10.5 The characteristicsof an ATF35076 transistorafter modification over the passband3.5-4.5 GHz (lumped-elementcircuit) [1]

Frequency (GHz)

F I l

r

2.00 3.00 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50 5.00 6.00 7.00 8.00 r0.00 12.00 16.00 18.00

3.62 2.47 2.08 2.02 r.96 1.90 L84 1.80 1.75 t.7l l.68 t.64 l.6l 1.47 1.30 t.l7 1.07 l.0l 1.05 t.t2 l l4

Go

G.

Gr

(dB)

(dB)

(dB)

(dB)

(dB)

(dB)

12.17 12.24 12.41 12.45 12.49 12.52 12.57 12.60 12.63 12.65 12.68 t2.70 12.'?3 12.82 12.90 1 3l.4 13.58 13.62 t2.44 I 1.07 10.65

12.17 12.24 t2.4r 12.45 12.49 12.52 12.57 12.60 12.63 12.65 12.68 r2.10 12.73 12.82 12.90 1 3l.4 13.58 t3.62 t2.u I1.07 10.65

6.18 7.69 8.21 8.30 8.39 8.47 8.55 8.62 8.68 8.74 8.79 8.85 8.89 9.09 9.34 9.49 9.63 9.42 9.13 8.58 8.53

E.66 8.40 8.39 8.39 8.40 8.40 8.41 8.41 8.41 8.41 8.42 8.42 8.43 8.46 8.44 E.56 8.82 9.1I 9.29 9.94 10.26

3.23 4.63 5.09 5.17 5.24 5.3t 5.38 5.44 5.49 5.54 5.59 5.64 5.68 5.88 6.16 6.42 6.78 7.03 1.33 7.81 8.07

r.52 1.30 1.22 1.21 l.l9 l.l8 l.l7 l.16 l.l5 l,l3 Ll3 l.l2 l.l I L07 0.99 0.98 0.99 1.05 1.09 l.0l 1.04

t

I Frequency

Fopt

Go(z",_ol

(GHz)

(dB)

(dB)

2.00 3.00 3.50 3.60 3.70 3.80 3.00 4.00 4.10 4.20 4.30 4.40 4.50 5.00 6.00 7.00 8.00

0.442 0.466 0.478 0.484 0.480 0.486 0.482 0.488 0.493 0.489 0.495 0.500 0.496 0.514 0.543 0.579 0.614

1t.27 I 1.35 I 1.39 I 1.39 I1.40 l l.4l 11.42 I 1.43 I1.44 I1.46 tr.41 I1.48 I 1.50 I 1.56 I 1.48 11.29 n.26

M(z",_.p)

F^

6(2",_')

(dB)

0.12 0.12 0.l3 0.13 0.13 0.13 0.13 0.13 0.l3 0.13 0.13 0.13 0.13 0.l4 0.14 0 . 15 0.16

0.48 0.50 0.51 0.52 0.52 0.52 0.52 0.52 0.53 0.52 0.53 0.54 0.53 0.55 0.58 0.62 0.66

0.24 0.30 0.32 0.32 0.33 0.33 0.34 0.34 0.34 0.35 0.35 0.35 0.35 0.36 0.35 0.35 0.37

4tl


2.5:l 2.65 t.00 {.10 0.todl 0.fin

Shr_h Shr-h st_h SI_Co qf_0

{s.5t6 -t.t7t[-3 lt.{x? 1.216[-3 -6.9{9[-3 nl.% 0.3t? 0.[b -0.991 18.37

l.7l 5.8t

1.n

-o

Ftgure 10.28

D-

Im a6'

a.a r?t'

{ro

a.@ 3.t1'

i.e 0.6'

The topology of a more realisticmodificationnetwork for the transistorin Example 10.2 with the associatedperformance(electricalline lengthsspecifiedat 4.5GHz).

The S-parametersand the optimum noise impedanceassociatedwith the distributedmodificationnetworkaredisplayedin Figure10.29. It is importantto realizethat a distortedpicturecanbe obtainedonly if the Smith Chartresultsareinterpreted.As mentionedabove,the actualterminations with the associated of the modifiedtransistorwill not be 50Oandthe impedances actual terminationswill be different. The performanceassociatedwith the actual terminationshouldbe evaluatedandtargetedduringtheoptimizationprocess.The to the actualterminationsof optimizationresultslistedin Figure10.28correspond interest. "Vswrl" valueslistedin Figure10.28definethe rangeof the Notethat the input VSWR valuesassociatedwith the optimumnoisematch and a conjugate matchon theoutputsideof thetransistor.TheinputVSWR will vary between2.57 and 2.66 over the passbandif the relevantmatchingproblemscan be solved perfectly. Similarly, the outputVSWR (with the optimumnoisematchingnetwork " in placeandbeforematchingtheoutputside)will vary between4.00and4.10.The "VsNMa" valuesare the output VSWRSwere calculatedfor a 50Q load. The relativetothephysical VSWRvaluescalculatedforthe optimumnoiseimpedances terminationfor the stage(50O in this case).TheseVSWRsserveasa measureof the degreeof difficulty of the noisematchingproblem.

