-
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)
Characterization and Analysis of Linear Circuits at RF and Microwave Frequencies
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
characterization and Analysis of Linear circuits at RF and Microwave Frequencies
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
characterization and Analysis of Linear circuits at RF and Microwave Frequencies
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
Design of RF and Microwave Amplifiers and Oscillators
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
Design of RF and Microwave Amplifiers and Oscillators
+ 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.
"-
Characterization and Analysis of Linear Circuits at RF and Microwave Frequencies
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 {
Characterization and Analysis of Linear Circuits at RF and Microwave Frequencies
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{^a^ - 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
Design of RF and Microwave Amplifiers and Oscillators
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
Design of RF and Microwave Amplifiers and Oscillators
_ =-+
'), '
- 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) -
characterization and Analysis of Linear circuits at RF and Microwave Frequencies
,[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.
*.
Characterization and Analysis of Linear Circuits at RF and Microwave Frequencies
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'
{
Characterization and Analysis of Linear Circuits at RF and Microwave Frequencies
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 -
Design of RF and Microwave Amplifiers and Oscillators
[ 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
Characterization and Analysis of Linear Circuits at RF and Microwave Frequencies
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 -
characterization and Analysis of Linear circuits at RF and Microwave Frequencies
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
Design of RF and Microwave Amplifiers and Oscillators
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
"
= a'(Jo) [r" - s''(7
(r.r24)
lcading to
S''(.1'or)S(7'ar)= 1,
( 1.12s)
'
in theseequationsthe superscript*' indicatesthe transposedconjugateof the relevantmatrix. It is clearfrom (1.125)thattheinverseof thescatteringmatrixof a losslessnetwork conjugate,that is, is constrainedto be equalto its transposed
s -'Ur) = S ' ' ( / r o )
(r.126)
A matrix whose inverse is equal to its transposedconjugateis called a unitary and sufficientconditionfor a matrix to be unitaryis that its columns mtix. A necessary (or rows) shouldbe mutually orthogonalunit vectors[3]. In termsof the elementsof the scatteringmatrix,this impliesthat the following equationsmustbe satisfied: ,v
) = 6,.r )si (,rco I su(7
(r.r27)
&
r {
t I
Characterization and Analysis of Linear Circuits at RF and Microwave Frequencies
I
4l
I
I
= d,* (/o)s; (.rar)
I
(1.128)
I
I
=0,if 6* =1, if i=k). f "-*6,, istheKroneckerdelta(5,.1 i*k; magnitude of each row or column vector of the on the The unitary constraint I . two relationships on the elementsof eachrow and :-ring the following matrix forces t -mn, respectively: I I
If
N,
o
I - 1;ro;su(.ro)=|,1",r(;t)1"=t J=r
(r.r2s)
l
r
r
Nt
.p =, = !lsr,(rr,r)lI I :,, t rco)sr,(rro) J=t
(l.t30i
r
I
- magnitudeof eachelementof theS-matrixof a lossless(andalsopassive)networkis I :e boundedby unity;thatis, *I I I
II
11
/o))l<
(1.r3r)
By applying(l.l2g) and(1.130)to a two-portnetwork,it followsthat
I
I I tr'f'=l-1",,(rr)l' ur)l'= l-1s,,(7or)l'
(1.133)
I
= t-1s,,(7co)l' ico;12
(l'134)
IX ,.-
- | ''',,)l' ' t = I ls,r(7co)l'
(1'135)
It , E
r
l
io)s;(,r'ro)= -srt(;ro)sn(7ro)
(1.136)
= -s,r(;ro)sr(7ro) I t t;to;si,(-rro)
(1.137)
II
I
I
:.orning(1.133)and(1.134)yields
42
Design of RF and Microwave Amplifiers and Oscillatore
1",,("rt)l= l"r0r)l while (1.134)and(1.135)canbe combinedto showthat
( 1.13e)
l",r0t)l=l"r,(ror)l Equation(1.138)canbe extended to t
t E
stz 0'ol) = szr(/'c'l)
(1.140)
wheneverthe network consideredis passiveand reciprocal.This canbe provedeasily by usingthe reciprocitytheorem. By combining(1.137) and(1.140),it canbe shownthat
rz-ro'') s,,7,,l)= ",io.,r)
(1.141)
szl0o) Theserelationships will proveusefulin laterchapters. EXAMPLE 1.8
| |
Calculationof theS-parameters of a losslesstwo-port.
As an example,the ,S-parameters of the losslesstwo-port in Figure 1.22will be derived,andsomeof the relationshipsgivenabovewill be illustrated. Becausethe normalizingimpedances in Figure 1.22 arepurely resistive, it follows that
ir(s)=ffi=n,{-s'l [
; and the input reflection and forward transmissionparameterscan, therefore,be determined by using(1.68)and(1.69),respectively. Theequivalentcircuit correspondingto a2 = 0 andthe chosennormalizing impedances areshownin Figure 1.22. The input impedancenecessary for determinings,,(7ro)is givenby I
Z^(t) = J-+ uo2+.rL
sL
and s,,(-7'ro)is therefore given by
-
-
characterization and Analysis of Linear circuits at RF and MicrowavsFrequencies
Ro,
Eo,
43
Ro2
E02
c (a)
&'
E
Eo,
c (b) Ror
Ro, C
Eo,
(c) 1.22
(a) The losslesstwo-port underconsideration;(b) the equivalentcircuit usedto determine s,,(jo) and sr,("/o); (c) the equivalentcircuit usedto determinesrr(jo) and s,r(7'
=#M1,,=o srr(-r.,) .-! i,
I
Gor+jaC _ t
+ iaL- R",
(r.r42)
^+7'
- R0rR02C) _ (R02 Ror o1ICR02)+ /o(I R o ,+ R o , - a ' L C R , , + j a ( L + R o r R o 2 C )
t
,
(r.143)
Design of RF and Microwave Amplifiers and Oscillators
of (1.142)and(1.143)yieldsthat Comparison
.:
lr,,Ut)l=ltr(t't)l as expected. The forward transmission parameteris given by
= -2"[R &, !P-l sz,(,lro)
Eorl"z=o
'.:
+
=, J oo'^*
Ro, Ro,+ Ro, Ro,- a2 LCR,, + ja (L+ cnorRo2 )
(1.144)
/o' s,z(.lro)=-2\tRod; l- ^
t
Eorl"t=o
^ .Frd,
R,
Ror* Ro, Ro,- o2 LCR', + ja(L + cRorRo2)
(1.14s)
Equations(1.144)and(1.145)areclearlyidentiialand,therefore, = srz(J'ro) szr(,1'ro) as expected. The same result can be obtained directly by application of the reciprocitytheorem. It is a simplematterto showthat ")
")
l"rtl-=l-1"''1and that
lr'rlt= 1- l"rrl' t.5.f 0
Conversion of ,S-parametersto Other Parameters
The schematic representationof the S-parameterand related relationships in Figure 1.1t, can be used to derive expressionsfor the conversion of the normalized S-parametersto th< other S-parametersas well as to Z- ot /-parameters. The results are
-
Characterization andAnalysisof LinearCircuitsat RF andMicrowaveFrequencies
45
S = nol/25rPo-tlz
(1.146)
s' = zo-'SYZi
(r.r47)
; t = l z + z o 1 - t 1-zz i 1
( l .r 4 8 )
s ' '= - [ f + r o 1 - t J r - r o 1
(1.14e)
| = YolI,- s' 7[1,+.s21-t
(1.150)
IE
'' Zo-' rb
(1.151)
areall purelyresistiveandequal the normalizingimpedances
(1.152)
3-S'=Sz
EXAMPLE 1.9
Derivationof theexpressionfor thereflectedvoltagesof an N-port in termsof the incidentvoltages.
It follows by inspectionof the diagramin Figure 1.10that I=W is equivalentto I,-1,=Y(V,+V,) *hich is equivalentto
Zotv, - z;tv, = w, + w,
-: t*${$'{-& \:,
This equationcanbe manipulatedto
J
,
Y , = - ( y + r o ) - ' 1 -r y ; ) v ,
ilMf
which yields the requiredexpression. L
,
-
'
.
Designof RF and Microwave Amplifiers and Oscillators
- ,?.--.:I
REFERENCES
l. s-parameterDesign,ApplicationNote 154,PaloAlto: HewlettPackard,April1972. Amplifier and Oscillator 2. MultiMatch RF qnd Microwave Impedance-Matching, (Pty) Ltd; http:/iwww.ampsa.com.' West,RSA:Ampsa Software,Somerset Synthesis 1998. 3.Chen,W.K.,TheoryandDesignofBroadbandMatchingNetworla,oxford:Pergamon Press.1976.
SELECTEDBIBLIOGRAPHY Amplifiers,NewYork: JohnWiley andSons,1975. Carson,R.5., High Frequency
'i" t'
D 3. It5
* : IF
3r thr
t,: rb. tb ti" Je
aa,-
||l
{
CHAPTER 2 CIIARACTERIZATION AND ANALYSIS OF ACTIVE CIRCUITS AT RF AND MICROWAVE FREQUENCIES : I
INTRODUCTION
--Eracterization andanalysisof linearcircuits in termsof Y-,Z-, Z-, andS-parameters were . .sideredin Chapterl. Whenactivecircuitsaredesigned,thenoiseperformance andthe .:Dutpowerarealsoof interest. Noiseparameters areusedto characterize thenoisebehaviorof linearcircuitsatRF -.{ microwavefrequencies. Theseparameters will beconsidered in Section2.2.Thenoise linear follows easily parameters. of a circuit from the noise Therelevantequations fture " . alsobe derivedin Section2.2 . Noisecharacterization andanalysisin termsof equivalentcircuitsandcorrelation will also be consideredin this section. The effect of feedback and loadine on the :. j. parameters of a transistor can be established easily by using noise correlation Calculation of these effects will be consideredin Section 2.2.2. arices.
Thepowerobtainablefrom a linearcircuit(classA andclassB) is a strongfunction rt tm biaspoint andthe intrinsic voltageandcurrentassociatedwith eachofthe transistors rcrt- The power level at which the intrinsic output current and/or voltage startsto clip point of a linearcircuit I I ]. The mlly providesa closeestimateof the I -dB compression rps approachto calculatingthe maximumoutputpowerwill be consideredin Section . : anda new setof parameters (powerparameters) will be introduced[2]; thesecanbe .-',Jto simplifu calculationofthe expectedoutputpower. Thepowerparametersmapthe :'nsic voltagesin eachtransistorto the externalvoltagesand also map the intrinsic .:rrt currentto the intrinsicvoltages. The relationshipbetweenthe intrinsic load andthe externalload of eachtransistor -.cwseasilyfrom the powerparameters.The derivationwill be consideredin Section ' " Thepowerparameterscanalsobe adjustedeasilyto incorporatethe effect of feedback u ' .r'loadingor any changein the configuration(common-source, etc.). common-gate, :heseaspectswill alsobe consideredin Section2.3. A setofpower parameters is associated with eachtransistorusedin thecircuit.The Ence of eachtransistoris consideredwith the othertransistorsin the circuit assumed 47
4t
F
F
Design of RF and Microwave Amplifiers and Oscillators
to be id€al. The output power is mainly determinedby the stagein which voltage and/or currentclipping first occurs.Thegeneralcasecanfollow an approachsimilarto whenthe outputpower of a cascadeof power amplifiersis calculated.The interceptpoint andthe ldB compressionpoint of a cascadeareconsideredin Section2.3.2. A modelis requiredfor eachtransistorusedin the circuit in orderto calculatethe powerpaftlmeters. Themodelusedshouldprovidea goodfit overthe completefrequency rangeover which dataareavailableand shouldaccuratelyrepresentthe intrinsic part and the parasiticsof the actualtransistor.Conventionalsmall-signalmodelswerefoundto be dcquate for this purpose.
2.2 NOISE PARAMETERS Insteadof consideringthe noise contributionof eachphysicalnoise sourcein a linear network,its noisecontributioncanbe modeledin termsof equivalentnoisesourcesat its input and/or output ports or by using correlationmatrices.Both approacheswill be consideredhere.Therelationshipbetweenthe equivalentnoisesourcesor the correlation typically suppliedfor a transistor(F.in,f,_o',,andRn) matricesandthe noiseparameters will alsobe established. at thebiaspoint Thenoisefigureof a transistoris a functionof its noiseparameters of interest and the sourceimpedancepresentedto its input terminalsby the circuit. The dependence ofthe noisefigure on the sourceimpedancewill alsobe considered. The equivalentnoise sourcesor the correlationmatricescan be usedto find thc networksin termsof for parallelnetworks,seriesnetworks,or cascaded noiseparameters the noise parametersof the individual networks.The influenceof matchingnetworks or of a transistorcan filters or of addingseriesandparallelfeedbackon thenoiseparameters using the results. established easily by be networkswill alsobe considered. The noisefigure of cascaded
22.1 }
}
Modelingthe NoiseContributionof a Two-Portwith EquivalentCircuits
l}'
*
Tbc noisegeneratedby anytwo-port devicecanbe modeled with two equivalentpartially areshorm current,andvoltagerepresentations correlatednoisesources[3]. Thecascade, areequivalentto the Z-parameter,I-parameter,andL in Figure2.l. Theserepresentations prameter approaches, respectively. Thetwo sourcesused in eachrepresentationarepartially conelated.Thecorrelated od uncorrelatedpartsin eachcasearedefinedby the following setof equations: I,(t) = I -(t) + Y*,|/,(t)
(2.1I
Ir,(t\= Ir^(t)+ X,I,G)
(2: -
Characterization and Analysis of Active Circuits at RF and Microwave Frequencies
In(t)
II
Noiseless Two-port
49
Iz,(t)
o)
(c) Fgure 2.1
Equivalentcircuits for tle noisecontributedby a two-port device:(a) cascade,(b) current, and (c) voltagerepresentation.
Yr(t) = Yr,"(t)+ X "Vr,(t)
(2.3)
for the 8r' usingthedefinitionof thenoisefigureandtheseequivalentcircuits,expressions e figure in termsof the equivalentnoisesourcesand the correlationfactorscan be :cnved.This will be doneherefor the cascaderepresentation. The noisefigure of a deviceis definedasthe ratio of the total noisepowerat the ,..i to that which would havebeendeliveredto the loadif thedevicewasnoiseless,that 6.
,.
L
Q.4)
Pno-ide
: , 3
If the spot noise figure (narrowband noisefigure at a particular frequency)is 'idered,the noisepowerat the outputin the idealcaseis givenby u(f ) = krBG r(f )
(2.5)
Design of RF and Microwave Amplifiers and Oscillators
50
Kelvin), wheretisBoltzrnan'sconstant(1.38x10-23J/K),Iistheabsolutetemperature(in 8 is the bandwidth(in Hertz), andG, (/) is the transducerpowergainof the two-port at of 290K (room temperature) is typically used. @uency/ A noisereferencetemperature Whenthe spotnoisefigureis considered, theoutputpowercanbereferenced easily to the input side,in which case(2.5) becomes
r' - 1,-^,
(2.6)
Pni-"r-ide
where Pr-"" is the effectivenoisepoweravailableat the input side,and{,-"u ,0"is the noisepowerwhich would havebeenavailableattheinput sideif thedevicewasnoiseless. The availablepowerat the input terminalscanbe obtainedby terminatingthe noise source(s)in the conjugateof the sourceadmittance,asis illustratedin Figure2.2.
v"(t)
Y"=GjjB"
figure 2.2
The equivalentcircuit usedto calculatethe availablenoisepower at the input of a two-port (cascaderepresentation).
components canbeappliedto theuncorrelated It shouldbenotedthatsuperposition of the noisepower.In general,superpositiononly appliesto the voltageand currentin a circuit. Becauseof the above,the noisepowerresultingfrom /,(t) cansimply be addedto that resultingfrom V,(t) andthe correlatedpart of {(r) (y"", V"(t)). The correlatedfractionof the availablepowercanbe calculatedin the following way:
Yr = nV Ys l (' sY + Y ' \ + Vr Yc o l l ( 2 G \
= v( Y
+ Y \ l O' G \ cot' J'
4
Po-
G,
T
-
1 f v"tt) v:"Q)at T J
0
-
51
Characterization and Analysis of Active Circuits at RF and Microwave Frequencies
- V N
O
V C
= V V n n
(f" . f"J(f"
* I"or)' I (4G,2)
(2.8)
rereZ is the periodoverwhich the noisepoweris averaged. The uncorrelatedfraction of the outputpower canbe calculatedin termsof .I, and by using(2.1):
(2.e)
,J:
I) = I^"l^u + Y*,YJo,
Q.r0)
In - Y*, Vn
= I,,l(2G")
),
(2.11)
v^ lr:, = Iouliu| (4G,2)
= I^I: I 1+G,2)- vry;
Y*,YJ,| (4G"2)
(2.r2)
:ion (2.9) follows from (2.1)becauseInuandZ, areunconelated. The total availablenoisepower at the input terminalscan now be calculated:
ILI; J;
'n'n
4G,t
=-'^!-. 4G,,
-
(r".r."J(r" *1.o.).,"r; Y*,Y:',V,V; n'n -cor-coil +
4G,,
Y"Yl + 2 S(f*,f"') 4G,,
4G,,
,"r;
(2.r3)
--.rse the available noise power in the ideal case is simply
=V3-*,n=kTBIG,
(2.r4)
52
Design of RF and Microwave Amplifiers and Oscillators
it followsthat
k r B+ r J G"
F =
.
Y,Yl + 2I
4G:
(Y*,Y,.) ' n
4G"'
n
I/TB / G"
= | + G,i/ G" * R,u l(G, * G"o)'
+ (8" + B"o), - (G:,, * ajSl t c,
(2.t st
where
,J-
= 4KTBG,,
,, r;
= 4|TBR-
Y*r=G*, + jB*, and Y,=G"+iB,
Q.le
For any givenvalueof G, thereexistsanoptimumvaluefor B. thatwill minimizr the noisefigure.Takingthe derivativeof (2.l5) andsettingit equalto zeroyields AFIAB,=0=0+0+0,,2R*(8,+B*)lG,
Q'2(
from which it follows that 4-opt
= -8.o, Note that the optimum value of B" is the same for all values of G".
The value of G" that will minimize the noisefigure can now be obtained (2.15),afterreplacingB" with its optimumvalue.Takingthe derivativeyields
(2.2f
I {
characterization and Analysis of Active circuits at RF and Microwave Frequencies
Fgure 2.3
53
An example of the constantnoise figure circles associatedwith a linear two-port device.
rr is,
(2.23) '
=
,-oF
Substituting(2.21)nd (2.23)into (2.15) yields ..r=l
+zR*(Q_opt*q"J
(2.24)
The inverserelationshipsand an expressionfor the noise figure in terms of F.,n, . o6,4- op,andR, canbe derivedeasilyby using(2.21), (2.23), and(2.24)andmakingthe - -'cessary in (2.15): substitutions (F,i, - l) / 12rR-) - Gr-ont
(2.26)
-4-opt
F.in * ft- - -F *** mln G,
(2.2s)
- G,-oor)t + (B - B,-op)zf I G, [ (G,
lr" - r"-o,lt
(2.27) (2.28)
By inspectinC Q.27) it is clearthatthe loci of constantnoisefiguresarecirclesin '-e linearadmittanceplane,asis illustratedin Figure2.3.
54
Design of RF and Microwave Amplifiers and Oscillators
N32400AA 6:1:1S t3Jl:2
O
2.@Hz
+
l@€Hz
A
5.W
O
6.m
O o.4roo.3r&a +
0J200.32&B
a
ot$o.w8
o
0.a&0.&B
(a)
(b) Figure 2.4
An example of constantnoise figure circles displayed on a Smith Chart.
lt will be shown in Chapter l0 that the constantnoise figure contoursare also circles on the Smith Chart. Someof the constantnoise figure circles for a low-noise transistoraredisplayedin Figure2.4. -
Characterization and Analysis of Active Circuits at RF and Microwave Frequencies
2.2.2 Noise CorrelationMatrices
'
::
consideredin the A correlationmatrix is definedfor eachof the threerepresentations revioussection[4]: -ascade Representation
'.'=lv-ttll rl';t't';atlrrzn> 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)
-17^amI,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
Design of RF and Microwave Amplifiers and Oscillators
Co = 2kT R*
Y;,I Gri I R*l
f[L,'
Tbc sccondtern of Q.32) is derivedas follows: I,(t)=I^uQ)+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
F^in-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
Design of RF and Microwave Amplifiers and Oscillators
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(Z^u)
F,(Z^q*tt)
G^(Z)
G*(Z^u)
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
Design of RF and Microwave Amplifiers and Oscillators
Fe""(zJ = Frct(Zouts,t)
llG,#(2")
Go-^o(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)
v^I^ 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.
{
Characterization and Analysis of Active Circuits at RF and Microwave Frequencies
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.5cos
70
Desigr of RF and Miqowave Amplifiers and Oscillators
= a3a3[lcostor* lcos3or]
Q.76\
Notethat if a, in(2.76)is negative,thethird-ordercontributionat thefundamental It is alsoclear the signallevel;thatis, thegainwill becompressed. frequencywill decrease in(2.76)). beverysmallwhenthesignallevelis low (c3-term thatthethird-ordertermswill If the contributionof thehigherordertermsis ignored,the l-dB gaincompression point canbe estimatedby setting 3 s arv* - ! atvt, = atv,"l1-lno from which it follows that 2 vi"
4ar(l- 16-taol 3a.
Q.77\
J
Whena two-tonesignalis used,the input signalis givenby v,=a[coscol/+coso2t]
(2.78)
In this case, fundamentaltone componentsUr; fr) with amplitude (2.25ata') are generated.The third harmoniccomponentsgeneratedat eachfrequencyare of the same amplitudeasin the singletonecase(0.25ata")' Apart from thesecomponents,additionalcomponentsareganemtedat(2f, f) 7nd is (0.75ata')' Qft D in the two-tonecase.The amplitudeof thesecomponents at a3:l ratewith increase products will the two-tone logarithmically, If displayed termscanbe order higher of any as the contribution as long that is increasingsignallevel, can compression long as the as a l:l rate increase at tone will neglected.The fundamental be neglected. The two-tone interceptpoint canbe estimatedby using the resultsobtained:
qlYip3 =
z
vio3 '
3 ; +
t
a3v ipl
4at
Q.7e)
I I
Jal
Equations(2.79) and(2.77) canbeusedat this point to find the relationshipbetweenthe point: third-orderinterceptpoint andthe 1-dBcompression
I
I l
? * {
Characterization and Analysis of Active Circuis at RF and Microwave Frequencies
'
2 vo3
=
2 v.-
:':
1
7l
(2'80)
= 9.195= 9.6 1- 1 g - t / 2 0
Note that the contributionof the higherorderterms(5f,7f,...) wasignoredin this :erivation.The estimation,however,is goodenoughto be of practicaluse. Note that becauseof the fixed slopes(at leastat lower signallevels),the level of :hird-ordercomponentsassociated with any signallevel canbe calculatedeasilyfrom the ttird-order interceptspecification: P"{n
P
= Po1(X1p)- (P,t(Xrn)-G -n
,,i=o.*
(2.81)
P (X)=Por(Xs)-3(Pa-G-n = -2P6(X1p) * 3(X*G) = -2P6(X1p) * 3P"t(X)
(dBm)
(2.82)
:re P,r(X) is the fundamentaltone outputpower at signallevel X, andP,r(X) is the .er at (2ft - fr) or (2f, - fr) at the samesignallevel (X). If the power is expressed asa numberandnot in dBm, (2.82)becomes
p1,(n .(X)=-# P;(Xrn)
i
(2.83)
nterestingresultfollows directlyfrom (2.83):
lw
Y,n) =
(2.84)
t- is, the third-order intercept point can be calculated from the fundamental tone and the s-:.i-order power level at any signal level. This result can be applied to calculatethe thirdder intercept point of a cascade.In this casePu, would be the fundamental tone power r . . at the load, and P6the total third-order contribution associatedwith that signal level, at the load.
Design of RF and Microwave Amplifien and Oscillators
t I j
Before deriving the result, it is useful to considerthe efflectof adding an ideal amplifier stageon the interceptpoint. If the operatingpower gain of the ideal stageis Gr, it follows from (2.84)that
P^-^(X,n) =
rc. P ,8)\3 = G, P"1(Xp) Gr Pd(n
l$
that is, the third-orderinterceptpoint is simply increasedwith the gain, as would be expected. At this point the third-order interceptpoint of a cascadecan be calculatedeasily. Considerthe amplifierchainin Figure2.13. Itfollows from (2.85)thattheinterceptpoint of eachstageascalculatedat the loadis increasedwith the gainof the stagesfollowing it. The distortioncomponent(power)contributedby stageTat the load is givenby (2.82) P*-jA(n
P:,-,"(n
(2.86)
Pit-'e\x',lt) wtrere all the power levels and the intercept point are referencedto the load. The normalizedvoltage contributedby stageTcan be obtainedby taking the squareroot of (2.86);that is,
(h ,, 'o3-iAt" t =
pl?tln
(2.87'
Po3-4(Xp)
z" P^(Z)
P"z(Zu)
Pon(Z.)
Zn-
Zn-
G.t
G-2
G-
zL
G.t
I
G"(^1y.. Gr1 (l)
I : Flgure 2.13
The circuit used to calculate the third-order rwo-tone intercept point of a cascade.
t
t
} {
Characterization and Analysis of Active Circuits at RF and Microwave Frequencies
73
The fundamentaltone power level at the load,P^-in(X), is the samefor eachstage and,therefore,it follows from (2.87)that the voltagecontributionfrom eachstageat the load(superposition)is inverselyproportionalto the modified(amplified)interceptpoint for that stage.By using(2.87)andassumingtheworstcasewhereall thethird-orderterms addin-phase,it follows that the interceptpoint for the cascadeis givenby
P],-n(E
Pl6rB\ P](,(h
+ . , , +
P ot-tn(Xtrt)
P]!',n(nl' |
P,.':,_,A(Xrn)l
I
W u"
t
I
+...+
I
(2.88)
P ot-nt(Xtpz)
lP"-rn(x,rt)
.1tls,
I = l * -(Xm) Pot_u(Xp)
I
. . . + P ot-*(X,rt)
(2.8e)
If the assumption is made that the ratios of the l-dB compression points and the :responding third-order intercept points will remain invariant, (2.89) leads to I
l
.3-T
Pro-rn
l
+ - + . . . +
Prou--
l
Pro-u
(2.e0)
"\ercP,*-, is the l-dB compressionpoint of stagei referencedto the load (that is, ' -:rsed with the operatingpowergainof the stagesfollowing it). If an infinite chainof identicalstagesis considered, (2.89)becomes " ' . , . )= P , ( X r n ) 0 - l l c )
(2.er)
(2.91)is a figure of merit similarto the noisemeasure[5].
The Cripps Approach to Estimating the Maximum Output Power Obtainable from a Transistor :mstant output power contours for a transistor are usually closer to ellipses than - s.This leadsto the assumptionthat eventhe linearpowergenerated by a transistor cr
up to the l-dB compression point) is strongly influenced by the nonlinear ..nents in the equivalent circuit. Cripps [] demonstratedthat the maximum linear
74
Ftgure 2.14
-
Deslgn of RF and Microwave Amplifien and Oscillators
approximated asthe The elliptical power contours oltarn-e{ for linear amplifiers can be intersectionofcircles when the intrinsic load line is considered[1].
ofthe intrinsic output power obtainablefrom atransistoris mainly determinedby clipping be approximated output voltage and current and that the elliptic form of the contourscan * tir" int"t."ction of two circles.This is illustratedin Figure2.14. to bebounded In thecripps approachtheintrinsicoutputvoltageis assumed the assumptions, these Under 1.*. by is bounded cunent to a maximum Z,*, *hil" the optimumintrinsic loadJineis givenby Yrop-i = G Lopt-i= I ^*
(2.e2)
I V^*
the cunent will be clipped and the voltage swing will be If R, , is smaller than Rroo,-,, possibleoutputpower'the .*""U; thanthe maximum'itlowed.Relativeto the maximum power(P,) is thengiven bY
(2.93')
po = (In,o: R) I (I^*2 RLopt)= Rr, / Rropt
trma
Figure 2.15
The equivalentcircuit usedin the Cripps approach'
a
{
Characterization and Analysis of Active Circuits at RF and Microwave Frequencies
I
75
In this case,reactancecanbe addedin serieswith the resistancewithout changing themaximum outputpower obtainableuntil the magnitudeof the load impedanceis equal lo R^*, at which point the currentandthe voltageareboth clipped. Whenthe intrinsicloadresistance is higherthanthe optimum,the voltagewill be clipped and the cunent will be lower than the maximum a)lowed - The output power reTative
:o the maximum (P,J is given in this caseby o., = (v^*2 G I (v^*2 G = G, I G Lopt Loor) L)
(2.e4)
Susceptandecan be addedin parallel with this resistancewithout changingthe outputpower obtainableuntil the magnitudeof the load admittanceis equalto Grorr,at whichpoint the voltageandthe currentareboth clipped. Equations(2.92) rfuough (2.94) can be used to calculatethe intrinsic power generatedor to find the contoursof constantoutputpower as a functionof the intrinsic valuesaregiven .oad.If the -XdB outputpowercontouris of interest,thetwo resistance h' R r _ r o * l X r o p=, 1 0 . 0 - o I
(2.es)
eaJ I
tx iigh/ Xr,opt= 10.00
(2.e6)
In order to find the constantoutput power contours in terms of the actual load hpedance,the transformingeffectof thetransistorparasiticsmustbe takeninto account. ts the simple equivalentcircuit shown in Figure 2.15 is used,the load admittance . -:espondingto any given intrinsic load admittancecan be calculatedeasily and the stantoutputpowercontoursfor theactualloadcanbeobtainedby adjustingtheintrinsic j line contoursappropriately. with a given intrinsicterminationcan Calculationof the externalload associated (parasitic is appliedto thetransistor.Any losses or otherwise) a major task if feedback bc -heoutputcircuit or a feedbackloop will alsobe a problem.
Li.4 -
Estimation of the Maximum Output Power of a Linear Network by Using the Power Parameters
iaurieof the simplificationsin theequivalentcircuitusedfor thetransistorin theCripps . 'roach,the intrinsicloadterminationcorresponding to anygivenexternalloadcouldbe . :ulated easily. With the intrinsic terminationandthe I/V constraintsknown, the output ; .ier could be estimated.The implicit assumptionthat all the intrinsicpowergenerated cndsup in the externalload is madein the process. by introducinga new setof parametersto The Crippsapproachcanbe generalized t:
76
;
Design of RF and Microwave Amplifiers and Oscillators
map the intrinsic voltagesto the externalvoltagesand to the intrinsic output cunent [2]' Any reductionin thepowercausedby lossesin theoutputcircuitor in anyfeedbackcircuit is automaticallytrackedwhenthis approachis followed' The assumptionthat the intrinsic output current and voltage are constrainedto a Theallowableareaonthel//-planecan rectangularar"uoith"IlZ-planecanalsobelifted. 2'16' be restrictedto the areadeiinedby four boundarylines instead,as shownin Figure operation breakdown, Ifthe goalis maximumpower,thelinescanbesettopreventvoltage current in the resistive area,andforwardconduction(field-effecttransistors(FETs)).The limit canusuallybe setslightly above16",' If the goal is linearity,the linescanbe setto boundthe areawherethe I/v cuwes areevenlyspaced. Becausethe purposeofthese mappingparametersis to calculatethe outputpower, by the they will be referredto as powerparameters.The power parametersaredefined following equations:
V1= MYr,+ NVzi PVu Vz=OVr,+ 12,=RVr,+SV2i
(2.e7) (2.e8) (2.ee)
In theseequations,21, and V2,are the intrinsic input and output voltages, an FET' respectively,while 1r,is the intrinsicoutputcurrent,asshownin Figure2-16fot V, andV, arethe input and output voltages,respectively' The power parameterscanbe usedto calculatethe intrinsic load associatedwith a givenextemh loaddirectly,aswill be shownin Section2.3.4.I . Theextemalvoltageand power iurrent associatedwith the maximumintrinsic voltageor currentandthe associated at the load canthenbe calculatedeasilyby using(2.97)through(2.99). be Similarly, the extemal load associatedwith a given intrinsic load can also This is usefulwhencontoursof constant calculatedeasilyby usingthepowerparameters. output power are generated. The main assumptionsin this approachare linearity and hard clipping of the approach intrinsic output currentand voltageat the boundarylines. The power parameter amplifiers' A point class of compression l-dB hasprovento be usefulup to at leastthe Thepowerparumetersa.eqrritegeneralandcanbemanipulatedto includethe effect parameters of anypassivenetworkin whichthetransistormaybeimbedded.A setof power in because, required is This network. snojA Uecalculatedfor eachtransistorusedin the the in used transistors of the general,the output power may be limited by clipping in any circuit.
itis assumedthat forany giventwo-portarecalculated, Whenthepowerparameters be present'The may no clipping will occur itr *y of the other active two-portsthat clipping first which in maximum (linear) output power is determinedby the two-port occurs,that is, ifthe otherstagesarenot closeto clippingtoo'
l
Designof RF and Microwave Amplifiers and Oscillators
7t
.
2.
Thebehaviorof thetransistoris essentiallylinearup to (ClassA) or around point. (ClassB) the l-dB compression
3.
The outputvoltageof the transistoris sinusoidal.
The last condition canbe approximatedin classB stagesby usinga push-pullstage or by short-circuitingthe harmoniccurrentsof eachtransistorby usinga resonantor lowpasscircuit. In orderto calculatethe linearbehaviorof thetransistorwith the loadtermination of interest,a linearmodelof thetransistoris required.This is a simplematterin the class A case(conventionalsmall-signalmodelshave been found to be adequate),but not necessarilyso in the classB case. Acceptableresultsareusuallyobtainedfor the classB caseby usingthe classA small-signal parametersat the rated class B output current and reducing the 9,, of the small-signalmodelby a factorof two. associated Typical small-signalmodelsareshownin Figure2.17.
Cca
L,
R,t
L,I
Y_tt g^ Vri
(a)
o) F$re
2.r7
Typical small-signalmodelsusedfor (a) FETs and (b) bipolar transistors.
-
Characterization and Analysis of Active Circuits at RF and Microwave Frequencies
79
2.3.4.1 calculating the Intrinsic Load Associatedwith a Given External Load -he
intrinsic load(Yt,:- Ir,/ vr) associatedwith any given external load can be calculated "' using the l-parameter expression for the voltage gain in terms of I, [2]:
!r,
l' v = L = \-
(2.100)
,rr.tt
touations (2.97) and (2.98) can be used to replace V, andV, above,leading to '-
O V r , + P V 2t i lzt - _ = A--Y MVr,+NVr, lzz * Yr
I
(2.101)
rn (2.101)cannow be replacedin termsof V, and/r,by using(2.99):
o(*L,-*r,r*PVzi - A
,((L^rr,-*nr*NVz,
Q.r02)
.1rc
next stepis to eliminate V2, and1r,in this expressionin termsof the intrinsic load :mittance(Yr,= -\, I Vr,):
2 L , , e - RY/ '>z 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
''
Desigr of RF and Microwave Amplifiers and Oscillators
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'
{
Characterization and Analysis of Active Circuits at RF and Microwave Frequencies
:':'
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
Desigr of RF and Microwave Amplifiers and Oscillators
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
Characterization and Analysis of Active Circuits at RF and Microwave Frequencies
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:
Design of RF and Microwave Amplifiers and Oscillators
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 {
Characterization and Analysis of Active Circuits at RF and Microwave Frequencies
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)
Characterization and Analysis of Active Circuits at RF and Microwave Frequencies
= (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
Design of RF and Microwave Amplifiers and Oscillators
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
Design of RF and Microwave Amplifiers and Oscillators
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
Radio-FrequencyComponents
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
Radio-FrequencyComponents
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
Design of RF and Microwave Amplifiers and Oscillators
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
Radio-FrequencyComponents
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
Design of RF and Microwave Amplifiers and Oscillators
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::
Radio-FrequencyComponents
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@)= jaNAB^o 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 productB^of 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:
-
Radio-FrequencyComponents
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
Design of RF and Microwave Amplifiers and Oscillators
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
Radio-FrequencyComponents
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 t^o = 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
Design of RF and Microwave Amplifiers and Oscillators
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
)<) 256 255 242
300 300 300 300
Thecapacitance,k-factor,andoptimum d/cratioforcoilswiththe//Dratiosuseci in Table3.6 aretabulatedin Table3.7 for convenience. Whenresonantcircuitswith high Q-factorsaredesigned,the unloadedp-factors of the coils and capacitorsusedmust be as high as possible.In orderto determinethc maximumrealizableunloadedp possiblefor a single-layerair-coredcoil, it is necessan.
111
Radio-FrequencyComponents
to determinethe optimum UDratio.Because,in(3.24),
Q=krr[7 (see(3.24)) the length-to-diameterratio influencesthe unloadedQ directly through the ofthe limit thatexists valueofthe constant,ft,andindirectly(throughr) because associated coil. ofthe on the self-capacitance The maximum radiuscorrespondingto a particularl/D ratio canbe determinedby using(3.29). Table 3.7 The self-capacitance,d/c ratio, optimum value of t(ft"r), p€r centimeter for coils with different llD ratios and the ratio ofthe & and the self-capacitance
{ .l
c
dlc
(pF/cm)
0.6 1.0 2.0 4.0
0.44 0.45 0.53 0.68
0.55 0.55 0.63 0.63
kor
d I I I
HC
(HzJcm)
(pF|ttz)
0.088 0.100 0.1l5 0.t33
0.200 0.222 0.216 0.196
By substitutingthe value for r as given by (3.29) into (3.24),the maximum p ing to a particular l/D ratio is found to be
-/oo=(k^lC)JfC^*
(3.30)
,here k^ is the maximum value of & correspondingto the particular I/D ratio, C is the per centimeterasgivenby the curvein Figure3.3,andC.* is the maximum ,rpacitance ,lueof the self-capacitance asdeterminedfrom the specifiedresonantfrequency. The influenceof the l/D ratio on the unloadedQ is clearly limited to the first term (3.30). r The HC ratiosfor differentl/D ratiosarecomparedin the lastcolumnof Table 7.It follows from this comparisonthatthehighestQ will beobtainedwhenthelength-to:ameterratio of the coil is equalto l. At this stage,the highestQ realizablewith a single-layersolenoidalair-coredcoil :n be determinedfor any particular inductancevalue if the operatingfrequencyand the ldf-resonant frequency are specified. The following procedurecan be followed in order to letermine the Q.
Esign Procedure for Maximum Q and Specified Inductance l.
C h o o s eU D : 1 .
Designof RF and Microwave Amplifiers and Oscillators
ttz 2.
is Determinethe maximum value of the self-capacitance(c.*). If the coil be can frequency the self-resonant circuit, to be usedin a parallelresonant chosencloseto the resonantfrequencyofthe circuit' Calculatethe maximum allowable radius of the coil by using the equation /,o = C'u* / (2C) = c^* I 0'g (cm)
(3'31)
with C,", specifiedin picofarads(pF)' if the value of the radius is unrealisticallyhigh, reduceit to an value. acceptable
3.
Determinethe maximumrealizableunloadedQ by using(3'24):
r.* f Q^^= 0.10 4.
I t L,
l
(3.32)
Calculatethe requiredthicknessof the wire: N z = 7 1 . 2L l r ^ o
(3.33)
wherethe inductance(I) is specifiedin pH and/'o in centimeter'
'
c=l l(N -l)=2r^^ /(N-l)
(3.34)
d=0.55c (cm)
(3.35 t
If the wire thicknessturns out to be unrealistic,changethe radius.
EXAMPLE 3.2
Designinga singleJayersolenoidalcoil for maximumQ.
with selfThe highestpossibleQ will be determinedfor a coil of 10pH at 5 MHz above' outlined procedure the following by ."roo-t frequencyat l0 MHz 1.
UD__T
2.
C,o= 1/ [(2nx10x106)2l0xl0-6] :Z5.3pF /.o =25.3/0.9 :28.1 cm
3.
Q*
:0.10 r.*(/)tz = 6286t
113
Radio-FrequencyComponents
N 2 : 7 1 . 2 x t 0t 2 8 . 1 : 2 5 . 3
4.
N :5.0 : 2 x 2 8 . 1i ( 5 - l ) = 14.1cm d = 0 . 5 5c :7.73 cmt If the coil size is limited to 3 cm by 3 cm by 3 cm, the maximum realizableQrvill be 335. t 1.3.7 The Design of Inductors with Magnetic Cores naller inductors with less parasitic capacitancecan be designedby using magnetic aterials. The corecanbe a rod, a toroid,a balun,or stackedtoroids(seeFigure3.9). ,,;, "Rodsareoftenusedif theinductoris to betuned,while toroidsandbalunsareused tr fixed-valueinductors.Stackedcorescanbe usedasan alternativeto a balun. The type of materialusedis a function of the frequencyrangeover which the - luctor is to be used,the desiredunloaded Q,lhe availablespace,and the temperature .19e.
II (a)
3.9
ffim
HH (d)
Different types ofmagnetic cores:(a) rod core, (b) toroid, (c) balun core, and (d) stacked toroids.
.+j"o
The unloadedp is determinedby the flux densityin the coreand,therefore,by the r.:rnumvoltageacrossthe inductor,the numberof turns,andthe frequency. Materialswith a highrelativepermeabilityareusuallyvery sensitiveto changesin -'rature. With the materialandtype of coreselected,the sizeof the coremustbe determined.
114
Design of RF and Microwave Amplifiers and Oscillators
The coremust be largeenoughfor the flux densityto be sufficiently low to ensurethat the desiredunloadedQisrealizedandthattherequirednumberof turnscanbeaccommodated. The selectionof the minimumcoresizefor toroidal(singleandstacked)inductors will be discussedin the next two sections.If a balunis to be used,the resultsfor stacked corescanbe appliedto get an ideaofthe sizerequired.
3.3.7.1
The Designof an Inductor with a SingleToroidal Core
The inductanceofa toroidal core inductor is siven bv L=Oilai=poprN2A/l
(3.36'
Thevalueof the productp,R,/L canbe determinedfrom theunloadedQ (Q) by usingthe following equation: Ir,Ro/ L = p,aRo / @L) =2np,f Qu
(3.37)
The flux density correspondingto this ratio (8.*r) canbe determined,wheregiven, from the manufacturer'sspecifications(seeFigure3.7 for an example). The flux densityin the coreis givenby B,no = V^o /(aAN)
(3.38)
The flux density in the coremust be lessthan or equalto the maximum allowable value B.*r. If thenumberofturns in (3.36)is replacedby using(3.38),theproductofthe crosssection area of the core (l) and the mean path length (/) is found to be
AI =l.p,,tto / &rB2^*)ltrl* t 1o,t7
F
(3.3e)
The requiredcore size can now be found by comparingthis product with that of availablecores. If a core with the required.,4/-productis not available,a core with a laryer Alproductcan be chosen.The numberof turns requiredmust then be calculatedby using (3.36).The alternativeis to usemore than one core (smaller)to obtainthe requiredl/product. Withthe coredimensionsknown,the numberofthe turnsrequiredcanbe found br using(3.38). Thefollowing procedurecanbefollowedto designaninductorwith a toroidalcore A DesignProcedurefor an Inductor with a Toroidal Core l.
Selecta suitablematerial.Take the frequencyrange,temperaturerangc
115
Radio-FrequencyComponents
r
requiredunloadedQ, andinductor size into account.
d. os d
2.
Calculatethe p,RrlI ratio at the lowestfrequencyby using(3.37): It,Rol L
--2np,f
Q,
whereQuis the desiredvalueof the unloadedp. J.
4.
Find the flux densitycorrespondingto the calculatedp,Ro/L ratio from the manufacturer' s specifications. Calculatethe requiredl/-productby using(3.39): Al = lt,$o t 6A2^^,4f(*
I @L)
5.
Comparethis productto that of availablecores.Selecta corewith an l/productequalor closeto it. Ifthe differencein l/-product is significant, choosethe corewith anr4l-productgreaterthanthat required. Altematively,smallercorescanbecombinedto obtaintherequired l/-product (seeSection3.3.7.2).
6.
Calculatethe requirednumberof tumsby using(3.36): L=p,IroN2A/l
7.
Check if there is enoughspaceto accommodatethe required number of tums of the conductorwith the requiredthickness.If the coreis too small, a largertoroid mustbe used.
Table 3.E A list of typical magneticcore sizes A Qtm') 12.5 31.5 37.5 65.0 97.s
I 2 3 4 f
EXAMPLE
3.3
/(mm) JO
57
/) 92 92
Al (1mr)
Size (mm3)
0.44 1.80 2.81 5.98 8.9'1
14x9x5 23"14x7 29\19x7.5 36"23'10 36x23x15
Finding the core size required for an inductor.