412


Table 10.6 The performanceassociatedwith the distributedmodificationnetwork [l] Go

Frequency (dB)

(GHz)

2.00 3.00 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50 5.00 6.00 7.00 8.00

Frequency (GHz)

;

2.00 3.00 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50 5.00 6.00 7.00 8.00

14.55 13.46 13.35 13.34 13.34 13.35 13.35 13.36 13.37 t3.37 13.38 13.37 13.37 1 3l.8 12.72 12.38 12.38

2.18 1.92 t.74 1 . 7| 1.68 1.65 1.62 1.59 t.56 r.54 t.52 1.50 1.49 1.45 1.39 1.37 1.27

Foo, (dB)

0.305 0.386 0.398 0.405 0.4t2 0.409 0.415 0.412 0.407 0.413 0.408 0.414 0.410 0.443 0.483 o.527 0.565

(dB)

(dB)

t4.55 t3.46 13.35 13.34 13.34 13.35 13.35 13.36 13.37 t3.37 13.38 13.37 13.37 13.18 t2.72 1238 t2.38

8.t5 8.97 9.41 9.49 9.58 9.66 9.74 9.82 9.88 9.94 9.99 10.04 10.08 10.19 10.14 9.98 9.94

Go(Z--opr)

M(Z,n_q)

Gr

(dB)

(dB)

(dB)

12.5 t I1.09 10.78 t0.74 10.70 10.68 10.65 10.64 10.63 r0.62 10.62 r0.62 t0.62 10.53 10.34 10.36 10.60

5.30 6.88 7.r9 7.25 7.31

1.23 t.t2 1.04 1.03 r.02 l.0l 1.00 0.98 0.96 0.95 0.93 0.92 0.91 0.87 0.81 0.80 0.81

I.J I

7.43 7.49 7.55 7.60 7.65 7.70 7.75 7.94 8.22 8.51 8.84

F,

6(7.r-"pJ

(dB)

(dB)

13.31 12.50 12.30 12.27 12.26 12.25 12.24 12.24 t2.25 12.27 12.28 12.29 12.30 12.22 I 1.87 I l.5l 11.47

NF (soQ)

G.

0.0E 0.10 0.10 0.10 0.1I 0.10 0 . 1I 0 . 1I 0.t0 0.1I 0.10 0 . ll 0.1I 0.1I 0.13 0.14 0.15

0.32 0.41 0.42 0.43 0.44 0.43 0.44 0.44 0.43 0.44 0.43 0.44 0.43 0.41 0.51 0.56 0.60

0.35 0.35 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.33 0.32 0.36 0.45

Note that while the outputVSWR in this caseis a measureof the mismatch betweenthe outputimpedanceof thetransistor(2"") anda 50Qload,it canalsobe usedasa measureofthe differencebetweenthe actualload (50Oin this case)and


413

ATF35076 7: 1:1990 15:19:43

o slt + s2'l a s22 O SnoDtr

S2l tlAX:?.75d8 SI2MAX: -27.24€dB FREOUENCYRANGE 3.5000 -,a.s({xlcHz

Figure10.29

Rot: R02l

50.00 50.00

The S-parameters and optimum norseimpedance associated with the distributed modification network[].

in this case).If interpretedin this theloadrequiredby themodifiedtransistor(^Z"",' way, the VSWR becomesa measureof how difficult the associatedmatching problemwill be. With a predefinedpassband, this approachusuallyyields good results. The alternative is to calculate the exact eain-bandwidth constraints with the matchingproblem. associated

FSt4120C 7:1:t999 16:8:28

O 6tltv + s2tw A S L o s22f

s.00 50.00

Figure 10.30

The optimum power terminationand small-signalgain for a TexasInstrumentsfoundry FET [] (I/,",= 0.55V;R,",= 1.86Q;Ry'".*: 100kQ: n/" .tn; Biaspoint:8V, 180mA).


Table 10.7 The estimatedoptimum power termination of the foundry FET with the associatedoutput power and small-signaloperatinggain [] Frequency

Load termination

(0)

(GHz)

9.00 9.25 9.50 9.75 10.00 10.25 10.50 10.75 I1.00

Outputpower

22.6r+ jr5.52 22.17+ j15.54 2t.71+ jt5.55 21.25+ j15.54 20.80+j15.52 20.37+ j15.50 19.94+ jts.46 +jl5.4l 19.53 t9.t2 + j15.35

EXAMPLE 10.3

Power gain

(dBm)

(dB)

27.910 27.9t8 27.925 27.933 21.941 27.949 27.957 27.965 27.974

12.541 12.300 12.059 I L903 I 1.750 I 1.530 I1.303 I l.163 I1.041

Modiffing a power transistorto improve its stabilityand the VSWRsassociated with an optimumpowermatch.

The optimum power terminationandthe associatedsmall-signaloperatingpower gain for a TexasInstrumentsfoundryFET (without modification)are shown in to the Figure 10.30and listed in Table 10.7.Note that the tracescorresponding optimum power match (Sr) and the optimum gain match (s2r.') are far apart.The with increasingfrequency.The operatingpowergain(sr,, trace)is alsodecreasing transistoris alsopotentiallyunstable(referto the top panelof Table I 0.8). Themodificationnetworkusedis shownin Figure10.3I . Theelectricalline lengthsof the padsusedarespecifiedat I lGHz. The optimumpowertermination andthegainaftermodificationareshownin Figure10.32.Thenumericalvaluesare listedin Table10.9.

70.EO 2.22"

70.80 8.30"

241f) 70.EO 0.93'

Figure 10.31

70.80 8.30"

70.EO 2.22"

FS|4120C 70.EO 0.92'

The modification circuit designedfor the foundry FET [].

415


Table l0.E The stability and gain of the foundry FET before (top) and after (bottom) modification [J

Frequency (GHz)

2.00 3.00 4.00 5.00 6.00 7.00 8.00 8.50 9.00 9.2s 9.50 9.75 10.00 10.25 10.50 10.75 l1.00 15.00 20.00 25.50

0.21 0.26 0.34 0.40 0.47 0.52 0.58 0.61 0.64 0.66 0.68 0.69 0.69 0.71 o.73 0.74 0.75 0.96 1.04 1.26

Frequency (GHz)

6.00 7.00 8.00 8.50 9.00 9.25 9.50 9.75 10.00 10.25 10.50 10.75 I 1.00 11 . 5 0 12.00 13.00 14.00

L43 1.44 1.45 1.45 1.45 1.46 1.47 1.46 1.46 1.46 |.4'l 1.45 1.44 1.46 l.4l L40 l.4l

MAG

MSG

Go

G.

Gr

(dB)

(dB)

(dB)

(dB)

(dB)

infinity infinity infinity infinity infinity infinity infinity infinity infinity infinity infinity infinity infinity infinity infinity infinity infinity infinity 9.03 6.28

t9.47 17.75 16.54 I 5.59 t4.E5 14.23 l3.68 13.44 13.22 l 3 . lI 13.01 t2.90 12.80 12.71 t2.62 12.s3 12.44 I 1.33 9.03 6.28

13.96 11 . 6 5 9.68 8.05 6.67 5.49 4.4s 3.99 3.54 3.34 3.15 2.94 2.75 2.57 2.40 'r'))

21.42 18.54 16.19 14.41 12.93 I 1.68 10.61 10.06 9.59 9.34 9.09 8.88 8.68 8.45 8.21 8.03 7.86 5.05 2.42 -0.79

13.48 11.21 9.21 7.52 6.06 4.78 3.61 3.08 2.57 2.33 2.r0 1.85 1.62 L40 l.l8 0.97 0.76 -2.17 - 5.16 7.98

MAG

MSG

Go

G^

Gr

(dB)

(dB)

(dB)

(dB)

(dB)

9.63 9.38 9.15 9.04 8.88 8.77 8.66 8.63 8.59 8.50 8.41 8.39 8.36 8.16 8.21 7.9"1 7.60

9.63 9.38 9.l5 9.04 8.88 8.77 8.66 8.63 8.59 8.50 8.41 8.39 8.36 8.16 8.21 7.97 1.60

5.03 4.t2 3.29 2.94 2.56 2.39 2.23 2.O7 l.9t 1.76 l.6l 1.48 1.35 1.06 0.83 0.38 -0.01

8.69 8.23 7.74 7.50 7.20 7.04 6.87 6.75 6.62 6.46 6.30 6.19 6.07 5.73 5.55 4.97 4.3r

4.87 3.85 2.87 2.43 1.96 1.74 1.53 1.32 l.l0 0.90 0.7r 0.52 0.33 -0.07 -0.42 - l.l l -1.75

70{ - 0 . 1I - 1.80 -3.09

416


Table 10.9 The optimum powertenninations of the modified foundry FET with the associatedoutput power and small-signalgain [] Frequency

Load termination

Outputpower

Powergain

(GHz)

(o)

(dBm)

(dB)

9.00 9.25 9.50 9.7s 10.00 10.25 10.50 10.75 l1.00

22.8 + j18.3 22.0 + j18.2 2 1 . 5+ j 1 8 . 0 21.0 + j17.8 20.6 + jr7.6 r9.7 + j17.4 19.3+ j17.2 18.9+ 116.9 t8.3 + j16.7

27.2

8.69 8.59 8.48 8.44 8.38 E.30 8.20 8.r6 8.13

,,a .