As an example of the application of the procedure outlined here, the core size for a masnetic-cored inductor with 31.40 reactanceat 2 MHz, and loss resistance
Desigrr of RF and Microwave Amplifiers and Oscillators
116
equalto 3920, will be determined.The maximumrms voltageacrossthe inductor will be 20V and4C4 materialis available.Note that p,: 120. The p,R,/ L ratio for the inductoris VrRp L
120x392
= l . 8 8 x 1 0 l os - l
31.41(2nx2xl06)
By using the 1.6 MHz curve given for 4C4 material in Figure 3.7,the s-r is foundto be (B^*.f) productcorrespondingtoa p,Ro/L ratio of 1.8x1010 2x104Tllz. Themaximumallowableflux densityin the coreis, therefore,0.01T. Thel/-product ofthe requiredcorecanbe foundby using(3'39)' The reis I .53x l0-6 m3. quired.,4/-product By comparingthis valueto thosein the list of somel/-products given in Table 3.8, it can be seen that the core with l/-product equal to 1.8 pm3 (l = 3l.5pm3;l:57mm;23x14x7mm3)canbe used' The numberof tums requiredis
tr.
L l l ( 1 t o 1 t . , A )= 5.5
The selectedcore can accommodatethe requirednumberof turns with ease.
3.3.7.2
The Design of an Inductor with a Stacked Toroidal Core
The designof an inductor with a stackedtoroidal coreis similar to that of an inductor with area(l) usedin the previous a single core, exceptfor the fact that the cross-sectional of toroids used (an even number is the N" where as N" A, taken sectionmust now be The meanpathlengthis that toroid. of a single area number)andI is the cross-sectional of a singletoroid. The inductanceof a stackedcore inductor is given by the equation . " . :
(3.40)
L = lt,FoNz(N"A / l) The maximumflux densityis B,n* = V^
(3.41)
lfr,l (N"l)Nl
andthe requiredl/-product is obtainedfrom N"Al =[p,po / @B'^ ))V]*
t1((I't7
(3.42)
Radio-FrequencyComponents
3.4
tl7
TRANSMISSION LINES
areusuallycoaxialcables(flexible (F) or Thetransmissionlinesusedat radiofrequencies of these semi-rigid(SR)),microstriplines,or twistedpairs.The importantcharacteristics power-handling capability' theinsertionloss,andthe impedance, linesarethecharacteristic of coaxial cables,microstriplines, and twisted pairs will be The characteristics discussedbriefly.
3.4.1
Coaxial Cables
The characteristicimpedanceof a coaxial cableis given by
(3.43)
Zo = (138/ .lt,) logro(b/ a)
andb is theinnerdiameter wherea is theouterdiameterofthe innerconductor(centimeter) of the outerconductor(in centimeter). The attenuationofthe cableis givenby [10]
o=(3.615I Z) (Kt I a+ K, I fi r,{f +9.r2tfr[4
tan}
(3.44)
where
a is the attenuationin decibels/l00m; K is the squareroot of the ratio of the resistivityof copperto that of theparticular conductor; /is the operatingfrequencyin megahertz; Celsius; f:p+0.0039(t-20)lt/2,where/istheoperatingtemperatureindegrees a is the inner conductorouterdiameterin centimeter; b is the outerconductorinnerdiameterin centimeter; tan 6 is the losstangentofthe innerconductorinsulation' The attenuationis increasingwith frequencybecauseof the skin effect and the ossesin the dielectricmaterial, The power-handlingcapability of a coaxial cable is limited by the maximum etlowable temperature.This is a fi,rnction of the insulation used (200"C for the diameterof the cable,andthe environmentaltemperature. :olytetrafluoroethylene), alongsomecoaxialcablesaregiven power-handling capabilityandattenuation The power-handling capabilitywith increasing decrease in the n Table 3.9 tlll. Note :requency. inthe VHF transformers Semi-rigidcoaxialcableis oftenusedfortransmission-line rd UHF ranges. of 50Oand25Q arefreelyavailable. impedances Coaxiallineswith characteristic canbe obtainedby connectingcablesin parallel,while higherimpedLowerimpedances
, I
118
Designof RF and Microwave Amplifiers and Oscillators
Table 3.9 The attenuationand averagepower-handlingcapabilitiesofsome coaxial cables(F:Flexiblecable;SR:Semi-rigidcable)at different frequencies
c (dB/m) (P (uD) Cable Type
@t]ll[Hz
@r0MHz
0.04(lk) 0.o3 (lk)
0.14(300) 0.08 (800)
@l00MHz
@ 500MHz -
50Q; 500; 50O; 50Q; 500;
1.7mm (F) 2.8 mm (F) Ll mm (SR) 2.2 mm (SR) 6.4 mm (SR)
0.44 (90) 0.27 (2s0) 0.35 (68) 0.18(330) 0.ll (1.17k)
0.7s(32) 0.43(140) 0.25(515)
I
F
anoescan be obtainedby connectinglineswith lower impedances in series(usingsemirigid cable). By doing the laffer, the effective capacitanceis decreased.The series connection is shownin Figure3.10.
Inn€rconductor
t
t
Figurc 3.f0
3.4.2
Increasing the characteristicimpedanceof a coaxial cable by connectingtwo cablesin series.
MicrostripTransmissionLines
(Z,UD andtheeffectiverelativedielectricconstant(e,-"u(/)) Tbecharacteristic impedance of a microstripline is a functionofthe width+o-heightratio(Wlh),theconductorthickness (r), coverheight(H), andfrequency(,f). The characteristic impedanceis alsoa function of the effectivedielectricconstant. The characteristic impedance (ZoU) and effective relative dielectric constant(€,-"rr("f))canbe computedby usingthe following setof equations[12, l3]:
_osh[(:)' .(#']] wn=ty+;i,. rn4
(3.4s)
Radio-FrequencyComponents
119
The geometryof a microstrip line.
Fgure 3.11
-1ffi)'-"1 f(wIh)=6+(2n-6)E>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)
fiW.n'
= 8.68R-n1]{ ffi U ( 2 n ) < ( twh \ < 2
*)]} ==ru^!!;* i^l2Tc'xp(ry.' {ry. lw,n* I h
w.ut@D 1 ,* ,orr,
(3.68)
W,rrl(2h)+0.941
c" in decibelsper centimeter.
'='-lY*1'
(3.6e)
l+n )
-h/wdr.hl"T-t/h)
(3.70)
122
Design of RF and Microwave Amplifiers and Oscillators
(3.71)
R"=
where o is the conductivity of the strip conductor. At high frequencies,the copper lossesare higher than those predicted by the equationsabove.This is dueto the coarseinterfacebetweenthe dielectric materialandthe conductor.Theselossesareincludedin the dielectriclossesby somemanufacturers. to the With the losstangent(tan6) known,the attenuationconstantcorresponding the following equation dielectriclossesin a microstripline canbe calculatedby using [4, 161:
.rr=zz.rffi
(dB/cm)
(3.72)
where l.e is the operatingwavelength. Materialswith dissipationfactorsof 0.00085at I MHz and0.0018at l0 GHz are available.With suchlow valuesfor the dissipativefactor,the dielectriclossesareusually small comparedto the conductorlosses.Silicon is an exampleof a materialwherethe dielectriclossescannotbe neglected. As an exampleof thedissipativelossesin a microstripline, the insertionlossof an 8-in 50O line on an Epsilam-I0@ substrateis specifiedby the manufacturerto be 0.1dB at 100MHz and0.21dB at 500MHz (0.19dB/wavelength). approximately capabilityofamicrostriplineis afunctionofthe insertionloss, Thepower-handling thebreakdownvoltageof the dielectricmaterial,andthemaximumallowabletemperature of the line. If the thermalresistanceof the substrateis known as a function of the line width, the maximumpower-handlingcapabilitycanbe computedeasily. I
3.4.3 TwistedPairs
I I
Transmissionlines with a wide rangeof characteristicimpedancescan be realizedby twisting lengthsof wire together. whenthickerwire Thecharacteristicimpedancesof thesetwisted-pairlinesdecrease (AWG) enamel20 impedanceis 35O whenNo. is used.For example,the characteristic insulatedwire with threetwists per centimeteris usedandis 120QwhenNo. 30 vinylcoatedwire (0.05cm outsidediameter)with 3.6 twistsper centimeteris used[17]. the characteristic Increasingthe numberof twists per centimeteralso decreases impedance obtained lines.For example,thecharacteristic impedanceof thesetransmission from approximately42O wirestogetherdecreases by twistingtwo No. 20 enamel-insulated to 30Owhenthe numberof twists is doubledfrom two to four [17]. impedancecanbe obtainedby twisting two No. 22 A line with 50O characteristic wirestogetherto have2.5 twistsper centimeter. enamel-insulated lowerthanlOQareoftenrequiredin theHF range.These impedances Characteristic
Radio-FrequencyComponents
r23
impedancescan be realizedby twisting togethermany wires (using rwo-wire lines) with smallerdiameters. thedielectric linesbecause It is difficult to calculatethelossesin thesetransmission losses,skin effect, proximity effect, and the fact that the current is flowing in both directionsalongthe line mustbetakeninto account.It is, therefore,easierto determinethe attenuationof theselinespractically. The lossesin twisted-wiretransmissionlines are usually not a problem below 100MHz.
REFERENCES Krauss,H. L., W. B. Bostian,and F. H. Raab,Solid StateRadioEngineering,New York: JohnWiley andSons,1980. TheRF CapacitorHandbook,AmericanTechnicalCeramics,1979. "RF
& MicrowavePorcelainCapacitors,"Cazenovia,NY: DielectricLaboratories, Inc.,1998.
"Di-Cap Microwave CeiamicCapacitors,"Cazenovia,NY: DielectricLaboratories, lnc., 1998. Hardy, K. H., High FrequencyCircuit Design, Reston,vA: PrestonPublishing Company,1979. "High FrequencyResistanceand capacity of Single-Layer Medhurst, R. G., Solenoids,"WirelessEngineer,March 1947,p' 35. "High-PerformanceChip Coils for the WirelessCommunicationIndustry,"Franklin Park,IL: Stetco,Inc., 1998. "On the Designof H. F. widebandPowerTransformers(ECO6907)," Hilbers,A. H., Eindhoven:PhilipsC.A.B.Group,1969. New York: Johnwiley and sons, skilling, H.H, Fundamentalsof Electric lV'aves, 1948. l. CoaxitubeSemi-RigidCoaxialCable,North Wales,UK: PrecisionTube Company, Inc.,(n.d.). I . Welsby,Y. G'The TheoryandDesignof InductanceCoils,London:MacDonaldand Co.Ltd., 1960.
124
Design of RF and Microwave Amplifiers and Oscillators
t2. Marc[ S., "Microstrip Packaging:Watch The Last Step,"Microw(nes,December r9 8 1 . 13. Pues,H. F., andA. R. van de Capelle,Electron.Letters,Vol. 16,November6, 1980, w.870-872. "A 14. Bahl,I. J.,andD. K. Trevedi, Designer'sGuideto MicrostripLine," Microwaves, May,1977. "Losses 15. Pucel,R. A., D. J. Masse,and C. P. Hartwig, in Microstrip," IEEE Trans. "Correction MicrowaveTheoryTech.,Yol. MTT-16, June1968,pp.342-350; to Lossesin Microstrip,"Ibid., (Conesp.),Vol. MTT-I6, December1968,p. 1064. "Losses in Microstrip TransmissionSystemsfor 16. Welsh, J. D., and H. J. Pratt, Integrated MicrowaveCircuits,"NEfuEMRec, Vol. 8, 1966,pp. 100-101. "Designing 17. Krauss,H. L., and C. W. Allen, Toroidal Transformersto Optimize WidebandPerformance ," Electronics,August 16, 1973.
SELECTEDBIBLIOGRAPHY Snelling,E. C, Sofi Ferrites: Propertiesand Applications,London:Iliffe Books Ltd., 1969. Howe,H. Stripline Circuit Design,Dedham,MA: ArtechHouse,1974.
}
i,i
...-.
.
:
CHAPTER 4 NARROWBAND IMPEDANCE.MATCHING WITH LC NETWORI$
I.1 INTRODUCTION to certainrequired networksareusedfor transformingimpedances lmpedance-matching When yalues,which may or may not betheconjugateof the sourceor theloadimpedance. power is transferred (i.e., maximum Zr: Z"'), to a load I sourceis conjugatelymatched bctweenthem.This is importantwhenthe powergainof a transistoris low, asis the case sith mosttransistorsat higherfrequencies. networksarealsooftenusedto contol Apart from matching,impedance-matching rhegain,the noisefigure, or the outputpowerof the different stagesin an amplifier. These Etworks aremoreaccuratelyreferredto ascontrolnetworks(gain,noisefigure,or power networks. control),but they arealsogenerallyreferredto asimpedance-matching When a matching network is designed for maximum power transfer, the rminations areusuallyknown. The terminationsto be usedwhena controlnetworkis . cnedaredeterminedby theparameter to becontrolled.This aspectwill be considered --.rhapter10. Independentof how the terminationsareestablished,the designprocedurefor the . urching network remainsthe same.The designof nanowbandimpedance-matching .rorks,mostly for maximumpowertransfer,will be consideredin this chapter. Narrowbandimpedancematching is done with t'wo or more components.Where :rie 'H "ornponentsare usedto bring about an impedancetransformation,the matching .i'ork matchingnetworksare usuallyT- or PI:. is called an L-section.Three-element :ons.The namesaredescriptiveof the configurationformedby the reactiveelements' The designof L-, T-, and Pl-sectionswill be discussedin this chapter.Trans'-:ation of real andreactiveloadswill be considered. When T- and Pl-sectionsare used,it is possibleto bring about the required r-sformation andto control the bandwidthof the network.Although the 3-dB bandwidth du L-sectioncanbe determinedeasily,it is not a designparameter. to know the bandwidthresultingfrom a transforming It is sometimesnecessary . r:onmore accuratelythan is possiblewith the approximationmethodthat is usually . : In thesecases,as well as in instanceswhere a bandwidthother than the 3-dB : . iwidth is of interest,the procedureoutlinedin Section4.9 canbe used.
r25
126
Designof RF and Microwave Amplifiers and Oscillators
It was shownin Chapter3 that losslessreactivecomponentsdo not exist. For this networkswill havesomeinsertionloss.Theselossescan all impedance-matching nsason, quite pronounced whenthe bandwidthof a circuit is very narrow.A simpleprocedure be for calculatingthe insertionloss causedby a cascadedLC networkwill be outlinedin 4.8. Section impedance-matching networks Apart from matchingandtransformingimpedances, aresometimesalsousedto rejectunwantedsignalsoutsidethe passband(this practiceis networks are designed).The not recommendedwhen widebandimpedance-matching networkswith high rejectionrequiredcanoftenbe obtainedby usingimpedance-matching great. too required is not rejection that is, ifthe Q-factors, The rejection obtainableby using parallel and seriesresonantcircuits will be consideredin Section4.2. Whenthe requiredrejectionbecomesvery high, the p of the components,their temperaturestability, and any tuning requiredcanbecomea problem.If the associated insertionloss can be toleratedand the filtering occursat low power levels,the required rejectioncan often be obtainedby using surfaceacousticwave(SAW) devices,ceramic arevery stableandcanprovideextremelysharp filters,or crystalfilters.Thesecomponents presented by thesedevices,some(low Q) impedance rejection.Becauseof theimpedances matchingis usuallyalsorequired.
4.2
PARALLEL RESONANCE
A parallel resonantcircuit is shown in Figure 4.1. Although it is not an impedancematchingnetwork,it is of interestherebecauseof its frequencyresponse. The frequencyresponseof this circuit is determinedby the zeroat the origin, the zeroat infinity, andthe two poles.That is, V,(s)= 71511
= I / U l R r +s C + 1 / ( s Z )
+ V" RL
Figure 4.1
A parallelresonantcircuit.
Narrowband Impedance-Matchingwith LC Networks
sLI s2LC+sL/Rr+I
t27
(4.1)
From Figure4.1 it is obviousthat the highestpossibleoutputvoltagewill occur where coZ= 1/ (coC) that is, when
(4.2)
oo=l1.,1rc
The 3-dB frequenciesof the circuit canbe determinedby using (a.1) and (4.2). Thesefrequenciesoccur where +| / (iaoL)l I t n, + jrlrC+| | (jo,l)l= {Z lt I R, + irl,oC
=Jitn,
(4.3)
After somemanipulation,the solutionsof this equationarefoundto be ,-------03dB= rrro{t+ | / gQ\
tl I (2RC)
(4.4)
Therefore,the bandwidthof the circuit is -o:os-r =I I (RC\ B = (Dros-z
(rad/s)
(4.5)
It can be seenfrom (4.4) that the circuit responseis not symmetricalaroundthe resonantfrequencyoo. It canbe provedeasily,however,that the resonantfrequencyis the geometricmeanof the two cut-off frequenciesby multiplying the two solutionsgiven by 4.4),that is, oo =
Jo:as_r
.o:ae_z
(4.6)
The p-factor of the circuit is defined as the ratio of the center frequencyto the :andwidth; that is, Q = a o /B = ooCR
(4.7) (4.8)
{
128
Design of RF and Microwave Amplifiers and Oscillators
(4.e)
= R / (cooZ)
A high Q, therefore,implies a very small relative bandwidthand, in the caseof with low impedance comparedto thatof theloadresistance. parallelresonance, reactances The reactances areshownin Figure4.2 for a Q of 10. Whenthe Q of thecircuit is high, the arithmeticandthe geometricmeanof the cutoff@uencies areapproximatelythe same(see(4.4)).
i)
Figure 4.2
|"rroo #-.rroo
A parallel resonantcircuit with p:
100()
10.
Extremely sharprejection can be obtainedby using a parallel or seriesresonant to be ideal,it canbe shownthat the ratio canbe considered circuit. Whenthe components (P,',J andthat at any other frequency load at resonance power to the transmitted of the the following equation: (P"(f )) canbe calculatedby using
+6=
+(roo / r,r)21 1-ze\ +Q'IG't oo)2
(4.10)
of the circuit, the attenuationis As an illustrationof the rejectioncharacteristics givenin Table4.1 asa functionof thenormalizedfrequency(f lfo) for differentvaluesof becausethe response areconsidered, aboveresonance the circuit Q. Otnythe frequencies curvelevelsoff to a singleis closeto symmetricalwhenthe Q is high.Wheretheresponse pole response, no moreentriesweremadeinto the table. therateofrejection,a-30-dBqualityfactory(Q-r) is defined In ordertoappreciate hereas
r I'
I
F i ;
Q+o=.folB-so
(4.1l)
"bandwidth" of the circuit (in Hertz). whereB-rois the -30-dB p-factorsusedin Table4.1me0.315(0 : l0), the three for The 30-dBQ-factors (Q:250), (Q: respectively. 100),and7.90 3.15 It follows by observationof the resultsobtainedthat the -30-dB Q-factorof a resonantcircuit is relatedto the 3-dB Q-factorin a simpleway whenthe 3-dB Q-factoris greaterthan l0:
Q_n=0.0315Q
(4.r2)
129
Narrowband Impedance-Matchingwith LC Networks
The normalized -30-dB bandwidth of the circuit is therefore given to good approximationby the following equation: (4.13)
B-to=31.75/Q
It canbe shownthat the two normalized-30-dB rejectionfrequenciesare given to good approximationby the following equation:
(4.r4) By using (4.13),the Q-factorrequiredfor a specified-30-dB bandwidthcan be calculatedeasily.
Table 4.1 The frequenciesat which the output signalofan ideal parallelor seriesresonantcircuit is attenuatedas listed for somevaluesofthe circuit quality factor
(/fr) frequencies Normalized
Attenuation(dB)
8: r0 0 -3 -10 -20 -30 -40 -50 -60 -'t0
EXAMPLE 4.1
1.0000 1.0512 1.1615 1.615
,:
Q=t00 1.000 1.005 1.015 l.051 l .l 7 l 1.620 3.46 4.24
Q:250 1.000 1.002 1.006 1.020 t.065 1.22 t:,
Establishingthe Q-factorrequiredfor -30-dB rejectionat two specifiedfrequencies.
As an exampleof the applicationof (4.13),the p-factor necessaryto provide -30-dB rejection at 40 and 60 MHz with a parallel resonantcircuit will be determined. The resonantfrequencyofthe circuit is fo = "[40 "60 = 48.99MH2
Design of RF and Microwave Amplifiers and Oscillators
130
' '":! The normalized-30-dB bandwidthis B-so= (60 - 40)I 48.99 = 0.4082 The requiredQ is obtainedby using(4.13): (4.15)
Q = 3 1 . 7 5 1B - t o = 7 7 . 7 6
of the parallelresonantcircuit have Up to this point the lossesin the components been ignored. When the requiredQ of the circuit becomesof the sameorder as the unloadedQs ofthe componentsused,this cannotbe done. When the lossesare taken into account,the effective load resistance(-Rr)at the resonantfrequencyis then given bY (4.16)
I I Rr = | I R, +l I (QLx) +l I (QcXc)
where X. andXrare the reactanceof the inductor and capacitor,respectively, at the resonantfrequency(seeFigurea.3).Qtffid Q, arethe unloadedQ-factorsof the inductor andcapacitor,respectively.
RL
Figure 4.3
A parallel resonantcircuit with lossy components.
areequal,and(4.16)canbe the capacitiveandinductivereactances At resonance simplified to X L / R r = X t l R , + [ 1l Q r . + I / Q c )
.
(4.r7)
The lastterm in this equationis definedasthe unloadedQ (Q) of the circuit: I l Q u = 1 /Q r + l l Q "
(4.18)
The effectiveQ of thecircuit (0"") is thereforegivenby llQ"n=IlQt +l/Q, areassumedto be lossless. whereQ, is the p whenthe components
(4.1e)
Narrowband Impedance-Matchingwith LC Networks
131
The highest Q obtainablewith a parallel resonantcircuit is limited by component lossesandthe temperaturestabilityof the components.
I 4.3 }
SERIESRESONANCE
The resultsobtainedfor a parallel resonantcircuit can be applieddirectly to a series :esonantcircuit by usingthe principleof dualism. According to this principle, for every circuit thereis anothercircuit for which ,,vhatever appliesto thecurrentof onecircuit,alsoappliesto thevoltageofthe othercircuit, rnd vice versa. Thisequivalentcanbeobtainedbyfollowingtheprocedureillustratedin Figure4.4. \ nodeis placedin everyloop of the first circuit, aswell asin the spaceoutsideit. These rodesarethenconnectedby passingfrom oneloopto anotherthroughthe componentsof :he different loops. Inductorsare replacedwith capacitors,capacitorswith inductors, :esistorswith conductors,andconductorswith resistors.The valuesassignedto the new :omponents(H, F, O, S) arenumericallyequalto thoseof the originalcomponents. The output voltageof the parallel resonantcircuit in Figure4.4 is given by the rcllowingequation: j',,= I I / R + s C + I / (s L)] = 0,2/ 10.5+ 3s + I / (5s)l [1
ir
llgure 4.4
N2
I-;
(4.20)
N3
The principle ofdualism appliedto a parallelresonantcircuit.
a
The output of the seriesresonantcircuit is obtainedby replacing V,,uirthIo, I sith4 R with G, Cwith L,and Z with C. Thus. 5 3 s +1 / ( 5 s ) ] I , = E / [ 1 l G + s Z + 1 / ( s C ) ]= 0 . 2 1 [ 0 . +
(4.21)
It follows from Figure 4.4 that the output current of the seriesresonantcircuit is eedgivenby this equation. By applyingtheprincipleof dualismto theresultsdeducedin theprevioussection,
132
Design of RF and Microwave Amplifiers and Oscillators
'ti-?l-
':n:..
.t1
I
s
l, ?..f;
Efrt
.,
r
.r,. V,,l
{.5
.
u
The seriesresonantcircuit.
b
the following equationsarefoundto applyto the seriesresonantcircuit of Figure4.5: Q=aoLl R=1/(rooCft)
(4.22)
o il
o 3 @= , o " l t * u
(4.23)
e1
GA)
l-Rl(2L)
r( B= Rl L (radls)
(4.24)
Rr=Rr+XrlQt+XrlQ"
(4.2s)
I I Q"n = Rr / X t + (l I Qt +l I Qc)
(4.26)
L
It follows from (4.26)that similarto the parallelresonantcircuit, the unloadedp for the seriesresonantcircuit is givenby l l Q " = l l Q t + L /Q "
(4.27)
The reactancefor a seriesresonantcircuit with O: l0 are shown in Figure4.6 atthe to theloadresistance. valuesarehighcompared rcsonantfrequency.Notethatthereactance With the sameloadedp, the frequencyresponseof the seriesresonantcircuit is identicalto that ofthe parallelresonantcircuit.
FEcrc 4.6
cfucuitwith O: 10. A seriesr€sonant
a
133
Narrowband,Impedance-Matchingwith LC Networks
4.4 L.SECTIONS An L-section is a two-elementmatching network. The fow possibleconfigurationsare shownin Figure4.7. Dependingon the positionof the first component(asviewed from the load),the oadresistancecanbe transformedupwardsor downwardswith an L-section. When the first reactivecomponentis a seriescomponent,the transformationis upward;andwhenit is a parallelelement,the transformationis downward. caused Thesecondelementin theL-sectionis usedto removetheresidualreactance .v thetransformationelement(i.e.,the first element).This secondelementis thereforethe . ompensating element. The basicprincipleusedin narrowbandimpedancematchingis thatthe resistance .f a complexload is not the samewhenviewedin impedanceor admittanceform. This is .lustrated in Figure4.8. When a reactive element (X,) is added in series with a resistor (R) and the ,'quivalentparallel combination is considered(seriesto shunt transformation),the -csistance increaseswith a factor
D r = r +0 l
(4.28)
.i here
(4.2e\
)1= X1/ R
When a reactive element (Xt) is added in parallel with a resistor (rR) and the :quivalent series combination is considered (parallel to series transformation), the 'esistancedecreaseswith the samefactor (D,). In this case,however,the Q-factor in (4.28) s defined by
ffi
n'-
-T-r*1
jX'
trx,
fn
^'-
(a)
(c) r*glrr.e4.7
The four possibleconfigurationsfor an L-section.
\rt
I (b)
(d)
"
Design of RF and Microwave Amplifiers and Oscillators
134 js0o
F13lrc 4.t
A complex impedancedisplayed in impedanceand admittanceform.
Qr=-Rl Xr=
(4.30)
g r1 6
The ratiosdefinedin (4.29)and(a.30)aresimilar in form to the Q-factorsof the series or parallel resonant circuits, respectively.These ratios are referred to as transformationps. is caniedoverto the transformationQ. The sign of the reactanceor susceptance It follows that the transformationQis positive when the effectiveseriesreactanceis is capacitive inductive (impedanceformat) or when the effective shunt susceptance (admittanceformat). The reactancechangesby a factor
\=r+U Q?
t
.'
I
(4.31)
in the tansformation step. As is the casewith the resistance,the reactanceincreasesafter a seriesto shunt whena shuntto seriestransformationis considered. transformationanddecreases The reactanceof the first elementused in an L-sectionis determinedby the (R) to the valuerequired(R')' tusformation Q requiredto transformthe loadresistance Tb Q valuecanbe calculatedby usingthe relationship
R ,= D r R = ( l + Q I R
(4.32)
A positive or a negativesign can be assignedto the transformationQ. level. The secondelementin the L-sectionis usedto achievethe desiredreactance of this elementis givenby If a purelyresistiveinput impedanceis required,the reactance
Xt=-Xll+tlQI=R'lQr
(4.33)
if the first element is a shunt element, and by
X z = - X t / ( 1 + ll Q I = - R ' Q r
t I
(4.34)
if the first elementis a shuntelement. Equations(a.33) and (4.34) can be verified easily by using the relationships Z=l I YandY:1 /trespectively.
Narrowband Impedance-Matching
t35
with LC Nefworks
The formulasrelevantto the designof an L-sectionaresummxizedinTable 4.2. of anL-sectionnearthe Whenthetransformationp is high,thefrequencyresponse resonantfrequencywill be similarto that of a simpleseriesor parallelresonantcircuit. The circuit Q of the L-sectionis approximatelyequalto half the transformationQ. If a more accuratevalue for the Q of the circuit is required,the procedureoutlined in Section4.9 canbe followed.
EXAMPLE 4.2
propertiesof an L-section. Illustrationofthe transformation
propertiesof anL-sectionwill be illustrated Thetransformationandcompensation hereby usingtheL-sectionshownin Figure4.7(a)asanexample.In this example,
=," jlo -.-T--r\-/-l Addition of the transforming elernent
*'n
The parallel equivalentofthe seriescombination(Y:1 | 4
Cancellationof the residualreactance
The transformed resistance
flure
4.9
lllustration of the transformation and compensationproperties of an L-section.
Desigr of RF and Microwave Amplifiers and Oscillators ' Table 4.2 FormulasRelevantto the Designof L-Sections Downward transformations
I,,'-rl,,n,
R ' = Rt ( l + Q I
(4.3s)
x'=xt/Q+llQh
gs6)
Xz= Xin+Q,R'
(4.37)
Upward transformations
l/R'+jBin 1
pF
9X
--r-
I
l,*,,l-_l_
I vtP,2
sX
trR
n,=R(r+Q?)
(4.38)
x'=x(l+r/Q?) g.3s) Bz= Bin+ Qt I R'
(4.40)
the resistanceand reactanceat the transformationfrequencyaretakento be R:lfl olZ: lQ llroC:2Q The first elementin the matchingnetwork(seeFigure4.9) is a seriesinductorof is alsoequalto 1Q,thetransformationp is equal 71Q. Becausethe loadresistance
Narrowband Impedance-Matchingwith LC Networks
137
is transformedupwardswith a factor I + 12: 2 to to +1, andthe I Q loadresistance a valueof 20. ThetransformationQhasthe samemagnitudebeforeandafterthe transformation(the sign of the Q changesin the process),and the reactancein parallelwith the transformedresistanceis therefore +72O(still inductive).This reactanceis removedby resonatingit off with a capacitor(-72O reactance),after which the original 10 resistorhas beentransformedto 2Q at the frequencyof interest. element,whilethecapacitor Notethattheseriesinductoris atransformation element. is a compensation
EXAMPLE 4.3
Designingan L-section.
An L-sectionwill be designedto transforma loadof 50Oto 250Oat 50 MHz asan above. exampleof the applicationof the theorydiscussed Becausethe transformationis upwards,the first elementof the L-section mustbe a serieselement.The diagramin Figurea.l0(a) applies. It follows from the diagramthat the transformationQ must be equalto 2; equalto it follows that a seriesinductoror capacitorwith reactance X=QR=2x50=1000 is required. If the first elementis chosento be an inductor,the requiredinductanceis I = 1 0 0/ ( 2 n x 5 0 x 1 0 6 )= 0 . 3 1 8p H Theparallelequivalentofthe transformingsectionis therequired250Qin parallel with a reactance ,t-:
X'=R'lQr=25012=125{l
p200O
0 . 3l 8 p H
2500 -t s50Q
(b)
(a) rgure4.10
(a) The transformation diagram and (b) the L-section relevant to Example 4.3.
138
Designof RF and MicrowaveAmplifiers and Oscillators
Note that the Q-factorsof the seriescombinationandits parallel equivalent mustbe equalin magnitude(thesignof theQ changeswhenthe transformationis done). Thecapacitance requiredto removethereactivepartofthe inputadmittance of the resistorandinductorcombinationis C = l / | 2 5 ( 2 n x 5 0x 1 0 6 )=l 1 2 5p F
4.5
o
a tl
't
The designed networkis shownin Figure4.10(b).
ir
PI-SECTIONSAND T.SECTIONS
4
Pl-sectionsand T-sectionsare three-element matchingnetworks.A Pl-sectionhas two parallelelements,andthe T-sectionhastwo serieselements,asshownin Figure4.11.
T F
ut
ri
1:
r
("'
i
{
I
t I
ftrrc
4.1f
Topology for (a) a Pl-sectionand (b) a T-section.
a I
3 The first two elementsin thesesectionsaretransformingelements.One of these elementscausesthe resistance to increase.while the othercausesit to decrease. The reactancelevel is set by the last elementin the section(the compensating elmt). Becausethe resistanceis transformedtwice, there are two transformationQs in bb sections.ThehighesttransformationQ canbechosento haveanyvaluehigherthan that requiredin an equivalentL-section. As in the caseof L-sections,the bandwidthof Pl-sectionsandT-sectionsarealso &mined by the transformationQs. Wherethe two Q-faclorsaredifferent, the Q of the R'
R'
{
F
G F
t
t
R'I
(a) ftrrc
{f2
r
{
R'
O)
process An alternative viewof thetransformation in (a)a PI- or (b) a T-section.
rf GI
Narrowband lmpedance-Matchingwith LC Networks
139
network will be approximatelyequalto half of the highesttransformationp. the bandwidthof a Pl-section Becausethe highesttransformationp is adjustable, or a T-sectioncanbe controlled. Thetransformationpropertiesof a Pl-sectionor a T-sectioncanalsobe considered, as illustratedin Figwe 4.12. The fact that the sourcetermination(R") and the load termination(R) mustbe transformedto the sameintermediatevalue(R') is consideredin this case.Both terminationsaretransformeddownwardsin a Pl-sectionandupwardsin a element T-section.Thesecondelement(ascountedfromtheloadside)is thecompensation in this case.The bandwidthis determinedby the side with the highesttransformationQ.
4.5.1
The Pl-Section
approach)areilllstrated in causedby a Pl-section(cascade transformations Theresistance Figure4.13. The resistanceis first transformeddownwardsby a factor I + U and then upwardswith a factor | + Qi . Q, is the first transformationQ and is associatedwith the load resistanceandthe first elementof the network.Q2is the secondtransformationQ. in series Thesecondtransformationp is equalto theratioofthe effectivereactance with the transformedresistance(R') andthe transformedresistanceitself.
r+Qr'
Figure 4.13
(a) Upward transformationof the load resistancewith a Pl-section and (b) downwmd transformationofthe load resistancewith a PI-section.
The transformedresistancewill be lower than the load resistancewhen the first :ransformationQ is higherthanthe second.An upwardtransformationrequiresthe second :ansformationQ to be higherthanthe first. by therequiredbandwidth Thevalueof thehighesttransformationQ is determined ,f the network.The Q of the networkis approximatelyequalto one-halfof the highest :ansformationp when the transformationQ factorsare sufficiently different. Thetransformationof a l0Q loadto 50Oby usinga Pl-sectionis illustratedin detail r Figure4.14.
Design of RF and Microwavc Amplifiers and Oscillators
r.r0
-jl0o
A designedPl-section
]J,* +
Shunt to series transformation
-jl0o
-j'a
€ Seriesto shunttransformation
=,," rcactancc of theresidual Canccllation
Ittrc
4ff
The transformationof a l0O load to 50Q with a Pl-section.
The formulasrelevantto the designof a Pl-sectionaresummarizedin Table4.3.
EXAMPLE 4.4
A Pl-sectionexamPle.
A matchingnetworkfor transforming50Oto I 2.5O will bedesigned.Themaximum transformationQ will be takento be 5. Becausethe transformationis downwards,the first tansformation p will highest. the be The next stepis to choosethe networktopology to be used.The network is arbitrarily assumedto have an inductor asthe first element,while the other componentsarechosento be capacitors.(It is not possibleto chooseboth ofthe first two componentsto be inductors or capacitors.If this is done, the second transformationQ will be the highest.)
Narrowband Impedance-Matchingwith LC Networks
t4l
Table 4.3 Formulasfor designinga Pl-section
Qt=Q^o=2Q
(4.4r)
R ' = R/ ( t + Q ? )
(4.42)
I+fi=R"lR'
(4.43)
Y1=Q1R l
(4.44)
Xz=R'(Qr+Qz)
(4.4s)
Y t = Q 2R I "
(4.46)
Increasing the load resistance
R
Qz=Q^*=2Q
(4.47)
R , = R , l,( t * d )
(4.48)
l + Q l = P 1P '
(4.4e)
Yt=Q1/R
(4.44)
Xz=R'(Qr+Qz)
(4.4s)
Yr=Qt I R"
(4.46)
Becausethe first transformationp is Qr=
-5
the reactanceof the inductor must be xr:50/5: l0o The secondcomponentmustchangethe transformationQto 2.35.Tlre Q must be positive (inductive) if the last componentis to be a capacitor:
r42
Design of RF and Microwave Amplifiers and Oscillators
+ (-5)) = -5.1o Xz = R'(Qr+ Q) = 1.92(2.35 is of thelastcomponent Thereactance X3= -R" I Qz= -12.512.35= -5.3C1 networkis shownin Figure4.150). Thedesigned
-j5.lo t+Q22:12.51t.92 Q2:+2.35
12.5il
st.92
o)
(a)
(a) The transformation diagram corresponding to Example 4.4; (b) a Pl-sectionfor matching50Q to 12.5Q.
Figure 4.15
4.5.2
The T-Section
Thedualof a Pl-sectionis a T-section.Therefore,the formulasfor designinga Pl-section can also be usedto designa T-section.In orderto do this, it is necessaryto replacethe respectively. resistanceandreactancein theseformulaswith conductanceandsusceptance, The terminationsusedmust also be inverted(i.e., if the actualterminationsfor the Tsectionare 50Q terminations,the terminationsfor the equivalentPl-sectionshouldbe
l/s0o). The reactanceresultsofthe Pl-sectionapply directly to the T-sectionifthese are requiredfor a Pl-section To illustratethis,ifthe components interpretedto besusceptances. requiredin the T-sectionare7l0S,-i5S, andi3S. are7100,-i5O and73O,the components
12.55
Figure 4.16
-r -15.33
An exampleof finding the dual of a Pl-section.
NarrowbandImpedance-Matching with LC Networks
143
Table 4.4 Formulasfor designingT-sections Decreasingthe load resistance
Qz = Q.* =2Q
(4.s0)
n =n " Q +Q ? )
(4.s1)
I* 4 = R'/R
(4.s2)
Xr = QtR
(4.s3)
Yz= (Qr+ Q) I R'
(4.s4)
X, = QrR"
(4.s5)
lncreasingthe load resistance
t*Qr'
Qr=Q^^=2Q
(4.s6)
n'=R(t+01)
(4.s7)
1 + Q l = P ' 1P "
(4.s8)
Xr=QrR
(4.s3)
Y z = ( Q t + Q )I R '
(4.s4)
Xt=Q, R"
(4.ss)
This approachis usefulwhena programto designPI- andT-sectionsis developed. Theprogramcanbe written to designPl-sectionsonly, andby enteringthe specifications correctly it can also be used to designT-sections.When the designis not done by computer,it is betterto follow the procedureoutlinedin Table4.4.
4.6 THE DESIGNOF PI.SECTIONSAND T-SECTIONS WITH COMPLEX TERMINATIONS Theprocedures outlinedin theprevioussectionscanbe extendedeasilyto thegeneralcase *'herethe loadandsourceimpedances arecomplex.The approachis illustratedin Figure 4.t7.
t44
!
Design of RF and Microwave Amplifiers and Oscillators
The reactivepartsof the load andsourceimpedance(T-section)or admittance(PIsection)are ignoredinitially, andthe networkis designedto matchthe load and source resistanceto each other. The first and last componentsare then changedto take the imaginary parts of the load and sourceimpedanceor admittanceinto account. to a new seriesvalue,the Becausea T-sectiontransformsa seriesloadresistance load and the required input impedancemust be specifiedin seriesform, that is, as for a Pl-sectionmustbe of parallelform. The first stepin The specifications impedances. desiping a matchingnetworkwhenthe terminationsarecomplex,therefore,is to getthe terminationsin the right form. The following equationsapplyto Figure4.17:
xi=Q,R"
(4.se)
Xt=QrRt-Xt
(4.60)
Bi=Qr/R'
(4.6r)
Br=QtlRL-BL
(4,62)
R'+jxb
llR'+jBio
Figure 4.17
The design of (a) a T-section and (b) a Pl-section when the terminations are complex.
EXAMPLE 4.5
Designinga T-sectionwith complexterminations.
A T-sectionfor matchinga l0 +710Qloadto 50 +740O(seeFigure4.19)with a maximumtransformationQ equalto 5 will be designed.Thesespecificationsare in impedanceform, asrequiredfor a T-section.
Narrowband Impedance-Matchingwith LC Networks
145
p260 *Q22:260150 Q2:2.0s
Figure 4.18
The transformationdiagramfor the T-sectionof Example4.5.
The transformationdiagramfor this problem is shown in Figure 4.18. Becausethe transformationis upwards,the first transformationp must be the highestin this case.The secondtransformationQ mustbe equalto 2.05. With the p-factors known, the next stepis to choosea topology.If the network shown in Figure 4.19 is chosen,the bandwidthof the circuit can be calculatedas was done before. Since the Q-factorsof the load and source impedancesarelow comparedto the maximumtransformationQ, predictableresultscanalsobe obtainedwith othertopologies.Theonly majordifferencewill be in the rateat which the slopeoutsidethepassbandlevelsoff becauseof thehigher numberof poles. The componentvalues of the chosennetwork can now be determinedby usingthe valuescalculatedfor the transformationQ's. In orderto havea transformationQ of 5 at the load,a reactanceof+/40O must be addedto the existing+/0O; that is, Xr=(5Q)-10=40O After the first transformation,the transformationQ is still equalto 5. In orderto changeit to 2.05,a capacitorwith susceptance d,
Y z = ( Q r + Q ) / R ' = ( 5 + 2 . 0 5 )/ 2 6 0 = 2 7 . l m S
740O
jl03o
j40a
(Dr740o.+
ftrrc 4.19
A T-sectionfor transforminga l0 +jl00 load to 50 + j40Q.
jl0o
Design of RF and Microwave Amplifiers and Oscillators
must be used;that is, both Qr andp2 must be positive. It is not possiblein this caseto usea capacitorwith susceptance
Y= (Qr-lQrDt n' sincethe last componentof the network was chosento be an inductor. and Thelastcomponentofthe T-sectionmustremovetheresidualreactance changeit to the requiredlevel of +740O.In orderto do this, a 14.3Oinductoris required. The designednetworkis shownin Figure4.19.
4.7 FOUR.ELEMENTMATCHING NETWORKS Whenfour elementsareused,the controloverthe frequencyresponseof the impedancematchingnetwork increases.The bandwidthcan be lower or higherthan that of an Lsection. One possibleapproachto designinga network to have a very high Q is shown in Figure4.20. The two low-Q sectionstransformthe load impedanceandthe sourceimpedance (R < Rr,R < Rt with thenetwork,asshownin Figure4.20),and to havethesameresistance the high-p sectionsetsthe reactancelevel andprovidesthe requiredrejection. Strictly speaking,only one downwardtransformingsectionis requiredin this network.Whentwo downwardtransformingsectionsareused,however,it is oftenpossible it is oftenpossibleto use to decrease the insertionlossof the circuit.This follows because p when this is done. in the circuit components higher
High-p section
;
i(RQ-4-x) R+j4 Low-Q section ftrre
4.20
R+iXz Low-p section
network. A high-Q, easilytunable,four-elementimpedance-matching
When the approachillustratedin Figure4.21 is followed,the bandwidthcan be to be smallerthanR'). In this wider thanthat obtainablewith an L-section(jR,is assumed aretransformedto their geometricmeanby using casethe sourceandthe load resistance two L-sections.
It pn un
147
Narrowband Impedance-Matchingwith LC Networla
Low-p section
Figure 4.21
4.8
Low-Qsection
A widebandfour-elementimpedance-matching network.
CALCULATION OF THE INSERTION LOSS OF AN LC IMPEDANCE.MATCHING NETWORK
was shown in Chapter3 that the ideal componentdoesnot exist. For this reason,all -acticalcircuitswill havesomeinsertionloss.If the insertionlossis to be kept low, the r loadedQ-factorsofthe components mustbesignificantlyhigherthantheQ ofthe circuit. The insertionlossof anycascaded LC networkcanbe computedby following the :ocedureoutlinedbelow: l.
Model eachreactivecomponentin thenetworkasan idealcomponentwith a resistorin seriesor in parallelwith it, dependingon whetherit is a series or shuntelement,respectively. The valueof this resistance canbe determinedfrom the unloaded estimated for the component. Q-factor Unloadedp-factorsfor capacitorsandmagnetic-core inductorscan usually be found by using the datagiven by the manufacturer,while those of air-coredsolenoidalcoilscanbe determinedby following theprocedure outlinedin Section3.3.6. The unloadedQ of a componentmay be a strongfunction of the frequency.
2.
Assumethat the powerdissipatedin the loadis equalto lW.
J.
If the first componentof the network(asviewedfrom the load)is a series element,calculatethe powerdissipatedin it by usingthe equation Pn=(Ro/R)PL
(4.63)
whereRo is the seriesresistance associated with the elementandR, is the (effective)loadresistance. P, is thepowerdissipatedin theload( I W in this case). If the first componentis a parallel element,calculate the power dissipatedin it by usingthe equation
Desigr of RF and Microwave Amplifiers and Oscillators
14t
(4.64)
Po=(Go/G)PL
where Gq is the parallel conductanceassociatedwith the component,and of the load.P, is the powerdissipatedin G1is the (effective)conductance the load. 4.
Add the power dissipatedin the first componentto that dissipatedin the load:
(4.6s)
P, = Pt+ Po
consider the first componentto be part of the load and calculatethe new (effective) load admittanceor impedance. 6.
Repeatsteps3 to 5 until the power enteringthe matchingnetwork (Pp)and the effective input impedanceof the network (Z) ne known.
7.
Calculate the transducerpower gain of the network (Gr) by using the equation
G,=(r-lt"l')+
(4.66)
- 'jl'l,,,,, -l':" =1, 1z^*z,l
(4.67)
)
|
4RinR"
PLI Pr
( R , n+ R " ) 2 + ( X , n + X , ) z
(4.68)
whereS"is the input reflectionparameterwith Z"asnormalizingimpedance, Zrn= Rir+ iXin
(4.6e)
and
(4.70)
Zr= Rr+ iX,
whereZ"is the internal impedanceof the sourcedriving the network.