27.2 )1 )

27.2 )1 7 zt,5

27.3 27.3

The optimum powermatchis now muchcloserto the optimum gain match. Note that the gain is now very flat (althoughit is on the low side).Themaximum power obtainablehasdecreased by I dBm. The modifiedtransistoris inherently stable(referto the bottompanelofTable 10.8).

FS14120C t: l:1999 '10:5'li5l

o

Sllw

+ s21W A S L o

!

s22W.

52tllAX:6.69d8

FREOUENCI RAI{GE 0.0000- 11.OOoGHI

F

Figure 10.32

Roll RO2:

50.00 50.00

The optimum power termination and small-signal gain for a foundry FET (T€xas lnstrumentsFSI4l20C) after modification [l].

ilc The Design of Radio-Frequencyand Microwave Amplifien and Oscillabrs

10.9

417

DESIGNING CASCADE AMPLIFIERS

At this point the basicknowledgerequiredto designsingleor multistagecascadetype amplifiersare in place.A typical designcycle is outlinedin the flow diagramshownin Figure10.1.Whenthis approachis followed,the designcycleproceedsfrom theloadside towardthe source,or vice versa.A low-noisedesignis usuallydoneby startingthe design at the input side.Whenthe outputpoweris moreimportant,the designis usuallystarted at the load side. the designcanbe Whena multistagehigh dynamicrangeamplifieris synthesized, startedat both sidesandthetwo sectionscanthenbe linkedup with aninterstagematching case,theload networkcan first be network(referto Figure 10.12(c)).Inthe single-stage inputnetworkcanbedesignedtolevel designedformaximumoutputpowerafterwhichthe thegainwith thenoisefigureaslow aspossible(thiscanbedoneby choosingtheoptimum noisefigure pointson the relevantconstantgaincircles). The designof eachstageconsistsof selectinga transistorfor the stage,modiffing it appropriately,andsynthesizinga losslessgain,noisefigure,or powercontrolnetwork for it. If the associatedmatchingproblem is too difficult to be solved properly, the transistorshouldbe modifiedmorestronglyor a differenttransistorshouldbe used. Whenthe controlnetworkfor eachstageis designed,the performancearoundthe relevantconstantgain, noise figure, or output power circle shouldbe evaluated.The optionsto matchto a specificpoint on eachcircle(a pointmatch)or to anyarbitrarypoint in a narrowregion is only acceptable on the circle (circlematch)exist.If the performance a point-matchshouldbe enforced. on the circle circumference, The performanceof a transistorarounda constantnoisefigure circle is displayed in Table 10.10.The following valuesare listed as a function of the anglearoundthe constantnoisefigure circle (Smith Chartcase)in this table: 1.

The reflection coefficient at the point of interest(fr--*o*,lr--J;

2.

The availablepowergain (G,);

3.

The outputpowerif the ouput sideis conjugatelymatched;

4.

The differencebetweenthe actualsourcetermination(50O in this case)and asa VSWR; the sourceterminationrequiredexpressed

5.

The sensitivityof the noisefigure to changesin the admiuancepresented at the input of the modifiedtransistor(6");

6.

The sensitivity of the available power gain to changesin the source admittance(0,);

7.

The sensitivity of the output match to changesin the sourceadmittance (6"").

418


Table 10.10 The performanceofa modified transistorarounda constantnoise figure circle [l] Go

Power

VSWR

6,

68

(o/o)

(%)

Ir.o

6o

lr-*

c)

(dB)

(dBm)

0.0 25.0 50.0 75.0 100.0 125.0 150.0 175.0

10.54 10.86 I 1.56 t2.37 12.87 t2.94 12.73 12.42

1.63 1.58 1.59 l.7l 1.90 2.01 1.98 t.99

9.05 l0.96 il.51 10.26 8.r4 6.18 4.72 3.77

0 .l 7 0.22 0.25 0.24 0.20 0.16 0.l3 0.10

1.85 2.30 2.26 1.55 o.73 o.47 0.58 0.64

0.02 0.02 0.03 0.03 0.03 0.03 0.03 0.02

0.80 0.83 0.84 0.82 0.78 0.72 0.65 0.58

36.3s 40.90 45.82 50.65 54.89 57.97 s9.t2 57.47

225.0 250.0 275.0 300.0 325.0 350.0

11.73 I 1.40 I 1.09 l0.81 10.59 10.50

2.03 1.97 1.90 1.83 1.75 r.66

3.06 3.24 3.70 4.74 6.20 8.r8

0.08 0.08 0.08 0.09 0 . 1I 0.15

0.69 0.73 0.81 0.95 1.21 1.64

0.02 0.02 0.01 0.01 0.01 0.02

0.51 0.53 0.58 0.65 0.72 0.78

44.72 37.05 32.05 30.46 31.65 34.75

f)

Note: The highlighting is usedto indicatethe optimum point on the circle

If matchingto any point on a circle is acceptable(circle match),the equivalent passiveproblemcanbe definedfor the circle asdescribedin Section10.7.Matchingto a specificpoint may alsobe required.Thehighlightingin Table10.l0 is usedto indicatethe optimumpoint on the circle circumference for both cases.

Yr-plane

Tolerance circle

G. Gr-.in G.*

Constant operating power gain circle

Ftgure 10.33

Calculationofthe sensitivityfactor associatedwith the operatingpower gain (6.) [l].


419

The sensitivity factor is calculatedby consideringthe changein the parameterof interestwhen the controlling admittancechangesby l%. Calculationof the operating in Figure10.33.The lowestandhighest powergainsensitivityfactor(6,) is demonstrated andG._.u*,respectively.Thesensitivity gainassociated with thetolerancecircleareG,_,1n factor(6,) is calculatedasthe maximumof

6,r = ABS [(G._,oo- G^) I G,]

(l0.l3l)

and 6^, = ABS [(G. 'in - G') I G,]

(10.132)

High values for any of the sensitivity factors are undesirable.Note that the sensitivityfactorscalculatedareindicationsofthe sensitivityofthe problemto be solved.

EXAMPLE 10.4

3'3-4.4 GHz) LNA (passband An exampleof a single-stage

trl. design,considerthe amplifier shownin Figures As an exampleof a single-stage and10.36. 10.34,10.35, The transistor(NE32484A;optimumnoisebias point) was modified by usingseriesandshuntloadingnetworkson the outputside(0.7pF in parallelwith 165Qand 102Oin serieswith a line). The structureto the right of the transistor position in Figure 10.35 was designed to accommodatethe parallel combination(a gapcapacitoranda chip resistor). capacitor/resistor with Themodificationwasdoneto leveltheavailablepowergainassociated VSWRs.Thetargetfor the anoptimumnoisematchandto improvetheassociated input VSWR wasaround2.5, andthatfor the outputwasaround8.0.Note thatthe input VSWR targetwasthe actualVSWR expectedif the definednoisematching problemcouldbe solvedexactly.TheoutputVSWR calculatedis a measureof the degreeof difficulty of the output match,as discussedabove(the actualoutput VSWR will be 1.0if the outputmatchingproblemcouldbe solvedperfectly). Themodificationnetworkwasalsousedto improvethestability.However, with thenetworkdesigned.The inherentstabilityis not obtainedat all frequencies to obtain inherentstability at all was used output circuit 2.9kO resistorin the frequencies. The step after device-modification was to design the input matching network for the optimumnoisefigure. The input matchingnetworkdesignedis shownin Figure10.36(a). With the input network designed,the output impedanceof the modified transistoris known. The output matchingnetwork was usedto matchthis impe-


gn 8 S

tir g*

q : ild

g 8+

j a z €la q o

eE

' Figure 10.34

E r i es ra, - -; u

HH

The schematicdiagramof the single-stageamplifier of Example 10.4 tll.


421

t6t tl

It

E

Ft l

til l

tl

ll

6 Flgure10.35

is locatedatthe The artwork of the single-stageLNA of Example 10.4Ul. Thetransistor position indicatedwith the mousecursor.

50.00 5.0"

l00O 36.1'

l00O Lll'

48.80 s.29"

(b) Figure 10.36

(a) The input matchingnetwork usedin Example 10.a.@) The outputmatchingnetwork usedin Example 10.4 [l]. The electricalline lengthsare specifiedat 4.4 GHz.

422

Designof RF and MicrowaveAmplifiers and Oscillators

LNA3P5 7.1:1W 17i25,32

o alt rsl a s22 o

SnoCf,

&tMd:10.81d8 St2Md: -26.0sd8 NGE FREOUENCY

R01: R02:

3.m-4.lmH:

Figure 10.37

50.m 50.m

of the amplifier in Example 10.4displayedgraphically[l]. The ^S-parameters

danceto the 50O load. The output matchingnetwork designedis shownin Figure 10.36(b). The final step in the design was to removethe input network and to redesignit for the bestinput matchinsteadof thebestnoisefigure.Thenoisefigure increasedslightly whenthis wasdone. The artworkof the amplifieris shownin Figure 10.35.