EXAMPLE 4.6
Calculatingthe insertionlossof a Pl-section.
As an exampleof the applicationof this procedgre,the insertionloss of the PI-
149
Narrowband Impedance-Matchingwith LC Networks
section designedin Example4.4 will be calculatedat the centerfrequency.The unloadedp-factors of the capacitorsare assumedto be 500, while that of the inductoris takenas 100. associated with the inductoris lmS, the seriesresistance Theconductance with the associated andtheconductance capacitor is 0.010, with the first associated is 0.38 mS. secondcapacitor Thepowerdissipatedin theinductoris 50 mW. Thepowerenteringthe last sectionof the networkis therefore1.05W.The input impedanceof this sectionis
Z =2.0+ i9.6O Thepowerdissipatedin thefirst capacitoris 5 mW. Thepowerenteringthe networkat this point is therefore1.055W.The input admittanceat this point is .
Y = (84 - 7186)mS
:'
The power dissipatedin the last capacitoris also 5 mW. The total power enteringthe networkis therefore1.06W.The input impedanceis
zi n = r 1 . 9 - j o.4 5 o powergainofthe networkwascalculatedto be 0.94,thatis, Thetransducer an insertionlossof 0.3dB. :
{.9
: ! .
CALCULATION OF'THE BANDWIDTH OF CASCADED LC NETWORKS
ne bandwidth of a network can be found iteratively if its transducerpower gain is :erminedas a function of frequency.The transducerpower gain of any cascadedLC :rvorkcanbe foundby following the procedureoutlinedin the previoussection. Becausethe cut-off frequencies(3-dB) of L-, PI-, and T-sectionsare known to ,ffi 'tood apptoximation(f-rau: fo+fo I Q^), the exactbandwidthof thesecircuits can be :rerminedquickly by following this procedure.
EXAMPLE 4.7
The 3-dB bandwidthof a matchingnetwork.
By following the proceduredescribed,the 3-dB cut-off frequenciesofthe network in Example4.6 arefoundto be 83 and 130MHz, thatis, if thecenterfrequencyis selectedas 100 MHz. The exactQ of the circuit is therefore2.3 insteadof the 2.5. estimated If a bandwidthother than the 3-dB bandwidthis required,it can be found easilyby following the sameprocedure.
I
150
Desiga of RF and Microwave Amplifiers and Oscillators
. ]: SELECTEDBIBLIOGRAPHY RF PowerTransistorManual,Somerville,NJ: RCA Corporation(Solid StateDivision), 197t.
CHAPTER 5 COUPLED COILS AND TRANSFORMERS (.I
INTRODUCTION
Thcn the parasiticscanbe ignored,transformersarc ideally suitedfor impedancescaling. :ealandpracticaltransformerswill be consideredin this chapter. A practical transformer differs from the ideal in that it has leakageflux, finite izing inductance, losses, and parasitic capacitance, all of which degrade its
-.rnance.Severalequivalentcircuitsfor practicaltransformers will bepresented here. is required widebandtransformation ofresistance Transformers areoftenusedwhen possible,this is usuallya betteroptionthanusingLC networks).It will be shown of a transformeris mainlydeterminedby thecoupling thatthewidebandperformance . This is alsothereasonwhy stackedtoroidsor baluncoresareusuallyusedto realize a transformer. Although the finite magnetizing inductance and the leakage inductance are in a wideband transformer,they can be put to good use in narrowband *nhing networks. Several narrowbandmatchins networks using transformerswill be :ered in detail.
Becauseit is importantto adjustthe couplingfactor of atransformerto the required when narrowbandmatching networks are used,methodsto measurethe coupling will be alsobe consideredin this chapter.
:
TIIE IDEAL TRANSFORMER
:o uivalentcircuit of the idealtransformeris shownin Fisure 5.I . The numberof turns ..in . primaryand secondarysidesof the transformerare,respectively,n, andn2. The idealtransformerhasthe followine characteristics: The magneticflux in the two windings is the same.Therefore,there is no leakageflux. The voltageinducedby the changingflux in eachwinding is given by Faraday'slaw: 151
F t52
Design of RF and Microwave Amplifien and Oscillaton '
R
r
h I
l
+
I
Irr
:
Figure 5.1
T
v2
R2
The ideal transformer.
V=n,AAlAt
(5.1)
wheren, is thenumberofturns inthe winding underconsideration.Because the flux coupling eachwinding is the same,the ratio of the primary to secondaryvoltageof thetransformeris \ ll/z = n, / n,
(5.2)
This relationshipis moresignificantif it is written in the form V1ln1=Vr/n,
(s.3)
The voltage per turn, therefore, is the same for both sides of thc transformer.
2.
to establishtheflux inthe idealtransformer Theprimarycurrentnecessary is negligible.The input impedanceof the transformerwith the load oper circuited,therefore,is infinite.
3.
Thereareno lossesin theidealtransformer.The averagepowerdissipated in the load,therefore,is exactlythe sameastheaveragepowerenteringtb transformer. Becausethe ideal transformerhas no reactivecomponents,tb powerdissipatedinthe loadis alsoequalto theinstantaneos instantaneous power enteringthe transformer;that is, vlt = v2t2
l
(-i 'l t
By using (5.2) to replace the voltages in this equation, the relationshig between the primary and secondary currents is found to be nrl, = nrl2
(j j I
CoupledCoils and Transformers
153
Thedemagretizingforce(magnetomotiveforce)ofthe currentin the secondarywinding is, therefore,balancedby that of the current in the primarywinding. By using this equationand (5.3), the relationshipbetweenthe primary and secondaryimpedancesis found to be
Zt =Vt I I, =fn, I nrlzZ,
(s.6)
The impedanceratio is, therefore,only a function of the tums ratio of the transformer. 4.
of theflux density Thepermeabilityofthe idealtransformeris independent is a perfectly linear the ideal transformer implies that the core. This in device.
From an impedance-matchingviewpoint, the ideal transformer is very by any factor. usefulin that it canbe usedto scaleimpedances
5.3 EQUIVALENT CIRCUITSFOR PRACTICAL TRANSFORMERS I practicaltransformerdeviatesfrom the ideal in the following ways: 1.
Thereis someleakageflux and,therefore,leakageinductance.
2.
The magnetizinginductanceis finite.
a
Therearelossesin thewindingsof thetransformer(copperlosses),aswell as in the core(hysteresisandeddycunentlosses).
4.
with signallevel Therelativepermeabilityofthe magneticmaterialchanges anddc current(saturation),aswell aswith frequencyandtemperature. Apart from the effect ofthe leakageinductance,the high-frequencyrebetweenthe sponseis degradedby the presenceofparasitic capacitance windingsandtums of eachwinding.
A circuit model for the practical transformer,ignoring the capacitanceand inlinearities,is shown in Figure 5.2 Ul. The two dots indicatethe sidesof the two .rndingsthat havethe samevoltagepolarity. represent the leakageflux, the seriesinductanceto the Thetwo seriesinductances 'rt togetherwith the shuntinductancearethe magnetizinginductances, r1 the resistance
154
Design of RF and Microwave Amplifrers and Oscillators
i lLrr-u/n1 i r,
Figure 5.2
Ir/,
transformer. An equivalentcircuit for a pra.ctical
and 12 representthe copper losses,and Rorepresentsthe lossesin the magnetic material. The mutual inductance Mcan be determined as a function ofthe magnetizing inductances .Lrt and Lrz by using the re\ationship
I : I
L I t f
s
t
M = k(LtrLzz)tt2
whereft is the coupling factor of the transformer. The symboln in Figure5.2canhaveanyarbitraryvalue,but it is usuallychosento be equalto thetumsratio ofthe two windingsofthe transformer.However,a betterchoice for it is
n=-
tr h
ft
l
(s.7\
r lE -
(s.8)
k\ Lu
Ifthe lossesin the magneticmaterialcanbe ignored,the equivalentcircuit for two oorryledcoils (seeFigure 5.3(a) can be usedfor the transformer.The circuit shownin Figure5.3O) is equivalentto the coupledcoil circuit [2]. This canbeprovenby settingup tbe Z-parametermatricesfor the two circuits. The transformationratio shown in Figure 5.3(b)is that for the impedances. of thetransformerareimmediatelyevidentfrom the The following characteristics oquivalentcircuitin Figure5.3(b): l.
The load impedanceis transformedto the primarysideof the transformer as ZL =lLr, / &2 Lz)lfZ, + rrl
(5.9)
that is, as long as
IahzQ- k')l' <
l
This equationcanbe changedto a moreusefulform by substitutinELzzby using(5.9):
Ialrr(l- k' ) / t' l' ..lzLl'
155
CoupledCoils and Transformers
rl
L22Q-t?)
L ! / Q CL 2 ) : 1 (a) Figure 5.3
(b)
Two equivalentcircuits for a practicaltransformer(Roneglected).
[al"r(r- k')I tt']' ..lzLl' leadingto
lr / k2 -Ll<
(s.1 r)
and,if the coupling This inequalitywill alwaysapply at low frequencies, betweenthe windingsis good,it will alsoapplyat higherfrequencies. factorat low It followsfrom (5.9)thattheimpedance transformation frequenciesis always
(s.r2)
n2 = Ln I (k2 L2z)
2.
The high frequencyresponseof the transformeris limited by the leakage reactanceaLrr(l - E).
3.
The low-frequencyresponseof the transformeris limited by the magnetizing inductanceI,,.
4.
The input inductanceat low frequenciesis equal to the magnetizing inductance211.
r2 o
zL
Lnl(E L2) i I Ilurc
5.4
An equivalent circuit for the practical transformer at low frequencies.
156
Designof RF and MicrowaveAmplifiers and Oscillators
If the frequencyis low enoughfor (5.11)to apply,theequivalentcircuit of Figure 5.3(b)canbe simplifiedto that shownin Figure5.4. By usingthis equivalentcircuit,the low cut-offfrequencyis foundto be
(s.r3)
0 r r d s= R " l ( 2 L r 1 )
:
This equationapplieswhenr, andr, canbe ignoredandthe load resistanceR; is R. transformedto be equalto the sourceresistance It follows that,underthesecircumstances, therequiredpri-ary inductancecanbe for the cut-offfrequencyandthe sourceresistance. determinedby usingthe specifications The requiredsecondaryinductanceis givenby the equation ,
L11
- R, Lrr. -
(s.14)
------;-
R" k'
This inductanceis clearlya functionof the couplingfactor,which is usually not known at the designstage. The easiestway to overcomethis problemis to ensurethat the couplingfactoris closeto unity, if possible. Very goodcouplingcanusuallybe obtainedby usingmaterialswith high relative permeabilities. Thecouplingbetweenthewindingsofthe transformeris alsobetterifbalun or stackedtoroidal cores,insteadofa singletoroidalcore,areused.
5.4 WIDEBAND IMPEDANCEMATCHING WITH TRANSFORMERS is possiblewith transformers Impedancematchingover very wide bandwidths(decades) when the coupling betweenthe windings is good and when the parasitic capacitance betweenthe windingsandturnsof eachwinding is small. canbeignored,the3-dBbandwidthof a transformercan If theparasiticcapacitance poles finding the of the equivalentcircuit in Figure5.5.This equivalent determined by be one in Figure5.2by settingn equalto l. can be derived from the circuit poles,a zeroattheoriginand 1 at infinity. Thepoles has two Theequivalentcircuit
Lzz-M
+ vo
Figure 5.5
A T-sectionequivalentfor the hansformer.
157
CoupledCoils and Transformers
canbe found by settingthe impedancesin either of the two loops of the equivalentcircuit equalto zsro.lt this is donefor the first loop,the following equationis obtained:
- stut+ (R, +rr)l = Q (& +",) +s(2,, - ItI)+sIrIlltsLzz
(s.l s)
Thisequation simplifiesto s2LrrLrrTl- k') + sfLrrR!+ LtlRLl+R;R;= 0
(s.r6)
The poles,and thereforethe cut-offfrequencies,canbe obtainedby solvingthis equation. Whenthe loadresistance andthe sourceresistance arematched,asis generallythe case,the following equationapplies: RL=Rr.+r, = (k2Lr2l Lt).R:
(5.17)
Becausethe couplingfactor is usuallyunknownat the outsetof the problemand is usuallycloseto unity at low frequencies, the transformeris normallydesignedfor
(s.18)
R'r=(Lrrl L1)'R: If this valuefor Rf is substitutedin (5.16),it simplifiesto s' Lrrlr(l
(s.re)
- k') + 2sL z2R:+ (L2zlrr r ).Rjxj= 0
This equationcan be written as ld,,(l+fr)+Rjllslrr(l-fr)+.l?jl=0
.
..
.
(s.20)
Thecut-offfrequenciesare,therefore,givenby the following equations: or = Rj /[Lrr(l+ k)]
(s.2r)
o n = R J/ I L I ( I -
(s.22)
k)l
thatis, when(5.18)applies. It is clear from theseequationsthat the low cut-off frequencyof the transformeris &termined by the effectiveresistancein parallelwith the magnetizinginductanceandthat thehigh cut-off frequencyis a strongfunctionof the couplingfactor. The relativebandwidthof the transformerwith the load and sourceimpedances matchedis given by
ar/aL=ll+kl/U-k|
(s.23)
The relative bandwidth of the transformer,therefore,is only a function of the coupling
r ;
1s8
Design of RF and Microwave Amplifiers and Oscillators
factor, that is, when the load and source impedancesare matched and the parasitic capacitancecan be ignored. andcloseto unity at the lower Becausethe coupling factor is frequency-dependent (5.23)canbe simplifiedto frequencies,
(s.24)
(Drl|d^r=21[1-f]
,
EXAMPLE 5.1
Designinga transformerto transforma 50o loadto 300Q.
The primary inductance,secondaryinductance,and coupling factor required to transforma load of 50Oto 300Owith 3-dB cut-off frequenciesat I and 20 MHz will be determinedasan examPle. Assumingthat the copperlossesin the windingsandthe hysteresislosses ratioofthe transformer in themagneticcorecanbeignored,therequiredinductance (5.18): canbe obtainedby using L1 I Lzz= R, / Rr= 50 / 300= 0'1667 The requiredmagnetizinginductance111canbe obtainedby using(5.21): Zrr = 300.0/[2.0nx 1.0x 105(1+1)]= 23.9pH The secondaryinductance,therefore,is L 2 2 =L 1 1 1 6 = 4 ' 0 l t H The requiredcouplingfactorat 2}lvlElzcan be obtainedby using(5'24):
L - k = 2 / 1 2 0/ l l = 9 . 1 '
k=0.9 The couplingfactorrequiredto obtainarelativebandwidthof 20,therefore,is 0.9.
5.5
SINGLN.TUNTD TRANSF'ORMERS
The single-tunedtransformershownin Figure 5.6 canbe usedto stepthe load impedance up or down and to obtain a frequencyresponseidenticalto that of a parallel resonant circuit. If the coupling betweenthe windings of the transformeris good, the leakage inductanceofthe transforrnercanbe ignored' The requiredmagnetizinginductanceof the transfonnercanbe found in terms o -specification, by usingthe equation andthe sourceandloadresistance, the Q
|lir#-
Coupled Coils and Transformers
159
LrlQ? L2) i't Figure 5.6
The single-tuned transformer.
(s.2s)
,toLr= (R,llni)/ Q=R, I (2Q) -,r{rereR;' is the load resistancereferencedto the input side and Qthe circuit Q. is necessary to provideresonance The capacitance
, =t t @tLn)
(s.26)
Becausethe coupling is assumedto be good,the requiredsecondaryinductanceis eiven by the equation - l : := ( R r / R , ) L , I k 2 = ( R r ./ R , ) L n
(s.27)
The next stepin the designof the single-tunedtransformeris to selecta suitable aagneticmaterialandto determinethe typeandsizeof thecorerequired(seeChapter3). If necessary,the requirednumber of tums aroundthe core can be found by -:casuringthe inductanceof a few turns of wire aroundthe coreto be used.Becausethe rJuctanceis proportionalto the squareof the numberof tums, the numberof turns :rquiredcanbe found easily.
5.6 TAPPED COILS .b tappedcoil (Figure5.7) is a very usefulnarrowbandmatchingnetwork.Independent : ':':stment of the center frequencyand the transformationratio of the transformeris it, n
Q ll&fi
$rxt. ;grrc 5.7
nT: nt+n2
The tappedcoil resonantcircuit.
160
Design of RF and MicrowaveAmplifiers and Oscillators
possible.It is alsovery easyto manufactureonceit hasbeendesigned. The frequencyresponseof this transformeris usuallysimilarto thatof the singletunedtransformer. Analysisof the tappedcoil is possibleby consideringit to be two coupledcoils. It follows from (3.23)that the self-inductance of the uppersectionof the coil is givenby
Lt=rn?ro'tlzzslzL*zs.4f
(s.28)
and that of the lower section by
L2= rnlro4 rlzzsl;.^
^l
(s.2e)
where/, is the total lengthof the coil, n1the numberof windingsin the uppersection,n, the numberin the lower section,andn.the total numberof turns(n7: nr+ n). The inductanceof the coil as a whole is given by the equation
Lr=r n?to*tlzz.slz + zsa)
(s.30)
If viewed in terms of its componentparts,the total inductancecan also be written as Lr= Lt+ Lr+2M
( s.31)
By usingthis equationtogetherwith (5.7), M = k(LrLz)'t2 the coupling factor is found to be k =flr - Lt- h\/[z(LtL)tt2]
(s.32)
(5.28)through(5.31)into (5.32),thefollowingcanbe deduced: By substituting 1.
The couplingfactoris dependent on the length-to-radius ratio ofthe coil, as well asthe relativepositionof the tap-point.
2.
The couplingfactoris independent of the total numberof tums (nr).
The couplingfactor of the tappedcoil is given in Table 5.1 as a functionof the relativepositionof the tap-pointfor a numberof lr lr ratios.Thecouplingfactorsarenot shownfor relativepositionsgreaterthan0.5, sincethey arethe mirror image(arithmetic) of the lower values(0.8 conespondsto 0.2).
161
CoupledCoils and Transformers Table 5.1 The coupling factor ofthe tappedcoil as a function ofthe 1,./r ratio ofthe coil and the relativeposition ofthe tap-point
l7/r = |
0.1 0.2 0.3 0.4 u.)
0.543 0.535 0.530 0.529 0.526
lr/r = 1.5
o.449 0.438 0.431 0.426 0.425
=2 17./r
l7/r = 3
l7/r = 4
0.386 0.372 0.363 0.358 0.357
0.304 0.288 0.277 0.272 0.270
0.253 o.235 0.225 0.219 0.217
l7/r = J
0.218 0.200 0 . 18 9 0.183 0.182
It canbe seenfrom Table 5.1that the couplingfactor is not a strongfirnction ofthe relativepositionof the tap-point. The input admittanceof the coupledcoil canbe foundby usingthe equation
'
'l irLr*r*r, - iaLz*zu1[t, = ["-l
l-i^Lr*, i.'Lr+nrlLlrl-LoJ
(s.33)
By applyingCramer'srule to this equation,it follows that
-=IrlE - j-a l r-* 'r -l ,' l t - ll t L l0 jaLr+ Rrl = (Rz + jo,lr) I fjaLr*r*ru(R, + jaLr) - jalr*u jalr*uf After somemanipulation,it follows that
,=
-_t) Lz!r!L_+ _ ", t^r!l_-q't,,tr(t2 _ +{z(f2 !\R^L R? at LltTrltt'- r)' +,nl:r(. a' L'rLl&' - l)' + Lzr
(s.34)
- :becoupling is perfect,this equationreducesto =|tRr-i/(al7)
cxpected.
(s.35)
162
Desigr of RF and MicrowaveAmplifiers and Oscillators
Whenthe couplingis not perfect,theresistivepartof the input admittancewill be frequency-independent if
a2fi tl1t2 - r)' .. I]rR?. that is, if ')]' << (L,Rr)' [aL,Ir(l - t
(5.36)
Whenthis is true,the input conductance will be
(s.37) It canbeseenfrom (5.34)thattheactualconductance ofthe coil will be I 0% higher thanthe valuepredictedby this equationif lroltLz (k' - l\1' = 0.1( Lr R)2 For the parallelinput inductanceto be approximatelyLr.,it is necessary that both (5.36)andthe following equationapply:
a 2L rL r l l - k 2 1 < < 4n l
(s.38)
It follows that the equivalentcircuit of the tappedcoil canbe simplifiedto that shovynin Figure5.8 whenthesetwo equationsapply.
LT
Figure 5.8
c
R;=ttG;
A simplified equivalentcircuit for a tappedcoil resonantcircuit.
rry.F
!P
' e*$':
flF163
CoupledCoils and Transformers
Table 5.2 as a function of the //r ratio and the relative position of the coil tapped the of factor The fransformation tap-pointfrom the upperend ofthe coil K LIRL
0.l0 0.20 0.30 0.40 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95
lTlr =2
lTlr = 4
l.l8 1.46
l.l8 1.46 1.92 2.68 4.00 5.07 6.62 8.99 12.80 19.60 33.00 64.70 166.00 505.00
r.92 2.66 4.00 5.08 6.64 9.04 12.90 | 9.70 33.00 64.00 160.00 732.00
l7/r = 6
Ll8 1.48 L94 2.70 4.00 5.05 6.56 8.86 12.60 19.10 32.10 62.80 162.00 808.00
(5'36) It shouldbe notedthat,from a practicalviewpoint,it is moreimportantthat be can circuit the of frequency rlies ratherthan (5.38). This is becausethe resonant (the slightly coil the r:;tedeasilyto the desiredfrequencyby stretchingor compressing 'ing factoris only a weakfunctionof thellr ratio of the coil)' The transformationfactor of the load (as given by (5.37)) is a function of the variables are Uance Lr,theratios LrlL7, LrlLrandthe coupling factot k. The lastthree the coil' The JctemrinJ by the relative position of the tap-point and the lrlr ratio of
rrmation factor itself is ihereforeonly a function of the llr ratio of the coil and the :-r'epositionof the taP-Point. was ihe transformationfactor for different //r ratios and positionsof the tap-point andis tabulatedin Table5.2.This tablecanonly be usedwhen(5'36)applies' Lb it doesnot, the transformationfactor canbe calculatedby using (5.28)to (5'30), . and(5.34). only It canbe seenfrom Table5.2thatthetransformationfactor ofthe tappedcoil is factor transformation the that evident is also It function of thellr ratio of the coil. r'.esvery sensitivewhenthe coil is tappedcloseto its groundpoint' a A tappedcoil resonantcircuit .-1" designedto matcha sourceto a load with below' ::d Q by following the procedureoutlined sr Procedurefor a Tapped-CoilResonantCircuit
:
Assumethat the input inductanceof the tappedcoil will be equal to the inductance(I, ) of the coil itself (referto Figures5.7 and 5.8). Find the requiredinductanceby using(5.25):
Design of RF and Microwave Amplifien and Oscillators
lQ= R,/tzQl @Lr- tR'llft"l This equationwill only apply if the unloadedQ-factorof the inductor and capacitorusedarehigh comparedto the specifiedcircuit Q.
2.
J.
Designthe inductor(Lr) to havethe requiredunloadedp as describedin Chapter3. Checkif the self-capacitance of the designedcoil is lower thanthe capacitance requiredfor resonance. At this stageLr, lr, r, andnr areknown. The next stepis to determinethe relative position of the tap-point. If the inequality(5.36) f(oL,Lr(l - k' )l' << (L, R r)' applies,Table5.2canbe usedfor this purpose.If it doesnot apply,thenit will be easierto determinethe tap-pointpractically.
EXAMPLE 5.2
Designinga tappedcoil resonantcircuit to transforma 50O loadto 10000.
A tappedcoil resonantcircuitwill bedesignedto meetthefollowing specifications, if possible:fr: l0 MHz, R, = 50O,.R-: 1000Q,Q :20, andIL < l0o . losses,the lossresistancemust be higher In orderto havelessthan 10%o (P:It'lR); that is, than (approximately)ten timesthe transformedloadresistance Rror.= 10.0x 1.0x 103= 10kQ This impliesthat the unloadedQ of the inductormustbe at least O, = Rror,/(roL) = lOk / 25 = 400 It is assumedhere that the lossesin the capacitorcan be ignored.The to usea ver-r requiredp, however,is high in this designandit will be necessary goodcapacitorandevena parallelcombinationof smallervaluesto obtaina p that is high comparedto that of the inductor. Assumingthe l, Ir ratioof thecoil to be equalto 2.0 anda d/c ratioof 0.55. the requiredradiusof the coil is foundto be (using(3.25)) r = Qullk(.f)'''l=1.26 cm
i
The requiredlengthof the coil is 2.53 cm(l7lr = 2). The numberof tums
CoupledCoilsandTransformers
165
canbe found by using(3.27): n =lL (22.9.l I r + 25.4)I rlttz = 4.75 The requiredwire thicknessis (using(3.28) 4 =(l I D)x(d /c)xZr l(N - l) = 0.37cm '
I Becausethe thicknessof No.12 SWG wire is 0.264cm, the requiredwire thicknessis unrealisticandaQ of 400 cannotbe obtainedwith a standardvalueof the wire diameter. to 12= 1.01 cm, the requiredwire If the radius of the coil is decreased thicknesswill be 0.26 cm. Theratio of the formerradius(r,) to that ofthe requiredradiuswasobtained iterativelyby usingthe equation r
I
-
-
.
1
Ll ^lL-r t N, 'l l=1tt - 1/N,l ,rl\r,
(s.3e)
dz
This equationcanbe derivedeasilyfrom (3.27) and(3.28). The numberof turns requiredis Nz =frrl ,r.lt'' Nr = 5.3
(s.40)
Becauseof the reductionin radius,the unloadedQ of the inductorwill to 317.The insertionlosswill thereforeincreaseto approximatelyl3%. decrease The parasiticcapacitance of the coil (0.51pF) is much smallerthan the (637pF). requiredto provideresonance capacitance This tap-pointcanbedetermined. With thecoil designedandrealizable,the 5.2,thatis,ifinequality(5.36)applies.Itfollowsfrom canbedonebyusingTable thelylr: 2 columnthat the coil mustbe tappedwhere Nt/Nr=0.75 that is, where ilr =40 to calculate In orderto establishwhethertheinequalitydoesapply,it is necessary Z, dndL"randtodeterminethe couplingfactorof the the valuesof the inductances tappedcoil by using(5.28):
Design of RF and Microwave Amplifiers and Oscillators
rcurlzzgLY*rr.ol ,n,
L1
|
I
= 0.269pH
'
By using(5.29),the valueof Z, is foundto be L2= 46'6 nH' The valueof the couplingfactorcanbe determinedfrom Table4'1 (or by valueis 0.37. using(5.32)).Its approximate Because laLrLr(l-
k')l' = 462.1x10-15
''
i
and [LrRLl2=400'0x10-12 inequality(5.36)doesapply,andthe tap-pointasdeterminedfrom Table5.2 will be accurate.
5.7 PARALLELDOUBLE.TUNEDTRANSFORMERS Thenarrowbandcircuits discussedup to this point matchthe loadconjugatelyto the source at a singlefrequencyonly. With the double-tunedtransformershownin Figure5.9, it is If thesefrequencies possibleto matchthe sourceto the loadat two differentfrequencies. canbe designedto be small. arecloseenoughto eachother,the ripple in the passband transformeris shownin The shapeof the frequencyresponseof the double-tuned Figure5.I 0 for differentvaluesof thecouplingfactor.Whenthecouplingfactoris smaller thana certaincritical value(k"),the sourcecannotbe matchedto the load. transformeraresuperiorto those ofthe double-tuned The rejectioncharacteristics of a simpleparallelresonantcircuit. The differencebecomesvery pronouncedwhenthe circuit Q becomeshigh.
Figure 5.9
A parallel double-tunedtransformer.
167
CoupledCoils and Transformers
'rgure 5.10
Threetypical responsesofthe double-tunedtransformer.
The transformercanbe analyzedbyusing the equivalentcircuits shownin Figure r.ll.
with the It is possibleto matchthe loadto the sourceat two differentfrequencies ^aralleldouble-tunedcircuitby splittingthe circuitup, asshownin Figure5.12(or Figure 'l3). In Figure 5.12, L2' and Cr' are transformingelements,while Ztt and C1 are to mpensating elements.The two transformingelementscanmatchthe loadresistance to r, at two differentfrequencies, be designed can elements while thetwo compensating :ancelthe residualreactanceat thesefrequencies. L220-P)
c2
RL
Lr/Q? L2) | |
L z ': L r r ( l l l C - l )
cz' = lln /QcL2)lc2 r:':re 5.11
Rr'= lL" /QeL)l RL
transformer. Two equivalentcircuitsfor a paralleldouble-tuned
168
Design of RF and Microwave Amplifiers and Oscillators
Figure 5.12
An equivalentcircuit for determiningthefrequencieswheremaximumpoweris transferred.
Equationsfor theripplein thepassband ofthe double-tuned transformer(seeFigure 5.I 0) canbeestablished by breakingup theequivalentcircuit,asshownin Figure5.I 3. The capacitorand inductorof the left-handmatchingsectioncausethe seriesresistance R, to decrease at high andlow frequencies, respectively.R, is equalto the sourceresistance R. at the resonantfrequencyofthe inductorand capacitor.The capacitorin the right-hand sectioncausestheseriesresistance R.,todecrease monotonicallywith increasingfrequency. If Rf is smallenough,therewill be two frequencies wherethe seriesresistance of the lefthandsectionis equalto that ofthe right-handsection. Becausethetransformedresistance variesoverthepassband, theremustinevitably be someripple in thepassband. Thesizeofthe rippleis a functionof theratio of the source resistance R"to the seriesresistance ofthe right-handsectionat the resonantfrequency
do=l /JLtpl
L,'
, U
t'l','- R , lr, .
&:> f
I
&-
T
+
t
l*' l^,'
,
RL'I (l+Qoz)
too: ll(L1Cr)tt2 roo: ll(LrrCr)tn Figure 5.13
The series resistance of the two impedance-matching
(r)
sections as a function of frequency.
CoupledCoils and Transformers
169
A DesignProcedurefor a Double-TunedTransformer The two maximumpowertransferfrequencies fa ndf,,z, the load resistance.rRr,the sourceresistanceR., and the transducerpower gain (Gr) at the centerfrequency.
Specifications:
1.
is a functionof As might be expected,the minimumripplein thepassband the bandwidthrequired.It can be shownthat, for two given maximum powertransferfrequencies minimumvalueof thekansducer ,fo,,andfo,r,the powergain in the passband will alwaysbe lower than Gr,^rn =
4.f^zI f^t (f^z I f^t)' +2f^z I f^t + 1
(5.41)
Thefirst stepin thedesignprocess,therefore,is to establishwhether conbe realized. the specifiedG7.61n : , .
2.
Calculatethe resistanceratio r (r = X" / lni (l + Q|\D for the required passbandripple:
I+lt-c,.lr'2
f
3.
4.
rr
5.
=
-------.1--------
r- lt- c,l'''
(s.42)
Calculatethe value of Q, (i.e., the second transformation Q of the transformingsectionin Figure 5.12) at the two desired maximum power transferfrequencies:
d.--r=r f^r/ f^r-l
(s.43)
Q ?^ z = rf ^ r / f - t - l
(s.44)
Solvethe following two equationsforthevalues of L2' andC2':.
-a.rI4.+(/ Cl)/a.,=lh_^,1R,l$+$,_^r1
(5.45)
*a . z I4 -Q/C l )/a ^r=l Qr-^rl n,lU+ Q;- ,zl
( 5.46)
Calculatethe valueof .R7.':
n,1 RL= 0+ Q-l-) t 1a2^,Cf
(s.47)
170
Design of RF and Microwave Amplifiers and Oscillaton
6.
Calculatethe value of the input susceptance (8,r, Bn) of the right-hand sectionin Figure5.13at the maximumpowertransferfrequencies .f,r md .f^2. Thesesusceptances are given by the equations
(s.48)
B^r=lmag
ja ^rLi +
G; + ja ^rC) I
B^, =lmag
ja ^214.+
7.
8.
(s.4e) Gi + ja ^rC)
Solvethefollowingtwo equations for thevaluesof 2,, andC,: (l / Li / a ^, - 0 ^tCt=lB^rl
(5.50)
(l / Lt) I a -, - a ^rC,= -lB^rl
(s.s1)
Calculatethe valuesof k, Lrr, andCrby usingthe following equations:
k=I/(l+Li/Lrr)ttz
(s.s2)
L22= (Lrr/kz7n, / ni
(s.s3)
C, =[L11/(k'Lrr)]Ci
(5.54)
9.
If the components of thetransformertum outto be unrealizable,it is often possibleto obtaina practicalcircuitby impedancescalingof theresults.Lsectionscanthen be usedto transformthe load resistanceandthe source resistanceas required.The altemativeis to designa seriesdouble-tuned transformer.
10.
Check the insertion loss and the frequencyresponseby following the proceduresoutlinedin Sections4.8 and4.9.
EXAMPLE 5.3
Designinga parallel double-tunedtransformerto havea passband ripple of lessthan0.5 dB.
Theprocedureoutlinedabovewasfollowedto designaparalleldouble-tuned trans-
t7r
Coupled Coils and Transformers
former with perfectmatchingat 9.0 and 11.0MHz and a passbandripple of less The load resistance is 50Q, and the source than 0.5 dB (Gr: 10-05/10:0.89). is 5000. resistance The resultsof the calculationsin the different stepsaxerepeatedhere: {t
l.
G7,n,6= 0.99
2.
r =I.9925
Q)l
|'
d -s = 0.630 (Qz-s= 0.739)
b
4.-n =1.435 (Qz-tt= 1.198)
t. .t I
4.
Li =19.5ttH C) = 13.2PF
5.
Ri = 5.9kCl
6.
B'r = l'49mS B^z= -236m5 Cr = 157PF Ln = I.7IPH k = 0.284
The transducer
Table 5.3 gain of the double-tunedtransformer as a function of the
Frequency(MHz)
4.00 5.00 7.s0 8.00 9.00 9.95 I1.00 12.00 15.00
Gr(dB)
-23.00 - 19.00 -5.50 -2.40 -0.06 -0.53 - 0.05 -3.70 - 19.00
172
Design of RF and Microwave Amplifiers and Oscillators
Iz = 0'18PH
9.
F
F
The transducerpower gain of the double-tunedtransformeris given in wereassumedto Table5.3 asa functionof the frequency(thecomponents be lossless).
EXAMPLE 5.4
Illustration of the rejection obtainablewith a parallel double-tuned transformer.
As an illustration of the good rejection characteristicsof a double-tuned transformer,the -30-dB quality factor of a double-tunedtransformerwith maximumpowertransferfrequenciesat 5.97 and 6.03MHz (0.l-dB ripple) was calculated. The -30-dB cut-off frequenciesfor the designed transformer are approximately5.915and6.082MHz. Theresulting-30-dB Q-factor,therefore,is Q-n=36 If the sameresults were to be obtainedwith a simple parallel resonant circuit, the required 3-dB p-factor would have been 1026! The highest transformeris 60. transformationQ in the double-tuned transformeris approximatelyequal The- 3dB p-factorfor thedouble-tuned to 7. Theratio of the - 3-dB and- 30-dBO-factorsof thetransformer,therefore,is Q-tolQ=36/7=5.1 This ratio is 145timesthat of the equivalentsimpleparallelresonantcircuit.
5.t
SERIESDOUBLE-TUNEDTRANSF'ORMERS
Although it is not the dual of the parallel double-tunedtransformer,the responseof the seriesdouble-tunedtransformershownin Figure 5.14 is similar to that of the parallel double-tunedtransformer. The insertionloss of the seriescircuit will be lower than that of the equivalent prallel circuit whenthe loadresistance arelow. andsourceresistance Impedancetransformation canbeobtainedwith thecircuitshownin Figure5.14by scalingthe componentson eachside of the transformerto the requiredlevelswhile the couplingfactorremainsunchanged. to matchthesourceconjugatelyto the Theseriestunedtransformercanbe designed
r73
CoupledCoils and Transformers
cl
RL
Figure5.14
nansformer. A seriesdouble-tuned
loadat two specifiedfrequenciesby following the procedureoutlinedbelow,
A DesignProcedurefor a SeriesDouble-TunedTransformer qnecifications:
l.
The two maximumpowertransferfrequencies,J,,andf,,r,the load powergain(G1) andthetransducer thesourceresistance, resistance, at the centerfrequency.
Calculatethe valueof the productkp:
k e = r t . l G , . t / o- t,
(5.55)
,t is the requiredcouplingfactorand Q is the quality factorofeach ofthe two sidesof the transformerwhenthe couplingbetweenthem is equalto zero, 2.
Determinethe couplingfactorrequiredby solvingthe following equation: * 4 + t c 2 1 1 t 2 )M o '-4(kQ)21+V-
M 2 l ( k Q ) n= 0
(s.s6)
where
14= f3'+ 'f3' .f^r.f^z 3.
(5.57)
Determinethe requireduncoupledQ-factorandthe circuit components:
Q= (kg)/ k
(5.s8)
fo =l.f^r.f^r(t- kr)t,rlt,,
(s.se)
'
In4
Designof RF and Microwave Amplifiers and Oscillators
c t = l / [ o r oR "9 ]
(5.60)
Lrt=QR,/ao
(s.61)
L2r=QR7/ao
(s.62)
Cz=llltcooRtQl
(5.63)
If someof thesecomponentsare unrealizable,the specificationscan be changedto be more realistic,or, in somecases,the componentscan be scaledto morerealisticvalues.In the lattercase,L-sectionscanbe usedto transformthe sourceandloadterminationsto thoserequired. 4.
The insertion loss and frequencyresponseof the transformercan be determinedby followingtheprocedureoutlinedin Sections4.8and4.9.The transformercanbecalculatedby 3-dBbandwidthofthe seriesdouble-tuned usingthe equation
B = J b z+ 2 b- l f o t Q
(s.64)
where
(s.6s)
b=k/k" and
(s.66)
k"=r/Q EXAMPLE5.5
transformerto transforma Designinga seriesdouble-tuned 50Oloadto 20O.
As an exampleof the applicationof the procedureoutlinedabove,a transformer at27 and28MHz.The wasdesignedto havemaximumpowertransferfrequencies wereG7': 0.89,R": 50O,andR, : 20O. otherspecifications The resultsobtainedin the differentstepsareasfollows:
r.
kQ= r.4rrs6
2.
M:2.00r32 : 0 ts -0,02101 I( + 7.93123 k:0.05246
175
Coupled Coils and Transformers
3.
Q:27.43 fo = 27.48MH2 Cr: 4.2pF L 1 : 7 . 9 41 t H L22:3.181tH C, : 10.6PF
4.
If the transformeris assumedto be lossless.the 3-dB bandwidthis B:1.97 MHz. The -3-dB and -30-dB p-factorsofthe transfonneraxe Q= 13.98 Q4s:2.81 The ratio of the two p-factors is Q 4 o /Q : 0 . 2
i.9
MEASUREMENT OF'THE COUPLING FACTOR OF A TRANSFORMER
:curatemeasurement ofthe couplingfactorofatransformeris sometimes required.Three rys to determinethe couplingfactorwill be discussed here. In the first method, the open-circuit and short-circuit input inductancesof the -rnsformerare measured,and the coupling factor is derivedfrom thesevalues.This ethodcanonly be usedwhenthe lossesin the transfonnerarenegligible. In the secondmethod,the couplingfactor is estimatedby measuringthe openruit voltagegain of the transformer.An oscilloscopeor voltmeterwith a high input rpcdance (comparedto the leakagereactanceof the transformerat the measuringfre-ncy) is requiredif reliableresultsareto be obtainedwith this method. T\e Z-pararretersof the transformerareusedin the last method.It is usually more ovenient to measurethe S-parametersof the transformer. These pararneterscan be 'overtedeasilyto Z-parameters by using(1.148).
i.9.1 Measurement of the Coupling Factor by Short-Circuiting Secondary Winding
the
1c influence of short-circuiting the secondary winding of two coupled coils can be
:Llishedby usingtheequivalentcircuitshownin Figure5.3(b).Theresistivelosseswill r:urned to be negligible. The input admittanceof the transformer, with the secondarywinding short. :ed,is given by the equation
t76
Design of RF and Microwave Amplifiers and Oscillators
| / (aLr) = 1/ (corrr) + | / l( Ltl / 1*2tt 17atrr(l - k' )l = l / ( t r , L r)+l /[ro Z ,,(1 I kz -D l
(s.67)
This equationcanbe rewrittento obtainthe valueof the couplingfactor asa function of the othervariables:
(s.68)
&=[l- Lr/Lnltz
It is thereforepossibleto determinethecouplingfactorof thetransformerby measuringthe open-circuitand short-circuitimpedancesof the transformer(i.e., if the lossesin the transformercanbe ignored).
5.9.2
Measurement of the Coupling Factor by Measuring the Open-Circuit Voltage Gain
Whenthe equivalentcircuit of Figure5.3(b)applies,the open-circuitvoltagegain of the transformeris given by the equation Yz= xU a k(LnLn)ttz lv, / (rr + 7cor,r )
(5.6e)
From this equation,the couplingfactorcanbe obtainedas
v, / V, * =Krl * r' 4) / 1tl2LrrLrrllttz
(5.70)
The couplingfactor can,therefore,be determinedby measuringthe open-circuitvoltage Z ,, andL22canbe determinedby measuring gainof thetransformerr, andtheinductances of the transformer. the open-circuitedprimary andsecondaryinput impedances It is importantfor the input impedanceof the voltmeteror oscilloscopeusedto be muchhigherthanthe leakagereactancela'L22(l- F). Becauseof this, it is a goodidea to definethe low-impedancesideof the transfornerasthe secondaryside.
5.9.3
Deriving the Coupling Factor from S-Parameter Measurements
These of the transformercanbe measured. If the equipmentis available,theS-parameters by using standardconversionformulas. parameterscan be convertedto Z-parameters are If the equivalentcircuit of Figure5.3(b)applies,thetransformerZ-pararrreters givenby the equation
z =l',+iaL" | ljaM
tiaM
f
rz+ i@Lzz)
(s.7r)
CoupledCoils and Transformers
L77
With the Z-pararreters known,it is a simplematterto determinethe copperlosses,aswell asthe primary and secondaryinductances. The coupling factor can be determinedfrom the mutual inductanceM and the magnetizinginductances Ztt andI22by using(5.7).
REFERENCES 1.Skilling,H.H., ElectricalEngineeringClrczifs,NewYork:JohnWiley andSons,1965. "A 2. Van der Walt, P. W., Simple Procedurefor Designing Impedance-Matching Networks with Loosely CoupledTransformers,"ResearchNote, University of SouthAfrica. Stellenbosch.
CIIAPTER 6 TRANSMISSION-LINE TRANSFORMERS 6.I
INTRODUCTION
The high-frequencyresponseof a magneticallycoupledtransformeris limited by the leakageinductanceandthe parasiticcapacitance ofthe transformer. The leakageinductancecanbe decreased significantlyifa balunor a stackedcore (insteadof a toroidal core)is used.It is moredifficult, however,to decrease the parasitic capacitancebetweenthe two windings andthe tums of eachwinding. Ifthe outerand centerconductorofa coaxialcableareusedas the primary and secondarywindings of a 1: I transformerand connectedas shownin Figure6.1(b),a I :4 impedancetransformationcan be obtained(similar to that obtainedwith the autotransformershown)andtheparasiticcapacitance betweenthewindingscanbe controlled. One would expectthe performanceof this transmissionlinetransformerto be .ptimumwhenthecapacitance betweenthewindingsis low,thatis,whenthecharacteristic rmpedance of the line is high. Fortunately,this is not true; rather,thereis an optimum performance characteristic of the transformer impedancefor the line. Thehigh-frequency :sthereforeimprovedby the transmission-line effect. transformercan be consideredto be a At low frequenciesthe transmission-line :onventionaltransformerwith excellent coupling. Becauseof the transmission-line :apacitance,however,this model is not valid at high frequencies. In fact, the magnetic :ouplingbetweenthewindingscanbe removedcompletely(by not usinga magneticcore rndstraightening performance of thetransformerwill not theline), andthehigh-frequency .e influencedat all (i.e.,if the losseswerenegligible).
F{ure 6.1
(a) A conventionalauto-tansformerand (b) a l:4 transmission-linetansformer.