Table 10.11 of the amplifier in Example 10.4 Ul The S-parameters

(GHz)

3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50

szz

Jrz

Frequency

(dB) -9.52 -11.25 -12.25 -12.45 - 12.05 - I 1.53 - I l.12 -t0.92 - 10.93 - I 1.20 - l 1.53 - l r.69 - 11.19 -9.84

(') 329.7 302.4 273.1 244.0 218.7 196.1 178.2 160.7 143.1 124.3 101.6 75.0 44.4 13.4

(dB)

(")

-28.95 -28.50 -28.16 -27.89 -27.67 -27.49 -2't.29 -27.08 -26.86 -26.63 -26.39 -26.18 -26.04 -25.99

I l.E 357.8 344.6 331.8 319.7 t07.9 296.6 285.3 273.8 262.1 249.8 236.7 222.8 207.7

(dB)

(")

10.48 71.3 10.66 56.6 10.72 42.6 10.73 29.0 10.71 15.8 10.68 2.9 10.66 350.3 10.66 337.8 10.69 325.r 10.73 312.1 10.17 298.5 10.81 284.4 10.79 269.2 10.67 253.2

(dB)

-24.99 -30.29 -32.83 -29.87 -27.06 -2s.30 -24.27 -23.93 -24,35 -25.84 -29.65 - 45.40 -28.92 -2r.02

339.2 311.3 255.2 215.6 198.5 189.1 181.8 1' 75.3 169.2 163.4 158.1 179.4 3l1.5 304.9

The Designof Radio-Frequencyand Microwave Amplificrs and Oscillators

LNA3P5

423

7: l:l0S 1fti68

dB

r.ooo

6.000

0.9(x)

5.500

0.800

5.000

0.700

,t.5OO

0.600

4.m0

N.FIG

vswR-l

0.400

3.000

0.300

2.500

0.2m

2.m

0.'t0o

t.500

0.m0

1.mo 3.7m

3.900

4.t00

FREO(GH:)

Figure 10.38

'

The noisefigure and the input VSWR of the ampliffer consideredin Exanrple10.4.

The S-parameters of the final amplifierarelistedin Table10.11and are displayedgraphicallyin Figure 10.37.The noisefigure andthe input VSWR are displayedgraphicallyin Figure10.38. Thegain of the amplifieris closeto 10.7dB overthewholepassband. The noisefigure is lower than 0.7 dB. The input VSWR is below 1.8andthe output VSWR below 1.15.TheRollettestabilityfactoris largerthanL I overthecomplete frequencyrange.The expectedl-dB compression point variesbetween-2.8 dBm and 1.2dBm over the passband.

EXAMPLE 10.5

Designinga two-stageamplifier.

As an example of designing a multistage amplifier, a distributed two-stage amplifier will be designedover the passband2-6 GHz by designinga lumpedelementnetworkand usingthe Pl-sectiontransformationtechniquedescribedin Chapter9 to convertthe matchingnetworksto distributedform. In orderto usethis technique,the impedance-matching networksdesignedwill be constrainedto containlow-passPl-sectionswheneverpossible.The^S-parameters ofthe transistor usedarerepeatedin Table 10.12. Becausethe gain-bandwidthconstraintsresultingfrom theinput andoutput impedancesof the transistoraretoo severe,it was decidedto usea voltage-shunt feedbackmodificationnetworkin orderto reducetheseconstraints. More feedback wasusedon thetransistorof the first stagebecausea low input VSWR is required

424


and the constraintsassociatedwith the input impedancesof the FET are mone severethan thoseassociated with its outputimpedance(this is usuallythe case). The feedbackcomponentsareshownin Figure 10.39(a).

Table 10.12 The S-parameters of the Dexcel I 503A'GaAs transistor(chip)

Frequency

(dB) ( ")

(GHz)

-0.26s -22 -0.630 -31 -t.olz -42 -1.412 -53 -t.938 -68

2.0 3.0 4.0 5.0 6.0

'

Jru

stt

(dB) () -30.5 -28.0 -24.4 -23.r -21.9

18 76 69 66 56

szz

(dB) e) 9.99 9.48 9.40 9.48 9.25

159 r 50 143 134 122

(dB)

( ")

-2.270 -2.384 -2.734 -2.975 -4.013

- l0 - 13 - 16 - 19 -22

The specificationsof the output matching network are shown in Table 10.13.Becausea good output matchis required,the operatingpower gain was chosento be as high as possible.The minimum gain of the five-elementoutput matchingnetworkdesignedis 0.955andthedeviationfrom thedesiredresponseis thereforevery small. The specificationsof the interstagematchingnetworkareshownin Table 10.14.Thedesignednetworkis shownin Figure10.39(a).Themaximumdeviation from the specifiedgainresponsewas0.25dB. Thespecifications of theinputmatchingnetworkareshownin TableI 0.I 5. is shownin Figure10.39(a).Thecalculatedtransducerpower Thedesignednetwork gain of the amplifier is 18.65* 0.35 dB, and the input and outputVSWRs are

Table 10.13 The specifications for the initial ouput matching network of the two-stage amplifier designed

Frequency (GHz)