179
rt0
Desigrrof RF and Microwave Amplifiers and Oscillators
I+Ie-iF I
I e-fr f
It+Ve-F
Figure 6.2
The voltagesacrossandthe currentsin a transmission-linetransformerat high frequencies.
by assumingthecurrentsin thetransmission Thatthis is possiblecanbeappreciated line to be balancedand the line to be short enoughfor the phasedifferencebetweenthe voltagesacrossthe line andthe currentsin the line to be small.This is illustratedin Figure 6.2. It has been assumedin Figure 6.2 that the currentsin the tansmission line are balanced.If the currentswere perfectlybalanced,as shownin Figure6.3(a),the output voltage would havebeenzero, which is not the case.The currentsin the line must be unbalancedfor the output voltageto be non-zero. Because (6.1)
Yo= 2sLrl I ru=2 jaLr(. I tu
In (6.1)L, is the inductanceper the unbalancedcurrentis very smallat high frequencies. line, and 4, the unbalanced the length ofthe unbalanced current, 0 length for the unit two conductors of the line. in each of the componentof the current performance transformeris of the transmission-line Althoughthe high-frequency response is seriously not affectedby the removalof themagneticcore,the low-frequency degradedwhenthis is done.The reasonfor this is the increasein the unbalancedcurrent. This currentincreasesapproximatelyinverselywith frequency.Whenmagneticmaterial currentis reduced.Ifthe is use{ the inductancein (6.1) is increasedandthe unbalanced ofthe transformerwill bevery currentis smallenough,thefrequencyresponse unbalanced goodat both high andlow frequencies. The bandwidthobtainablewith a transmissionJinetransformeris significantly bcfier than that obtainablewith a conventionaltransformer.
- V,,+
: (a)
Egprc 6.3
O) asa functionof (a)thebalanced transformer Theoutputvoltageof thel:4 transmission-line currents in theline. and(b) unbalanced
Transmission-Line Transformers
181
Becauseof the wide bandwidth,transrnission-linetransformersare often usedto transformresistance.When only one transmissionline is used,the only transformation ratiosthat canbe obtainedare l:1 and 1:4.Whenmorethanoneline is used,it is possible to realizetransformerswith othertransformationratios. Apart from impedancematching,transmission-line transformersarealsousedto performvariouscombiningandsplittingfunctions. The mostcommonlyusedconfigurationswill be presentedin the next section. The analysesof the l:4 transmission-line transformerandothertransmission-line transformers will be discussed in detailin Section6.3.It will be shownin this sectionthat the basiccomponentof any transmission-line transformeris an unbalanced transmission line with increasedinductancefor the unbalanced currentsin the line. This simplifiesto a balancedtransmissionline at high frequenciesand a l: I transformerwith magnetizing inductanceat low frequencies. primarilyrequiresthatthetansformer Thedesignof transmission-line transformers may meetsthelow-frequencyspecifications. Compensation at low and/orhighfrequencies alsoberequiredto extendthebandwidth.Thevariousstepsin designingthesetransformers will be consideredin detailin Section6.4. Transmission-line transformers areusedextensivelyin RF poweramplifiers.The designof impedance-matching networksfortheseamplifierswill beconsideredin Section 6.5.
6.2 TRANSMISSION-LINETRANSFORMER CONFIGURATIONS Transmission-line transformers areusedto changeresistance levelsin impedance-matching networksandamplifiers,aswell asto performcertainsplittingandcombiningfunctions. The transformationratiosobtainablewith thesetransformersarelimited to those shownin Table6.I , that is, if lessthanfive transmissionlinesareused. Thenumberof linesusedin a practicalapplicationis limitedby theavailablespace. To ensurea good low-frequencyresponse, eachline mustbe woundarounda magnetic @re.
More than oneline cansometimesbe wound aroundthe samecore.Thepolarity of lhe voltageinducedby the flux in the coreandthe relativesizeofthe voltageacrosseach windingmustbe takeninto accountwhenthis is done. The configurationcorresponding to a particulartransformationratio canbe found by usingthetechniqueillustratedin Figure6.4 [ ]. Whenthistechniqueis applied,theratio of the input to output voltagechangesfroln x I y tolx l(X+ y)l.The impedance - ,nsformation ratiois changedfrom (x I y)" to fx I (x + y)]'. In the simplestcase, x=l=! rnd the applicationof this techniqueresultsin the configurationfor the 1:4transmissionIre transformer.
Design of RF and Microwave Amplifiers and Oscillators
182
Table 6.1 Transformation ratios obtainable with transmission-linetransfonners as a function of the numberof lines used Number of lines
l:1.00(l/l)' l:4.00 (li2f Transformation ratios obtainable
2
I
t:2.25(213f l:4.00(r/2f l:9.00 (l/3)'
_
J
l:1.78 (3/4)' l:2.78(315f r:6.25(2/5)' l:16.0(l/4)'
:
4
l:1.56 (4/5)' 1:1.96(5171 l:2.56 (5/8\' 1:3.06(4/7)' r:5.44(3/7)2 l:7.1l (3/8)' r:r2.3 (2/7)2 l:25.0(l/5)2
If the techniqueis applied to the l:4 transformer,the l:9 transformershown in is obtained. 6.5(a) Figure thehigh impedancesideof the l:4 transformerto be the inPut,the considering By in Figure6.5(b)is obtained. shown 1:2.25transformer (3/5)2transformerscanbe (ll4)2 andthe(215)2, The configurationsfor the(314)2, 4:9 and 9:4 transformers, the 9:l and 1:9 and to the found by applying the technique areshownin Figure transformers 4:25 and the I : 16 respectively.Theconfigurationsfor the 6.6.
Figure6.,f
The influence of adding an extra line on the impedance transformation ratio ofa transmission-linetransformer.
183
Transmission-Line Transformers
- 2Y+
3V
lsR (a)
leR o) Fgure 6.5
Derivation of the configurationsfor the (a) l:9 and (b) 4:9 transmission-linetransformers.
The transformersmost often used in power amplifiers are the l:4 and 1:9 ransmission-linetransformers.The high cut-off frequencyof the I :4 transformercanbe increased considerablyif two linesinsteadof only oneareused,asshownin Figwe 6.7 l2l. - 3 V +
l,-
- 3 V +
(a)
- 3 V +
J,,^
- 3 V +
(b) figure 6.6
The configurationsofthe (a) l:16 and (b) 4:25 transmission-linetransformers.
Desigr of RF and Microwave Amplifiers and Oscillators
Zo:3tn N2
fhe configuration of a I :4 transmissionline transformer that has no high cut-off frequency (theoretically).
Figure6.7
Impedancehansformationbetweena balancedsourceanda balancedloadis often transformers required.The configurationsfor the balanced1:4 and 1:9transmissionJine areshownin Figure6.8(a)and (b), respectively.
-V Figure 6.E
I
-2V
(a)
-3v
-V
rb)
The configurationsfor the balanced(a) I :4 and (b) 1:9 transmission-linetransformers.
the l:1 transformershownin When eitherthe load or the sourceis unbalanced, transformation. Figure6.9 canbe usedto providethe requiredunbalanced-to-balanced
Figure6.9
1:l transmission-linetransformer. Theunbalanced-to-balanced
Transmission-Linc Transformers
185
I : I balanced-to-unbalanced transformer
Figure6.l0
lllustration of the equivalence betwecn the l:4 balanced and the l:1 balanced-tounbalancedtransmission-linetransformers.
the frequencyresponseof the I : I transformeris exactlythe At high frequencies, szrme asthat of a transmissionline terminatedin the sameload. Becauseof the symmetry,the high-frequencyresponseof the l:4 balanced transformeris identicalto that of the 1:I transformer.The equivalenceis illustratedin Figure6.10. transformation canbeobtainedby combiningthe I : 1 A 4: I unbalanced-to-balanced and 1:4 balancedtransformers,as shownin Figure6.1l(a), or by using the transformer shown in Figure 6.11(b). The latter transformeris less sensitiveto nonoptimum characteristic impedances thantheformer,althoughit hasa lowercut-off frequencywhen theoptimumcharacteristicimpedanceis used. Theoutputcurrentsof thetwo transistorsin a push-pullclassB amplifierareoften
-21/ (b) Figure 6.11
(a) Combination of a I : I transformer and a 4: I transformer to obtain an unbalanced-to. balancedtransformation.(b) A l:4 unbalancedtransmission-linetransformer.
Designof RF and Microwave Amplifiers and Oscillators
186
OA .V (b) Figure 6.12
(a) I :4 and (b) I :9 transformersfor combiningthe currentsof the transistorsin a classB amplifier into a single load.
combined (at lower frequencies where the conduction angle is 180
")
by using either of the
shownin Figure6'12. l:4 or l:9 transformers Although they areusedfor different purposes,it canbe seenthatthe configurations of the I :4 transformershownin Figure6.12 aresimilarto that of the I :4 unbalanced-tobalancedtransformershownin Figure6.11(b)' planeof the I :4transformerin Figure6.12(a)(asshown By redefiningthereference ofboth areidentical. in Figure6.13),it becomesclearthat the frequencyresponses
I
2V Figure 6.13
The configurationofthe l:4 transformerfrorn Figure6.12with a redefinedreferenceplane (ground)'
Transmission-LineTransformers
187
(a) Fgure 6.14
(a) A transformerfor combining two in-phasesignalsinto the sameload; (b) the same transformerusedas an in-phasepower splitter.
The combinershown in Figure 6.la(a) is often usedto combinetwo in-phase sipals at radio frequencies. As indicatedin the figure,the voltagedrop acrossthe l:l transformerusedin the combineris equalto zerowhen the two input signalsareequalin amplitudeand are inthetwo sourceswill be isolatedfrom eachother abase.Whenthe signalsareunbalanced, by the transformer.This is illustratedin Figure 6.15 for the casewhereE2 = 0. If the transformeris assumedto be ideal, and R,' = 2R" = 4Rr, no currentwill flow in the resistance R"r. At low frequencies,the isolation obtainablewith this transformeris a function of tb magnetizinginductanceof the 1:1transformer;that is,
(6.2)
$ = [ 4 r o ! 1/ ( R , / 2 ) ] 2+ l Rr=2R"
Fgure6.15
Illustration of the isolating action of the hybrid nansformer shown in Figure 6.14(a).
188
Design of RF and Microwave Amplifters and Oscillators
where.Sis the ratio of the powerdissipatedin the load (R,,: R.l2) andthe power dissipated (R.), whenEz:0. in the sourceresistance Theisolationat high frequencies is a functionof theelectricallengthof theline and the characteristicimpedance. The combinercan be changedto a splitterby connectingit as shownin Figure 6.14O). As in the caseof the combiner,thevoltagedropacrossthe 1:1transformerwill be zeroas long asthe loadsarebalanced.If not, the transformerwill causethe powerto be distributedmore evenly betweenthe two loadsthan would be the casewith a direct connection.
6.3 ANALYSISOF'TRANSMISSION.LINE TRANSFORMERS transformeris anunbalanced transmission Thebasicbuildingblock of a transmission-line line (seeFigure6.16).The line canbe woundaroundmagneticmaterialor canbe shaped asa solenoidalcoil. The lattercanbe doneby usingsemi-rigidcoaxialcable. transmission-line arenotequal, Thecurrentsin thetwo conductorsof anunbalanced but are relatedby the equation
I
Ir(x\ = /, (x) + /o(x)
(6.3)
Because the effective current entering the line at any point (d6 = d (r) - Ir(x) : - 1o@))must be equal to the current flowing out of the line at any other point further along the line (see Figure 6. I 7), the unbalancedcurrent (10(x))is independentofthe distance(x) along the line. Therefore, (6.3) simplifies to
It
(6.4)
Ir(x)=It(x)+Io
It(x):Irn@)-Io/2
Ir(x)=Iro@)+Io12
Figure 6.16
The balancedand unbalancedcomponentsofthe currentin a transmissionline.
189
Transmission-Line Transformers
10ft) =/r(x) -
Figure 6.17
Io@)= Ir(xr) - I,(xr)= Iogr)
The unbalancedcurrenton atransmissionline as a function ofthe distancealong the line.
The effect of the magrreticmaterial(or solenoidalshape)is to increasethe impedance associated with the unbalancedcurrentsin the line. with the balancedcurrents Becausethereis no extemalmasneticfield associated , { H dl = I ro- I n =0), thesecurrentsie not influencedby themagneticmaterialusedor :y the form in which the line is wound.
Ir(x)
IzQ) Flure 6.lE
The equivalentcircuit of an unbalancedtransmission-line.
If the influence of the magneticcoreor the coil form on the unbalancedcurrentsis ignoredinitially, the equivalentcircuit shownin Figure6.18appliesandequationsfor the ; oltageon and the currentin the unbalancedtransmissionline can be derivedin a way ;imilar to the balancedtransmissionline case(referto AppendixA). The resultsof the Jerivationare shownbelow: l,{.r; = -Io / 2+ 1"-rx * '"+rx
(6.s)
/1tx) = Io / 2+ 1"-rx * B"+fx
(6.6)
llrx) = rr(0) - z0 / 2-(A- B) + zo I 2.lAe-'* - B.t"l 1..j?1!t
+sLuxIo/2
.
(6.7)
f 90
Design of RF and Microwave Amplifiers and Oscillators
Y 2 ( x=) V r ( 0 )+ Z o/ 2 ' ( A - B ) - z o/ 2 . l - A e - "- B " t " l +sLuxIo/2
(6.8)
VrzG\=4@)-Vr(x)=Zoltre-r"- Ber'1
(6.e)
where
f =,tie s=ia Jztc
(6.I 0)
Zo="|ffi
(6.1l)
L andC arethe inductanceand capacitanceofthe line per unit length,respectively,andr is the position of the point of intereston the line (relativeto the LHS). Note that the inductancefor the balancedcurrents(I) and that for the unbalancedcurrents(2,) are not the samebecauseof the magneticcouplingbetweenthe two conductorsof the line. When magneticmaterialis used or the line is shapedas a coil, the reactance associated with the unbalancedcurrents(IoI 2) mustbe changedfrom sZ,/ to Xu = 2sL,
(6.r2)
where I11 is the inductanceassociatedwith each conductorof the line when the other conductoris open-circuitedand/ is the lengthof the transmission-line. The inductanceassociatedwith the unbalancedcurrentin eachconductoris twice the expectedvalue(Zrt) becauseof the excellentcouplingbetweenthetwo conductorsof the transmissionline. When currentis flowing in only oneconductor,the voltageacross the lengthof the otherconductorwill be equalto that of the first, providedthat thereare no resistivelossesin the conductors.Thecouplingfactor,therefore,is very closeto unity. when magneticmaterialis usedor the line is woundasa coil, (6.7)and(6.8)musr be changedto
Yr(x)=vr7g)-(zo /2)(A- B) +(Zo lZ)lAe-r' -B"t'l + s ( 2 L r r ) ( x/ l ) I o / 2
F
(6.7b,
and
I/r(x)=vr1g)+(zo /2)(A- B) +(zo lz)IAe-r'*B.t'l + s(2Lrr)(x / l) Io/ 2
(6.8b
' Before(6.5)through(6.9)canbeusedto determinethevoltageson andtheculr€nl, inanyparticulartransmission-linetransformer,theconstantsl,B,l, Z,(0),andZr(0)mus. be determined. These constantscan be determinedby using the boundary conditions for the transformerunderconsideration.
Transmission-LineTransformers
19l
Whenthevoltagesandcurrentsareknown,the powergainandthe input andouQut of the transfofinercanbe determinedeasily. impedances it is sufficientthat lossless, line canusuallybeconsidered Becausethetransmission rheinput impedanceof a transformeris known(in thelosslesssnse,the magnitudesof the input and output reflection coefficients are equal, and the magnitudeof the transducer power gain is only a function of the reflectioncoefficient).The input impedanceof a transformeris a functionof the frequency,the load impedance,andthe transmission-line lengthandcharacteristicimpedanceof thetransmissionline used.The expressionfor the rnputimpedanceis thereforeusuallyquitecomplex. Although the complexityis not a problemwhen a computerprogramis usedto transformer,it is possibleto simpliff theequationfor theinput atnlyznatransmission-line rmpedanceat low and high frequenciesby making appropriateassumptions.At high tiequencies,the reactanceassociatedwith each conductoris high comparedto the impedanceof the line, andthe approximation characteristic sLul>> Zs
(6.13)
rn be made. Under this approximation, the input impedanceof the transformer is only a ..nction of the balanced currents in the line. As far as the impedance is concerned, the -rnsmission line can then be consideredbalanced. At low frequencies,the line is electrically very short and the approximation
(6.14)
:f I _1 - I
rn be madeand the input impedanceof the transformeris independentof the length and -.echaracteristic can transformer line.Thetransmission-line ofthe transmission impedance :enbe consideredto be a conventionaltransformer. transformerreduces It follows that thebasicbuildingblock of a transmissionJine '' a balancedtransmissionline and a conventional1:1 transformerwith magnetizing rductanceZllathighandlowfrequencies,respectively.ThisisillustratedinFigure6.l9.
EXAMPLE 6.1
Theinput impedanceofa I :4 transmissionJinetransformer.
transformer(seeFigure transmission-line of a I :4unbalanced Theinput impedance of (6.5)to (6.11). asan exampleof theapplication 6.20)will be determined The boundarvconditionsfor the transformerareasfollows:
vr(0) = g
(6.15)
Vr(l)=Yt1g1
(6.16)
Vr(l)=Z t11171
(6.r7)
F
t92
Designof RF and Microwave Amplifiers and Oscillators
Iu@)-1, Ib@)+1, Unbalancedtransmissionline (a)
I{r)
Balancedtransmission
I
(b)
Ltr l:l Ideal l: I transformer with magnetizing inductance (c) Ftrrl
*,
(a) The basicbuilding block of a transmission-linetransformersimptified at (b) high and (c) low frequencies.
6.19
Theseconditionswill be usedto find two independentequationsfor the unbalancedcurrent1oin termsof A andB. In this way, the relationshipbetweenI and B can be establishedandthe input impedancecanbe found: ,r,(0)= VnQ)= ZofAe-r' - Ber*1=7oU- B)
?.4.1
.1t
Figure 6.20
I
a..
v{0)
V'(l)
1'(o)
Ir(D
V'(o)
Vz(D
Iz(o)
Ir(D
The l:4 unbalancedfansmission-linetransformer.
.:
Transmission-Line Transformers
193
v2(l) = 0 + 0.5z rfA - B) + s L, t I 0 I 2 - 0.5Z ofAe-il- BerI f ((0) andY2(l)areequalthesetwoequations canbeusedto obtain Because thefollowing an equationfor 1oin termsof I andB. After somemanipulation, equation is obtained:
sLlIo = z0/ (sL,t)'LA-B)+Zo/ GL,t)'fAe-''- B"''l
(6.r8)
The second equation is establishedby using the constraint imposed by the load: Vt(l) =Vr(o\ -0'5 Z0[A - B]+ sl,l Io l2 +0'5ZolAe-r-t- Bertl and
zLI{t) = Zrl-Io /2+ Ae-rt+BeF/l it followsthat By equating thesetwo equations, fZ, + sLulll' =2A[Zre-n -0.5 Zo@-rt+l)] + 2BfZrert+ 0.520(er/ + l)l
(6.1e)
The relationshipbetweenI and B can now be determinedby using (6.18) and (6.1e):
B _ Z0E2U+ZL / G L, t)l-2[Z Le-rt- (ZoI 2) E2] A Z,Eil+ ZL / @L,l)l+2lZrert+ (Z0l2)EJ
(6.20)
where Er=l+en and Ez=l+e-rt The input impedanceof the transformeris givenby the equation Z,^ =V1(0)/[1r(0) + I2Q)] "
I_B/A Ez+@lA).El
(6.21)
194
Design of RF and Mioowave Amplifiers and Oscillators
If the approximation etr/ = I is used,the equationsfor the ratioBlAand the input impedanceof the transformer simplifiesto
B _ 2ZosLrl + Z rZo - Z rs Lul A 2ZosLul + Z LZo+ Z,s Lul
(6.22)
and 1-B /A _ (ZL/4).sL,l12 -
7 -_ / 7 t o' zi' \Lo', , rfrEn
(z/ 4)+ sL"u,
(6.23)
If magaeticmaterialis used,the reactancesZ,/ in theseequationsmustbe replaced with (22,,)s.The input impedanceis then
'v' " -_ 7
-
(Zrl4).sL, \-L
't
(6.24)
--ll
17r14y+sL, At high frequencies,the approximation s Lul >> Zo can be made,and the expressionfor the ratio B/A simplifies to B _ZoU+e-ftl-ZLe-ft A Zol+ "*t'l + Z r"*''
(6.2s)
The impedanceis still givenby (6.21). The transducerpower gain for the transformercanbe determinedby using the equation (6.26)
where Z" is the impedance of the source driving the transformer.
EXAMPLE 6.2
The input impedance of a l:l balanced-to-unbalanced transformer.
transmissionlinetransformer Theinput impedanceof a I : I balanced-to-unbalanced (seeFigure6.21)canbe determinedby usingthe following boundaryconditions:
Transmission-Line Transformers
195
a
Ziot
:F
Znz -
Figure 6.21
The l: I balanced-to-unbalancedtransmissionline transformer.
t / t ( t ) = z L Il t)
(6.27)
vr(l)= g
(6.28)
V{0) = -Y210)
(6.2e)
By using (6.27), the unbalancedcurrentis found to be Is / 2 = Ae-ft1l- Z0 I Z Ll+ Ber/[ + Z0 I Z L]
(6.30)
for determiningthe When (5.28)and (6.29)used,the secondequationnecessary ratio BlA is found to be
s L uI I o / 2 = ( Z o/ 2 ) ' f A e - r t - B " ' ' ]
:
(6.3 r)
TheratioB/A cannowbeobtainedby usingthesetwo equations: B
e-t'[l - zo I z L]-lzo I (2sL,l)le-rt
7=-
(6.32)
When B/A is known, the input impedances Z,n and Zin can be determined. These impedancesare given by the equations.
7.= -'nr
Zin2 =
zo[1.-BtA] -lz,r (sL,t)l [e-tt- @ I A) e''1+z1r+B I A]
z o [ r -B I A 1
(6.33) (6.34)
+ l z 0 l ( sL , t ) l [ e - r r- ( B I A ) e r t l + 2 0 + B I A l of (6.33)and(6.34) It is clearfromthedifferentsignsin thedenominators
that the two input impedancesare not equal at low frequencies.
196
Desigrr of RF and Microwave Amplifiers and Oscillators
When sI,/ 27 Zo, the two impedances are approximately equal, independentlyof the characteristic impedancevalueof the line. Furthermore,the input impedanceof the transformeris identicalto that of a balancedtransmissionline terminatedin the sameload impedance(Zr). By using this equivalence, it follows that the input impedanceof the 1:1 balanced-to-unbalanced transmission-line transformerwill be purely resistiveat high frequenciesif Zo: Rr. Because of the symmetry, the same applies to the l:4 balanced transmission-line transformer.
;
6.4
DESIGN OF TRANSMISSION LINE TRANSFORMERS
The designof transmissionlinetransformersconsistsof the following:
,
l.
Determining the characteristicimpedanceand the diameterof the transmissionline to be used;
2.
Determiningthe minimum value of the magnetizinginductanceof the transformerat the lowestpassband frequency;
3
.
Selectinga suitablemagneticmaterial(if needed);
4.
Determiningthe type andsizeof the coreto be used;
5.
Calculatingthe line lengthandthe correspondinghigh cut-offfrequency of the transformer;
6.
Compensating the transformerfor nonoptimumcharacteristic impedance,
, ,,)- 7.
Extendingthe bandwidthby using LC impedance-matching networks,if necessary.
Eachof thesepointswill be discussedin detailin the following sections.
6.4.1
Determining the Optimum Characteristic Impedance and Diameter of the Transmission Line to Be Used
At high frequencies, the input impedanceof a transmission-line transformeris a function of the characteristic impedanceof the transmissionline. Theoptimumcharacteristic impedancecanbeestablished by takingtheratio of the
197
Transmission-Line Transformers
voltageacrossoneendofthe transmissionline andthecurrentpassingthroughit. Thebasic buildingblock of the transformeris thenconsideredto be an ideal l:l transformer. transformeris The applicationof this rule to a l:4 unbalancedtransmissionJine illustratedin Figure6.22.
R=2V/l
r igurc 6.22
Determiningthe optimumcharacteristicimpedanceof an I :4 unbalancedtransmissionline transformer.
If a line with any other characteristicimpedanceis used, the input reflection -oefficientof the transformerwill be affectedadversely. The effect of the characteristicimpedanceon the cut-off frequency of the -rnsformerwill be discussedlater. When the optimum characteristicimpedanceis known, the type of line to be used rnustbe chosen. impedancearefreelyavailable.A Coaxialcableswith 25Qand50Ocharacteristic newith a 12.5Qcharacteristicimpedancecanbe obtainedby connectingtwo 25O lines : parallel,while l00O canbe obtainedby connectingtwo 50Olines in series. can be obtainedby twisting together A wide rangeof characteristicimpedances arerequired(less very low impedances When with various diameters. conductors nirs of together. can be twisted with smaller diameters ran l0O), manyconductors ofthesetwistedlinesareinfluencedby thediameter impedances Thecharacteristic : thewire used,aswell asthe numberof twistsper unit length. Apart from the characteristicimpedance,it is also necessaryto decideon the jiameterof thecableto beusedwhereapplicable.This is determined by thelossesthatcan -e toleratedandthe powerto be transmittedthroughthe line. The attenuationof bifilar or multifilar transmissionlinescanbecomea problemat -,:ghfrequencies, asmentionedin Chapter3.
6.4.2
Determining the Minimum Value of the Magnetizing Inductance of the Transformer
\t low frequenciesthe transmission-linetransformercan be consideredto be a ..rnventional 1:I transformerconnectedin a specialway.
198
Ii I
Design of RF and Microwave Amplifiers and Oscillators
When this model is used,the input impedanceand power gain versusfrequency responseat low frequenciescanbe determinedby usingKirchhoffls voltageand current laws on the simplifiedequivalentcircuit. If the loadconsistsof a singleresistor,only theinputimpedanceofthe transformer Thepowerdissipatedin theload(andthereforethepowergain)can needsto bedetermined. be foundby usingthe equation
(6.35)
PL=v]"G.tr /2
where Vo is the maximum (peak) voltage acrossthe effective parallel input resistance (R"6= l/G"6) of the transformer. When the input impedanceand the transfer function are known, the minimum canbe determined' inductance(2,1)requiredto meetthe low-frequencyspecifications l:4 unbalancedand 1:l the of inductance The minimumvalueof the magnetizing as examples. transformerswill be established unbalanced-to-balanced
& l-l lrrwT"l
4&
f-ff(a)
I
&
4&
LI
(b)
&
LI
&
(c)
Flgure6.23
Simplification of the equivalent circuit of the l:4 unbalancedtransfornier atlow frequencies.
t99
Transformers Transmission-Line
EXAMPLE 6.3
The magnetizinginductancerequiredin a l:4 transmissionline transformer.
With the transmissionline replacedby a l:l transformerwith magnetizing inductance,the equivalentcircuit of the 1:4transmissionlinetransformercanbe simplified asshownin Figure6.23. Ifthe cut-offfrequencyis to be the 3-dB cut-offfrequency,it is obvious Z' mustbe suchthat from Figure6.23(c)thattherequiredmagnetizinginductance (6.36)
tDLr,= ft, /2
If this transformeris to be usedin a power amplifier, the magnetizing inductancemustbehigh enoughfor thespecifiedminimumallowableripple in the passband to be achieved. Becausethe power dissipatedin the load is given by (6.35),the output poweris directlyproportionalto the effectiveparallelinput resistance' if the effectiveload is reactive The efficiencyof the amplifieris decreased of thetransistoris assumed (referto Section2.3.3),thatis, if theoutputimpedance by a factor is it decreased purely Specifically, resistive. to be T't,= | /[ + (R.u I X"u)']U'
(6.37)
whereX"6 is the effective parallel input reactanceof the transformer. BecauseR.6is equalto the optimumvalue(R")in this particularproblem, the power transmittedthroughthe 1:4 transformeris also equalto the optimum value,that is, at low frequencies. The relativeefficiencyis givenby I, = 1/[[l + (R" I (aLrr)121v2 the magnetizinginductancemustbe suchthat If rl, : 0.95is acceptable, = 3R" roZ11
(6.38)
(o211is often chosento be equalto 4X).
R L +
RL
(b) (a) Fryure 6.24
The l: I unbalanced-to-balanced transmission-line transformer at low frequencies.
200
Design of RF and Microwave Amplifiers and Oscillators
EXAMPLE 6.4
The magnetizinginductancerequiredin a l:l transmission-line transformer.
The equivalentcircuit for the 1:l unbalanced-to-balanced hansformeris shownin Figure6.2a@). By transformingthe load on the secondaryside of the transformerto the primaryside,the equivalentcircuit of the l:l unbalanced-to-balanced kansformer canbe simplifiedto that shownin Figure6.24(b). By using this equivalentcircuit, the input admittanceis foundto be rin =
R r + s L r r R r l [ R+, s l r , ] R, +2sL' R?,+ZsLrrR,
_
1 .l+R./(s2,,) 2R, 1+ R, / (2sLrr)
(6.3e)
It is clear from this equationthat the input resistancewill be equalto 2Rrif the magnetizingreactance is relativelyhigh. Therelativepowerdissipationin thetwo loadresistances canbedetermined by transformingthe parallelcombinationof oZ,, andR, in Figure6.24(b)to the equivalentseriesimpedanceshownin Figure6.25. Becausethe samecurrent flows through the two resistors,the ratio of the power dissipatedin each load is equal to the ratio of the resistanceof these resistors.If altt = 4.4Rr
(6.40)
the powerdissipatedin the two loadresistorswill differ by 5%. The input power to the transformerwill thenbe 1% higherthanthe designvalue,andthe relative efficiencvwill be 0.99.
Figure 6.25
The seriesequivalentof the impedanceof the circuit from Figure 6.24(b).
201
Transmission-Line Transformers
6.4.3
DeterminingtheTypeandSizeoftheMagneticCoretoBe Used
transformers.The sizeof the toroidal Toroidalcoresare often usedin transmission-line coreis determinedby the inductancerequired,the maximumflux densityin the core(and thereforethe allowablelosses),andthe line lengthrequiredto meetthesespecifications. It was shownin Chapter3 that if the inductanceandflux densityspecificationsare to be met simultaneously.a corewith
Folr, V3* ,, -- ---------:--
(6.41)
nt
aB'^* aLrt shouldbe used(see(3.33)). It canbe shownthattheline lengthwill alwaysincreaseif a corewith anll-product argerthanthat given by this equationis used. it is possiblethat the line lengthmight be shorter,at If the coresizeis decreased, .ea* initially. Whetherit will be shorteris a function of the extentto which the inductancemust .e increasedto meetthe lossspecification(theflux densityin the corewill be too high if :heinductanceis not increased), aswell asthe dimensionsof the core. the If the lossesin thematerialincreasesharplywhentheflux densityis increased, optimumcoresizewill alwaysbe that givenby (3.39). to providetherequired It is sometimespossibleto reducetheline lengthnecessary ragnetizing inductanceby usinga numberof smallertoroidalcoresinsteadof only one 3rgercore. The ratio ofthe line lengthfor a singlecoreto that of//" stackedcoresis given rpproximatelyby the equation 2wr+2ht+4t
l r ,_ 1,,
(6.42)
(At I A).lU + (1, I lt).(4w, + 4t)
l-
7-F T hl
--rl rurlr-
(a) f4rtre 6.26
;
TI
--1rI
l w 2
O)
of(a) a singletoroidalcoreand(b) a numberofstackedtoroidalcores. Thecross-section
;
202
Design of RF and Microwave Amplifien and Oscillaton
wheret is the outer diameterof the transmissionline used,/r the meanpath length of the largercore,/, the meanpath lengthof eachof the smallercores,11 the effectivecrossareaofeach ofthe sectionalareaofthe largercore, andA, the effectivecross-sectional smallercor€s.lolew2,hr, andft2aredefinedin Figure6.26. Equation(6.42)wasderivedby assumingthe inductanceandthe flux densitiesof the two inductorsto be equal. In order to havethe sameflux density,it is necessarythat (6.43)
Nt/\=N2/12
whereN1is the numberof tums usedwith the singlecoreand N, the numberof turns used with the stackedcore. The inductanceof the two inductorswill be the sameif
N?At/\=N"NlAz/lz
(6.44)
inductor. whereAf is the numberof coresusedin the stacked-core By using (6.43),(6.44) canbechangedto
.
Arlt= N,- A2l2
(6.4s)
andstackedIt follows from this equationthattheeffectivel/-productsof thesingle-cored coredinductorsmust be the same. Equations(6.45)and(6.43)canbe usedto determinethe numberof coresandthe numberof tums required,if usinga transformerwith stackedcoresis worthwhile(i.e., if the coredimensionsareknown). If a core with suitabledimensions(comparableto thoseof the stackedcore) is the line lengthofthe transformer. available,a baluncorecanalsobe usedto decrease
EXAMPLE 6.5
Comparisonof the line lengthsassociatedwith a stacked coreanda singlecoretransmissionlinetransformer.
As an o
l
4w+4t 1",l"z z1wt Ji) + (r t 2) -14(wz t Ji) + +tl 4w+4t
(6.46
Transmission-LineTransformers
203
From this it follows that the ratio of the line lengthfor the two differentcoresis
4:4/3 =r/0.655=r.5 la = l " z 2 4 2+ 2 1 3 therefore,can Becauseofthe reducedline length,thebandwidthofthe transformer, be increasedsignificantlyby usinga stackedcoreor a baluncore.
6.4.4
Compensation ofTransmission-Line Transformers for Nonoptimum Characteristic Impedances
impedanceis not available,it is possibleto Whena line with the optimumcharacteristic ofthe transformerby using response the high-frequency of for thedegradation compensate inductorsand/orcapacitors. compensating elements.Oneelementis usedin It is usually adequateto usetwo compensating to changethe input resistanceor load parallel(capacitor)or in series(inductor)wiitr the to therequiredvalueatsomefrequencybelowthecut-offfrequency,while the conductance otheris usedto removethe reactivepart of the input impedanceor admittanceat the same frequency. The compensationfrequencycanbechosensuchthatthe ripple in the outputpower is equal to a specifiedvalue. The optimum compensationfrequencycan be found iteratively. The compensationof somefrequentlyusedtransformerswill be consideredhereas examples.
EXAMPLE 6.6
Compensationof a l:4 unbalancedtransmission-line transformer.
asshownin The I :4 unbalancedtransmission-linetransformercanbe compensated Z,.oet. > I.35 Zo Figure6.27, thatis,if
Zo=2rR
Figure 6.2?
Compensationof the unbalancedl:4 transformer when Zo> l'35 4$'
; ;
204
Design of RF and Microwave Amplifiers and Oscillators
capacitorsaregivenby the equations[3] The valuesof the two compensation
Ct=
^
I
F F
'
- sin2(p/). r2]t/2 I + cos(p/)- {[1+ cos(p/)]2 0-*/Rsin(p/) - sin2(p/).12\v2 2cos(p/)- {[ + cos(p/)]2 40,*/Rsin(p/)
(6.47)
(6.48)
(6.4e)
r=Zol(2R)
Equations(6.47) nd (6.48) can be derivedby settingthe real part of the input admittanceof the transformerterminatedin a resistor(4R) in parallel with an unknowncapacitor(Cr) equalto llR. C, is usedto cancelthe reactivepart of the input admittance. rl-*l(2x)) is a function of the The compensationfrequency(f,*: power the and minimumefficiencyrequiredin acceptablevariationin the output It canbe found iteratively. the passband. The high-frequencyresponseof the transformer can be improved technique.This can be appreciatedby considerablyby using this compensation at the comparingthe electricallengthsof the line,with andwithoutcompensation, cut-off frequenciesof the transformer. The electricallengthsof theline atthe cut-off frequency,with andwithout arecomparedin Table6.2 for compensation, LPL
Zs=2rR
Figure 6.28
Compensationof the unbalancedI :4 transmission-linetransformerwhen Zo < | .3szo'pt
205
Transmission-LineTransformers
It can be seenfrom Table 6.2 that the high cut-off frequencycan be value,independently increasedby morethanan octaveaboveits uncompensated impedance. of the characteristic Thehigh-frequencyresponseofthe transformercanalsobe improvedwhen zo <1.35zo,opt elements. This canbe doneby usingan inductoranda capacitorascompensating the transformer, end of the high-impedance in series with The inductor is used side,asshown in parallelwith thelow-impedance while thecapacitoris connected in Figure6.28. cut-off frequencycanbe increasedby an octave.The WhenZs: Zo,optthe canbefoundby usingthefollowing equations: requiredinductanceandcapacitance a nL = 0.7558Zo
(6.50)
$ sC = 0.8961fo
( 6.s1)
@H=ao.z+74],
(6.s2)
Exact equations for the values ofthe inductance and capacitancecan be derived by following the same procedure as before.
Table 6.2 The electricallength of the transmissionline usedin an l:4 unbalancedtransmission-linetransformerat the compensationand cut-off frequencies,with and without compensation
r = ZoI (2R)
2.0 2.5 3.0 3.5 4.0 4.5 5.0
EXAMPLE 6.7
Line length without compensation
0.070 0.050 0.036 0.032 0.027 0.024 0.021
Line length at the comPensation frequency
0.144 0 . 1l 6 0.099 0.085 0.075 0.068 0.061
Line length with compensation
0.160 0 . 13 0 0 . 1t 0 0.095 0.084 0.075 0.068
transformer. compensationof a I : I balanced-to-unbalanced
(and I :4 balanced)transmissionJinetransformer The I : I balanced-to-unbalanced
206
Desigrof RFandMicrowaveAmplifiersandOscillaton can be compensated for characteristic impedances that are too high, as shown Figure 6.29. Zo: rR
Figure 6.29
transformer(r > l). Compensationofthe l: I balanced-to-unbalanced
The capacitanceofboth capacitorsis given by the equation t - [r - (r2 - t) tanz(Pl)]ttz " _ o.o/Rtan(P/)
(6.s3)
frequency. wherer = Zs I Zs,or,= Zo / R, andto.* is the compensation
Table 6.3 transmission-line The electical length of the transmissionline usedin the I : I balanced-to-unbalanced transformerat the compensationand cutofffrequencies,with andwithout compensation Line length without compensation
(r.)
0.5 0.6 o.7 0.8 0.9 1.0 l.l 1.2 1.3 1.4 1.5 1.6 1.8 2.0 2.5 3.0 3.5 4.0 5.0
Line length at compensation frequency
(r)
Line length with compensation
(i.)
0.073 0.080 0.089 0.109 0.170 0 . tl 9 0.092 0.075 0.065 0.058 0.042 0.041 0.031 0.025 0.021 0.018 0.013
*t 0.094 0.075 0.075 0.081 0.081 0.056 0.038 0.044 0.044 0.031 0.031
0.,to 0.155 0.135 0.123 0 . 1l 6 0 .l 0 l 0.085 0.066 0.058 0.048 0.043 0.035
Transmission-LineTransformers
207
The allowable ripple in the output power and the minimum effrciency should be takeninto accountwhenthe compensation frequencyis calculated. The cut-off frequency(APt > 77%o, 11"> 95%) of the l:1 transformer,with and withoutcompensation, is shownin Table6.3asa functionof thecharacteristic impedance of the line. It canbe seenby inspectionof this tablethat the cut-off frequencyof the I : I transmission-line transformer(andthereforetheI :4balanced transformer)is moresensitive to deviationsin the characteristic impedancethanthe unbalanced1:4transformer. Compensation of thetransformerhasa significanteffecton the cut-offfrequency.
EXAMPLE 6.8
Compensation of ahybrid coupler.
The optimumcharacteristic impedanceof thehybridtransformershownin Figure 6.30cannotbe determinedby usingtherule givenin Section6.4.1.Thereasonfor this is that the voltageacrossthe end ofthe line is equalto zeroin the balanced case. It can be shown(by deriving the exactequationsfor this transformer)that the optimumcharacteristic impedancefor this transformeris the lowestavailable characteristicimpedance. The hybrid transformer can be compensatedwith an inductor and a capacitorasshownin Figure6.30.Thecompensation frequency(aswell asthecutofffrequencyofthe transformer,with andwithoutcompensation) is givenin Table 6.3 as a functionof the characteristic impedance. The componentvalues aregivenby the equations 0 J y L = 0 . 3 0 1Z5L t l 2
(6.54)
| / (a rC)=1.8005ZLt / 2
(6.5s)
wherethe compensationfrequency(fr) carrbedeterminedby using Table 6.4.
r
o3
rarrc 6.30
Compensationof the hybrid divider shown in Figure 6.14(b).
208
Design of RF and Microwave Amplifien and Oscillaton
Table 6.4 The electricallength of the transmissionline usedin the hybrid divider (and combiner)at the compensationand cut-off frequencies,with and without compensation
7^lzu
Line length without compensation
(r)
Line length with compensation
0.283 0.173 0.093 0.048 0.034
0.367 0.303 0 . 13 9 0.075 0.057
(r.)
0.212 0.121 0.066 0.029 0.020
o.25 0.50 1.00 2.00 3.00
6.4.5
Line length at the compensation frequency
(r)
The Design of l{igh-Pass LC Networks to Extend the Bandwidth of a Transmission-Line Transformer
transformercanbe extendedconsiderablyby using The bandwidthof a transmission-line a high-passmatchingnetwork to compensatefor the effectof the magnetizinginductance. A network that can be usedfor this purposeis shown in Figure 6.31(a).It is smetimes alsopossibleto useits dual,which is shownin Figure6'31(b). of the networkaregivenin Table6.5 of the components The optimumreactances arenormalizedfor a Thereactances asa functionof the allowableripple in the passband. loadresistanceof 1O. When this compensationtechniqueis used,the magnetizingreactancerequiredto Becausethis impliesa shorterline, the decreases. meetthe low-frequencyspecifications high cut-off frequencyof the transformerwill increase. The exact amount by which the bandwidth will increaseis a function of the rcceptableripple in the output power,the magneticmaterialused,and the lossesthat can be tolerated. The new line lengthcanbe determinedby usingthe informationin Section6.4.3. h shouldbenotedthatthemaximumvoltageacrosstheline is slightlymorethanthatwith-
il
in tlltl of thereactance values Tbenormalized
in thepassband frequency atthelowest uo.r1a,
as a function of the ripple in the outputpower
4.0
5.0
Ripple (%)
0.5
1.0
2.0
3.0
X.(o)
0.30
0.36
0.45
0.50
0.55
0.59
xr(O)
1.80
t.52
l.3l
t.20
l.l4
1.09
Incrcase in bandwidth
l.4E
l.6l
1.73
1.82
r.87
1.90
#
209
Transmission-Line Transformers
"reR :-il-} o)
(a)
Figure 6.31
networksthatcanbe usedto extendthe bandwidtb Two low-frequencyimpedance-matching of a transformer.
transformstheresistance Thecapacitorin serieswith theloadresistance outcompensation. slightly upward,andthe voltageacrossthe line mustthereforebe higherthanthat across in parallel theloadresistance in orderto deliverthesamepowerin the effectiveresistance with the magnetizinginductanceasin the load.This is illustratedin Figure6.32.
0+Qt1R (a)
Figure 6,32
PL
PL
(b)
Illustration of the increase in the voltage across the magnetizing inductance ofthe transmission-linetransformerwith low-frequencycompensation.
Becauseof the decreasein the reactanceof the magnetizinginductanceand the ncreasein the effectiveresistancein parallelwith it, the unloadedQ of themagnetizing nductance will alsochangeif the lossesareto remainthe sameaswithout compensation. .'hemaximumallowableflux densityin the core,therefore,will alsochange(it must be essthanbefore). Despitethe increasein the voltageacrossthe magnetizinginductanceand the decrease in themaximumallowableflux densityin thecore,it is usuallyworthwhileto use technique. rhiscompensation -j7.37rJ
-j0.5899O
-17.37Q
o-l 12.5Q
Figure 6.33
network of Figure6.31(a)if the passbandripple The low-frequencyimpedance-matching is 5% and (a) Rr = lQ and (b) Rt= 12.5Q.
zto
r
Designof RF and Microwave Amplifiers and Oscillators
6
Ifthe lossesin the corecanbe ignoredandthe coresizeremainsthe sameasbefore the improvementin the bandwidthwill be that given in Table6.5, aslong compensation, asthe transformeris designedto havethe samelow cut-off frequencyasbefore. that Along with improvingthe bandwidth,this techniquehasthe addedadvantage the requiredfrequencyresponsemight be obtainedby usingan air-coredsolenoidalcoil inseadof magneticmaterial.If this canbe done,the increasein thevoltageper turn is not doesnot apply. a problemandthe flux densityconsideration
EXAMPLE 6.9
transLow-frequencycompensationof a 1:4 unbalanced mission-linetransformer-
technique,the As anexampleofthe applicationofthe low-frequencycompensation for a l:4 unbalanced required magnetizing inductance and capacitance transformerwith Rr: 12.50 ar,d/,:2 MHz will be determined. transmission-line The passbandripple is to be 5%. The compensationnetworksfor Rr = lO and R, : 12.50 are shown in fromTable6,5. in Figure6.33(a)wereobtained Figure6.33.Thecomponentvalues capacitorsin the l:4 transformerare The positionsof the compensating ofthe transformeris thesameasthat shownin Figure6.34(a).Theinputimpedance of the equivalentcircuit shownin Figure6.34(b),which is of the sameform asthe matchingnetworkin Figure6.33(a). By using this equivalence,it follows that if the requiredhigh cut-off frequencyof the transformeris low enough,the magnetizinginductancecan be rcalizedby usingan air-coredsolenoidalcoil.