2 J

4 5 6

Sourceimpedance

(0) 86.98- t22.r3 95.97- J28.85 88.90- j44.97 88.33- j52.29 79.8s- j48.70

Load impedance

Transducer power gain

(0) 50.0+70.00 50.0+70.00 50.0+J0.00 50.0+/0.00 50.0+/0.00

1.000 r.000 1.000 1.000 t.000


425

0.l4pF -t2.05nH 3.26nH

0.54pF

0.l9DF

1s0.00 5 5 ."1

0.1pF

Z.02nH

t -t z n n

150.0r) 1 0 . "1

0.66nH

0.3,lpF

150.00 98.6' 150.00 9.6"

(

3 .I l n H

L90nH

0.54pF

0.05pF

rs0.00 51.3"

25JQ 2t.9"

t50.Ocl 28.4"

Figure 10.39

(a) O) (a) The lumped-elementtwo-stageamplifier designedin Example 10.5 [Gr: 18.59+ 0.34 dB; input VSWR < I .8I , outputVSWR < L731and(b) a distributedequivalent[Gp= 18.65* 0. l9 dB; inputVSWR < I .81; outputVSWR < I .861.

_;&, 426


Tsble10.14 The specifications of the interstagematching network of the two-stage amplifier designed Frequency (GHz)

2.0 3.0 4.0 5.0 6.0

Source impedance

Load impedance

(o)

(o) 75.08+"10.84 8r.22+j2.98 81.94- jr.52 -jr.40 85.15 81.44- jl.l9

83.16-j135.9 s3.02- jr02.9 35.56- j77.55 39.93- j68.64 22.69- j46.rr

Transducerpower gain

o.7462 0.8874 0.8802 1.0000 0.8605

Table10.15 for the inputmatchingnetworkof thenro-stageamplifierdesigred The specifications Frequency

Source impedance

Load impedance

(GHz)

(o)

(o)

2.O 3.0 4.0 5.0 6.0

49.95- j1.57 49.89- j2.3s 49.80-73.13 46.69- j3.90 49.56- j4.67

80.13-jI3.83 1 3 9 . 0-07 2 l . l l 102.s0- j79.36 68.13- j64.62 41.80- j37.01


0.9383 0.9685 0.9672 1.0000 0.9783

theoutputVSWR is too high,the Because smallerthan l.8l and2.24,respectively. specificationsin Table 10.l6 wereusedto redesignthe outputmatchingnetwork. of thedesigned shownis theactualoutputimpedance Thesourceimpedance two-stageamplifier. The designedoutputmatchingnetworkis shownin Figure

Table 10.16 The specificationsfor the final output matchingnetwork of the two-stageamplifier designed

Frequency (GHz)

2.0 3.0 4.0 5.0 6.0

Sourceimpedance


(0)

(0) 122.613r.6120.4rr7.0 93.1-

Load impedance

j42.09 j36.89 j3s.73 j34.32 j16.88

50.0+j0.00 50.0+70.00 50.0+j0.00 50.0+j0.00 50.0+.70.00

0.944 1.000 l.000 l.000 1.000

TheDesignof Radio-Frequency andMicrowaveAmprifiersandosci'ators

427

10.39(a).Thetransducerpower gain of thefinal amplifieris lg.6 + 0.34dB, the input vswR is smallerthan r.69, andthe outputvswR is smallerthanr.72. this stagea distributedequivalentcanbefoundby usingthe techniques outlinedin chapter9. The distributedamplifieris shownin rigur"er0.39(b).The electricalline lengthsare specifiedat 6 GHz.The transduc"r-po*". g;i, irii" amplifier is 18.60+ 0.34 dB, andthe input and outputvswR; are smallerthan 1.80and 1.86,respectively. Thehigh impedance capacitance stubsinthedesignedamplifiercanbeneglectedwithout significantlydegradingthe performance. EXAMPLE 10.6

A three-stage LNA (3.7-4.2 GHz;NF : 0.65dB)

A three-stage amplifierdesignedfor thepassband 3.7-4.2 GHz will be considered in this example[]. Notethat it is usuallya goodideato overdesignan amplifierin bandwidth. In this casethepassband wasextendedto 3.5-4.5GHz.Adding 100MHz on each sideis usuallyadequate. The artwork (soft substrate;biasingdetailsnot shown)of the amplifier is shownin Figure 10.40andthe schematicis shownin Figure 10.41.The transistor usedwastheNEC NE32484A(optimumnoisefigurebiaspoint).In Figure 10.41, the input stageis shown first, followed by the other stages.The samedevicemodificationtopology was usedin all threestages(differentcomponents).The initial input matchingnetworkwasdesignedfor optimumnoise.Theothercontrol (matching)networkswere designedto level the overall gain (MAG). The final (output)matchingnetworkwasdesignedto minimizethe outputVSWR. Note that the device-modification in the secondandthird stageswas only donewhen the designof the previousstage(s)was completed.The actualsource impedancepresentedto the relevantstageand the performanceof the stage(s) alreadydesignedwerethereforetakeninto accountwhenthemodificationnetwork was designed. i , :

n

ttll

l\-_n -

Figure10.40

6

l

l

;i l ll\+l

- t

F

l l l l silHcon ERJH ei l

. :

lls

dt

rJ lla tl

l

u

The microstrip arhvork of the LNA consideredin Example 10.6 (biasing detailsnot shown).