6.5
I
CONSIDERATIONS APPLYING TO RF POWER AMPLIFIERS
The desigrrof RF and microwavepower amplifiersdiffers from that of small-signal amplifien in the designof the output circuit. While the output circuit in small-signal the amplffiersis usuallyconjugatelymatchedto theloador usedto taperthegainresponse, loadimpedanceof a poweramplifiermustbechosenin sucha waythattherequiredpo*'e: caobe obtainedandthat the efficiencyis ashigh aspossible. The outputpowerobtainablefrom an amplifierstageis limited by the limitatiors ofthe transistorusedand/orthe outputcircuit designed.Thedevicelimitationsstemfrom the finite voltage,current,andpowerratingsofthe transistorandits saturationvoltageor saturationresistance.The saturationvoltageor resistanceof a transistordeterminestb voltageacrossit (ideallythevoltageshouldgo downo lowestvalueof the instantaneous zero). Saturationvoltagesof a few volts for bipolartransistors,andsaturationresistancc of fractionsof anohmup to a few ohmsaretypicalfor FETs.It wasshownin Section2.3 I that the maximum output power obtainablefrom a classA or a classB stageat RF frequenciesis givenby
ztl
Transmission-LineTransformers
c/4 r2.sQ (b)
(a)
Figure 6.34
_ -Pmax
(a) The positionsof the compensatingcapacitorsin the unbalancedl:4 transmissionline transformerand (b) the equivalentcircuit for the input impedanceof the transformer.
(v' - Y")'
RL
(6.56)
2(R, + 0R.u,) R, + crRru,
.,rtrereZ"is the supplyvoltage, 2,",the saturationvoltage,and.R*,the saturationresistance. r is equalto 2 for classA amplifiersandequalto I for classB amplifiers.In derivingthis -'quation(seeFigure6.35),it wasassumedthatthe outputsusceptance of thetransistorwas ':movedby the outputmatchingnetworkandthattheoutputpoweris voltagelimited (i.e., re outputvoltageclips beforethe outputcurrent). Thesaturationresistance ofbipolartransistors is usuallynegligiblysmall,while the saturationvoltagefor FETs canbe neglected. It follows from (6.56)that,in orderto obtainthemaximumpossibleoutputpower ':om a transistor,it is necessary to usethe highestsupplyvoltagepossible,andto choose possible (the saturationresistance :re loadresistanceas small as is usually significantly 'mallerthan the load resistancerequired).The minimum value of the load resistanceis :eterminedby themaximumdc andRF cunentsthatcanbetoleratedthroughthetransistor. The optimum load for an RF power transistor is usually specified by the ranufacturer. When this is not done, the optimum terminationscan be determined ^:acticallyat each frequencyof interestby usingstubtuners.The optimumterminations .an also be estimatedby using a large-signalmodel, if such a model is available. iltematively,the small-signalS-parameters at theratedcurrentandthe dc 1/Z-curvescan -e measured,and the power parameterapproachoutlinedin Section2.3 canbe usedto ::nerateload-pullcontoursfor the transistor.
Matching Network
rgure 6.35
The matching problem associatedwith the output circuit of a power amplifier.
2t2
Designof RF and Microwave Amplifiers and Oscillators
The optimum load is often specifiedin terms of the equivalentoutput impedance the device underthe assumptionof a conjugateoutputmatch.It is importantto realize of that this impedanceis not the sameasthe actualoutputimpedanceof the transistor.The terminationsrequiredfor the optimum power matchare usuallynot the sameas those requiredfor a conjugatematch(low outputvoltagestandingwaveratio (VSWR)). The efficiencyof a poweramplifieris a functionof the classof operationandthe included).When in theoutputcircuit(thetransistorsusceptance effectiveshuntsusceptance thevoltageacrossthe outputterminalsof thetransistoris purelysinusoidal,the efficiency will alwaysbe lessthan 50% for classA amplifiers(the conductionangleof the current throughthe transistoris then 360'), while that for classB amplifiers(180' conduction angle)is constrainedto below 78.5%.Higher efficienciescanbe obtainedwith classC in a narrowerpulse,thepeak the snmepowermustbeconcentrated amplifiers,but because Thedevicespecifications transistorincreases astheefficiencyincreases. currentthoughthe for a classC amplifier,therefore,aremoreseverethanthosefor classA or B amplifiersof the sameoutputpowerwith the samesupplyvoltage.A classC amplifieralsocannotbe useddirectly for linearapplications. Whenthe effectiveload (the outputsusceptance of the transistoris consideredas part of the load) of a powerstageis reactive,the efficiencydecreases by a factor
(6.s7) /Gr)' (Br). becauseofthe increasein the supplycunentcausedby theeffectiveshuntsusceptance In optimizingtheefficiencyof anamplifier,it is thereforeessentialto removetheeffectof of a transistor. the outputsusceptance In orderto obtaintherequiredoutputpower,thephysicalloadof a poweramplifier (usually50Q)mustbetransformedto a lowervalue.It will be shownin Chapter8 thatthis to very low resistancevaluesis often cannotbe donewith LC networks(transformation usually required and the insertion loss may also be a problem). Consequently, transmission-linetransforrnersare usually used for this purposeat RF frequencies. Combinersandsplittersarealsorequiredin a balancedor a push-pullconfigurationor to connecttransistorsin parallel for higher output power.The cancellationof the output susceptance of a power transistoris carried out with an LC network betweenthese transformersandthe transistor. The gain taperingrequiredin a poweramplifieris bestdoneon the input sideof eachkansistorwith an RLC matchingnetwork.It will be shownin Chapter8 that these networkscan be usedto level the gain (the operatingpower gain in this case)withottr thepowerrequiredfrom the reactivemismatching.Any reactivemismatchingwill increase driver stages. It should be noted that ferrites are usually not requiredin transmission-line transformersat the higherRF frequencies(typically 100MHz andabove). This sectionconcludesby consideringtwo poweramplifierexamples.In the first example,designingthe output matching network of a narrowband(225-260 MHz) television broadcastamplifier will be considered.The design of a broadbandinput
Transmission-Line Transformers
2r3
matchingnetwork (2-30MHz) for a push-pullclassB amplifier will be consideredin the secondexample.
EXAMPLE 6.10
Designing an output matching network for a power amplifier.
As anexampleofthe designof anRF poweramplifier,anoutputmatchingnetwork will be designedfor thebalancedamplifiershownin Figure6.36overthepassband 225-260MHz. The networkwill be designedfor an outputpowerof 165W.The of eachtransistor supplyvoltagewill be takenas28V, andthe outputcapacitance as130pF. can for eachtransistor valuefortherequiredloadresistance An approximate be obtainedfrom (6.56).Thesaturationvoltagewill betakenas3V, andsaturation resistanceis assumedto be negligible.Applicationof (6.56)yields -Q8-'2 165/2=P, ' 2Rt leadingto Rt = 3 J 9 t ) (baluns)in theoutputcircuit andthe transformers The quarter-wavelength andthe sourceimpedance the load impedance are used to transform input circuit (12.512) and alsoservesasa combiner for each transistor, to approximately6.25O power. can power the input The actualimpedances anda splitterfor for the output of a the input impedance be obtained easily by the standardequation for (the and dividing the currentis very smallin this case) unbalanced transmissionline resultsby 2to getthe loadfor eachtransistor.The impedancethusobtainedis the
: gure 6.36
The configurationof the balancedpower amplifier in Example6.10.
214
Design of RF and Microwave Amplifien and Oscillaton
Table 6.6 The specifications for the output matching network of the power amplifier of Example 6 . I 0 Frequency (MHz)
225 230 235 240 245 250 255 260
Sourceimpedance
Load impedance
(o)
(o)
2.55- jt.78
z.sr- jr.79 2.48- j1.80 2.44 - jl.8l 2.41- jt.82 2.38-71.83 2.33 - jt.84 2.30 -jl.8s
6.31- jI.03 6.28 - j0.72 6.27 - j0.43 6.25- j0.t6 6.25 + j0.20 6.27+ j0.49 6.29 + j0.80 6.32 + jl.l0
Transducerpower gain
1.0 1.0 1.0 1.0 1.0 t.0 1.0 1.0
network to be designed. load specificationfor the output impedance-matching The sowce impedancefor the output matchingnetwork is simply equalto the load resistancerequiredto obtainthe specifiedoutputpower,in parallelwith the output capacitanceof each transistor.Becausea conjugatematch to this impedanceis required,the transducerpower gain requiredfor this matching problemis equalto l. The specifications for the matchingnetworkto be designed aresummarizedin Table6.6.This matchingproblemcanbe solvedby first designing anL-sectionto providea conjugatematchatthehighestfrequency,afterwhich it can be optimizedfor the bestperformanceover the passband.The solution impedanceby usingthe transformation-Q shownin Figure6.37 wassynthesized matchingtechniquedescribedin Section8.4.3.The deviationfrom the specified performanceis negligibly small.
3.28nH
fyo=24l9MHz
Figure 6.37
The output matchingnetwork designedfor the power amplifier of Example6.10.
2ts
Transmission-Line Transformers
MRF406
Figure 6.38
The input matchingnetworkdesignedfor the push-pullpower amplifier of Example6. I l.
EXAMPLE 6.11
Designinga inputmatchingnetworkfor a powertransistor.
Designinga wideband(2-30 MHz) input matchingnetworkfor a push-pullclass B stage(40W peakenvelopepower(PEP))will be considered in this example.No attemptwasmadeto level the gainresponsein this case.The input impedanceof the transistorused(MRF406)is listedin Table8.1. Thematchingnetworkdesignedis shownin Figure6.38.The50Obalunon the input sidewas usedto obtaina balancedsignal,afterwhich the balanced4:1 transformerwas usedto transformthe 50O sourceresistanceto 12.50. The LC networkwas designedto matchthis resistanceto the input impedanceof the two transistors(theinputsareeffectivelyconnected in series)by usinganearlierversion ofthe programLSM FORTRANprovidedon thedisketteaccompanying thisbook (referto Section8.4.1). The balancedmatchingproblem can be transformedto a single-ended problemby replacingeachofthe two capacitors usedwith two capacitors connected in seriesandby usingthefactthatthecenterpointsarevirtualgrounds.Thesingleendedmatchingnetwork was designedto match half of the output impedance presentedby the 1:4transformer(approximately6.25A)to theinput impedanceof a singletransistor.Thegain(Gr) of the LC networkobtainedvariedbetween0.85 and 0.95over the passband. With the outputpowerhigherthan6W, the input VSWR of the amplifier wasmeasuredto be betterthan2.6 overthecompletepassband[4], which is close to the expectedvalueof 2.3.
REFERENCES Rotholtz,E.,"Transmission-Line Transformers," IEEE Trans.MicrowaveTheoryTech., Vol. MTT-29,No.4, April 1981. DuttaRoy, S. C., "A Transmission-Line TransformerHavingFrequencyIndependent Properties,"Int. J. Circuit TheoryApp.,Vol. 8, January1980,pp. 55-64.
216
Designof RF and Microwave Amplifiers and Oscillators
"On the Design of HF Wideband Transformers(ECO 6907)," 3. Hilbers, A. H., Eindhoven, Netherlands: PhilipsC.A.B.Group,1969. 4. Abrie, P. L. D., ImpedanceMatching Networlrsand BandwidthLimitations of ClassB Power Amplifiers in the HF and VHF Ranges,Master's Thesis,University of Pretoria,1982.
SELECTED BIBLIOGRAPHY "Designing Klauss,H. L., andC. W. Allen, ToroidalTransformers to OptimizeWideband Performance,"Electronics,August 16, 1973. : "Some Transformers Ruthroff,C. L., Broad-Band ," Proc.IRE,August1959. "Evolution of a 4:l ImpedanceTransformingBalun," IEEE Trans. Van Nierop, J. H., AntennasPropag.,Vol. AP-30,No. 4, July 1982.
CHAPTER 7 FILM RESISTORSAND SINGLE-LAYER PARALLEL.PLATE CAPACITORS ..I
INTRODUCTION
he distributednatureoffilm resistorsandparallel-platecapacitorscannotbe ignoredat 'ricrowavefrequenciesandwill be consideredin this chapter. modeledby consideringtheresistor Thebehavioroffilm resistorscanbeaccurately *r be a lossytransmissionline. Film resistorswill be consideredin Section7.2. The Single-layerparallel-platecapacitorsareoftenusedat microwavefrequencies. 'nfigurationscommonlyusedin hybridcircuitsareshownin Figure7.1.Metal-insulator:etal (MIM) capacitors(seeFigure 7.2) are extensivelyused in MMICs (monolithic ::icrowave integratedcircuits).
o)
(c) 7.1
(a) A series connectedparallel-plate capacitor, (b) a vertically mounted capacitor, (c) a parallel-platecapacitormountedon a groundplane,and (d) a gap-capacitor.
217
218
Designof RF and Microwave Amplifiers and Oscillators
At low frequenciesthesecapacitorscouldbetreatedasideallumpedcapacitors,but their distributed naturemust be takeninto accountat higher frequencies. When the capacitoris mountedon a groundplane (bottom plate connectedto ground;seeFigure7.1(c))andthe excitationcanbe takento be uniform acrossthe width of thecapacitor(narrowwidth, ribbonor multiplebondwire cases),theparasiticbehavior of the capacitorcan be modeledfairly accuratelyby consideringit to be a open-ended transmissionline. This case will be consideredin Section 7.3.1.The general case (microstripcapacitors)is consideredin [1]. Analysisof the verticallymountedparallel-platecapacitor(Figure7.1(b) is also straightforward. Thiscapacitorcanbeconsidered to bea seriesconnected open-ended stub. The sameresonancesencounteredin an open-endedstub are also encounteredin this configuration.Fortunately,theresonant behavioris sharplyreducedby anycapacitorlosses (this is importantwhena capacitoris usedfor widebandcouplingor decoupling). Analysis of the seriesconfigurationshownin Figure 7.1(a)provesto be more challenging.If the capacitance densityof the capacitoris high comparedto that of the associatedmicrostrip line (which is usually the case)and the behaviorat frequencies significantlylowerthanparallelresonance is considered, thesecapacitors canbeaccurately modeledas a lumped capacitorcascadedwith a transmissionline on both sides(linecapacitorJinemodel) [2]. In this casethe transmissionJine behaviorof the capacitoris essentiallythat of the microstripline. The line-capacitor-lineapproachis very practicaland is adequatein most cases. Modelingofparallel-platecapacitorsin thiswaywill beconsidered in detailin this chapter. The generalcasecan be handledas describedin [2, 3]. The model usedfor the capacitorin [2] is instructiveand is shown in Figure 7.2(b).Note that the magnetic couplingbetweenthe capacitorplates(21) andcapacitance to the groundplane(C1s,C2e) areincludedin the model.
l Qrrc
7.2
The distributedmodel usedfor a parallel-platecapacitorin [2].
Film Resistorsand Single-LayerParallel-PlateCapacitors
219
Parallel-plate capacitors exhibit series and parallel resonant behavior as the frequencyis increased.Theseefflectsare very pronouncedin high Q capacitorsand are important when designingcoupling or decouplingcapacitors.The parallel-resonant behavioris not evidentin the line-capacitor-line model. will beestablished Thebasicreasonfor theparallelresonance in overlaycapacitors by consideringtheparallel-platecapacitorin freespace.It will be shownin Section7.3.3 thata moreaccuratemodelfor the capacitorwouldbeto usetheline-capacitor-line model with a frequency-dependent value for the capacitance. The analysiswill be done by consideringthe seriesconnectedparallel-platecapacitorto be anunbalanced transmission line, as was done with transmissionJinetransformersin the previous chapter.This approachcanbe extendedto handlethe microstripcase,too [4].
7.2 FILM RESISTORS Thin-film techniquesareoftenusedto manufactureresistorsat microwavefrequencies.By keepingthedimensionsof theresistorsmall,theassociated capacitance andinductancecan be minimized.The capacitance can be reducedfurtherby depositingthe thin film on a substrate with a low dielectricconstant. A film resistor(seeFigure7.3) canbe modeledasa lossytransmissionline. The relevantequationsareasfollows r=RrlW
(7.1)
n,,r="/E*-
(7.2)
C =l/(up,.Zo_rc)
(7.3)
L = ZsJc lvd,
(7.4)
(7.s)
o, = 6.51n* tanl liot lr)l + (aL)z . [ c o s 0+, ; s i n 0 , ]
= - = cr + /0 = .,/roC'(r + ja11.L)
(7.7)
o: = -o.5tan11rt1at11 '.e,
Fgure 7.3
(7.6)
A film resistoron microstrip.
0---; h
220
Designof RF and Microwave Amplifiers and Oscillators
r+ jal jaC
(7.8)
_
((r) Zosinh cosh te0llv,1
lr,l=f
lt, I lsinh((t) / Z,
cosh((f ll1, J
(7.e)
v,rhereZs_rgis the characteristicimpedanceof a losslessline with identical dimensions,Z per the width (in meters)of the film resistor,/ its length(in meters)andi?"the resistance square.The angles01and 0r arespecifiedin radians. of 10Qto 10000per squareareavailable. Films with resistances The influence of the skin-effect on the resistancecan be incorporatedinto the resistanceper square,R". The skin effect can usually be ignoredbecauseof the high resistivityof the film material. The transmissionmatrix equationfor a seriesresistoris definedin (7.9). V, andI, in this equationarethe input voltageandcurrent,while Vrand Irare the loadvoltageand the load current,respectively. to the circuitby a film resistor(or anytransmissionline The impedancepresented with aloadZlcan be derivedfrom (7.9) with serieslosses)connectedin series(cascaded) andis given by Z 7.cosh(l + Zs sinh(/ -.m- Zr.Yo.sinhl(+cosh((.
- o i
z
"
ZTcosh(l+Zs sinh(/
-
ZTsinh(l+Zocosh(l
+ Zotarhcl = Z "n . Z L Zo+Zrtanh(l
7.3
SINGLE.LAYERPARALLEL.PLATE
(7.r0)
CAPACITORS
considered in Section7.1. TheconfigurationsofsingleJayercapacitorstypicallyusedwere The capacitorcan be mountedon the groundplaneor on a microstrip line (conductive epoxy is usedfor this purpose).When mountedon microstrip,the seriesconnectionis usuallyused(seeFigure 7.1(a)),but verticalmountingis also an option. Standingaxial (Figure beamleadsareusuallyusedwhenverticalmountingis required.A gap-capacitor 7.1(d) hasthe advantagethat no bondingwires or ribbonsarerequiredwhen it is used. This capacitorconsistsbasicallyoftwo parallel-platecapacitorsconnectedin cascade. Parallel-platecapacitorsareusedfor filtering,impedancematching,coupling,and decoupling.
Film Resistorsand Single-LayerParallel-PIateCapacitors
221
When decouplingto ground is required,the capacitoris usually mountedon the groundplane and connectionto the circuit is madewith bond wires or a ribbon. The parasiticinductanceassociated with aribbonwill usuallybelowerthanthatassociated with bonding wires. Severalbonding wires can (and should) be used in parallel, but the inductancewill not decrease proportionallywith the numberof wiresusedbecauseof the couplingbetweenthem. Discreteparallel-platecapacitors areavailablein differentsizes.Typicalwidthsare l0 mil (D10),l5 mil (Dl5),20 mil (D20),andsoon. The capacitance valuesobtainablefrom [5] arelistedin Table7.1 as afunctionof thewidth (50V breakdownvoltage).ClassI materialsareusedwhenhigh p capacitorsare required(filtering andimpedancematching),while classII materialsareusuallyusedfor resonance-free couplinganddecoupling.
Table 7.1 The capacitancevalues(pF) obtainableas a function ofthe capacitorwidth Capacitorwidth Dl0
Dl5
D20
D25
D30
D35
ClassI
0.05-4.7
0.05-12
0.08-l8
0.2-33
0.3-39
0.4-68
ClassII
l.E-68
3.3-l80
3.9-220
l0-470
12-s60
20-1000
The lengthof thesecapacitorsis a functionof the dielectricmaterialusedandthe layerthickness.To provideanideaofthe lengths,upperboundson thelengthsareprovided in Table7.2 for variousdielectricmaterials[5] with a dielectricthicknessof 4 mil (50V breakdownvoltage).Thevalueswerecalculatedby consideringonly theplatecapacitance and neglectingany fringing capacitance.The dissipationfactors [5] for the different materialsarealsolisted in the table.The secondgroupof materialsareclassII materials. Accurate information on the exact size of a capacitor can be obtainedfrom the manufacturer. With the physicaldimensionsof a capacitorknown,the associated characteristic impedance andelectricalline lengthcanbedetermined (verticalmountingis assumed here; seeFigure 7.1(b). The electricallength can also be estimatedby measuringthe first parallelresonantfrequency(opencircuit) ofthe capacitor. The first parallel resonantfrequency and the characteristicimpedanceare not independent for a given capacitance value.This follows from the following equations:
GG o=roJ',--! C
C
(7.rr)
222
Designof RF and Microwave Amplifiers and Oscillators
Yo= Cr -+tlg'-"n
(7.r2)
'(
tt {t'-'u ( . ( . .I =r.' Je' - "n' = = (u rn 2n . / / AQ ? u 2 n . ' / fi = r. c c
l
n = o oJ " ' -
(7.t3)
n
I c " io=;-7=-.-
l Y o
17.i41
2 ,l''-,u'( 2 C'
whereC1is the capacitance, )zothe characteristic admittance(I'o = llz), Athe capacitor length,andfr the first parallelresonantfrequency.
Table7.2 Upper boundson the length requiredper picofaradfor different dielectricmaterials (dielectriclayer thickness:4 mil) Material (DF)
Length per picofarad(mm) Dl0
Dl5
D20
D25
D30
D35
cF (0.6vo) cG (0.7vo) NR (0.25olo) NS(0.5%) N U( 1 . 5 % ) }w(1.2%)
2.0616 0.6479 0.2926 0.r 463 0.0756 0.0454
1.3744 0.4320 0.195r 0.0975 0.0504 0.0302
1.0308 0.3240 0.I 463 0.0732 0.0378 0.0227
0.8246 0.2592 0.I 170 0.0585 0.0302 0 . 0 l 8t
o.6872 0.2160 0.0975 0.0488 0.0252 0 . 0 1I5
0.5890 0.1851 0.0836 0.0418 0.0216 0.0130
rc(20'6)
0.1134 0.0181 0.0082
0.0756 0.0121 0.0055
0.0567 0.0091 0.0041
0.04s4 0.0073 0.0033
0.0378 0.0060 0.0027
0.0324 0.0052 0.0024
BH (2.5o/o) BU (2.5%)
It follows from (7.12) and(7.14)thatthe first parallelresonantfrequencyandthe characteristicimpedanceassociatedwith a given capacitancevalue is completely determined by theproduct\E;./. prc,huclis kept constant,the frequency-dependent If thenGll.Z behaviorof differentrealizations(differentvaluesof e, ) of the samecapacitorvaluewill be identical (i.e.,ifany differencein the dissipationfactorsis ignored). Equations(7.11) and (7.14) canalso be combinedto give an expressionfor the capacitancein terms of the Is andfr: C r = Y o/ Q f o )
(7.1s)
Film Resistorsand Single-Layerparallel-plateCapacitors
The electrical performanceof a parallel-platecapacitordependson the way connected.The differentcaseswill be considerednext.
7.3.1
Parallel-Plate Capacitors on a Ground plane
Theequivalentcircuit of a capacitormountedon a groundplaneis shownin Figure7.4. Thisequivalentcircuit is valid ifthe excitationcanbe considered to beuniform acrossthe $'idth of the capacitor.This canbe ensuredby usingseveralbondwires in parallelor by singa ribbon insteadof the bondwires. The bondwire (or ribbon)inductancecanandshouldbe minimizedby keepingits engthas shortaspossible. With the equivalentcharacteristicimpedanceand the resonantfrequencyof the rrallel-platecapacitorknown, the impedancepresentedby it to the circuit can be ,lculated.Note that becauseonesideof the capacitoris directlyconnected to the ground ane,thetransmission-line inductance couldbereducedby up to one-halfcomparedto the -'rticallymountedcase(this effectwill be reducedby couplingeffects).A slight change thecapacitance shouldalsobe expectedbecause ofthe differencein the fringing fields. Theequationsderivedin Section7.2for a thin-film resistor(transmission line with ,'rieslosses)also apply to this case.If the parasiticedgecapacitancein Figure 7.4 is - rored,Z, in (7.10)is an opencircuit and(7.l0) simplifiesto
" = Io tafuql - v
(7.16)
tanhul +tanh(ipl) I + tanh a/.tanh(ipl)
- Y o tanhq.l.+ jtan9[. l+ j tarlt':.rl.l'tanpl
Microstripline
zo,
Lb
zn "c4c
"cdsc T Fgurc 7.4
T
The equivalentcircuit for a capacitormounteddirectly on a ground plane.
224
Design of RF and MicrowaveAmplifiers and Oscillators
The generalcasecan be handledby using the following equation: iroC.ar".cosh(/ + Io sinh(/ . Yin=jaC"or" wv'v+ Yo-" jaC"t . ".sinh(/ + I/ocosh(/ wtrereC*r" is the parasiticcapacitanceat eachopenend. Ifthe excitationis at thecenterofthe capacitorinsteadofat the edge,thecapacitor canbeconsideredto consistof two transmission linesconnected in parallel.Theexcitation mustbe uniform acrossthe width for this to be the case.
7.3.2
Parallel-Plate Capacitors Used as Series Stubs
Theequivalentcircuit for a parallel-platecapacitorusedasa seriesconnectedopen-ended stubis shownin Figure7.5.If thefringingcapacitance at theopenendis ignored,theseries admittancepresentedto the circuit by the capacitorcanbe calculatedby using(7.17). The insertionlossassociated with the capacitorcanbe calculatedby using(7.17) and(l.l l): .4G"GL
Gr= ,+\)(tzr+YL)-lnlzr
(I" +
'4G,GL
YL)-eYi)eYi")
Microstrip line
Figure 7.5
The equivalentcircuit for a parallel-platecapacitorusedas a seriesstub.
225
Film Resistorsand Single-Layer Parallel-PlateCapacitors
'4p,Gt
(7.1e)
where { is the admittanceto the left of the stub and 1, is the admittanceto the right. With f" : Yo= Ytand I" = G"andYr= Gu (7.19)simplifiesto
(7.20)
in decibels,this becomes Expressed
I tyo l . r r ^ l '= -roto*,rlt.;fr = -10loe'olt+-rel cr(dB)
(7.21)
Substitutionof the expressionfor It" yields that the insertionlossof the capacitoris given
(7.22) Theinsertionlossat the series(p0= Qn + l)' n/2) andtheparallel(pQ=2n ' n/2) resonant nequencies areof interest.Substitutionofthe relevantvaluesfor B/-inQ.z2)yieldsthatthe nsertionlossat the seriesresonantfrequenciesis givenby
o*'/ i=2otos,nl,*l=]L -'"1 2 Yoc
(7.23) I
is givenby frequencies whilethatattheparallelresonant
IL=Zotos,olt4# #;
(7.24)
Becausetanha0 is small when c0 is small, it follows from (7.23) that the insertion loss will be small at the series resonantfrequencies when a0 is small, as expected.It follows from t7 .24) thatthe insertion loss will be severeat the parallel resonantfrequencieswhen cr0is surall, again as expected.
t 226
Designof RF and MicrowaveAmplifiers and Oscillators
The attenuationat the parallel resonantfrequenciesis decreasedsharply with increasingcr0.In contrastwith this, the attenuationat the seriesresonantfrequencies increasesslowly with increasingc0. It follows that a resonance-free low-impedance connectioncanbe obtainedby usinga capacitorwith significantlosses. It is also clearfrom (7.23) and(7.24)that the insertionloss at the seriesand the parallel resonantfrequencieswill be decreased as the characteristicimpedanceof the capacitoris decreased.The ideal coupling capacitor,therefore,will have the lowest possiblecharacteristicimpedancewith sufficientlossesto removeany resonance effects. The characteristicimpedancevalues claimed for the capacitorsconsideredin Section7.3 I5l rangefrom 0.4Q to 50O (capacitance range:800 to 0.05 pF; forange: 1.5-200GHz; 50V breakdownvoltage). The ideal capacitorfor a filter or an impedance-matching networkwould be one with negligible lossesand with the s'eriesresonantfrequency("foI 2) far outsidethe passband.When a coupling capacitoris required,the seriesresonantfrequency(^ /2) shouldbe chosento be insidethe passband, ifpossible.
7.3.3
SeriesConnectedParallel-PlateCapacitors
A seriesconnectedparallel-platecapacitor(seeFigure7.1(a) canbe consideredto be a cascade connectionoftwo transmission linesseparated by a lumpedcapacitor,asexplained in Section7.1.Thebasicreasonfor thismodelwill beillustratedin this sectionby deriving the /-parametersandtheassociated modelfor thecapacitorin freespace(no groundplane; seeFigwe 7.6).Theresultsobtainedcanalsobeusedto refinetheline-capacitor-line model
- jakLrrdxlr(x)+
G&
Cdx
fic'x ,i: :. , .. ., i:
Figure 7.6
,;..
Ltr& iakLldtl{x)
An equivalent circuit for the free space capacitor based on [2].
, ., .
Fikn Resistorsand Single-LayerParallel-PlateCapacitors
227
by replacing the capacitancevalue with that obtained in this sectionfor the free space capacitor.In doing so, the parallel resonantbehaviorexpectedis also obtainedin the modifiedmodel. An equivalentcircuit for the capacitorbasedon [2] is shownin Figure7.6.Instead of using this equivalentcircuit, the analysiswill be donein termsof the balancedand unbalancedcurrents on the line, as was done for transmission-linetransformersin currents to thebalancedandtheunbalanced Chapter5. Theeffectiveinductancepresented will also be different in this case.The relationshipcan be establishedby using the equivalentcircuit for two coupledcoils (seeFigure5.3(a). The effective voltage drop acrossthe inductanceandthe mutual inductancefor an incrementalsectionin the top plateis givenby 62,(r) = jaLrrdx . Ir(x) - jakLudx. Ir(r)
= jalrrdx.(1r(r) - k lr(x)) = jaLrrdx.{Ua(x)- 1,(x)l- kllu@)+/,(x)l} 'l+k
= j@Lt. (l - k) . dx .lI o@)-
I "(x))
;
'
= 7'ro[(1- k) hldx . ItG) - /'o Kl + k) Lnldx. I"(x)
(7.2s)
while that on the bottomplateis givenby 6Vr(x) - ja Lrrdx.Ir(x)- ja Lrrdx.Ir(x) = 7or[(l - k) Ltr]dx . Io@)-iro [(l + k) Lnl& ' I,(x) where
' a : .
!
(7.26)
: . .
It(x)= Iu@)- I"(x)
(7.27)
Ir(x)=Jo1r)+1,(x)
(7.28)
l,, , in theseequationsis the (magnetizing) inductanceper unit length of one of the capacitor plates with the other plate open-circuited (zero current). It follows from (7.25) and (7.26) that the inductance presented to the balanced currents is decreasedby a factor (l - fr) becauseofthe coupling between the lines, while the inductancepresentedto the unbalancedcurrents is increasedwith a factor (l + fr). The inductance used when the characteristic impedance of a transmission line is calculated is the inductance per unit length associated with the balanced currents
228
Design of RF and Microwave Amplifiers and Oscillators
(Lo= |-klltt). this valueby
The inductancepresented to the unbalanced currentsis givenin termsof
l+h L"= (l+k)hr = fito
(7.2e)
The equivalentcircuit shownin Figure 7.6(b) cannow be modified asrequired.The new equivalentcircuitsareshownin Figure7.7.
I,(x) (b) Figure 7.7
The equivalent circuits used to calculatethe influence of (a) the balanced and (b) the unbalancedcurrentson a transmissionline.
The final equivalentcircuit is shownFigure7.8O). Ldx n this figwe shouldbe interpretedasexplainedabove. At this point the free spacecapacitorcan be analyzedby consideringit to be an unbalanced transmission line.Theinputandoutputcurrentandvoltageofthe capacitorwill first be established, afterwhich the l-parameterswill be calculated. It follows from Figure7.8(b)that /r(0) : 0 andd(0) = 0. Sincethe capacitoris in freespaceandno otherpathis availablefor the current,it follows that I2Q) = -1t (o)
(7.30)
The current enteringthe top plate ofthe capacitoron the left is thereforeleaving it at the RHS of the bottomplate. The currentson the two capacitorplatesare given by
"*
Fikn Resistorsand Single-LayerParallel-PlateCapacitors
I I
I
t
vl0)
II 1
(b)
vz(o)
(c)
l
jaL,,'QI4
Figure 7.8
The equivalentcircuit for a single-layerparallel-platecapacitorin free space'
(7.3r)
/1(x)= -+. Ae-Q+ BeQ md
(7.32)
IzG)=!*e"-u +BeU
2 wtrered(x) is the current on the top plate,-I2@)the currenton the bottom plate, andIs I theunbalancedcurrenton the line. By applying(7.31) atx: 0andconsideringthatthe currentat this point is zero,an expressionfor the unbalancedcurrent is obtained: f-
I{l)
=O = -**
z
rt
1"-t't a g"t't
1
t"
230
Designof RF and Microwave Amplifiers and Oscillators
(7.33)
b= 1"-t't + Beet 2 It (7.32)is appliedatx = 0, a secondexpressionfor.Isis obtained: f
1 ' ( 0 )= 0 = 9 + A + B 2 -
: f
9= 2
(7.34)
-(A+ B\
A relationshipbetweenu{andB is obtainedby combining(7.33)and(7.34): -(A+ B) = tr"-ee+ Beel A(e-ct+l)=-31"4*t, ""=r*l A= -B e-q' + I
(7.3s)
The expressionfor the cunent enteringthe top plate of the capacitorcanbe simplified by usingthis expression:
(7.36)
/ , ( 0 )= - b + l + n 2 = ( A + B ) + ( A + B ) = 2 ( A +B )
(7.37)
=. ga-1c--elc e-et + I
Becauseof the relationshipbetweend(0) and/2(0),the expressionfor 1d0)follows immediatelyfrom (7.37):
"-'!r, "1.t I2e)=-1,(o)= -28 ' ' e
(7.38)
+l
With B known, both the input andthe output currentsareknown at this point. The voltageson the two platesare given by (6.7) and (6.8), while the voltage differencebetweenthe two platesis givenby (6.9).Since1ois knownin termsof A andB,
-G,
Film Resistorsand Single-LayerParallel-PlateCapacitors
231
point' andI is known in terms of B, all the voltagesare also known in terms of B at this these derive to ofthe capacitorcannow be calculated.In order The y-parameters -l* (!n: !n: !zr, nd !zz: !n)' onlyy, r mustbe calculated parameters, currentandthe inputvoltageare input the for expressions to calculatey,,, In order required.An expressionfor the input currenthasalreadybeenderived'Derivationofthe expressionfor the input voltagefollows:
(7.3e)
Vll) =Vrr(l) = Zo(Ae-ct - Beqt) Substitution.ofV{\ in the expressionby using(6'7) yields
+ stLul-@+ B)l 0 = I{(0)- ht n - D - hre"-et - Beet) 2 2 ' from which it follows that (Ae-qt ,r,(0)= ?rn - B)+sL,t(A+ $ +Z-a-
IilIN,. i,ft dilu t6
#
ffi #]
Beet)
=nl! o*"-(')+r,t)-tl? u*"<\-*,tf ---t *L-l+, *"-('). r,tf-alh-s*"q1-* r) t# =-B Zo(eet +r)+sLu!.
(7.40)
With the input currentandthe input voltageknown,the desiredexpressionforytl cannow be derived: a
a
ln=
_rt 5"
-a:-
rt
w
1+ e-t'
e-e(-e1( -Z^(ee( -u' - t . -sL,,(.|s ( r \ v +l\+ l+e_1t
(7.4r) L,,(. Zo 1t+ e(1;1t* "-(') "r - T 2 2 eet-e-C(
t 2J2
Desigr of RF and Microwave Amplifiers and Oscillators
}
to belossless, thesecondtermin the denominator If thecapacitoris assumed of (7.41)canbesimplifiedasfollows:
aL
u _ -Zo 1t+ e(r;1t+ "-(')
I
SL,
ry
e\r -e-rl
2
+e :Zs-\ "'l-
(
L
sr
+e(t +\ -{\t
q\
_Zo e-i9t+ei9t +2 2 "i0t _ s-iFt
Zo 2cos0+2 2 cos0+7sin0- (cos0-7sin0) _ (cosO+l)/2 +7ro sino
Q'42)
Substitution ot (7.42)in (7.41)yields I llt=
{ul - T
(7.43)
2
jYoc
sin0
(1+cos0)/2
With ytt known, the other l-parameterscan be calculatedand the l-parameter matrix for the parallel-platecapacitorin freespaceis known. In the losslesscase,thesel-parametersleaddirectlyto theequivalentcircuitshown in Figure7.8(c).It follows thattheparallel-platecapacitorin freespacecanbe considered to be purely lumped,with fixed inductanceand variablecapacitance. As expected,the capacitance at low frequenciesreducesto C{: Ys=j2Ysg
sinp/ I + cosp/
(7.44\
a "i)f^.W * 2 - | -
.a"l(2L,\.C.1
, 1 2L u
=a.(Q\
(7.4s)
232
Desip of RF and Microwave Amplifiers and Oscillators
If the capacitoris assumedto be lossless,the secondterm in the denominatorof
(7.41)canbesimplified asfollows: u _ -Z_o- 1 t + e ( / ; 1*t " - 1 t ) e\t -e-\'
2
_ Z o l + e - u + e c t+ l "(t _"-Ct
2
_T
e - i \ t + e i v (+ 2 _ Z s -lp't-i$erYu _ g
rY"
=z-, 2
2cos9+2 cos0+7sin0 - (cos0-7sin0)
(cos0+1)/2 +7Io sinO
(7.42)
SubstitutionotQ.a\ in (7.41)yields 1 llt=
d..t 2 4
! -
,u /'oc'
1 sin0 ( l + c o s o )/ 2
(7.43)
With y,, known, the other f-parameters can be calculated and the l-parameter matrix for the parallel-plate capacitor in free spaceis known. In the losslesscase,these lz-parameterslead directly to the equivalent circuit shown in Figure 7.8(c). It follows that the parallel-plate capacitor in free spacecan be considered to be purely lumped, with fixed inductance and variable capacitance.As expected, the capacitanceat low frequencies reducesto C0:
Yg=/-Ysg;ffi
(7.44)
=izYocry
=a.(Q)
(7.4s)
Film Resistorsand Single-Layer Parallel-PlateCapacitors
233
The series inductance obtained in (7.43) is also significant. Since aL,: aL11'(1 + f), it followsthat
s L . . L - l + k ^(
;=s4,
z
from which it follows that, if the magneticcouplingbetweenthe two capacitorplatesis tight (this is usuallythe case),the seriesinductancewill beapproximatelythe sameasthe uncoupledinductance(zerocurrentin the othercapacitorplate)ofone ofthe capacitor plates.Insteadof interpretingthis inductanceasthetotal inductanceof oneplate,it would bemoreaccurateto seeit asthe sumof the inductanceof thetop platefrom the LHS edge to the centerandthe inductanceof the bottomplatefrom the centerto the RHS edge(see Figure7.8).Note that the inductanceof a microstripline identicalto the bottomplateand with the top plateabsentwould alsobe 2,,'0. The significanceof this will be appreciated whenthe microstripcaseis considered. frequencies In orderto establishthe influenceofthis inductanceon the resonance (7.43) in terms of Zo..This by replacing Z, mustbe simplified of the free spacecapacitor, (7. canbe doneby using(7.29)and I 5):
Zoc =
2Lo C
= ,z"? ' Y o- z o 2Lb.l.=23.C
(7.46)
2"fo Zfo
:rom which it follows (by using (7.15) that r , l= + k Z o
"
(7.47)
- . , ' L
l-k
4fo
Substitution inQ.a! yieldsthat of thisexpression I
. l+k Zn"
"t*w - -
= jYoc
l- k 8 fn
sin0
,u
"trOC
(7.48)
1"rgAE€
2 sinO
(l+cos0)--tI '9'16g
(7.4e)
234
I rr' t' I
It L
!
t 5
I
Designof RF and Microwave Amplifiers and Oscillators
The effectiveadmittancepresentedby the freespacecapacitorcanbe calculatedby usingthis expression.In interpretingthis expression,it shouldbe kept in mind that the characteristic impedance ofthe capacitor(Iod is notindependent ofthe couplingfactorand will approachinfinity asthe couplingfactorapproaches unity. Theeffectivecapacitance of thecapacitorin theequivalentcircuitis determinedby thc sin0/[(l + cos0)/2] term inQ.a\. This functionis comparedwith the tangent firnction(tan0) in Figure7.9.It is clearlymuchmorelinearthanthetangentfunction,and seriesresonance only occurswhenthe electricalline lengthis I 80o,not 90' asin thecase of thetangentfunction.Seriesresonance in theactualcapacitorwill occursoonerbecause of theeffectof theseriesinductance. (opencircuit)occurswhentheline Parallelresonance lengthis 360". Theparallelresonance frequencyis not influencedby theseriesinductance. The combined influence of the inductanceand the capacitanceon the total admittanceof the freespacecapacitor(ascalculatedwith (7.49))is shownin Figure7.I 0 for a couplingfactorofzeroandone-half.Seriesresonance is clearlyaccelerated drastically by anymagneticcouplingbetweencapacitorplates. Fortunately,thisproblemis eliminated whenthe capacitoris mountedon a microstripline. Whenthe couplingis tight andthe capacitoris mountedon a microstripline, the inductanceof thebottomplatebecomes theinductance ofthe microstripline (aswasshown above), and this inductancecombines with the microstrip capacitanceto have a transmission-line effectup to the centerof the parallel-platecapacitor.The characteristic impedanceandphaseresponseof this line sectionis essentiallythatof themicrostripline. Similarly,the inductanceof thetop platecombineswith the seriescombinationof the capacitorcapacitanceand the microstripcapacitance to havea similarline effect(the
Figure 7.9
Comparisonofthe tangentfunction (tan 0) with the functionsI: sin 0 / [(l + cos 0)/2] and IZ= 0 (<.lCcase).
Film Resiston and Single-LayerParallel-PlateCapacitors
Y r0
-20.0
Fgure 7.10
Comparison of the tangent function (YlYo = tan 0) and the linear case (0) with the normalizedadmittanceof the free spacecapacitorwhen i = 0 and & : 0.5 (losslesscase).
:apacitanceof the capacitorusually acts as a short-circuitcomparedto the microstrip :apacitance becauseofthe relativedifferencein the dielectricconstantsandthe thickness rf the dielectriclayers). The free spaceanalysisclearlysupportsthe useof the line-capacitor-line model.
zot, t
4
-
1 0 ,
"
f,rre 7.f f
'2"_ I '' sinO. f ,._^ = '\,ro,' J@u"4 I --:1J ,, ( l -+ c o s o ' ' ) / 2 ;--
,.
A transmission-line modelfor a paratlet-plate capacitor mountedon microstrip(therz and c subscripts denotemicrostripandcapacitor quantities, respectively).
236
Design of RF and Microwave Amplifiers and Oscillators
1ac, =Jrrc
sin0. (l;;;%)t
0, Figure 7.12
A transmission-linemodel for a gap-capacitoron microstrip.
This model can be enhancedby replacingthe fixed capacitance valuewith a frequencydependentterm basedon (7.44).It shouldbe notedthat the differencein the modelswill be minor when the line length is short (the capacitancewill differ by 10% when the electricalline lengthis 60'). Theproposedmodelis shownin Figure7.I l. Notethatin anMIM capacitorthetop plateis smallerthanthe bottomplate. The model for a singleparallel-platecapacitorcanbe extendedeasilyto obtain a modelfor gap-capacitors too. The gap-capacitor modelis shownin Figure7.12. If the dielectricconstantof the capacitordielectricis muchhigherthanthat of the microstrip and the capacitoris thin comparedto the microstrip substrateheight, the characteristicimpedanceof the line sectionassociated with the gapcan usually also be estimatedto be that of the microstrip. It should be noted that, when possible,the widths of a gap-capacitorand the microsrip line shouldbe chosento be the same.The main reasonfor this is the parasitic effectof the stepdiscontinuitiesintroducedat the gapwhenthis is not the case.
REF'ERENCES l. wolff, I., andN. Knoppik, "Rectangularandcircular MicrostripDisk capacitorsano Resonators," IEEE Trans.MicrowaveTheoryTech.,Yol.MTT-22,No. 10,October 1974. 2. Mondal,J. P., "An Experimentalverification of a simple DistributedModel of MIM
Film Resistorsand Single-Layer Parallel-PlateCapacitors
237
Capacitorsfor MMIC Applications,"IEEE Trans.MicrowaveTheoryTech.,Yol. MTT-35,No.4, April 1987. "MIM CapacitorModeling: A 3. Giancarlo,B., F.Giannini, E. Limit, and S. P. Marsh, PlanarApproach,"IEEE Trans.MicrowaveTheoryTech.,Y ol.MTT-43,No. 4, April t995. 1. MultiMatch RF and Microwave Impedance-Matching,Anplifier and Oscillator West,RSA:Ampsa(PTY)Ltd; http://www.ampsa.com'' Software,Somerset Synthesis 1998. "Di-Cap Inc., 5. MicrowaveCeramicCapacitors,"CazenoviqNY: DielectricLaboratories 1998.