428

Design of RF and Micrmnrrc Amplifiers and Oscillators

(c) Flgure 10.41

The schematicsof(a) the input stage,@) the secondstage,and (c) the outputstageofthe amplifier consideredin Example 10.6.

With the basic designcompleted,the interstagematchingnetwork on the input side was resynthesizedto level the overall gain and to improve the input VSWR. ofthe LNA aredisplayedgraphicallyin Figure10.42and TheS-parameters numericallyin Table10.17.ThegainandtheoutputVSWR aredisplayedin Figure 10.43,andthe noisefigureis displayedin Figwe 10.44withthe inputVSWR' The Rollette factor for this amplifier is greaterthan 3l over the completefrequency range.


Ro,t: R02:

Figure 10.42

50.00 g).(x)

The S-parameters and the optimum noiseimpedanceof the LNA consideredin Example I 0.6 displayedgraphically.

LNA3P7

7: l:1999 l7:54:0

da

r.m

6.000

30.80

5.500

21.N

5.000

23.80

,1.5(x)

20.40

4.0o0

GAIN

vswR-o

13.80

3.m0

10.20

2.50()

8.o{to

2.000

3.400

1.500

0.m0

r.000

3.700

3.900

4.100

4.30,0

4.500

FREO(GHZ)

tr'igure 10.,f3

429

The gain and the output vSWR of the LNA consideredin Example 10.6.

430


LNA3P7

x - LFt

7: i:1099 17:tl:lg

A -\6YYRI

1.q)0

6.0(n

0.9(x)

5.500

0.E00

5.000

0.70,0

4.500

0.600

4.(X)0

N.FIG

VSWRI

0.400

3.m0

0.300

2.500

0.200

2.(x)0

0.100

L500 't.q)0

0.000 il.l00

3.900 FREO(GHZ)

Figure 10.44

The noise figure and the input VSWR of the amplifier consideredin Example 10.6

Table 10.17 The S-parameters of the LNA consideredin Example 10.6 [l]

Frequency

stt

(GHz)

(dB)

3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.10 4.50 4.60

- r4.05 - 16.83 - 18.80 - 19.70 - t 9.84 - 19.75 - 19.76 - 19.95 -20.46 - 21.01 -2r.04 -t9.72 - 17.03 - 13.84

Jrz

(") 265.7 243.6 2t8.8 192.3 168.4 148.2 t 30.7 I14.0 96.3 74.6 46.2 13.3 342.8 317.8

(dB) -85.07 -83.75 -82.62 - 8 l. 6 7 -80.85 -80.r2 -79.46 -78.80 -78.17 -77.49 -76.78 -76.05 -75.36 -74.79

Jzr

(") 87.5 59.6 33.1 8.1 344.3 321.4 299.4 278.2 257.1 236.0 214.6 t92.2 168.6 143.2

szz

(dB)

(')

32.28

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Design of RF and Microwave Amplifiers and Oscillators

Design of RF and Microwave Amplifiers and Oscillators

rf and microwave oscillator design

RF and Microwave Transmitter Design

Design and Control of RF Power Amplifiers

Handbook of RF and Microwave Power Amplifiers (The Cambridge RF and Microwave Engineering Series)

Microwave and Rf Design of Wireless Systems

Microwave Transistor Amplifiers: Analysis and Design

Microwave Transistor Amplifiers: Analysis and Design

Microwave and RF Engineering

RF and Microwave Transistor Oscillator Design

Microwave and RF Design: A Systems Approach

Design of Linear RF Outphasing Power Amplifiers

RF The Rf And Microwave Circuit Design Cookbook

High Efficiency RF and Microwave Solid State Power Amplifiers

RF and Microwave Microelectronics Packaging

Nonlinear microwave and RF circuits

RF and Microwave Radiation Safety

RF and Microwave Microelectronics Packaging

The RF and microwave handbook

Nonlinear Microwave and RF Circuits

RF and Microwave Radiation Safety

RF and Microwave Radiation Safety

The RF and microwave handbook

The RF and Microwave Handbook

Nonlinear Microwave and RF Circuits

Microwave and Rf Design of Wireless Systems - Book

Switchmode RF power amplifiers

Modeling and Characterization of RF and Microwave Power FETs (The Cambridge RF and Microwave Engineering Series)

Essentials Of RF And Microwave Grounding

RF Power Amplifiers

Design of RF and Microwave Amplifiers and Oscillators