CHAPTER 8 TIIB DESIGN OF WIDEBAND IMPEDANCE-MATCHING NETWORKS 8 . 1 INTRODUCTION .Animpedance-matchingnetwork usually matchesa load to a sourceinside the passband ad may alsobe usedto attenuateunwantedsignalsoutsideit. Whenthe load impedanceandthe soruceimpedancearepurely resistive,inductorcapacitor(LC) networkscanbe designedrelativelyeasilyto fulfill the filter specifications n widebandmatching networks.It is difficult, however,if not impossible,to scale inpedancesover a widebandby using only a limited numberof inductorsand capacitors. '!tis transformationfunction can only be done with transformerswhen the bandwidth,"- .sformationproduct becomeslarge. If the requiredbandwidthis relatively small, bwever, it is possibleto transformresistance overlargedistanceswith LC networks. When the load impedanceor the sourceimpedanceis reactive, part of the hpedance transformationfunction ofthe matchingnetworkis to removethis reactiveness. Tbeextentto which this canbe doneis a functionof the loadimpedanceitself, aswell as t Fansducerpowergain versusthe frequencyresponserequired. The limitationsof a specificloadimpedancefor a specificfrequencyresponsecan bc determinedin at least three ways. Fano'sset of integral equationscan be used to j': ' :rminetheseconstraintsI while Youlaformulatedtheconstraints in termsof Laurent I ], . :s expansions[2]. Carlin advancedan iterativeprocedurefor this purpose[3]. Becauseof its relativesimplicity,only the iterativetechniquedevelopedby Carlin " . bepresentedhere. While theunderlyingtheorywill not beconsidered here,theintegralconstraintson resistor-inductor(RL) and resistor-capacitor (RC) networkslead to simple and -.-:ul upperlimits on the gain [4]. Thesegainlimits will be considered in Section8.3.3, r-og with the Youla gain-bandwidthconstraintsassociated with a parallelRC or a series ; :oad(Chebyshevresponse). With the limitations of a particular load (or source)known, a network that will idetherequiredpowergainversusfrequencyresponse canbedesignedby usingdirect resisor iterativetechniques. Bothoftheseapproaches will bediscussed in thischapter. Networksfor matchinga complexloadto a complexsourceareoftenrequired.A tr,.rctical approachto solving this classof problems was developedby Chen and Satyana-
239
240
Design of RF and Microwave Amplifiers and Oscillators
rayana[5], andmorerecentlyanaltemativeandsimplifiedtheorywasintroducedbyCarlin and Yarman[6]. Carlin and Yarmanalsodevelopediterativetechniquesfor matchinga complexloadto a complexsource[6, 7]. Becauseof its relativesimplicityandits superior results[8], only iterativetechniquesfor matchinga complexloadto a complexsourcewill be considered. It is often possibleto designmatchingnetworksfor complexterminationsby arethenabsorbed initially assumingtheterminationsto bepurelyresistive.Thereactances the effortrequiredto parasiticallyinto thenetworkwhenthedesignis completed.Because designa networkin this way is minimal whenit canbe done,this approachwill alsobe consideredin this chapter. networksare often requiredto providea transducerpower Impedance-matching LC networkscanbe with a positiveslopein thepassband. gainversusfrequencyresponse There is, however,the fulfill this requirement. or directly to interactively designedeither in the frequencies at the lower be mismatched will inevitably thatthe source disadvantage in the to changes gain tend to be sensitive slopesalso passband.LC networkswith RLC impedance-matching componentvalues.Becausethis doesnot necessarilyapplyto networks,the designof thesenetworkswill alsobe examinedin this chapter.
8.2 '
F'ITTING AN IMPEDANCE OR ADMITTANCE FUNCTION TO A SET OF IMPEDANCE VBRSUS FREQUENCY COORDINATES
When impedance-matchingnetworksaredesigned,impedance(or admittance)functions that will approximatea setof discreteimpedanceversusfrequencycoordinatesareoften input impedanceof a transistoror mightbe themeasured required.The setof coordinates (admittance)of a networkto be input impedance output or it could be the &tenna, or dcsigned. It is sometimespossibleto approximatethemeasuredimpedanceof a devicewith rimple RC, RL, or RLC equivalentcircuits.This canusuallybe donewhenthe resistive pt of the impedanceor admittanceis moreor lessconstantoverthe frequencyrangeof interest. The componentsof suchan equivalentcircuit canbe determinedby settingup an ofthe networkchosen,andequatingits real oqgationforthe inputimpedanceor admittance od imaginarypartsto the measuredvalues.Althoughthis techniquecanbe used,more techniquesareoftenrequired. sophisticated A major problem in finding an impedancefunction that will fit a given set of coordinatesis its realizability.The functionobtainedmustbe positive-real. A techniquethat usuallygivesgoodresultsis basedon the fact that the reactance (admitiance) functioncanbedeterminedwhenthe (susceptance) of a minimum-impedance is known [9]. The equivalentcircuit of a minimum-impedance rcsistance(conductance) functionthathasa parallelcapacitoror inductorasthelastelementis shownin Figure8.1. is known,it follo'*'s canbe determinedwhenthe resistance Becausethe reactance
The Designof WidebandImpedance-Matching Networks
241
that the impedanceitself is known when its resistivepart is known. In termsof equations, if
= IEc,/(ro -
R0/ zl
(8.1)
2n
then
(8.2)
Z ( j a ) = Z Z C i / ( r o- o r r )+ i t o 2n
The poleso)r,G)2,..., @2n arethe first and secondquadrantpolesof the resistance frrnction(which is an evenfunction),while C1,Cr, ..., Crnarethe residuesof thesepoles. Thepolesrrl1,rrl2,....drn andtheresidues dr,dr,...,er, aretheconjugatesofthefirst andsecondquadrantpoles and the residues,respectively.Rois a real-valued(positive) constant.
z(ja) -
Z(ja) -
(a)
Z(ja) -
o) Figure 8.1
Equivalentcircuits for (a) minimum-impedanceand (b) minimum-admittancefunctions,
If a rational function for the resistivepart of the minimum impedancefunction can "e found,the impedancefunctionitself canbe foundby using(8.1) and(8.2). If the resistancefunctionis assumedto be of the form ?(ro)= a2o I 7or^a2- + ar^-ra2^-2+ ... + ao)
(8.3)
eeunknowncoefficientsin the function canbe found by usingthe equationsavailablefor :ning a polynomialto a given setof coordinates with least-square error. Equation(8.3) canbe changedto havethe following form f(or)=r32P/R(@) = a2^(a2)^ + ar^-r(a2)^-r + ... + ao
T
j 242
Design of RF and Microwave Amplifiers and Oscillators
i
1 = b^x' + b^,-rx^-l+ .,. + bo
(8.4)
a
where x = co'. BecauseR(o) andthe numberof zerosat the origin areknown (the designermust choosethe numberofzeros),the functionl(ar) is alsoknown at discretefrequencies. The coefficientsbo,br, ..., bo,cannow be determinedby solvingthe following set ofequations:
sobo+s,0, + ... * s^b, = ts
i
I
slbo+ s2Dl+...+ s^*rb^= t1
sr*rbo + sn+2bli...*
s2.b^ = t^
(8.s)
I
where so=h sr = lx, s2^= LXi
2m
(8.6)
and /o = II(o,) tr = I T(a,)x,
t^ =27(a,)xi
(8.7)
whereft is the numberof coordinates(x,, Z(to)). With theseequationssolved,the coefficientsin (8.4)and,therefore,thosein (8.3) areknown. The minimum-impedancefunction itself cannow be determinedby using(8. I ) and (8.2). In order to use these equations,the poles of the resistancefunction must be determined. The impedancefunction fitted will bepositive-realif careis takento ensurethat the approximationfunctionhasno real zerosin the o-plane. It is possiblethat the input impedance,as given by the synthesizedminimum This situation impedancefunction,will deviateslightly from the initial setof coordinates.
The Design of Wideband Impedance-MatchingNetworks
243
can be improved by addinga pole at the origin or infinity to the impedancefunction. Altematively,eithertheminimum-admittance functioncorresponding to the setof impedance coordinates canbedeterminedor a computeroptimizationprogramcanbeused to improvethe matchbetweenthe two setsof impedances. An approximationfunction for the resistivepart of an impedancefunction can be found by using the programPLNM FORTRAN (this programis providedon the diskette accompanying thisbook).With R(tl) known,theminimum-impedance functionassociated with the resistancefunctionwill alsobe foundby this program. The programzvR FORTRANcanalsobe usedto find the minimum impedance or minimum-admittancefunction associated with a specifiedresistanceor conductance function (the discretetargetvaluesfor the resistance/conductance to be fitted arespecified in PLNM). It happensoccasionallythat the polynomialdeterminedby the programpLNM is not positive-real.Therearetwo reasonsfor this. First,thenumberof coordinates specifiedin areaswhereZ(ro)approaches zeromay beinsuffrcient.This canberemediedeasilyby specifuingmorecoordinates in theseareas. Second,the increasein Z(ol)may be too slow at high frequencies. In sucha case thepolynomialwill havea zeroon therealaxisof theo-plane.This,in turn,impliesa real polein the resistancefunctionR(al),andthereforea pole on thej
EXAMPLE 8.1
Fitting a functionto a setof resistance coordinates.
As an exampleof the applicationof the programsPLNM FORTRAN and ZVR FORTRAN,anequivalentcircuitfor theinputimpedance of theMotorolaMFR406 powertransistorwill be determined. The input impedanceof the MFR406,asspecifiedby the manufacturer, is tabulatedin Table8.l. It canbe seenfrom this datathattheresistance approaches a constantat low frequencies.The approximationfunction for the resistance, therefore,shouldbe of low-passform; that is, the functionmustbe of the form R ( c o )= l l f a r ^ a z - + a r ^ - r { D 2 ' - 2 + ...+aof In polynomialform, this becomes
(8.8)
Design of RF and Microwave Amplifiers and Oscillators
244
r(co)= t / R(o) = b^(a2)^ + b^-1(a2)^-r+'..+ bo
(8.e)
The resistancefunction and the associatedminimum impedancefunction will be fitted by using the programPLNM. ASCII The data for the program must be enteredin a text file (only "plnm.dat", is file the data of areallowedin thefrle). Thedefaultname characters but any othername(DOS compatible)may be specified. "MRF406.dat"which can Thedatafor this problemwerespecifiedin a file "\plnm" on the diskettesuppliedwith this book.The source be found in directory codefor this programcanalsobe foundin this directorywith a WATCOM make file. An executableversionof this programis providedin themaindirectoryof the "PLNM.EXE". TheDOS4GWDOSextenderis requiredin orderto run disketteas theprogramsprovided,anda copyof thisextenderis providedin themaindirectory of the diskette. The following datamust be specifiedin the datafile: l.
Any arbitrarytitle line (thelengthofthis line canbe up to 79 characters);
2.
The degreeof the PolYnomial(z);
3.
Theresistanceot conductanceto befitted atthe frequencies(Hz) ofinterest.
The datamust be specifiedas shownin the examplefiles. When a new problem is defined,the bestapproachis to makea copy of one of the datafiles provided(usea differentnamefor the frle) andedit the valuesin the new file. If zeroson therealaxisarerequiredin thefunctionfitted,the optionto add an exfia datapoint (asexplainedabove)will be provided' specifiedareshownin Table8'l' The resistances The user is promptedfor the numberof zerosrequiredin the resistance functionat the origin duringexecutionof the program. The polynomialobtainedfrom the programis
Table 8.1 The input impedanceof the MRF406 as a function of the frequency Frequency(MHz)
,, 5 l0 l5 20 25 30
Input impedance(O)
7.5 - t2.6 5.2 - t2.4 3 . 1- j I . 9 2.3 - it.8 1.7 - jt.1 r.3 - jt.4 1.0 -jr.0
245
The Design of Wideband Impedance-MatchingNetworks
+0.4284x 10{ ro2- 0.1181x 10-troa T(a) =Q.1448 x l0-13(D6 + 0.1841 The resistancefunctionobtainedis 6] R(ro)= I / [0.1448+ 0.4284x 104 ro2- 0.1l8l x l0-8ora + 0.1841x 10-r3ro The polesof this functionare a =+204.92+ jgl.483 000.000t jss.704 Therearefour polesin the complexo-planeandtwo poleson the imaginaryaxis. After multiplication of the numeratorand the denominatorby -j, the functionobtainedfrom the programis minimum-impedance Z(ja) =
-0.3969x l03ro2+0.94728x 105rro+ 0.19368 x 10E - jrt -0.238669x l03or2+ 0.605525 x l0s7
The impedancefunctionis givenasa functionof s by the equation
-\-/
x 108 +0.9473xl05s+0.1937 0.3969x 103s2
/e In\
s 3+ 0 . 2 3 8 7 x1 0 3 s+20 . 6 0 5 5 xl 0 s s + 0 . 2 8 0 51x0 7
The impedanceasgivenby this equationis comparedto the measuredimpedance in Table 8.2. The equivalentcircuit associated with this impedancefunction is (without shown in Figure 8.2. The network the 6.6 nH seriesinductor) was extracted by continued fractionation from the impedancefunction (Cauer development).
33.78nH
Figure,8.2
An equivalentcircuit for the input impedanceof the MRF406 transistor.
246
Design of RF and Microwave Amplifiers and Oscillators
Table 8.2 The input impedanceof the MRF406 comparedto the impedanceas given by (8.l0), as well asthe input impedanceof the equivalentcircuit shown in Figure 8.2
Frequency
(MHz)
2.0 f.u
10.0 15.0 20.0 25.0 30.0
Input impedance
Impedanceas
oftransistor
givenby (8.I 0)
(o) 7.s - j2.6 5.2- J2.4 2.6- jt.8 2.3- jt.8 1.7- jt.'t r.3- jl.4 1.0-J1.0
Input impedanceof the equivalent circuit shownin Figure 8.2
(o)
(0)
6.6- it.4 5.4- i2.8 3.4- i3.2 2.2- iz.e t.7- j2.6 r.3 - j2.4 r.0 - i2.3
6.6- jl.3 5.4- j2.6 3.4- j2.8 2.3- j2.3 r.7- jl.8 1.3- jt.4 1 . 0- j l . l
givenby (8.10) correlates It canbe seenfrom Table8.2thattheimpedance well with the input impedanceof the transistor.The match can be improved, however,by addinga pole at inhnity (seriesinductor)to the impedancefunction. The resultingnetworkis shownin Figure8.2. If necessary,the correlationbetweenthe impedanceof the transistorand that given by the new equationcan be improved by enteringthe network into an optimization program.The alternativeis to fit a minimum-admittancefunction to the measureddata.
8.3 THE ANALYTICAL APPROACH TO IMPEDANCE MATCHING Impedance-matchingnetworkscanbe designeddirectly (analytically) or iteratively. The directapproachwill be discussedin this section. of animpedanceandthesourceimpedance In its simplestform theloadimpedance matchingproblemarepurely resistiveand equal.In this casethe matchingnetworkhas only a filtering function. Whenan explicit function for the requiredtransducerpower gainversusfrequency r€sponseis specified,the network requiredto meet the filtering specificationscan be designedeasilywith thewell-knownDarlingtonsynthesistechnique.Darlingtonsynthesis in Section8.3.1. will be discussed networkis designedby usingDarlingtonsynthesis,the source Whena band-pass oftendoesnot havetherequiredvalue.When of thenetworksynthesized or loadresistance the bandwidthis relativelynarrow(lessthan two octaves)and the networkcontainsLsectionsconsistingonly of inductorsor capacitors,it is sometimespossibleto transform
Networks The Design of WidebandImpedance-Matching
247
the sourceor the load to havethe requiredvalue by using LC transformers.The designof thesenetworkswill be discussedin Section8.3.2. The only solution possiblewhen the required bandwidth is large is to use conventionalor transmission-linetransformers.Although transformationover a wide transformeris used,the transformation bandwidthis possiblewhen a transmission-line ratiosarelimited(114,ll9,...). widebandtransformation ofresistanceis possiblewhen At microwavefrequencies, linesareused. taperedor stepped-impedance transformation by designing It is oftenpossibleto eliminatetheneedfor resistance powergainversusfrequency transducer or semi-high-pass a networkwith a semi-low-pass response(i.e., matchingnetworkswithout transmissionzerosat the origin or infinity, respectively).This techniquecanalsobe usedto providematchingbetweenunequalload andsourceterminationswhenthe transformationbandwidthis smallenough. It should be noted that the transformationdistanceobtainableover a given networksis limited. This follows from the fact bandwidthwith LC impedance-matching thata high transformationQ is requiredwhenthetransformationdistanceis large.A high transformationQ, in turn, impliesa high networkQ andthereforea narrowbandwidth. networksare also limited The gain-bandwidthproductsof impedance-matching can whenthe load or sourceimpedanceis reactive.The extentto which this reactiveness be removedis limited becausenegativeinductorsandcapacitorsdo not exist. When only the load (or source)impedanceis reactive, the gain-bandwidth constraintsimposedby the load (or source)on a particulartransducerpowergain versus formulatedby Fano, frequencyresponse canbedeterminedby usingtheintegralconstraints the Laurentseriesconstraintsof Youla,or the iterativeapproachof Carlin. art The constraintson severaltypesof loads,usuallyfor Chebyshevresponses, availablein the literaturein explicitform. Only thelimitationsimposedby simpleRC and here.in Section8.3.3. RL loadswill be considered The constraintsimposedby any otherload can be determinedby using Carlin's iterativeapproach,which will be discussedin Section8.4. Theanalyticaldesignof networksfor matchinga complexloadto a purelyresistive to solving impedancesourceis discussedin Section8.3.4.Two analyticalapproaches matchingproblemsbelongingto this classwill be considered. isused,thecomplexityoftheload WhenthetechniquediscussedinSectionS.3.4.l is immaterial.As long asthe specifiedtransducerpowergainversusfrequencyresponse is realizable,any load canbe matchedto a resistivesource. The parasiticabsorptionapproachdiscussedin Section8.3.4.2can only be used whenthe terminationscanbe modeledwith simpleequivalentcircuits. network,it is Whenthe load is parasiticallyabsorbedinto animpedance-matching to bepurelyresistive.A networkwith a suitabletopologyis thendesigned initially assumed power imposedby thereactiveloadonthetransducer constraints and,ifthe gain-bandwidth gain versusfrequencyresponsechosenwere taken into account,it will be possibleto absorbthe reactivepart ofthe loadinto the designednetwork. Although it is limited to simplerproblems,this techniquehasthe advantagethat lesseffort is requiredin designingthe network.Parasiticabsorptioncan alsobe usedto
248
Designof RF and Microwave Amplifiers and Oscillators
solve simple problemswhere both the sourceand the load terminationsarereactive. The principleof parasiticabsorptionis illustratedin Figure8.3. Although it is also possibleto match a complex load to a complex source analytically[5, 6], the relevanttheory will not be consideredhere,becausemuch better resultscanbe obtainedwith considerablylesseffort by usingiterativetechniques[8].
Network desiened
I
I
Reactive source
Figure 8.3
,
I
L-
I
i __.1
T
Reactive load
Illustration ofthe principle ofparasitic absorption.
The additional theory necessaryto design commensurate distributed networks will be covered in Section 8.3.6. fuchards'transformation, Kuroda andNorton's identities, and unit elements and their extraction will be considered. Under Richards' transformation all ofthe theory applicable to the design of lumped element networks also apply to commensuratedistributed networks.
8.3.1
DarlingtonSynthesisoflmpedance-MatchingNetworks
(seeFigure8.4),a networkthatwill Whena resistive loadis matched to a resistive source providetherequired powergainversus by transducer frequency response canbedesigned following the procedureoutlinedhere. ofthe networkdesignedby It shouldbe notedthat the source(or load)resistance following this procedurewill oftennot be equalto the specifiedvalue.Whenthe network designedcontainsband-pass L-sections,it is sometimespossibleto useLC transformers to adjustthe resistancelevel.Transforrners canbeusedto changethe impedancelevelsin widebanddesigns. Gr(a')
Network to be designed
Figure 8.4
P(s) -Zrr(s))
network when the terminationsarepurely resistive. The designof an impedance-matching
The Design of Wideband lmpedance-MatchingNetworks
249
If a networkwithout transmissionzerosat o = 0 or o)- 6 is designedandthe gainbandwidthlimitations are not a problem,the sourceresistancewill always have the requiredvalue.
Darlington Synthesis Specifications:
powergainversusfrequencyresponse Thetransducer requiredand the valuesofthe loadresistance andthe sourceresistance.
Replpceall the o2 terms in the specifiedtransducerpower gain function Gr(at), with -s2terms. Determinethe product p(s)p(-s), where p(s) is the reflection coeffrcients22(s)correspondingto the specifiedtransducerpower gain function: P(s)P(-s)=1-Gr(-s2) 2.
(8.l r)
The next stepis to determinep(s). Assignall theleft-handplane(LHP) polesof p(s)p(-s) to p(s).It is not necessary to assignonly LHP zerosto p(s).Any combinationof zeros canbe assignedto it, aslongastheyareassignedin conjugatepairsandthe relationshipbetweenp(s)and p(-s) is kept in mind. When a minimum phasenetwork(i.e., a networkwith minimum phasevariation in the passband)is required,all the LHP zerosmust be assignedto p(s). Whentheparasiticabsorptionapproachis followed,theright-hand plane(RHP)zerosmustbe assignedto p(s) if the sourceis reactive.When arereactive,it is usuallybestto try boththeloadandthesourceimpedances all possiblecombinations. The sign assignedto p(s) is often important. When low-pass as networksaredesignedandthe loadandthe effectivesourceresistance, viewedfrom the loadterminalsat o : 0, arenot equal,the sign of p(s) is determinedby its valueat the origin. The signmustbe suchthat
222(0>-RL o(o)= Zrr(0)+ R,
(8.12)
When the two resistance values are equal, a plus or a minus sign may be assignedto p(s). When the value of p(s) is plus one (open-circuit) at infinity, the first element of the network (as viewed from the load terminals) will be a series inductor. When the sign is negative (short-circuit), the first element will be a parallel capacitor. The sign of p(s) must be such that
250
Desigr of RF and Microwave Amplifiers and Oscillators
2,,(q\ - R,
O( @ ') = ----==--:-----:-
(8.13)
Zrr(a) + R,
Where the load resistanceis equal to the sourceresistanceand a networkwith a seriescapacitorasfirst elementis required,the signof p(s) must be suchthat P(o)= +l When P(0)= -l the first elementwill be a parallel inductor. The relationshipbetweenthe value of the reflection coefficient at zeroor infinity andthetopologyofthe networkis summarized in Table8.3 for low-pass,high-pass,andband-pass networks. The informationin this tableis usefulwhennetworksaredesigned to absorbthe reactivepart of the loadimpedanceparasitically. Findtheimpedance functioncorresponding to thereflectioncoefficientp(s) by usingthe equation
Zrz(t) _ 1+ p(s) RL I - p(s)
(8.14)
4.
Synthesizethe requirednetwork by using standardfilter theory, If the topologyis important,thetransmission zerosandthepolesat theorigin and infinity mustbe extractedin thepropersequence.
5.
Ifthe sourceresistanceofthe networkdoesnot havethe requiredvalue, transformers or LC transformers canbeusedto changetheimpedancelevel asrequired.
EXAMPLE 8.2
Darlingtonsynthesisof a Butterworthnetwork.
A third-order Butterworth network will be synthesizedas an example of the applicationof this procedure. R, : 1Q: R,,andtherequired3-dBcut-offfrequency is I rad/s.A low-passnetworkwith an inductorasthe first elementis required. 1.
The transducerpower gain function for the third-orderButterworth characteristicis
2sl
The Design of Wideband Impedance-MatchingNetworks
Table 8.3 The relationshipbetweenthe reactivepart ofthe output impedanceofa network and the value ofthe corresponding reflection coefficient prr(s) at the origin and infinity
Typeof network
p(s)
o(U)-
R", - R l. R"+R,
P(o) = I Low-pass P P _ r\' r\s o(o) = R"+R,
zrr(s)at ro--0
Zrr(s) at
Example
(t)-@
W
Resistive
Resistive
TYfT
uapac|llve
-1 P(oo)=
P(0) = I R.-Rz o ( o') = R"+R,
Capacitive
P(0)= -1
Inductive
Resistive
High-pass
p(oo)=#
Resistive
P(0)= I P(o) = I
Capacitive
P(0) = -l P(o) = 1
Inductive
P(0) = 1 P(o) = -l
Capacitive
Inductive
Inductive
Band-pass
P(0)= -1 P(-) = -1
T
Inductive
Capacitive Inductive Capacitive
Gr(r') = I /[1+o6] eacho2termwith-s3 thisb""o-es By substituting
T
252
Design of RF and Microwave Amplifiers and Oscillators
Gr(-t') = | / ll+ (-r')'l = I / u - s6l The product p(s)p(-s) cannow be determined: P(s)P(-s)=l-Gr(-s2) s6 "6-1 *s3
+s-
s 3+ 2 s 2+ 2 s + l
2.
-zs+r
-fr?
After calculationof the pole positionsandthe zeropositions,all the LHP p(s).Because Rr.=lQ:R" zerosareassignedto polesandhalfoftheT
3.
is givenby (8.14): of thenetworkto be designed Theouput impedance ZoG) _ l+p(s) RL I - p(s) 2 s 3+ 2 s 2+ 2 s + l 2 s "+ 2 s + l
4.
The network can now be synthesizedby continued fractionation (Cauer development)of the impedancefunction:
r yR L
=.rzr+
| zs+ r+1/1
I
I tLr+llG4
J'L;' + -
hastherequiredvalue,no tansformersor LC Becausethesourceresistance particularcase. in this are required transformers by The networkdesigned this exampleis shownin Figure8.5'
253
The Design of WidebandImpedance-Matching Networks
IH
IH
Ll
L3
lo
lo
2F G1
Figure 8.5
v2
A network with a third-orderButterworthresponse.
LC Transformers
8.3.2
The impedancelevel in a band-passnetwork containingan L-section consisting of capacitorsor inductorsonly canbe changedby replacingthe L-sectionwith a suitableTor Pl-section. Similar to the L-sectionsdiscussedin Chapter4, the output impedancewill be transformeddownwardwhenthe elementof the L-sectionto the left is a parallelelement andwill be transformedupwardwhen it is a serieselement. TheT- andPl-sectionequivalents fortheband-pass L-sectionsareshownin Figures 8.6and8.7. The maximumtransformationdistanceof thesesections(n2)is limited by the ratio ofthe seriesreactance andthe parallelreactance ofthe originalsection,ascanbe seenby inspection of(8.15)to (8.18). "\lctrans" TheprogramLCTRANSFORTRAN,which is providedin directory on the diskette,canbe usedto calculatethe components required.
nL!.ol@L"ao- Lo)
n2l"t1l-r1
1r*fl-'
TF t'l-i"'
( 8 .l 5 )
Col(-n)
-l-'-TCln *;r" t,:::_t
C"Crl(n'Cno- nCS
-T* * '" t,L_i",,,
Cr+"- C"/n
g1*-co-1-t < n
c"'
,"" Q-n)C"h?
(8.l 6) ' i::11
Figure 8.6
LC transformersyielding a downwardtansformation.
254
Design of RF and Microwave Amplifiers and Oscillators
(t-dlp
L"+Lp(t-n)
,,-TTTTTTTTTTTTTTTT*r, r,#r", ,,Grr, nL"t(n-t)
n2LS.ol1t"+1t-n1Lo1
1
(8.17)
c,
CsC' Kl-n)CN+Cr)
-F-r z,-
coi -
Col(nz-n)
Crln
z" (n-l\C,ln
((l-n)Cs+Cp)tn2
1.n.1*92
(8.l 8)
C,
Figure 8.7
LC transformers yielding an upward transformation.
EXAMPLE 8.3
Thetransformation propertiesof a second-order Chebyshev bandpass network.
A band-passnetwork with a second-orderChebyshevresponseis shown in Figure8.8.The ripple in the passband is 0.5 dB, the centerfrequencyisfr andthe bandwidthis B. The input impedanceof the network canbe transformeddownward(or the output impedancecanbetransformedupward)by replacingeitherof the trvo bandpassL-sectionswith an equivalentT- or Pl-section(The secondL-section is obtainedby moving the seriescapacitorto the left andthe shuntcapacitorto the right). The maximumtransformationdistance(r2) possibleby replacingeitherof thetwo band-pass L-sectionsin Figure8.8with its equivalentT- or Pl-sectioncan be determined by using(8.15)to (8.18): 0.50
o.7|7llB
Bl(0.707r@;)
B/(1.4029oo Figure 8.t
A band-passnetwork with a second-orderChebyshevresponse(center frequeircy: os radls; bandwidth:,Bradls).
The Desigr of Wideband Impedance-MatchingNetworks
255
. Table 8.4 The maximum transformationdistancefor an LC transformerin the network shown in Fieure 8.8 as a function of the relativebandwidthfolf,)
Transformation distanceof the LC transformer
Relative bandwidth
.,
9.0 3.0 1.7 1.2
J
5 l0
nz^o=u[r+ ry,OhG)' 2
=[ l+(ao I B)2
(8.1e)
The transformationdistanceobtainable.therefore.is a function of the ratio of the centerfrequencyandthe bandwidthof the network.(This ratio can be definedas the Q-factorof the network.) The maximum transformationdistancefor different valuesof the relative bandwidth(f, / ft) is shownin Table8.4.Thetransformationdistanceobtainable is largewhenthe bandwidthis narrowandis smallwhenthe bandwidthis wide. networkfor oo = 1.732radls,B :3 rad/s(fr/ft :3), The transformed = R, lQ, andR": l/6 O is shownin Figure8.9.Notethatthe inductorto the left of the replacedcapacitiveL-sectionis scaledby a factor of 3. This must be done becauseof the changein impedancelevel causedby the LC transformer. imposedby the Becauseofthe differencein theloadandsourceimpedances distanceof the ripple specificationandtheorderofthe network,thetransformation networkin Figure8.9 is twice that of the LC transformer.
E
Figure 8.9
A network for matching a lQ load to a sourcewith l/6Q intemal resistance(Chebyshev response,l/2 dB ripple, do= l.732radJs,B = 3 rad/s).
'
2fi
8.3.3
Desigr of RF and Microwave Amplifiers and Oscillaton
The Gain-Bandwidth Constraints Imposed by Simple RC and RL Loads
The constraints imposed by a parallel RC load on the gain-bandwidth product of a lossless network with reflection coeffrcient p(s) can be expressedin the form []
I d.o
(8.20)
If the RC time-constantin (8.20) is replacedwith Z/R, the gain-bandwidth constraintsassociated with a seriesRL loadcanalsobe determinedby using(8.20). The gain-bandwidthexpressionfor a seriesRC loadis []
f r-t ln Jo
r
d,
(8.21)
lP(,rot)l
Theconstraintsassociated with a parallelRL loadalsofollow from (8.21)by replacingRC wfthL/R. Becausethe valueof ln(l/ | p(7ct)l) is normallyhigh insidethe passband andclose to zerooutsideit, theseintegralequationsclearlyillustratethe trade-offpossiblebetween the gain and the bandwidthof the matchingnetwork.It follows that any increasein the bandwidthwill bring about a decreasein the gain when the gain-bandwidthproduct exceedsthe limit imposedby the load. Assuming that the matchingnetwork has an ideal response(Gr,.- inside the passbandandzerooutsideit), (8.20)and(8.21)canbe manipulatedto obtainthe absolute maximumtransducerpowergain associated with a bandwidthB (Hz). The upperlimit resultingfrom (8.20)is
(8.22)
Gr.,oo
-
A similarexpressioncanbe derivedfor theseriesRC case.Becausethis derivation is moreinvolved.the detailsareshownbelow:
f',-, ro
r"llla. lp(,,o)l
=ro ["r-, rrl-]-la. +r."r l', .'-zrnlI la, lp(,rar)l
lp(,rr)l
+ Ja[*H,-'r"l I la' lp(,rol)l
=l|'.-""l|- * I;;'-''h*j-
. [.co-2 0ao
The Design of Wideband Impedance-MatchingNetworks
257
=-rl#,-' l;;=rnlo",i,,l t# *J =nRC It followsfromthisresultthat _tcRCa La n
6n-aL
(8.23)
lP.tl=t BecauseGr = l- lolt, ttt" requiredexpressionfor the gaincannow be obtained: _2nRCata
a
0n-02
Grr;o=l-e
$.24)
It has been shown in [4] that these gain expressionscan be simplified to
(8.2s)
Gr,r,n =l-e-2tQ"lQ,
rn all four cases. Q"in(8.25) is the circuit Q andis definedby I
O-=
{0a02 aH-0t
-0-o B
(8.26)
rasusual),while Q, is the Q-factorof theloadatthecenterfrequency (asClR in theparallel RC case,l/(tloCn) in the seriesRC case,
t l0 l
fl - K,lt''
l'\ ,''';ll '1,, =,,i"r,{,,i* *(i'r-'aRC% )J
(8.27)
258
Designof RF and MicrowaveAmplifiers and Oscillators
wheren is the order of the network Curvesillustrating the relationshipbetweenthe maximum realizablegain (K") as a functionof theRCf"productaregivenin Figure8.l0 for a Chebyshev response with 0.5dB ripple in the passband[0].
K,
0.1
0.5
1.0
RCf" Figure 8.10
The maximum realizablepower gain (K,) of a parallelRC load as a function of the RCf," productofthe load(low-passChebyshevresponsewith 0.5-dBripple in thepassband) (after
tl0l).
It canbe seenfrom thesecurvesthat the maximumpowergain will be lessthan I whentheRQf productis greaterthanapproximately0.3.WhentheRQf productincreases abovethis value,the maximumrealizablegaindropsrapidly. If more elements(n) are used in the impedance-matching network, the gainbandwidthproductincreases. The improvement,however,is smallwhenmorethan four elementsareused.
8.3.4
Direct Synthesis of Impedance-Matching Load (or Source) Is Reactive
Networks When the
If the specifiedtransducerpowergaincanbe realized,a networkfor matchinga reactive loadto a resistivesorucecanbe designedby following the procedureoutlinedhere[0]. Similar to Darlingtonsynthesis,the sourceresistanceobtainedwill often not be equalto that specified.The requiredtransformationcan usually be obtainedby using transformersor LC transformers.
The Designof WidebandImpedance-Matching Networks
259
Gr(<^rt)
Network to be designed
I
I
Ftgure E.ll
P(s) -_ZnG))
Illustration of the designof impedance-matching networksbetweenaresistivesourceand a reactiveload.
DesignProcedure Specifications: 6.
Thesourceresistance, (Z(s)),andthetransducer theloadimpedance powergainversusfrequencyresponserequired(seeFigure8.1l).
Replaceall
2.
The next step is to determinep(s). Assign all LHP poles to p(s). If a minimumphasesolutionis required,assignall theLHP zerosto p(s).If not, any combinationof the zeroscan be assignedto it, as long as they are assignedin conjugatepairs,andtherelationshipbetweenp(s)andp(-s) is kept in mind. Whenthe load resistancediffers from the sourceresistanceand a low-pass or high-passnetwork is designed,the sign of p(s) can be determinedby determiningthe valueof p(s) asdefinedby (8.28)at <^r: 0 andwhen(r)* 6: respectively.
p(s)= zzzG)z!(:s.)nrn
(8.28)
l(s)= fIt" - s,l/[s+s,]
(8.2e)
Zrr(s)+ Z,(s)
i
wherel(s) is an all-passfunctionwith poles(s) equalto the openLHP poles(LHP poleswith thejar-axispolesexcluded)of the load impedance frrnction21$). A(s) ensuresthat all the polesof the reflectioncoefficient, p(s), will lie in the LHP by cancelingthe RHP polescausedby 21(s) in (8.28).
260
Design of RF and Microwave Amplifiers and Oscillators
222@)is the outputimpedanceof the networkto be designed' arepurelyresistiveandbandWhentheloadandsourceimpedances andeithersign thesignof p(s)is indeterminate passnetworksaredesigned, canbe used,that is, unlessa specifictopologyis required.This doesnot alwaysapplywhenthe loadimpedanceis reactive. Whether there are any constraintson the sign of p(s) can be determinedby considering(8.28)at o : 0 andwheno) - @.
3.
Determinethe output impedanceof the impedance-matchingnetwork as seenfrom the load terminals.This can be done by using the following equations:
rr(s)= 0.5lZ1$) + Z,(- s)l
(8.30)
function21(s). where4(s) is theeven(resistive)partofthe loadimpedance (8.31)
(8.32)
4.
Synthesizethe requirednetworkby usingstandardfilter theory.Whenthe topologyis important,theelementsof thenetworkmustbeextractedin the propersequence.
Darlingtonsynthesiswhenthe loadis complex.
EXAMPLE 8.4
As an exampleof the applicationof this procedure,a networkwill be designedto matcha loadconsistingof lQ resistorin parallelwith a L39F capacitorto a source powergainversusfrequencyfunction Thetransducer with 0.5Ointernalresistance. low-passChebyshevfunction with 0.5-dB ripple in the is to be a second-order passband. o.: I rad/s;K,: I The specifiedtransducerpower gain function is
Gr(roz1=
K
I+ezC|(allo") I
-4{o2+l) t+0.12202(4aa
The Design of Wideband Impedance-MatchingNetworks
261
(The ripple factor and the polynomiit Cr(to) were obtainedfrom standardfilter tables.) BecauseRCf" = 0.221,it is clearfrom Figure 8.l0 (r : 2) that the gain functionspecifiedis realizable. Step I
Gr(-s27=
l+0.12202(4sa +4s2+ l)
P(s)P(-s)= 1- Gr(-s2) "o +"t +0.2500 so+st +2.2988
Step 2 By assigningthe LHP poles of the productp(s)p(-s) to p(s), its denominatoris found to be p(s) = s2 + 1.4257 s + 15126 By assigningthe LHP zerosto p(s),the numeratoris foundto be q(s)=s2+0.5000 The reflection coefficient p(s) is therefore .
v\u,
-:--;--
"2 + 0.5000 s" + 1.4256s +.1.5162
A(s)=fllr-s,l/[s+s,] _ s 0.7914 s + 0.7914 ':' and,by using(8.28),
@ 262
Designof RF and Microwave Amplifiers and Oscillators
= z?(!)-?('o)4ry p(o) ZzzQ)+ Z,(0)
_ 0.5-1.0 0-0.7194 0 . 5 + 1 . 00 + 0 . 7 1 9 4
' t
=o'5 1
5
it followsthata positivesignmustbeassigned to p(s). Step3 Z{s)=l/[l+1.39s] 4(s) = 0.5lZ,(s) + Z,(- s)l = -0.5176 /l.t' -0.51751 F(s) = -1.0351 / ls + 0.719412 ZzzG)= F(s)/[A(s) - p(s)]- ZrG) t.4297s 0.0185s2 + 0.0129 s + 2.0360
0.0129F Figure 8.12
1.390F
A network for matchinga capacitiveloadto a sourcewith 0.5Q internalresistance(0.5-dB ripple, second-orderChebyshevresponse).
The Design of Wideband Impedance-MatchingNetworts
263
Step 4 zzz(s) =
0.0129s +
Cr+
0.707ls+ -l2.0
I ^ s L' , G + 3
Step 5 The sourceresistanceis equalto the specifiedvalue and,therefore,no transformer is required. The desienednetworkis shownin Fisure 8.12.
8.3.5
Synthesis of Networks for Matching a Reactive Load to a Purely Resistiveor a Reactive Source by Using the Principle of ParasiticAbsorption
Whenthe load impedanceor the sourceimpedancecanbe approximated with simpleRC, RL, or RLC networks,impedance-matching networksfor the reactiveterminationscanbe designedby at first ignoringthe reactiveness. Ifthe gain-bandwidth constraintsaretaken into accountand a network with a suitabletopology is designed,it will be possibleto absorbthe reactivepartsof the terminationsinto the designednetwork. The topology of the network is a function of the orderof the gain function chosen, its transmissionzeros,andthe signof p(s),aswasexplainedin Section8.3.1. Whenonlytheloador sourceimpedance is reactiveanditcanbeapproximatedwith a parallel RC or seriesRL network,the maximum gain in the passband(K,) can be determinedfor a Chebyshevtransducerpower gain versusfrequencyresponsewith a specifiedripple factore, by using(8.27). Although this equationgivesthe optimumK, corresponding to a specifiedripple factor,it givesno indicationasto which ripple factorwill causethe lowestinsertionloss in the passband;in otherwords,the optimumripple factoris not known. The optimumripple factorscorrespondingto somevaluesof the load or source qualrty factors at the highest frequency in the passband(2nRCf") were determined iterativelyby substitutingvariousvaluesfor theripplefactorinto (8.27).Thecorresponding (K,; K,l(l+e2)) are valuesfor the highestgain and the lowestgain in the passband tabulatedfor different values of the Q-factor and the number of elementsused in the networkin Table8.5. It follows from thetablethattheinsertionlosswill be approximately0.5 dB when Q:2.25 andfour matchingelementsareused(anidealtransformerwill alsobe required ifthe sourceor loadresistancediffersfrom that required). When both the sourceimpedanceand the load impedancearereactiveand can be approximatedwith parallelRC or seriesRL networks,the optimumvaluesfor the maxi-
264
Design of RF and Microwave Amplifiers and Oscillators
Table 8.5 The valuesof the highestand the lowesttransducerpower gain in the passband(K,; K, /(l + e2))of the optimum low-passChebyshevfunction as a function ofthe load or the sourcep-factor at the highest frequencyin the passbandand the numberofelementsused
o
K,; K,l(l+ez) n:2
n:3
n=4
n:5
0.25
1.0000 0.9998
1.0000 1.0000
1.0000 1.0000
1.0000 1.0000
0.50
0.9997 0.9969
0.9999 0.9994
r.0000 0.9998
1.0000 1.0000
0.75
0.9991 0.9876
0.9989 0.996t
0.9992 0.9981
o.9994 0.9988
1.00
0.9929 0.9703
0.995I 0.9876
0.9962 0.9925
0.9968 0.9946
1.25
0.98t 4 0.9459
0.9875 0.9729
0.9894 0.9816
0.9905 0.9856
1.50
0.9685 0.9165
0.9749 0.9527
0.9789 0.9655
0.9E05 0.9715
t.75
0.9515 0.8839
0.9589 0.9284
0.9626 0.9451
0.9665 0.9533
2.00
0.9319 0.8499
0.9419 0.9016
0.9453 0.9216
0.9492 0.9319
) )<
0.9rl l 0.8t 57
0.9206 0.8731
0.9256 0.8963
0.9294 0.9083
2.50
o.E877 0.7821
0.9004 0.8442
0.9046 0.8699
0.9082 0.8834
)1<
0.8666 0.7496
o.8776 0.8t 52
0.8828 0.8431
0.8861 0.8580
3.00
0.8440 0.7184
0.8548 0.7868
0.8609 0.8165
0.8638 0.832s
??5
0.8223 0.6887
0.8344 0.7592
0.8391 0.7903
0.84l5 0.8073
3.50
0.7997 0.6606
0.8126 0.7326
0.8177 0.7648
0.8195 0.7827
3.7s
0.7800 0.6340
0.7915 0.7071
0.7968 0.7402
0.8000 o.7587
4.00
0.7597 0.6090
0.7695 0.6827
0.7749 0.7165
0.7791 0.7355
Networks The Design of Widebandlmpedance-Matching
265
mum gain in the passband(K,) and the ripple factor (e) can be determinedby using the following setof equationsI l]:
(8.33)
X=U/Qr+I/Qrlsin! 2n
(8.34)
Y= f l l Q r - l / Q r l s n ! ! 2n I = sinh-rX
(8.3s)
= ln[X+ X z + l l
B = lnlr+ ",ly';
(8.36) (8.37)
=l lsirblnAl. sinholnB]* sinh:[nB] + o.s c = o.s sitth'lnAl
sintr4lnll
4
(8.38)
si*21n11
Kn = Lllc2 sinhz(nA\1
(8.3e)
Gr(a' ) = K^ / fl + e' C',(a)l
(8.40)
p-factors at the highest frequency in the Qt and Qz Ne, respectively, the source and load passband. When the load or source impedance can be approximated with series or parallel RLC resonant circuits, the inductance or capacitancecan be increasedto causeresonance at the center frequency ofthe passband(alo),andthe band-passproblem can be transformed to an equivalent low-pass problem (o" : I radis) by using the standard transformation formulas repeated in Figure 8.13. The optimum Chebyshev gain function can then be determined as described above. The low-pass p-factors conesponding to the band-pass Q (atthe cufier frequency) by using the equation be determined can
(8.41)
QL = Qa l(ao I B) This equation can be derived easily by using (8.44) and (8'45).
EXAMPLE 8.5
A gain-bandwidthexamplebasedon Table8.5.
The optimum valuesof K" and e (two-elementchebyshevmatchingnetwork) will for the loadof Example8.4(1O I I 1.39F). be determined
266
Design of RF and MicrowaveAmplifiers and Oscillators
Since QL=6"RC=1.39 it follows from the first column of Table 8.5 and linear interpolation ofthe data that
K." = 0.98t+
l?q-1?5
- r.2s 1.50
= 0.9742 and
- t:1?-|??p.s45s - o.e r65l ' +| + e ' =o.e45e 1 . 5 0- 1 . 2 5 = 0.9294 It follows by manipulationofthe lastequationthattheoptimumvalueofthe ripple factor is e:0.2196 Therefore,the optimum two-elementgain function is ?. - I \ - - ') =
Qr((t)-
0.97 l+0.0452C:(a)
-
The maximumvalueof the insertionlossis 0.32dB.
.1.:
L
C^t LBL ---trr_{F
Lm=L/B
(8.42)
Car.=ll1a2ofrr1
(8.43)
Cac=ClB
(8.44)
Lar=lt1rtlf,Cur)
(8.4s)
cBL
-'r \J
L^Figure &13
Formula for transforming a low-pass network (o"= I radls) to a band-passnetwork with center frequency oo and bandwidth B (rad/s).
267
The Desip of Wideband Impedance-MatchingNetworks
8.3.6
The Analytical Approach to Designing Commensurate Distributed Impedance-Matching Networks
By usingRichards'transformation[2], the analyticaltheoryapplicableto the designof lumped-elementnetworks also applies to commensuratedistributed networks (i.e., distributednetworks in which the line lengths are all equal). Open-endedlines are transformedto lumpedcapacitors, andshort-circuited linesto lumpedinductorsunderthis transformation. Unlike short-circuitedandopen-ended stubs,the seriestransmissionlinesusedin distributed designs have no lumped equivalentsunder Richards' transformation.The influenceof theseserieslines(unit elements)on thegainfunctionandtheirextractionfrom animpedancefunctionwhenanetworkissynthesized will bediscussed in Section8.3.6.1, togetherwith Richards'transformation. Theseriesshort-circuitedstubs,whichareoftenfoundin a networkdesignedby the use of Richards'transformation,are not realizablein planarform. When the designed networkis to be realizedin planarform, theseunwantedstubscanbe removedby using Kuroda'slow-passidentities. As in the caseof lumped networks,impedancescalingis often required for a designedimpedance-matching network.Thisimpedance scalingfunctioncanbeperformed by usingKuroda'shigh-passidentitiesandNorton'sband-pass identities. Kuroda'sandNorton's identitieswill be discussedin Section8.3.6.2.
83.6.f
Richards'Transformation
By usingRichards'transformation[2]
t ,S=.rO= jtan[04=;,an[i I L 2t o - J
(8.46)
open-endedandshort-circuitedstubsaremappedtocapacitorsandinductorsinthe S-plane.
jZorQ
jZotQ
jYor0
& (a) Figure 8.14
RL
(b)
(b) A lumped-elementequivalent for the distributed network in (a) underRichards' transformation.
Design of RF and Microwave Amplifiers and Oscillators
The inductanceand'capacitanceof the lumped equivalentsarerespectivelyequal to the characteristicimpedanceandadmittanceof the short-circuitedandopen-endedlines in the distributednetwork.This is illustratedin Figure8.14. The frequencyresponseof a commensuratedistributednetwork is comparedwith that of its lumped equivalentin Figure 8.15.Note that the responseof the distributed andthatthegainat theevenharmonics networkis periodic(p/ versustan p/ characteristic) (including o : 0) andthe unevenharmonicsof toois equalto that of the lumpedequivalent at o : 0 and o ' o, resp€ctively.The distributedresponseis simply a compressed, periodicversionof its lumpedequivalent.
(b) Figure E.15
(b) The change in the frequency responseof (a) a low-pass network with Richards' transformation.
Seriestransmissionlines are often usedin distributeddesigns.The transmission distributednetwork) line in a commensurate matrix for a unit element(seriestransmission is (" t'\ 'oU;,,J
f-
iv,,t'(;g,J
"(;-,Jl (rl l
jZosi
cos
matrix becomes the transmission By usingRichards'transformation,
(8.47)
'? The Design of Wideband lrnpedance-MatchingNetworks
'=r#["lr'f]
269
(8.48)
with N" unit The transducerpower gain of a commensuratedistributedcascadenetwork form [13] the of expression an given by elements,Nohigh-passllements,andorderN is
(8.4e)
Gr(st) contributes whereG]y(S2)is anNth degreepolynomialin s2.Eachunit element,therefore, a factor(f'- 5'; to the numeratorof the transducerpowergain function' network wittt ttr" gain function chosen,the input impedanceof the corresponding previouslyfor lumpedimpedanceasdescribed canbedeterminedUyOarlingtonsynthesis, thenetworkcanbesynthesized' established' impeAance matchingnetworks.With theinpui elements. ofunit extraction This can-bedoneasbeforeexceptfor the function when the evenpart impedance the A unit elementcan be eitracted from impedanceof the unit characteristic The of the input impedanceat ,S: I is equalto zero. elementis given bY
(8.s0)
zo=2.(Dls=r
determined With the unit elementextracted,the remaininginput impedancecanbe by usingthe exPression[12]
,i^=r,^Q)##2ft
(8.s1)
denominatorand This impedancefunction will alwayshavea commonfactor s2 I in its numeratorwhich canbe canceled. distributed with its lumped-elementequivalentknown, the designofthe required required' not is scaling impedance if matchingnetworkis completed,that is,
EXAMPLE 8.6
Extractionof aunit element'
the The extractionprocedurefor a unit elementwill be illustratedby synthesizing networkwith inPutimPedance. Ztn(S) =
7552+ 1255 1 . 5 ^+ S12. 5 S + 1 . 0
Design of RF and Microwave Amplifiers and Oscillators
[75^S'+] l25S [.5S'+1.0]+1.5S
8.3
Knr
ffser
Becausethe numeratorof the evenpart of Z;,(S) at S: I is givenby
=[7s^r2][1.s.92 -0 +1.0]-tl25.rltl.ssl NZ " " e v_l enls=l I ls=t
frec toc
Zin\) = 50 a unit element of 50Q can be extracted. The input impedancewith the unit element removed can be determined by applying (8.51):
7' _.7 /^s zh(l)- Zi^(S)
"
= sg
=50
s4"(s)- 2,,(l)
- 75^s375's2 75S'+50^S'-75^S-50
Figu
[s2 - l]75.t
ls2-11[7ss+s0] -
I
Zin
t/f?5sl.t/50
The synthesized networkis shownin Figure8.16.
!
Zin'
ll3rrc
t.l6
The network synthesizedin Example8.6.
The Design of Wideband Impedance-MatchingNetworks
8.3.6.2
Kuroda and Norton's Identities
271
r.:
Kuroda'slow-passidentitiesareshownin Figure8.17.Theseidentitiescansometimesbe usedto transformunrealisticimpedancesto more realistic levels, but they are more frequentlyusedto removeunwantedseriesshort-circuitedstubsfrom planardesigns. Kuroda'shigh-passidentitiesareshownin Figure8.I 8.Theseidentitiescanbeused to changethe impedancelevel in a matchingnetwork,asillustrated.The impedancelevel
_fi-: sooll z.
soo
U
(a)
(b) Figure E.17
Kuroda's low-passidentitiesfor commensurate networks,
zsztI (zs+z)
" _E;r- 3'
(4tz) zr4 t (2.+z)
50Q
1= 1l+z"/Z)2 (a)
z
lL_ 50O
J-
1= 1t+2"/Z)(b) Figure 8.18
Kuroda's high-passidentitiesfor commensuratenetworks.
272
Design of RF and Microwave Amplifiers and Oscillators
to the right of the transformedcomponentsis scaledwith a factor ft, while the input impedance(2,")remainsunchanged.
4 Figure E.19
Adding unit elementsto a commensuratenetwork with a resistivesource.
In applyingKuroda'sidentitiesit is usefulto know that whenthe load or source impedanceis purelyresistive,anynumberof unit elements(withZo: R.or Zo: ftr)can be addedin serieswith it without changingthe amplituderesponse.This is illustratedin Figure8.19. TheimpedancelevelinanetworkcanalsobechangedbyusingNorton's identities. Unlike Kuroda'sidentitiesin whichunit elementsarealwaysinvolved.Norton'sidentities
(4stzb)(Zk+4p,
,-(l+zktzb)2
k: (l+ 26tzc)2
Figure E,20
Norton's band-pass identities for commensurate networks.
The Design of Wideband Impedance-MatchingNetworks
273
are appliedto L-sectionsconsistingonly ofopen-endedor short-circuitedstubs.These identitiesare shownin Figure8.20.
8.4 THE ITERATIVE DESIGNOF'IMPEDANCE. MATCHING NETWORKS Insteadof following the analytical approach,impedance-matching networks can also be designediteratively.The "real-frequency"iterativetechniquesconsideredhere have a major advantage overanalyticalandothertechniques in thatequivalentcircuitsfor theload or sourceimpedances,or an analyticalexpressionfor the transducerpower gain versus frequencyresponse, is notrequired.Thenetworkssynthesized by usingiterativetechniques aregenerallysimplerin form with superiorgainproperties[6-8]. "real-frequency" Thefirst techniquewasintroducedby Carlin[3]. A complexload canbe matchedto a purely resistivesoluceby usingthis technique.In this techniquea piece-wiselinearapproximationof the outputresistance or conductance of thenetworkto be designedis optimizedby usinga least-square optimizationroutine. propertiesofthis techniqueare very good,and it can be used The convergence generallyto estimatethe gain-bandwidthconstraintsimposedby any complexload. It hasthe disadvantage, however,thattheresponse outsidethepassband is usually unnecessarilyconstrained,and occasionallyit will not give the bestresultsobtainable withoutconsiderableexperimentation with the responseoutsidethe passband. When band-passnetworksare designed,the reactanceof the networkmay not approximatethe expectedreactancewell becauseof the difficulty of detectingand approximatingthe narrowspikesthat canoccurin the resistance functionof a band-pass network.Inthesecases,theactualresponse will bepoorerthanthatexpectedfrom thelinesegmentresults[4]. Despitethesedisadvantages, the networkssynthesizedby usingthis techniqueare superiorto thoseobtainableby directapplicationof analyticaltheory.This techniquewill bediscussed in detailin Section8.4.1. Apart from matching a complex load to a purely resistivesource,the "realfrequency"techniqueintroducedby YarmanandCarlin in [7] canalsobe usedto match a complexload to a complexsource.In this technique,the numeratorcoefficientsof the input reflectionpararneter(str) of the networkterminatedin a purely resistiveload are optimized. Comparedto the line-segmenttechnique(whereonly one of the terminationsis complex), the reflection coefficient technique has the advantagethat it has no approximationstep. Initializationof the reflectioncoeffrcientprocedweis not assimpleasin the case of the line-segmenttechniquewherethe unknownoutputimpedanceof the networkto be synthesized is takento be equalto the resistivepartof theknownreactiveload.However, excellentresultscan usually be achievedif the resultsobtainedfrom the line-segment techniqueareusedfor initialization. Althoughthesolutionachievedmaynotnecessarily bethebestsolutionobtainable,
274
Designof RF and MicrowaveAmplifiers and Oscillators
it is, as a rule, much better than anything obtainableby direct application of analytical theory. in detailin Section8.4.2. The reflectioncoefficienttechniquewill be discussed In anothertechniqueproposedby Yarmanand Carlin to solvedouble-matching problems(i.e., problemsin which the load and the sourceare complex)[6], the output resistanceof the matching network terminatedat the input in a purely resistive load is optimized. Becausethis procedurehas no significant advantageover the reflection coefficienttechnique,it will not be coveredhere.The interestedreaderis refenedto [6]. The double-matchingproblem can also be solvedvery effectively by doing a systematicsearchon the transformationQ-factors(introducedin Chapter synthesis-based 3) to obtaininitial solutions,which canthenbe optimized.Insteadof optimizingonly the bestsolutionobtainedin the search,it is a goodideato storea numberof thebestsolutions obtained(10-25 areusuallyadequate)andthenoptimizeall of these. by the that initial solutionsaregenerated This approachhasthe distinctadvantage softwareand are not requiredfrom the designer. Ifthe systematicsearchis donethoroughlyenoughandenoughsolutionsarestored for optimization,the probability of finding the optimum solution to any impedancematchingproblemis very high. ofthis approacharethatmanysolutionsareobtained(not Othermajoradvantages jus one) and that transformersare never requiredin the solutionssynthesized.This techniqueis alsovery robustandcaneasilybe extendedto incorporatea greatvariety of constraints(topologyconstraints,constraintson the elementvalues,etc.). Synthesiswith this techniqueis overall topologies(if required),asis thecasewith "real-frequency" techniquesintroducedby Carlin et al. the The transformation-Qapproachcan also be extendedto form the basis of a algorithm for the designof distributedmatchingnetworks.This can be done without solutions,and resortingto Richards'transformation,being restrictedto commensurate havingto dealwith any short-circuitedstubsin the main line. When commensuratenetworks are designedby following this approach,the andthe line lengthsarefixed. The networks variablesarethe characteristicimpedances canalsobe generalizedsothat the samelengthis usedfor the differenttypes synthesized of lines(main-linesections,open-ended stubs,andshortedstubs)andnot all the linesare thatthe line lengths used[ 5]. In additionto its generality,this approachhastheadvantage with lumpedelements(mixed usedfor stubscanbeconstrained to beshortfor replacement lumped/distributed solutions)or differentstubs,if required[5]. impedances to beused If commensurate solutionsarenotrequired,thecharacteristic determined and the optimum lengths can be for the diflerent line typescanbe fixed [15]. impedance To approximatetheresultsobtainablewith lumpedsolutions,thecharacteristic assignedto the main-linesectionsand any shortedstubsshouldbe as high as possible. while that usedfor open-ended stubsshouldbe aslow aspossible. The lengthsof the linesusedcanbe reducedby usinglumpedelements,if spaceis a problem.Part of the line to be replacedshouldbe retainedas a pad for the lumped component.Again the transformation-ptechniquecanbe extendedeasilyto achievethis
tl 51. in detailin Section8.4.3. will be considered Thebasictransformation-Qtechnique
The Designof WidebandImpedance-Matching Networks
275
Table t.6 Comparisonofthe different iterativeimpedance-matching techniquesconsideredhere
Line-segment technique
Reflection coeffrcient technique
Transformation-p technique
Single-matching technique.
Double-matching technique.
Double-matching technique.
Topology independent,except for the number of transmission zerosat the origin which must be specified.
Topology independent,except for the number of transmission zerosat the origin which must be specified.
Topology independent.
Topology control only through the number of transmissionzeros at the origin and the sign ofthe reflection coefficient.
Topology control only through Topology control can be the number of transmissionzeros implementedeasily. at the origin and the sign of the reflection coefiicient.
Initialization by setting the output resistanceequal to the required load resistance,
Initialization by using the resultsofthe line-segment technique.
Initialization by synthesisbasedsystematicsearches.
The resultsobtained are usually degradedin the approximationstep presentin this procedure.
No approximationstep. Can be usedto optimize the solution obtainedwith the line-segmentapproach.
No approximationstep.
Excellent convergence properties,for low-passor high-passsolutions,but no guaranteeoffindingthe global optimum.
Strongly dependenton initial solutions.No guaranteeoffinding the global optimum.
Dependenton the search rangeand density used.Ifthe systematicsearchis done densely enough,it is highly likely thatthe global optimum will be obtained.
Ideal transformers may be required in band-pass solutions.
Ideal transformers may bc required in band-pass solutions.
Ideal transformers are never required.
Single solution for each set ofspecifications.
Single solution for each set ofspecifications.
Many solutionsto each matchingproblem.
Limited to lumped or commensuratedistributed networks.
Limited to lumped or commensuratedistributed networks.
Can be generalizedeasily to synthesizelumped solutions, commensuratesolutions(without any seriesstubs),non-commensurate distributedsolutions and mixed lumoed/distributedsolutions.
Easy to implement.
Easy to implement.
More involved.
Theadvantages anddisadvantages ofthe threetechniques considered in this section are comparedin Table 8.6 above.The first two techniqueswere implementedin the programsLSM andRCDM suppliedonthedisketteaccompanying thisbook,respectively. Thethird techniqueis implementedin [5].
276
Design of RF and Microwave Amplifiers and Oscillators
The Line-SegmentApproach to Matching a Complex Load to a Resistive Source
t.4.t
The gain-bandwidthconstraintsimposedby a reactiveload(or source)canbe determined iteratively by assumingthat the output impedance(or admittance)of the network is a (admittance) function(seeFigure8,21).Whenthisis done,theoutput minimum-impedance is known.The optimumresistance when theresistance of the networkis known reactance deviationbetweenthedesiredand canthenbe determinedby minimizingthemean-square the actualtransducerpower gain. the problemis well behaved with line segments, is approximated If the resistance optimizationroutine. andthe optimizationcanbe donewith a simpleleast-square A detaileddescriptionof the procedure[3, 16] and the mathematicsinvolved follows. This techniqueis implementedin the programLSM FORTRAN,which canbe found on the disketteaccompanyingthis book.The sourcecodeis providedin directory "\lsm" file is providedinthe rootdirectory with a Watcom[17] makefile. Theexecutable ofthe diskette.
The Line-segmentApproach 1.
of the optimum network is Assumethat the output impedance/admittance function. a minimum-impedance/minimum-admittance
2.
Assumeasa first approximationthattheresistivepart(R(cl))of the output impedanceof the optimum network is equalto the resistivepart of the e (Zt); that is, measuredload impedanc R ( r o , )= R r ( o r )
3.
Approximatetherationaloutputresistanceof thenetworkwith a piece-wise linearfunction,asillustratedin Figure8.22. Enoughincrementfrequencies(thefrequenciesat which the slope apof the linearfunctionchanges)mustbe chosento ensurea reasonable
Lossless impedancematching network
t
Figure t.21
(8.s2)
The impedance-matchingproblem under consideration.
1",,'
The Designof WidebandImpedance-Matching Networks
277
proximation of the unknown resistance.This can usually be done by choosingthe frequencies to ensurea goodapproximationof the measured load resistance. For the sakeof simplicity,the resistance R(cl) is assumedto equal zeto at frequenciesgreaterthanthe last incrementfrequency(ro"). Thelinearresistance functioncanbeconsidered to bethesumofthe semi-infinitefunctionsar(a), ar(a), ...,q,(a) shownin Figure8.23,each with an appropriateweight factor r*: R ( o r )= r , + l r o a o ( a )
(8.s3)
where
if 0 ( o & _ t c) - (D,. ,
ar(a) =
ot - or-r
if 0 r _ t ( 0 r (8.54)
if 0 r ( C 0
I and
ro=Z"(0)
(8.5s)
When the optimum low-passnetwork is determined, /o=R"
(8.s6)
whereR"is the sourceresistance asshownin Figure8.21. In all othercases,rois equalto zero.
R("1'
T It .o
ot
Figure 8.22
62
(,)3
tJ4
05
Approximation of the output resistanceof the matching network with a piece-wise linear function.
278
Design of RF and Microwave Amplifiers and Oscillators
ar(o)
Figure E.23
Illustration of the semi-inhnitefunctionsusedin the line-segmentapproach.
Sincethe resistanceR(7'o) is equalto zero when the frequencyis greaterthanthelastincrementfrequency(<1,),theincrementfactor(weight of the otherincrementfactors.The following factor)r, is not independent equationapplies: rn = -ffo + rt + rz +...+ rr-, ]
(8.57)
Whenthis valuefor r, is substitutedinto (8.53),it changesto R(ro)= [1 - a, (o)]ro *Zro[or
(r,r)- a,(co)]
(8.58)
In vectorform this U""orrrl, R(or)= [ - a,(ol)]ro+Ar (a\7'
(8.5e)
where a' @) = [a,(or) - a,(rl), az(a) - a,(a),..., a,-Ja) - a, (r,l)]
(8.60)
and
7'= I
rl
(8.61)
The Design of Wideband Impedance-MatchingNetworks
279
valueatanyparticularfrequency Theresistance canbecalculated by using (8.5e). Becausethe impedance function was assumedto be a minimumimpedance function, the reactanceassociatedwith the resistance.R(ftl) is known. It can be determined by using the equation
X(rrr)= t'1..o7r
(8.62)
where ...,b,(co)l a t(r) = fbr(a),b2(or),
(8.63)
and
bp(or)=
n[ro * - roo_,]
I:;"lffila
(8.64)
The value of the integral in the last equation is high when the arecloseto ol,andthevaluedecreases asthe relevantincrementfrequencies incrementfrequencies deviatefrom o. It follows from this and(8.62),that associated with theresistance at a particularfrequencywill be thereactance changesrapidly at nearbyfrequencies. high whenthe resistance Theintegralin (8.64)hasa simpleclosedform evaluation[9], which is usefulin determiningits value. If the dependence of r, on the other incrementfactorsis taken into
account,(8.62)becomes +6'1 1a17' X(crr)= [0 - b"(rrr)]ro
(8.6s)
where 6,(trl) - D,(ro)
4(ro) -4(at) b'(rrl) =
(8.66) 4-, (or)- 6,(ar)
to aparticularincrement andreactance corresponding Theresistance vector f ' , car,be determinedat any particularfrequencyby using(8.59) and(8.66). 4.
with the initial value(rs') Calculatethe transducerpowergain associated
280
Design of RF and Microwave Amplifiers and Oscillators
of the incrementvector f, atthe various frequenciesof interest.This can be doneby usingthe equation
= 1- 1",,l' G.(
- '1- ll 4 t r r@1)-. zo.r@1) l '
lRz(r,r)+ jX ,(a)l+ [R(ro)+ 7X(ro)] _
a R, (ro)l? (ro) lRr (or)+ R(co )12+ IXr (o) + X(co)12
where2,. (o) is the measuredload impedance. 5.
Determinethe optimum value of the incrementvector ;' iterativelyby minimizingthesumofthe relativedifferencein theactualtransducer power gain (G. (ro))andthedesiredtransducer powergain(GI(cr))squaredat the differentfrequenciesofinterest.This canbe doneby usinga least-square optimizationroutine. Therelevantequationsareasfollows:
B =le21r,,a,)
(8.68)
=y'[G,('"'r)-r1t 7t G,(a t) .l
(8.6e)
.f'(r)=W
(8.70)
de(R(ro),X(ro)) dR(rrr 1X(a) ) * de(R(ro).x(co)) 6R(ro) A-rd 0X(a) a-rt de(R(ro),X(ar))
de(R(ro),X(ro))
dR(rrr)
Ox(ot)
Z(co)+
a(r)
(8.71)
(8.72\
The Designof WidebandImpedance-Matching Networks
281
wherer]'is the currentinitial valueof the incrementvector,anal'1ro; is the gradientvectorassociated with the errorfunctione(r ', ro), where e(R(ro),X(ro))= Gr(R(ro),X(ro))I G, (a) - |
(8.73)
with Gr(R(ro),X(ro)) asdefinedin (8.67). e ( F ' , ai ) = e ( r s ' , ia) + j ' r @ , ) . 6
(8.74)
with 6 definedby the equation
(8.7s)
;'=d+6 where F' is the new initial value of the increment vector.
Ol e ( F ' , a , ) / =0
(8.76)
a6
lfj
'{r; i'r (aj) 5 = -I
e1-r5,cor)/'(or7)
(8.77)
j
With the optimum incrementvector known, the gain-bandwidth constraintsimposed by the load, as well as the output impedance (admittance)of the optimumnetwork,areknown. 6.
The next stepis to determinethe optimumnetwork. Sincetheoptimumincrementvectoris known,theoutputresistance of the network is known at any particular frequency. A rational approximation function and the correspondingminimum-impedance (admittance)functioncanbe obtainedby following theprocedureoutlined in Section8.2. The order of the network and the numberof zerosat the origin are variablesin the approximationstage. With theminimum-impedance andminimum-admittance functions known,the optimumnetworkcanbe synthesized easily. Whethera minimum-impedance or minimum-admittance function will be the bestsolutionto a particularproblemis usuallynot knownat the outset.Ifgood resultsarenot obtainedby usingone,theothercanbetried. When the load resistanceis higher than the soruceresistance,a minimum-impedance solutionoftenyieldsbetterresults. Whenthe gain-bandwidth productin a problemis a limiting factor,
282
Design of RF and Microwave Amplifiers and Oscillators
it will be found that the resultsare dependenton the position of the last incrementfrequency.Someexperimentation with this frequencywill then be necessary. Whenband-pass networksaredesigned,both the first andthe last incrementfrequencies havea significantinfluenceon the resultswhenthe gain-bandwidthproductis limited.
''r' .
The line-segment techniqueis implementedin theprogramLSM FORTRAN.This programcan be usedto solve singlematchingproblems(load complex,sourcepurely resistive). Theoptimumincrementvectorandthegain-bandwidth constraints associated with any reactiveload are establishedby the program,after which the resistancefunction (conductionfunction)is fitted to the numericaldataanda networkis extractedfrom the (or minimum-admittance) associated minimum-impedance function. The input datafor the programmustbe specifiedin an ASCII datafile. The file "lsm.dat" is providedon the disketteasan example,aswell asa templatefile. The input datafor the programconsistof the following: l.
A title line (79 characters maximum);
2.
Theloadimpedance(or admittance) powergain andtherequiredtransducer at the differentfrequencies of interest(f, R,X, Gr);
3.
The number of increment frequencies and the output resistance (or conductance) of the networkto be designedat to : 0;
4.
The initial valuesof the incrementsat eachincrementfrequency(rt, r);
5.
The numberof iterationsto be done:
6.
Theamountby whichthelastincrementfrequencymustbeincremented (in Hertz) and the numberof times this must be done.
The incrementvector is first optimizedby the program,after which the resistance function (or conductancefunction) is determined(the useris promptedfor the numberof zerosto be used at the origin beforethis is done). With the resistance(conductance) functionknownthe associated (minimum-admittance) minimum-impedance functioncan be found.The final stepis to extractthe matchingnetworkby continuedfractionation.
EXAMPLE 8.7
The gain-bandwidthconstraintsof a matchingproblem.
As an illustration of using the program LSM FORTRAN, the gain-bandwidth constraintsimposedon a low-passnetworkby theinputimpedance of thetransistor
283
The Design of WidebandImpedance-Matching Networks
Table 8.7 The optimum valuesofthe highestincrementfrequencyand the associatednormalizedleast-square error and maximum deviationfrom the prescribedtransducerpower gain for the MRF406 asa function of the transducerpower gain specified
Transducer power gain specified
1.00 0.98 0.96 0.94 0.92 0.90 0.88 0.86 0.84
Optimum value ofthe highest increment frequency (MHz)
Maximum deviationfrom the prescribed transducerpower gain(%)
36
23
) t
t) ll 8
J I
38 39 42 58 83 t20
) J
2 I I
in Example8.1 is established. The sourceresistance usedis 6.25Q.The goal is to achievea flat responseacrossthe 2-30 MHz passband. "MRF406a.dat" "\lsm" The datafile in directory on the disketteis usedto definethe matchingproblem.In orderto determinetheconstraints, thegainin this file is decreased progressivelyuntil theripplein thepassband becomesvery small. Thelastincrementfrequency(whichestablishes thezeroresistance or conductance point) is alsoadjustedin eachcaseto minimizethe ripple. The resultsobtainedaresummarizedin Table8.7. If the criterionof minimuminsertionlossacrossthe passbandis used,the bestresultswill be obtainedif the gain is chosento be approximately0.90.The insertionlosswill thenbe lessthan0.6dB (G?-.i,= 0.866).Thematchingnetwork associated with this gainlevel is shownin Figure8.24. Note that the sourceresistancein Figure 8.24 is not 6.25Q as required. Whenalow-passnetworkis synthesized, it shouldbepossibletosolvethisproblem by addingextradatapointsat the lower frequencies.
32.619nH
24.81lnH
4.946Q
Figure 8.24
The best solution obtained (highest minimum insertion gain) with the programLSM FORTRAN to the matchingproblem solvedin Example8.7.
284
8.4.2
Designof RF and Microwave Amplifiers and Oscillators
The Reflection Coefficient Approach to Solving Double-MatchingProblems
arereactive,impedance-matching andthesourceimpedance Whenboththeloadimpedance developedby Yarmanand by using the algorithm networkscan be designediteratively Carlin [7]. thelosslessmatchingnetworkis modeledasatwo-port In following this approach, network. When all the transmissionzerosare at the origin or infinity, the scattering parameters(S-parameters) of the networkaregivenby the following equations:
srr(s)=ft(s)/g(s)
(8.78)
srz(s) = szr(s) = lst / g(s)
(8.7e)
szz(s) = -(-I)r lh(-s) / g(s)l
(8.80)
whereft is an integerspeciffing the orderof the zeroof transmissionat the origin, andil(s) andg(s) arepolynomials. arerelatedby the equation Becausethe transferandreflectionparameters
( 8.8r )
s,,(s)s,,(-s) = I - sr,(s)srt(-s) it follows that
(8.82)
g(s)g(-s) = ft(s)ft(-s)+ (-l)t s2*
It is clearfrom this equation,andthefactthatthepolynomialg(s)mustbepositivereal, that SG) is a functionof lz(s)andthe orderof the zeroof transmissionat the origin ofthe networkarecompletelydeterminedby &(s),g(s),and only.BecausetheS-parameters & it follows that the networkitself is definedwhenft(s)andft are defined.The optimum network,therefore,canbedeterminedby finding theoptimumcoefficientsof thenumerator polynomialh(s) for a givenvalueof ft. In order to optimize the coefficientsof ft(s), an expressionfor calculatingthe the tansducerpowergainat eachrelevantfrequencyis required.In termsof S-parameters, gain is given by
Gr(o) =
Jr- lso1coll' ls, ltl-
r(co
(8.83)
1t- lsolroll'trl- lsrl'l lgLrt) - (-l)t,Scsr g(-.rro) soh(io)+ (-l)rsz h(- ja
(8.84)
The Design of Wideband Impedance-MatchingNetworks
285
where
,56 (or ) =
Z"(ol) + Ro
(8.85)
S , ( r o' -) Z t ( a ) - R o Zr(a)+ Ro
(8.86)
andRois the normalizingresistance of the S-parameters The coefficientsof ft(s) can be optimizedby using a linear least-square routine. Becausethe gain is not a simplefunctionof h(s),theproblemis morecomplexthanbefore andthe choiceof the initial valuescanbe critical. Goodresultscan be achievedif the resultsobtainedby designingan impedancematchingnetworkto matchthemorecomplexterminationto a resistivesourceareusedto determinethe initial values. Theoptimizationcanbe doneby usingtheprogramRCDM FORTRAN.Theinput datafor this programarereadfrom anASCII datafile. Theexamplefile "rcdm.dat"on the diskettecanbe usedasa templatefor anynewproblemto bedefined.Notethattherelative spacingsandthe text stringsin the file mustremainasthey are. The sourcecodeand a watcom [l7] makefile for this programareprovidedin *Vcdm" directory on the diskette.An executablefile is providedin the root directory. The input dataof the programconsistof the following: l.
A title line consistingofTgcharacters or less;
2.
Thesourceandloadimpedances to bematchedandtherequiredtransducer powergain at eachfrequency(f, RnXn Rr,Xr, Gr);
3.
The degree of the numerator polynomial ft(s) and the number of transmissionzerosat the origin;
4.
The initial valuesof the coeffrcientsft,, hr, ...,h, in that sequence;
5.
The normalizationresistance to be used(usually50Q);
6.
The numberof iterationsto be done.
Wheninitial valuesareassignedto thecoefficientsof &(s),it is importanttorcalizs thattherearesomeconstraintson the valuesof the coefficients. In thecaseoflow-passnetworks,thevalueofthe transmissionparametersr,(s) must be equal to I when <,r: 0 and the sameinput and output normalizing resistance(R) are used.
286
Desigr of RF and Microwave Amplifiers and Oscillators
Table 8.8 The constraints on the numerator (i(s)) and the denominator (g(s)) coeflicients ofp(s)
Network
h(s)
g(s)
Low-pass High-pass Band-pass
fro=o
8o= I 8r= |
hn=0 scale factor a for fr(s)
Since
rzr(s)= t-
."t
gts, *st
g n s n* g r - l J ' - l
(8.88) +...+go
and [ = 0 for a low-passnetwork, it is clearthat gemust equal I to ensurethat sr,(0) will be equalto l This restriction on the value of goimposestwo constraintson the polynomial /r(s). The first is that in determining initial values for ft(s), the numeratorand denominatorof the input reflectioncoefficients,,(s)mustbe scaledto enswethat 96will equall The secondrcstrictionfollows from (8.82)by settings = 0:
gfi=t8+l
(8.8e)
Sincego= l, it impliesthat &6mustbe equalto zero. Followingthereasoningabove,it canbeshownthatfor high-passnetworksy'1, must be equalto zero.The numeratoranddenominatorof the inputreflectioncoefficientmust alsobe scaledto enswethatg,: I wheninitial valuesfor the coefficientsaredetermined. problemis initialized,noneof thecoefficientsneedsto be equal Whena band-pass to zEro,but scalingis still required.It is obviousfrom (8.79)thatif&(s) andg(r) arescaled by a factora, the gain lsr,l2will changeby afactora2. An appropriatescalefactorcanbe determinedby using(8.79)to calculatethe gain at any frequencyin the passband without a scalingfactor.Therequiredscalingfactorcan thenbe takenasak / (lg(ja)l' - lh(jrl)l')". The constraints on the numerator and denominator coeflicients of p(s) are summarizedin Table8.8.
EXAMPLE 8.8
A double-matching example.
programsdiscussed, As an exampleof the applicationof the impedance-matching
287
Networks The Design of WidebandImpedance-Matching
a high-passnetworkwill be designedto mismatchthe sourceimpedancein Table 8.9 to a 50O load,asindicatedin the table.Theproblemwill first be transformed to low-passform in orderto solveit with programLSM, afterwhich the solution obtainedwill be usedto initializeprogram RCDM.
Table 8.9 The source impedance,load impedance,and transducerpower gain corrcsponding to the problem solvedin Example8.6 Frequency (MHz) 100 il0 120 140 160 180 190 200
R"+ jx"
(c))
146.0- jl r4.0
-jr r2.5 r38.s 1 3 1 . 0- j l l l . 0 t37.0-j103.0 144.0- j88.0 140.0- j88.0 136.5- J92.0 133.9- 796.0
RL+jxl
(c))
79.r 73.6 68.0 63.2s9.6 s7.5 5s.0 s3.5 -
j72.6 j68.7 764.8 j56.8 j47.9 j47.3 j41.9 j40.4
GT
(o) 0.224 0.262 0.299 0.400 0.559 0.709 0.764 0.818
The Single-MatchingProblen In orderto designthe requiredhigh-passnetworkwith programLSM FORTRAN, the specifications in Table8.9mustbechangedto thoseofthe equivalentlow-pass problem. This is done by using the transformations - 1/s. The new set of in the tablewereobtained specificationsis shownin Table8.I 0. The frequencies (a' = llro- f' = ll (4n2f)) by usinga scalefactor from the transformedfrequencies of 4n2 x l0e. Becauseof the transformation,the impedancesto be matched are the conjugates of thosein Table8.9.
Table E.l0 The sourceimpedanceand transducerpower gain correspondingto the equivalent ' low-passproblemin Example8.6 Frequency (Hz)
5.00 5.56 6.25 7.14 8.33 10.00
& (o)
x" (o)
133.0 140.0 144.0 137.0 131.0 146.0
96.0 88.0 88.0 103.0 lll.0 I14.0
0.818 0.'709 0.559 0.400 0.299 0.224
'@
288
Design of RF and Microwave Amplifiers and Oscillators
Table8.ll The resultsobtainedwith the programLSM FORTRAN in Exampleg.6 Frequency (Hz)
5.00 5.56 6.25 7.14 8.33 10.00
Input impedanceofthe network to be designed(Q)
64.7- j49.9 40.7- j56.4 30.r - j46.3 22.8- j42.4 rs.t - j34.6 12.6- 127.6
Transducerpower gain obtainable
0.835 0.677 0,541 0.427 0.291 0.225
A minimum-impedancematchingnetwork was designed. Note that as a rule, both options should be tried (the program can be modified to do this automatically). Thedatafile usedis storedon thedisketteas"lsm.dat".Notethatwhile the impedanceto be matchedis the sourceimpedanceof the problemto be solved,it is takento be the load impedancein programLSM. The resultsobtainedare summarizedin Table 8.11.The gain-bandwidth limitationsof the networkwerefoundto be insensitiveto the valueof the highest incrementfrequency.(Thiswasnotthecasewhena minimum-admittance solution wasattempted.) Becausea low-passsolutionis required,the numberof transmissionzeros at the origin wasspecifiedto be zero.Theresistance functionobtainedis - 0.1004x10-r0ct6 X(o): l/T(a)= l/(0.7730xt0-r5os + 0.3965xl0-7ct4 -0.3238x10-a ro2+ 0.1974t16-t; The polesof this functionare +j||.837 ar=+78.854 +24.688+i 13.630 The minimum-impedance functionobtainedis Z(ja) = -y0.1998xlOaro3 0.2562xl0e +70.1308xl08o -0.1018x106ro2 -0.7799x -70.5 0.5057x 107+70.1922x106rrl 104co2 094.102 a3 + t.OOO,oT As a functionof s, this becomes
289
The Desigl of WidebandImpedance-Matching Networks
Z(s) = 0 . 2 5 6 2x l 0 e + 0 . 1 3 0 7x l 0 8 s + 0 . 1 0 1 7 9x 1 0 6 s 2+ 0 . 1 9 9 8x l 0 a s 3
(8.e0)
x 107+0.1922 x 106s+0.7798 x l0as2+0.5094 x 102s3 +1.000s4 5.0565 The network obtainedfrom program LSM is shown in Figure 8.25. It was synthesizedby continuedfractionationof the impedancefunction.
Figure 8.25
The network associatedwith (8.90).
As a checkon the results,the transducerpower gain was calculatedat the in Table8.12. relevantfrequencies. Theresultsarecomparedto the specifications The variation in gain is less than 0.24 dB over the frequencyrange of interest.
Table8.12 Comparisonof the gain obtainedwith the network in Figure 8.25 and the specifiedgain Frequency (MHz)
Transducer power gain specified
Transducerpower gain obtained
5.00 5.56 6.25 7.14 8.33 10.00
0.81E 0.709 0.559 0.400 0.299 0.224
0.835 0.677 0.541 o.427 0.291 0.22s
The Double-MatchingProblem
:
Initial values for the design of a double-matchingnetwork can be obtainedby calculatingthereflectioncoefficientfor the networkobtainedfrom programLSM
290
Desien of RF and Microwave Amplifiers and Oscillators
0.9519H
1.6493H
Flgrc t.25
The matchingnetwork designedby using the reflectioncoefficienttechnique.
(referto Figure8.25).The reflectioncoeffrcientobtainedfrom LSM is
.r^^l.r|-= '22\r'
'';''
0.6552x l0-2 -0.6812 x l0-2s-0.5661x l0-rs2 -1.0773xl0{sr -0.9823x 10-7sa l.0oo0+0.4456x 10-ts+0.9661x l0-3s2+0.8929x l0-5 +0.9823x l0-7sa
The numeratorcoefficients were used as initial values in RCDM. Note that the from the loadsidetowardsthe input sidein LSM, while networkwassynthesized in RCDM. the reverseis donein RCDM. srr(s)in LSM is therefores11(s) "rcdm.dat" data ftle problem in the is defined The double-matching providedon the diskette.The sourcecodefor RCDM FORTRAN is providedin "\rcdm" with a makefrle. A Watcom[17] executablefile is providedin directory the root directoryof the diskette. The matchingnetworkconespondingto the impedancefunctionobtained is shownin Figure8.26.Themaximumdeviationfrom the specifiedgainresponse is 0.24dB. The resultsobtainedarecomparedto the specificationsin Table8.13. The final stepin this exampleis to transformtheresultsbackto high-pass form underthe transformations - l/s. Becausea scalefactorof 4n2x10ewasused to obtainthe frequenciesusedin Table8.10,the networkobtainedfrom RCDM
Table8.t3 Comparisonof the specifrcationsand gain obtainedwith the network shownin Figure 8.26
Frequency (MHz)
5.00 5.56 6.25 7.14 8.33 10.00
Transducerpower gain specified
0.818 0.709 0.559 0.400 o.299 0.224
Transducerpower gain obtained
0.801 0.700 0.560 0.414 0.295 o.222
The Designof WidebandImpedance-Matching Networks
291
must be frequency scaledwith the samefactor in order to obtain the matching networkrequired. The final networkis shownin Figure8.27.
l5.36pF
Figure 8.27
26.6lpF
The network associatedwith (8.90).
8.4.3 The Transformation-QApproach to the Designof ImpedanceMatching Networks 4 canbeextended technique described in Chapter Thenarrowband impedance-matching to increasethe number of elementsto an arbitrary number and to mismatch any complex load to any complex source by any specifredamount at any single frequency [ 5]. In order to do this, it will be shown in Section 8.4.3.1 that the locus of input impedancesfor which the source impedance of a network will be mismatched to the load by a specified amount is a circle in the linear admittance plane or the impedanceplane. The parametersof these circles will be derived here. The necessaryextensionsto the single-frequencytechniquewill be made in Section
8.4.3.2. Theextendedsingle-frequency matchingtechniqueformsanexcellentbasisfor the networks.The main reasonfor this is iterativedesignof widebandimpedance-matching thatthe rangeof eachtransformationp is limited sincehigh Q-factorswill inevitablylead to a narrowbandwidth. Becauseof this fact, it is feasibleto do a systematicsearchon the transformation Q-factorsin searchof the optimum combination,thus eliminatingthe needfor a good initial solution.Furthermore,if the searchis donethoroughlyenough,the probabilityof finding the optimum solutionwill be very high. With the systematicsearchcompleted,a numberof the bestsetsof Q-factorscan optimizationroutine,but betterresults be optimized.This canbe donewith a least-square in lesstime areobtainableby usinga simplegradientoptimizationtechnique. The mean-squareerror from the specifiedgain responsecan be used in thc systematicsearchandduringthe optimizationphase,but a betteralternativeis to usethe maximumrelativedeviationfrom theoptimumastheerrorcriterion.In doingso,theerror andnottheaverageresponse. in thepassband valueis determinedby theworstperformance by following this approachcaneasily The topologiesof the networkssynthesized form with no seriescapacitors. belimitedto low-passform,high-passform,or to band-pass
292
Designof RF and Microwave Amplifiers and Oscillators
The last option is usuallyattractivein hybrid circuitsat microwavefrequencies. The time requiredto solvea matchingproblemcan be reducedsignificantlyby constrainingthe gainat the frequencywherethe Q-factorsarecalculatedto be higherthan a specifiedminimum.Therequiredrun time is usuallyvery shortwhennetworkswith less thansix elementsaredesigned(afew minutesmayberequiredwhenfive-elementsolutions aresynthesizedfor a difEcult problemon a fastpersonalcomputer). Major advantages of thetransformation-Q techniqueoverthetechniquesdescribed previously arethat many solutionsinsteadof only one are obtained,that transformersare neverrequiredin the solutions,andthattheprobabilityof finding theoptimumsolutionto a matchingproblemis veryhighwhenthesearchis donethoroughlyenough.Thisapproach is alsovery robustandcanbe extendedeasilyto designmorecomplicatednetworks(like distributedor mixed lumped/distributed networks). The first advantageis importantwhenthe bestsolutionobtainedis not physically realizableor whena differenttopologyis required.The sensitivityto componentchanges will usuallyalsobe differentfor the solutionsobtained. E.4.3.1
Constraints on the Input Admittance of a LosslessImpedanceMatching Network If the Gain Is to Remain Constant at a Specific Frequency
The locus of input admittancesfor which the gain of a losslessimpedance-matching networkwill remainconstantcanbe derivedby usingthe expression
ls"l'=lffir
(8.e1)
--1r,".", l"r - "".It
(8.e2)
1
where S" is the input reflection parameterwith the actual source impedance(Z) as normalizingimpedance,andZin(IiJ is the input impedance(admittance)of the matching network. By substituting
l E l ' =r - G ,
(8.e3)
Yrn=G.rn+jBr,
(8.e4)
and
Yr=G,+jB"
(8.e5)
into (8.92), it follows that the locus of the input admittancefor which the transducerpower
The Design of Wideband lrnpedance-MatchingNetworks
293
gah (Gr) will remain constantis a circle in the admittanceplane.The parameters(center andradius)ofthis circle are
Go+ jBo =f2 I Gr -llc, - jB,
(8.e6)
and
Rvo= 2fl / Gr' - r / Grltt2G,
(8.e7)
The gain of a losslessnetwork will remainconstantfor all valuesof the input admittance that fall on the circumferenceof this circle.This is illustratedin Figure8.28. For all valuesof the input admittancethat fall insidethe constantgain circle,the gain will be higher than that on the circumference.The transducerpower gain of a matchingnetwork, therefore,canbe constrainedto be higherthana specifiedminimum at any particular frequency by limiting its input admittanceto the inside of the relevant constanttransducerpowergaincircle. Because(8.91) and (8.92) are of exactly the sameform, the locus of input impedances for which the gainof a losslessnetworkwill remainconstantis alsoa circle, andthe parameters of this circle canalsobe obtainedfrom (8.96)and(8.97)with G, +74 replacedwithn- + jX",andGo+jBo with,R0+ jxo.T\e resultingequations are
Ro+jXo=I2 / Gr - 1lR"- jX"
Figure 8.28
(8.e8)
The locus of the input admittancefor which the gain of a losslessmatchingnetwork will remain constantat a specifiedfrequency.
294
Design of RF and Microwave Amplifiers and Oscillators
and Rro = 2 f t / G 7
8.4.3.2
- t / G r 1 t / 2R ,
(8.ee)
Extension of the Transformation-Qlmpedance-Matching Technique
In animpedance-matching network,theresistance levelis changedby eachelementexcept the last,which only servesto adjustthe reactance or susceptance level.The changein the resistanceof a four-elementnetwork with the first elementa serieselementand no resonatingsectionsis illustratedin Figure 8.29. The resistanceis transformedin each transformationstepby a factor of the form
(8.l 00)
D,(rtt\=l+Q,2(a\ where
Q,(a) =
X"(or) + X,,(a\ ft,, (co)
(8.101)
Qr-
Qt'
Qn-
a'-
c1
RL w2
R"+jx,
PRin l
l+Qr'
(b) Figure 8.29
(b) Schematicillustrationof the changein the resistancelevel of (a) a matchingnetwork.
Networks The Design of WidebandImpedance-Matching
295
or o - ( c 'o ) -4 ( o ) + B - ( o t ) G,^(a)
(8.102)
or dependingon whetherthe transformationunder considerationis series-to-parallel parallel-to-series, respectively(seeFigure8.30). The factor p" (r,t) is referredto asa transformationQ. In (8.101),X, (ro) is the reactanceof the nth component,X^(a) the effective in serieswith it asillustrated theeffectiveresistance reactance to the right of it, andR, (<,1) in Figure8.30(a). of the nth component,B- (or)is the effective Similarly,B, (cl) is the susceptance in parallelwith it. This to the right of it, andG,,(co)theeffectiveconductance susceptance is illustratedin Figure8.30(b). resistance will alwaystransformtheassociated transformation A series-to-parallel upward,while downwardtransformationsare effectedwith shuntelements. p is positive It followsfrom (8.101)and(8.102)thatthesignof a transformation is inductive,or whenthe in serieswith theeffectiveresistance whentheeffectivereactance is capacitive. in parallelwith the effectiveconductance effectivesusceptance
6oL, yn? =
@&,
Gr(a) +jB,r(a)
zL
(b) Figure 8.30
Definitionof the symbolsin (8.101)and(8.102).
When the first element of an N-elementnetwork is a serieselement,the input by the expression[5; referto Chapter4] is given after(N - 1) transformations resistance
Ri*il= *,##...u+er-,| or
(8.103)
296
Design of RF and Microwave Amplifiers and Oscillators
l+e*-,
Rio,.nu= ^r#
(8.104)
l +A r-,
dependingon whether the last matchingelementis a shuntelementor a serieselement, respectively. When the first elementis a shuntelement,the input conductanceis given by the expression /= = GL l* Qr' Gi*N ftI ffi"'tr+Q'r-,1
(8.10s)
or GL l+Qr' /v=i * N - @ @
(8.l 06)
l+Q2*-z
T.
(8.103)and(8.105),aswell as(8.104)and(8.106),areof thesameform Because exceptthat the resistanceand conductance must be interchanged,it is only necessary to considerthedesignof matchingnetworkswith a serieselementasthefirst element.Exactly the sameprocedurecanthen be followed to designnetworksin which the first elementis a shunt element, after replacing all impedance specifications with the equivalent admittances. In low-passandhigh-passdesignsthe numberof Q-factorsis equalto the number of elementsin the network. In a band-passnetwork with N elements,but M resonating sections(seriesof parallelLC combinations), thenumberof Q-factorsincreases t o (N + M This is illustratedfor a three-element networkin Fieure8.31.
Qt(a)
L1
L3
Qr
cl
Qz=-Qr-
Missingcascade elements
o) Figure 8.31
&
Q o- f
Ln
Q,-lc,
RL
The influence of (a) seriesand (b) parallel resonatingsectionson the hansfonnationQfactors in a matching network.
The Desigr of Wideband Impedance-MatchingNetworks
297
When an element is absent,the associatedp is equal to the negative of the previousp. of By using(8.103)through(8.106),it is very easyto calculatethe input resistance any impedance-matchingnetwork when the p-factors areknown. In orderto designa matchingnetworkto havea specifiedtransducerpowergain(G7) to constrainthe lasttwo p-factorsto ensure at a particularfrequency,it is only necessary thatthe input impedance(if the lastelementis a parallelelement)will fall on the relevant gaincircle as derivedin the previoussection. Whenthe lastelementis a serieselement(seeFigure8.32),thenextto lastp should be constrainedto ensurethat the input resistance(R'") will fall in the range
(8.107)
Rin.rin SR,n 3R,n,,n*
:
Figure 8.32
.;-il
The constraintson the input resistanceof a matchingnetwork wherethe last elementis a serieselementwhenthe transducerpowergain shouldbehigherthanor equalto a specified minimum at a particularfrequency.
The boundson the nextto last Q follow easilyfrom the valuesof the previousQfactorsby using (8.104)in conjunctionwith (8.107).The resultingconstraintsare
ei-,=ok #... [r+Q'*-,:-r
(8.l 08)
and
-,=##... a?,
rr+Q'-,tN|
(8.l 0e)
298
Designof RF and Microwave Amplifien and Oscillators
Whenthe lastelementis a parallelelement,the next to last Q shouldbe constrained to ensurethat the input conductance (Gj will be within the constraintsimposedby the constantgain circle on the admittanceplane. The resulting constraintsare
-,>"1{;etr
#
...tt+etr -,t- |
and
<# Q'*-,
#
"'tr+Q?N -,t-|
( 8 . 1I l )
Theconstraintson thelasttransformationQ canbe derivedfrom Figure8.33.With the resistance(if the lastelementis a serieselement)or conductance (if the lastelementis a parallel element)in range,the reactance or susceptance mustbe suchthat the resulting impedanceof admittancefalls on the circumference of the gain circle. Whenthe lastelementis a serieselement,the reactance is constrained to
' ^'# x,n= xrt&, sinfco,
]
(8.1l2)
The equivalentexpressionwhenthe lastelementis a parallelelementis
o'h;or Bi,= Botnrosin[cor-'
(8.1 r3) ]
Figure 8.33
The constraintson the last hansformationp of a network if the gain is to be higher than a specifiedminimum.
The Desigr of Wideband Impedance-MatchingNetworks
299
Because.R;n or G1nis known,it is a simplematterto calculatethe Q corresponding to thesereactances or susceptances. From a single-frequencymatchingviewpoint, thereareno constraintson the first N - 2 transformationQ. lf the responseis also of interestover a n€urowpassband,any to be smaller beconstrained changein the p-factorsshould,asa first-orderapproximation, than tw'icethe Q requiredfor the circuit. The changein p from onetransformationto the next is given by
LQ=18,+Q*J
(8.1r4)
The reasonfor the positive sign is that the sign of a Q of transformationchangesunder a As an exampleof this, if the Q of a transformation. or parallel-to-series series-to-parallel seriescombinationis +3 (inductive),the Q of the parallelequivalentwill be -3 (again inductive). In summary,the following procedurecan be followed to designan N-element matchingnetwork to match or mismatcha complexload by a specifiedamountto a complex sourceat a particularfrequencyand to havea specified(approximate)quality factor. DesignProcedurc Specifications:
powergain G. Z",transducer Loadimpedance 2., sourcei mpedance (at frequencyf), andquality factor Q.
l.
If the first elementof the network is to be a shunt element,changethe to llZ.and llZt,respectively,andassumethat sourceandloadimpedances the first elementis now a serieselement.
2.
Chooseany values for the first N - 2 transformationQ-factorswithin the constraintthat all the transformationps mustbe smallerthan2Q,that is,
lQ,l
(8.1 Is)
of the If thelastelementis to be a parallelelement,calculatetheparameters constantgaincircle on the admittanceplaneby using(8.96)and(8.97). If the last elementis a serieselement,use (8.98) and (8.99)to plane. ofthe constantgaincircleon theimpedance calculatetheparameters 4.
Calculatethe minimum and maximum allowable values for the input or resistance by usingthe following equations,asapplicable: conductance G,n,.* -- Go + Rro
(8.r16)
Gin,*in=Go-Ryo
( 8 . 17 )
300
Designof RF and Microwave Amplifiers and Oscillators
R.,.*=lto+Rro
(8.1r8)
Rin,*in= Ro-Rro
(8.1le)
5.
Determinethe constraintson the next to last transformationQ by using (8.I 08)and(8.I 09)whenthelastelementis a serieselement.Otherwiseuse (8.1l0) and(8.1I l). Choosea valuefor Q1n-,within theseconstraintsandthat imposed by (8.11s).
6.
When the last elementis a serieselement,calculatethe two possible reactancevaluescorrespondingto the last transformationQ ({n,rin md use(8.1l3) to calculate theallowable {,,.J by using(8.1l2).Otherwise, susceptance values(B,n,r,n andB,n,r",). = l/R,." by using(8.103)or (8.104),as CalculateR'n, or Ginru applicable.Calculatethetwo possiblevaluesfor the lasttransformationQ by usingthe following equations: /\ Yil.mu
r.
_'-rn,mAx
= Q*.^rn '
(in,lv
(8.120)
v- 'n'm fiin,,v
(8.121)
8,n,.*
(8.r22)
or Q*,^o
G,n,t
Qy,^in=*
.
7.
(8.123)
Chooseeither of the t'wo possibleQ valueswithin the constraint imposedby (8.1l5) on eachtransformationp.
Calculatethe elementvaluescorresponding to the set of p values.The reactanceor susceptance ofeach componentat the frequencywherethe p valuesarecalculatedis givenby an expressionofthe form
x,=(Qn+Q,-)Ror
(8.r24)
The Designof WidebandImpedance-Matching Networks
B,=(Q,+Qn-)G^
301 (8.125)
dependingon whetherit is a seriesor parallelelement. In theseequationsR,, andG,,aretheeffectiveseriesresistance and to theright of thecomponentwhosevalueis effectiveparallelconductance to be determined,respectively(referto Figure8.30,if necessary). 8.
Ifthe first elementof thefinal networkshouldbe a shuntelement,consider all inductorsto be capacitors(i.e., 5 pH is 5 pF) and all capacitorsto be to the actualnetwork. inductors,andassignthesevaluesin sequence As an illustrationof this step,if elementvaluesof 3 pH (series element),9 nF (shuntelement),and7 pF (serieselement)wereobtainedby following the procedureoutlinedabove,the elementvaluesin the final circuit are 3 pF (shuntelement),9 nH (serieselement),and 7 pH (shunt element),respectively.
EXAMPLE 8.9
A transformation-pexample.
As an exampleof the application of the procedwe outlined above,considerthe designof a five-elementmatchingnetworkwith the first elementa serieselement, no resonatingsections,andthe following specifications: Zt=50+i50O
z, --20- j20a Gr = 0'89 fo =lDDlvtrIz Q=5
l.
et=5 (Le=4) Qz=5(49=10) Qt=-3
7.
(LQ=2\
With no resonatingsectionsand specificationsas above,the last element of the network will be a serieselement,and thereforethe constantgain planeis of interest.Applicationof (8.98)and circleon theinput impedance (8.99)yieldsthat the parameters of this circle are
Design of RF and Microwave Amplifiers and Oscillators
F
+i20.00O Ro+ jX o =12| O.8g-ll20 + i20= 24.94 and Rzo: 2Jtl 0.892- 1/ 0.892o = 14.91
3.
R'n,** =24'94 +14'91=39'85O Rin.-in=24'94 -14'91= l0'03O
Applicationof (8.108)and(8.109)yieldsthat
P'l'[##tt
-']"'='''n' +32)
P'l=[##tt
-']"'=u''*n + 32)
Qn-r=5(LQ=z) 5.
The two allowable values for the input reactanceare found by using (8.1l2):
t9.zt-24.941 "f -, --'-J + t4.9lsinl cos-' Xin=20 : :-: | 14.91 L I = 6.23{2:33.77 O wasfoundby applying(8.103): above,.R,n In theequation
Fq
t.,
TI
ca ca ap
<19.23< 39.85) Ri"= 19.230 (10.03 Thelasttransformation Q is simply Qn= 6.23I 19.23- 0.32 (LQ = 5'32)
M
as
frc X r = Q ,R r -X r = 5 ( 5 0 ) - 5 0 = 2 0 0 f () 3 1 8 .n3H )
thr un
303
The Design of Wideband Impedance-MatchingNetworks
yz=[ea+e]GLz= ##h
=7.6ems (tz.zapF)
Xt =lQt + QzlR rt=2(50)g
= l00O (1592nH)
Yq=fQ++ Q)GL+=,**L+#
= 4.0mS(6.37 pF)
1t xs =ler+eolRts= 5.3215sy
#=
(162.8nH) 102.3c)
The designednetwork is shownin Figure 8.34.The gain at l00lWHz is equalto 0.889andthe p of the circuit is equalto 6.0.
79.6pF
l62.8nH
l59.2nH
6.37pF
Figure 8.34
t.4.3.3
3l 8 . 3 n H
79.6nH
t2.24pF
50rl
The network synthesizedin Example8.9.
Optimization of the Transformation Qs of a Network
ThetransformationQ-factorscorrespondingto an initial solution for a matchingproblem canbeoptimizedby usinga linearleast-square optimizationroutine.Althoughgoodresults canbe obtainedby doing this, betterresultsare obtainablethrougha slightly different approach. The first improvementis to usethe maximumrelative deviationMRD)
MRD= pq"l c.ttl lGt-"',(t)
-,;
(8.126)
|
asthe error criterion insteadof the mean-squareerror. The main advantageof the MRD error criterionis that the maximumdeviation from the optimumratherthanthe averagedeviationwill be minimized. Becauseof this, the solutionwith the lowest insertionlosswill be obtainedwhenthe ideal gain is setto unity.
304
]
Design of RF and Microwave Amplifiers and Oscillators
The secondimprovement is to optimize the error by using the steepest-decent method.The resultsobtainedin doing this were superiorto thosecorrespondingto the leas-squaremethod. The gradient vector required for optimizing the Q values can be determinedby calorlating the changein MRD correspondingto a small incrementin eachQ. Thenew setof p values(p1s)canbe obtainedfrom thepreviousset(p"- ,) andthe currentMRD by usingthe equation
aMRDI AO,l Qn =Qtt-t-
(aMRDt OQ)' +...+(aMRDt OQi'
.
I l
(8.r27)
I aMRDt AQu)
wheretheoptimum valueof c canbedeterrrinediteratively by usingthe followingmethod
u8l.
Startwith a smallvalueof a (a,), andincreaseit duringsubsequent iterations(a) by usingthe expression c r r = o , [ + l + 1 2 + . . ./ ' - t ]
i = 1 , 2 , 3 , . . .l = 1 . 5
(8.128)
until the errorvalueincreases. This will resultin the situationdepictedin Figure8.35.A quadraticcurvecannow be fitted to the lastthreecoordinates, andthevalueofa (c.), for which the error will be a minimum,canbe estimatedby usingthe expression
MRD E*z fQa'
E*t
&n-2 dn-2
Figure 8.35
dn-l
dn
&n
Estimationof the optimum scalefactor in optimizing the MRD.
The Design of Wideband Impedance-MatchingNetworks
305
I o, =, +[cr,_r2-cr,,_,2]Ilm,D, [cl,-,2-ant]l\m,D,-, +[c,t -cr,_r2]Nm,Dn_, $.12g\ , [ c r ,- c r , _ r ] M R D ' t - l f a n _ z - a , _ , ] M R D , ' [ c t , _-rc l , ] M R D n _ + The actualvalueof theMRD at crcan now be calculated.Dependingon which of the four errors is now the largest,one of the four coordinatescan be eliminated and the procedurecanbe repeatedon the remainingthreepoints. The optimum value of s canbe determinedby continuingwith this procedureuntil the improvementin the errorvalueis negligible. Excellentresultswere obtainedby optimizingthe Q valuesof transformationas outlinedabove.
8.4.3.4
An Algorithm for the Design of Impedance-Matching Networks by Using the Transformation Q-Factors of the Network
A procedurefor designinga networkto matcha complexloadto a complexsourcewith a specifiedgain at a specifiedfrequencywas outlined in Section8.4.3.2.By taking the transducerpower gain to be the minimum expectedgain at the frequencywherethe e valuesare evaluated(usuallythe highestfrequencyin the passbandor the frequencyat which the gain requiredis a maximum),this narrowbandtechniqueforms the basisof an excellentapproachto solvingwidebandimpedance-matching problems. It wasshownthatthe first (N - 2) Q valuesin thesingle-frequency designcantake arbitraryvaluesandthe constraintsimposedon the lasttwo Q valueswerederived.Since therangeof possibletransformation-Q valuesis limited in a widebanddesign,it is feasible to do a systematicsearchon thesep valuesin orderto find solutionsthatyield goodresults over the whole passband.In this way, the dependenceon a good initial solution is eliminated. When the searchis completed,a numberof the best resultsobtainedcan be optimizedas describedin Section8.4.3.3.If the searchwasdonethoroughlyenough,the optimum solution to any matchingproblem will be obtained.A further advantageis that the local minima correspondingto other initial solutionswill also be obtained,and, consequently,a large choice betweennetworks with different element values and topologiesexist. An ideaof therequiredrangeof p valuescanbeobtainedfrom thedesiredQ of the networkwhen applicable,the maximumQ of the load and sourceimpedances, and the analyticallyderivedconstraintson simplereactiveloadsassummarizedin Table8.5. As a rule, a minimum valueof -4.2 anda maximumvalueof 4.2 yield excellent results.When someof the Q valuesof solutionsobtainedexceedthesevaluesand the optimumsolutionis required,theboundsmustbeextended.This will seldombenecessary whena widebandnetworkis designed.Incrementvaluesin the rangefrom 0.4 to 0.6 are used.
306
Design of RF and Microwave Amplifiers and Oscillators
The algorithmdescribedbelowcanbe usedwhenthis approachis followed.
Algorithm l.
Decideon the numberof elementsandthe frequencyat which the Q values areto be evaluated(/o). Estimatethe range of possibleQ valuesof transformationand specifythe incrementalvalueto be used. Estimatethe minimum gainexpectedat/q. setsto be storedduringthe Specifuthenumberof transformation-Q search(.11,1).
2.
Generatean allowableset of Q valuesby using the theory outlined in Section8.4.3.2.
3.
Synthesizethe equivalentnetwork and calculatethe gain enor (MRD). Comparethe results with the previous results obtainedand store the solutionif it is betterthantheM ,be$.solutionspreviouslystored.
4.
Optimizethe bestresultsobtainedin the searchas describedin Section 8.4.3.3.
EXAMPLE8.l0
Adouble-matchingproblemsolvedwiththetransformationQ technique[5].
As an exampleof the resultsobtainablewith the transformation-ptechnique, problemof Example8.8. considerthe double-matching With the gain setequalto the specifiedvalueof 0.818at 200 MHz during the systematicsearch,and using minimum, increment,and maximumvaluesof -4.4, 0.4, and4.4, respectively,for the transformation-Qvalues,the maximum for wasfoundto be 0.05dB (MRD : 1.23o/o) deviationfrom the specifiedresponse the best four-elementsolution synthesized.This network is shown in Figure 8.36(a). to this solutionne 2.399,'1.797, 1.039,and Tt e Q valuescorresponding -0.148,respectively. The secondbestsolutionobtainedis the networkshownin Figure8.36(b). The maximum deviation from the specified gain response is 0.06 dB (MRD : l.43yo),and the Q valuesare -0.777,3.422,2.360,and -0.130, respectively. The maximumdeviationsfor the othersolutionsshownin Figure8.36are dB (MRD :3.02Yo),and dB (MRD=2.58Yo),0.13 0.09dB (MRD :2.08Yo),0.11 =3.41o/o), respectively. 0.15dB (MRD
The Design of WidebandImpedance-Matching Networks
3.55pF
z"
307
0.l3pH
zL
0.24ytH (a) 0.I 8pH
0 . 1l p H
z" {
un'or-nr*"
zL
(b) 0.I 7pH
z,
6.07pF
3.01pF
zL
0.| 6uH
(c) ( ??nE'
0.2lpH
z"
0.84pH
0.3OpH
zL
(d) 3trH
z"
0.l4pH
l0.3pF
zL
l5.2pF
(e) 3.33pF
z,
0.271tH
0 . 1t p H
5.45pF
zL
(0 Figure 8.36
Some of the solutions obtainedwith the transformation-B technique for the matching problem of Example 8.8.
The best three-elementsolution obtainedunder the constraintthat the topology must be of high-passform is shown in Figure 8.37. The maximum deviationfrom the specifiedgainis 0.15dB (MRD : 3.4%) andthe Qvaluesare -0.185,-0.570,and-0.931,respectively. Thissolutionis basicallythe sameasthat obtainedwith the reflectionparametertechnique. Having severalsolutionsto choosefrom is an advantageboth from the viewpoint of topologyandsensitivity.
r#
308
Designof RF and Microwave Amplifiers and Oscillators
Figure 8.37
;
The bestthree-elementsolution (high-passtopology) obtainedwith the transformation-Q techniquefor the matchingproblemof Example8.8.
It is importantto notethatthe solutionwith the bestperformancewith the designvaluesmay not havethe bestworst-caseperformancetoo. It alsodoesnot necessarilyfollow that fewerelementswill be betterfrom a sensitivityviewpoint. In this case,the MRD of the bestfour-elementsolutionfound is increased is assumedto ftoml.23o/oto4,12o/o if the tolerancein all the lumpedcomponents be loh. A l% changein the componentvalues,therefore,leadsto a2.9o/oincrease in the MRD in this case. from 3.4olo to5.5Yo TheMRD ofthethree-element solutionshownincreases with 1% tolerancesin the componentvalues. The solutionsshownso far arepurelylumped.In practice,solderpadsare approachcanbe alsorequiredfor the lumpedcomponents. Thetransformation-Q extended to allow for this requirementtoo. A few of the best mixed lumped/distributed solutionsobtained[15] areshownin Figure8.38with thepads used.The padsdo not havea stronginfluencein this example,but will becomea factorasthe frequencyis increased. The influenceofthe padsis alsomoresevere whenthe dielectricconstantof the substrateis high. The MRD of the bestsolutionis 1.09%.This valueis increasedto 4.32o/o when l% tolerancesareassumedfor the lumpedcomponents. The MRD valuesfor the otherthreesolutionsshownin Figure 8.38 are Note that the 1.29%(4.29%),3.0% (5.67%),and 3.18%(6.78%),respectively. worst-caseperformance ofthe secondsolutionis fractionallybetterthanthatofthe first solution(4.29%versus4.32%). The electricalparameters ofthe padsusedfor the shuntinductorsare57O and0.38"(at}.2GHz),whilethoseusedfortheshuntcapacitorpadsare36.40and while 0.38".57Opads(0.23'or 0.36' long)werealsousedfortheseriesinductors, pads (0.23" 71.2Q or 0.36' long)wereusedfor theseriescapacitors. wasused(e,: 2.99; h : 0.381mm). The A low dielectricconstantsubstrate width of thesepadson the substrateusedare 0.75 mm, 1.5 mm, 0.75 mm, and 0.5mm,respectively. Note that the length usedfor the seriespadsshouldbe long enoughto ensuresufficient separationbetweenthe shuntcomponents.This is essentialto preventoverlap in the artwork and coupling betweenthe shuntcomponents.
309
The Design of Wideband Impedance-MatchingNetworks
0.l9FH
z,
o
0.llrH
tl4.8lnH
6.l9pF
ZL
tr F
Tl-l
H
(a)
90.59nH
z"
0.llpH
3.8lpF
0.22yH
zr.
nErtr]il
E (b) 3.l7pF
z"
0.28lrH
r6t D
0.l2pH
4.4spF
ZL
4ffi-
(c)
0.l7pH
g
6.03pF
"ti= __!
z"
zL
0.I ?pH
2.49pF
E
pl (d)
Figure E.3E
The four best mixed lumped/distributedsolutions obtained with the transformation-Q techniquefor the matchingproblem of Example8.8.
EXAMPLE 8.11
Matchinga 25O sourceto a l00O load (2-6 GHz).
A load resistance of l00Q will betransformedto 25Ooverthe passband 2-6 GHz in this example.The solutionsweresynthesized for a l0 mil microstripsubstrate with e,: 9.8by using[5]. The best solutionsto a purely resistivematchingproblem are usually obtainedwith a commensurate distributednetwork. Whenthebandwidthrequired
310
Design of RF and Microwave Amplifiers and Oscillators
is large, the line length should be 90" long at the center frequency(arithmetic mean). The commensurate solutionshownin Figure 8.39(a)was synthesizsdby settingtheline lengthequalto 90" at4 GHz. This solutionwasobtainedby setting the searchrangefor the transformationQs to the intervalf-1.2, 1.2],with a step sizeof 0.1. Note that a25Qpadwasusedto completethe inputjunction (thefirst wasa shuntelement). elementof the networksynthesized The input VSWR of this solutionis betterthan 1.06over the passband (lstr | < -30.58 dB; MRD = 0.08o/o, MRD* = 0.35o/o). Thesizeof this solutionis 11.63mm by 6.41mm. The noncommensurate solutionshownin Figure8.390) was obtainedby impedance ofthemain-linesectionsandtheshort-circuited settingthecharacteristic 73O. The characteristic impedance usedfor the open-endedstubswas stubsto l.5,withsteps 33.50.ThesearchrangeforthetransformationQswasfrom-l.5to of 0.25. The l00O line on the output side was usedto completethe junction with the open-ended stub. associated The input VSWR of this solutionis betterthan 1.25(lsttl < -19.16 dB; MRD : l.2yo,}dRD* = l.78Yo).The outlinesizeof this solutionis 6.88 mm by 4.133mm (a gapof 0.1 rnm wasusedfor the seriescapacitor). Thebestsix-elementlumped-element solutionobtainedis shownin Figure 8.39(c).The input VSWR of this solutionis betterthan l.l9 over the passband (lsttl < -2736 dB,MRD:0.71Yo,MRD*":1.24%).
73.00
0.77nH
F'igure 8.39
l.93pF
73.0Q
r00()
2.72nH
(six elements),and (c) The best (a) commensurate(four elements),(b) noncommensurate lumped (six elements)solutions obtained for matching a 25Q sourceto a l00O load (2-6 GHz).
311
The Design of Wideband Impedance-MatchingNetworks
8.5 THE DESIGNOF RLC IMPEDANCE-MATCHING NETWORI(S RLC impedance-matchingnetworksare often usedto compensatefor the decreasein the This canusuallybe donewithout gainof the transistorsusedwith increasingfrequencies. of thesenetworksover losslessnetworks. reactivemismatching,which is an advantage
Rr
& R2
&
vL
RL
(a) Figure 8.40
Rr
YL
R2
RL
(b)
Impedance matching with resistors.
RLC networksareusuallydesignedby usinga computeroptimizationprogramon a circuitwith a suitabletopology,afterinitial valueshavebeenassignedto its components. networkhavetwo functions:they The resistorsin an RLC impedance-matching provide the requiredattenuationat the lowest frequencyin the passband,and they match the load impedanceto the sourceimpedanceat this frequency.A minimumof one series andoneparallelresistorarerequiredin orderto do this. When only one seriesand one parallel resistorare used,initial valuescan be assignedto them by usingthe following setof equations: A2=(E /V,\2 =
oi['
4ft" GrRt
erz>f+ot[' ^)+c,lzc,
Gz=Ga #t
(8.130) (8.3 1l ) (8.132)
whereG. is the requiredtransducerpowergainat the lowestfrequencyin the passband, G;l/Rr, G2:llRr,and Ginis the requiredinput admittanceof the matchingnetworkat the lowestfrequency(if a perfectmatchis required,Gr": GJ. Equations(8.130)to (8.132)applyto Figure8.a0(a).The equationsrelevantto Figure8.40(b)canbe obtainedby replacingG. with G1.andG, with G in theseequations. In order to minimizethe insertionlossat the higherfrequenciesin the passband,
312
Design of RF and Microwave Amplifiers and Oscillators
the resistorsin an RLC network should be usedin parallel with capacitorsand in series with inductors,dependingon whetherthey are used in a seriesor a parallel branch, respectively. Apart from reducingthe insertionloss,the capacitorsand inductorsusedin the networkalsoserveto matchthe loadto the sourceat the higherfrequencies. The networkshownin Figure8.41is a typicalexampleof an RLC network.Note that the elementsthat arenot combinedwith resistorsareusedaslow-passelements. Initial valuescan be assignedto the losslesselementsof the networkchosenby consideringthe diflerentelementsto bepartof independent L-, T-, andPl-sections.As an exampleof this, C2andZ, in thenetworkshownin Figure8.41form a low-passL-section thatshouldbedesignedto matchtheloadto thesourceat thehighestfrequencyin thepassband.Zt andC' shouldbedesignedto ensurethattheinsertionlossat thehighestfrequency will be aslow aspossible.
&
L2 p ,,2
Zrn -
c2
L---^J ' Rl
L1
Figure 8,41
An exampleof an RLC impedance-matching network.
An altemativeway of assigninginitial valuesto thelosslesscomponents of anRLC networkis to follow theiterativeapproaches outlinedearlierfor designinga losslessbandpassnetwork that will match the sourceto the load at the intermediateand higher frequenciesin the passband. With initial values assignedto the losslesscomponentsand the resistors,an optimizationprogramcanbe usedto optimizethe network.
EXAMPLE 8.12
Exampleof a RLC matchingnetwork.
Theuseof (8.130)to (8.132)will be illustrated by applyingthemto thefollowing problem: Rr,= 7'500 R, = 6'254 Gr = 0'19
4' = 6'250
3r3
The Design of Wideband Impedance-MatchingNetworks
!_:
4rc.25\ 4R '--' =__s_1 -alo RrG,
0.19(7.5)
r z-q:g tz1 2 -4.191 0 C?1Z.OO t.r-r--------== - a6,12'L 7 .5 ) 7 .5 ' G r = 0 . 1 2 1 8 ;0- . 1 3 3 4 S G' " =
0.1218 | = 0.0965 6.25 7 . s [ 0 . 1 2+1r 8 /751
The initial valuesof the resistorsaretherefore Rr : 8.2OandRr = 10.4O
EXAMPLE 8.13
Matchingnetworksfor an HF poweramplifier
In this examplea 5-20 MHz poweramplifierwill be designedwith the Motorola MRF406 (20W peakenvelopepower (PEP))by using [1]. The operatingpower RLC networkon the input side gain will first be leveledby usingan series-shunt of the transistor.Mixed lumped/distributedmatching networks will then be designedto maximizetheoutputpowerandto providea goodinputmatchoverthe passband. thattheterminationsweretransformedto 12.5Owith It will be assumed transmission-line transformers. The load impedancerequiredto maximize the output power and the andoperatingpowergainareprovidedin thedatasheet associated input impedance valuesareshownin Table8.14. The estimated for the transistor. in Table to convert the impedanceandgainspecifications It is convenient coefficient input reflection The 8.14to anequivalentsetof unilateralS-parameters. to theinput impedance of thetransistorwhenthe optimum is chosento correspond Table8.14 The optimum load impedanceof the MM406, with the associatedoperatingpower gain and input impedance Frequency (MHz)
2.O 5.0 10.0 r 5.0 20.0 30.0
Input impedance
Load impedance
(o)
Operatingpower gain (dB)
7.s- j2.6 5.2- j2.4 3.r - jr.9 2.3- jl.75 r . 7- j l . 7 1 . 0- j l . 0
8.314- j4.263 6.212- j4.9r4 4.911- j4.476 4.47r- j4.028 4.272- j3.536 3.484- 12.445
20.93 20.14 18.44 r6.99 16.01 14.30
(o)
314
Dcsigr of RF and Microwave Amplifiers and Oscillators
load is in place, while the output impedanceis taken to be the conjugateof the optimum load impedance.The forwardtransmissionparameter(sr,) is setto the valuerequiredtoensurethatthemaximumavailablegain(MAG) ofthis equivalent transistoris thesameastheoperatingpowergainofthe transistorwith theoptimum loadin place.TheequivalentS-parameters for theMM406 areshownin Table8.I 5 in polar format(magnitudeandangle).
Table 8.15 The equivalentsetof S-parameters of the MM406 Frequency (MHz)
2.0 3.0 5.0 7.0 10.0 15.0 20.0 30.0
szr
Jrt
Jrz
(-; ")
(-;') 0.7398 186.1 0.7661 185.9 0.8120 185.6 0.8473 r85.1 0.8834 184.4 0.9t21 184.0 0.9343 183.9 0.9608 182.3
5.224 4.597 3.707 2.907 2.238 1.588 1.210 0.709
327.6 316.7 299.1 284.5 264.7 234.1 206.0 148.7
szz
(-; ")
(-;') 0.00010.0 0.00010.0 0.000r 0.0 0.00010.0 0.00010.0 0.000r 0.0 0.00010.0 0.00010.0
0.7167 0.7448 0.7809 0.8045 0.8205 0.8368 0.8433 0.8700
190.0 191.0 t91.4 190.9 190.3 189.3 188.1 185.6
With the equivalentS-parameters in place, the original targetscan be realizedby levelingthe MAG of the equivalenttransistor(lossysectionson the input side;no feedbackallowed)anddesigningmatchingnetworksto minimizethe input andthe outputVSWRs. The RLC networkdesignedto level the MAG of the equivalenttransistor is shownin Figure8.42.With thisnetworkin place,theMAG variesbetween10.94
36.2O44.lnH36.2Q36.2Q 36.20 12.6036.2036.2Q 0.11"
0.11. 0.08"0.1r"
0 . 1 1 " 0 . 0 8 "0 . 2 3 . 1 8 . 2 00 . 2 3 " 0 . 2 3 "
r-t;_llil
tr
36.2')26.7Q
I
tr tf E E Figure E.42
The RLC network usedto level the operatingpower gain of the MM406.
31s
The Design of Wideband Impedance-MatchingNetworks
25.0036.208.86nF36.2O
0.66nF36.2O 36.2Q66.7nH36.2Q
ff--{
0.3s' 0.15"
-. ^^ 0.15"0.15'
0.t5"
JO.Z\I
3 6 . 2 00 . 1 5 " 0 . 2 3 '
l-ol tl
t-t
m HLJ tl
LAJ Figure 8.43
The mixed lumped/dishibutedinput matchingnetwork synthesizedfor the MRF406.
and 11.14dB, and the input VSWR (relativeto the 12.5Osouncetermination) variesbetween1.61and3.06. Levelingthe MAG of the equivalenttransistorcorresponds to levelingthe operating power gain of the actual transistor terminated in the optimum load impedance. The mixed lumped/distributedinput matchingnetwork is shownin Figure 8.43(the electricalline lengthsarespecifiedat2}MHz). The input VSWR varies between1.24and1.46overthepassband (1s,,| < - 14.57dB).
0 . l 5 p H3 6 ,
36.2Q 2.1nF 36,
0,19"0.19" 36.20
0.291tH36.2Q36.2Q71.1nH36.2Q
0.19"
0 .l 9 '
0.r9" 0.19"
0.19'
0.44nF36.2e
36.2n
0.19' 0.35" 36.2Q
0 .l 9 '
0.19"
FI E
n
| [r]]l LI
E-
I
E LAJ Figure 8.44
The mixed lumped/distributedload network synthesizedfor the MM406.
316
Design of RF and Microwave Amplifiers and Oscillators
The load network synthesized(five-elementnetwork) to matchthe output of the equivalenttransistor(maximizethe outputpower of the actualtransistor)is shown in Figure 8.44 (the electricalline lengthsare specifiedat 25 MHz). T\e output VSWR in the equivalentcircuit variesbetween1.16 and 1.32 over the passband (ls,,l < -17.03dB). The artwork of the amplifier designedis shownin Figure 8.45 (the gap spacingsmustbe adjustedto accommodate the lumpedcomponents to be used). The final responseassociated with the equivalenttransistoris shown in Figure8.46andtabulatedin Table8.16.
g
t-ol tf
F , ,
|-hrrr I
PLa-r-rJ
LI
H
tr E
[h --E
fi
E
--t -
ft-
|
E
LAj
lej
E
l--l
LAJ FEg,c 8.45
The artwork of the amplifier synthesized.
Roi: M:
Figure 8.46
12.$ 12.S
The final responseassociatedwiththe equivalenttransistor. (Notethatthehighestfrequency on eachtraceis not marked;the Smith Chartshouldbe viewed asa polar plot only when the s2rtrace is interpreted.)
317
The Desigr of Wideband Impedance-MatchingNetworks
Table 8.16 The S-parameters ofthe final circuit (equivalenttransistor) Frequency (MHz)
2.0 5.0 6.0 7.0 8.0 9.0 10.0 I 1.0 t2.0 13.0 14.0 15.0 16.0 t7.o 18.0 19.0 20.0 25.O 30.0
Jrz
stt
(dB)
(")
-3.72 2r7.0 -14.57 r68.0 -16.67 174.8 -t7.48 185.6 - l7 .23 t94.4 - 16.51 199.3 -15.76 200.8 - 15.l8 200.2 -t4.78 t98.2 - 14.58 195.2 - 14.58 191.7 - 14.79 187.5 -t5.12 182.9 - 15.68 177.4 -16.52 170.9 -17.75 162.4 - t9.49 150.4 -14.59 0.2 -4.19 306.9
(dB)
szz
szt
(')
-110.56 10.0 -'76.96 237.0 -74.66 207.8 -72.59 l89.3 -70.65 r'75.2 -68.87 163.0 -67.25 152.0 -65.75 142.0 -64.37 132.5 - 63.10 123.4 -61.92 114.6 -60.8r 106.0 -59.74 97.6 -58.7r 89.2 -57.72 80.7 -s6.78 72.1 - 55.86 63.2 - 51.95 12.2 -50.45 306.6
(dB)
(')
- 1 6 . 6 43 1 1 . 6 10.91 120.0 t0.97 77.2 10.84 47.r 10.85 22.5 10.91 0.9 I1.00 341.3 I 1.00 323.0 10.98 305.7 10.95 289.3 10.93 273.4 10.91 25'1.9 10.88 243.0 10.87 228.2 10.87 213.5 10.88 198.8 10.9r 183.9 t0.80 r03.7 7.90 8.2
(dB)
(")
-0.01 133.9 -r7.16 336.3 -20.15 133.2 - 17.t4 t 00.3 - 18.16 72.5 -20.40 41.8 -22.41 3.9 -22.28 320.1 -20.64 288.6 -19.04 266.8 -r7.91 250.5 -t'l.24 237.0 - 17.03 224.8 - t7.17 213.2 -17.64 201.2 -18.45 188.1 - 19.58 172.4 -t7.14 44.1 -8.24 352.0
REFERENCES "Theoretical l. Fano, R. M., Limitations on the BroadbandMatching of Arbitrary Impedances,"Journal of the Franklin Inst.,Yol.249, January/February1950, pp.57-83,139-154. "A 2. Youla, D. C., New Theoryof Broad-BandMatching,"IEEE Trans.Circuils Syst, Vol. CT-l1, March1964,pp.30-50. "A New Approachto Gain-BandwidthProblems,"IEEE Trans.Circuits 3. Carlin,H. J., Sys/.,Vol. CAS-24,April1977,pp.170-175. "Calculator programfinds Fanobandwidth,"Microwaves& RF, September 4.Lev, J., 1985. "General 5. Chen,W., andC. Satyanarayana, Theoryof BroadbandMatching," IEE Proc. C (GB),Yol.l29,No. 3, June1982,pp. 96-102. "The 6. Carlin, H. J., and B. S. Yarman, DoubleMatchingProblem:Analytic and Real FrequencySolutions,"IEEE Trans.Circuits d!sr., Vol. CAS-30,No. 1, January 1983,pp. 15-27.
318
Designof RF and Microwave Amplifiers and Oscillators
"A 7. Yarman,B. S., and H. J. Carlin, Simplified'Real-Frequency'Technique Applied to Broad-BandMultistageMicrowaveAmplifiers,"IEEE Trans.MicrowoveTheory Tech.,Yol.MTT-30,No. 12,December 1982,pp.2216-2222. "On 8. Carlin, H. J., and P. Amstutz, Optimum Broad-BandMatching," IEEE Trans. CircuitsSys/.,Vol. CAS-28,No. 5, May 1981,pp. 401-405. 9. Bode H. W., Network Analysisand FeedbackAmplifier Design,New York: Van Nostrand,1945,pp.205-207, 3 19. 10. Chen,W. K., The Theoryand Designof BroadbandMatchingNetworks,London: Pergamon Press,1976. "Explicit I l. Levy, R., Formulasfor ChebyshevImpedance-Matching Networks,Filters andInterstages," Proc.IEEE,Yol. I I l, No. 6, June1964. 12. Richards,P. I., pp.2l7-2r9
"Resistor-Transmission-Line Circuits,"Proc.1R4 February1948,
13.BaherH., Synthesisof ElectricalNetworks,NewYork: JohnWiley & Sons,1984. 14. Yarman,B. 5., Broad-BandMatching a ComplexGeneratorto a ComplexLoad, DoctoralDissertation,Comell University,1982,pp. ll7-120. 15. MultiMatch RF and Microwave Impedance-Matching,Amplifier and Oscillator Synthesis West,RSA:Ampsa(PTY)Ltd; http://www.ampsa.com, Sofiware,Somerset 1998. "A 16.Carlin, H. J., andJ. J. Komiak, New Methodof BroadbandEqualizationApplied to MicrowaveAmplifiers," IEEE Trqns.MicrowqveTheoryTech.,Vol.MTT-27,No. 2, February1979,pp.93-99. 17. WatcomFortran 77(Iser's Guide,Ontario, Canada:WatcomIntemationalCorporation; http://www.watcom.on.ca, I 996. 18.Ha,T.T.,TheDesignofMicrowaveSolidStateAmplifiers,NewYork:JohnWileyand Sons,1981,pp. 178-179.
SELECTEDBIBLIOGRAPHY Abrie, P. L. D., Impedance Matching Networlu and Bandwidth Limitations of Class B Power Amplifiers in the HF and VHF Ranges, Master's Thesis, University of Pretoria, November I 982.
I
I
The Designof WidebandImpedance-Matching Networks
319
Fletcher, R., and M. J. D. Powell, "A Rapidly ConvergentDescent Method for Minimization,"ComputerJ, Vol. 6, 1963,pp. I 63- I 68. "Optimum Ku, W. H., and W. C. Peterson, Gain-BandwidthLimitations of Transistor Amplifiers as ReactivelyConstrainedTwo-PortNetworks," IEEE Trqns.Circuits Sys/.,Vol. CAS-22,June1975,pp. 523-533. "Computer-Aided Liu, L. C. T., and W. H. Ku, Synthesisof LumpedLossy Matching Networks for Monolithic MicrowaveIntegratedCircuits (MMICs)," IEEE Trans. MicrowqveTheoryTech.,Yol.MTT-32,No. 3, March 1984,pp. 282-290. "Tables Matthaei,G. L., of ChebyshevImpedanceTransformingNetworksof Low-pass Filter Form," Proc. IEEE, August1964. "Synthesis Mellor, D. J., and J. G., Linvill, of InterstageNetworksof PrescribedGain versusFrequencySlopes,"IEEE Trans.MicrowaveTheoryTech.,Yol.MTT-23,No. 12.December 1975. "Tables Pitzalis, O., and R. A. Gilson, of ImpedanceMatching Networks which Approximate PrescribedAttenuation versus FrequencySlopes," IEEE Trans. MicrowaveTheoryTech.,Yol.MTT-19,No. 14,April 1971. "Impedance Schoeffler,J. D., TransformationUsing LosslessNetworks," IRE Trans. Circuit Theory,CT-8,June1961,pp. l3l-137. Van Valkenburg, M. 8., Introduction to Modern Network Synthesis,New York: John Wiley andSons,1960.
CHAPTER 9 MICROWAVE LUMPED ELEMENTS, DISTRIBUTED EQUIVALENTS, AND MICROSTRIP PARASITICS 9.1 INTRODUCTION Impedance-matching networkscanbe realizedin lumpedform aslong asthe dimensions at the highest of the componentsused are small comparedto a quarter-wavelength frequencyof interest. The different types of inductorsand capacitorsusedat microwave will beconsidered in this chapter.Thedesignof microwaveinductorswill also frequencies be considered. When the dimensionsare on the order of lll2 of a wavelength,the phaseshift associatedwith the componentcan causea significant deviation from the expected characteristic impedanceis too low for inductors response. Furthermore,if the associated and too high for capacitors,the responsewill be degradedeven more by the parasitic Because it is oftenaproblemwhenhighimpedance respectively. capacitance or inductance, circuits are designed,the bounds imposedby the phaseshift acrossand the finite characteristic impedanceof practicalinductorswill be examinedin this chapter.In order to do this, the transformingpropertiesof a seriestransmissionline will be examinedfirst. cannotberealizedwith negligiblephaseshift andparasitics, Whenthecomponents matchingnetworksshouldbe realizedin distributedform whenpossible.Fabricationof or asmicrowaveintegrated distributednetworksusingmicrostrip,on thin-film substrates circuits(MICs), is relativelyeasy.The designeffort involvedis alsomuchlessthanthat requiredwhen lumped elementsare usedandthe parasiticscannotbe ignored. Excellent equationsfor the characteristicimpedanceand effective dielectric constantof microstriplineshavebeendevelopedby themanyworkersin thefield [] and discontinuitiessuch werereviewedin Chapter3. In MIC/IvIMIClayouts,transmission-line are and cross-junctions in width, right-angle bends, T-jwrctions asopen-ends, steps BapS, the effect ofthese incorporate it becomes necessary to At higher frequencies encountered. discontinuitiesinto designsin orderto obtaingoodresults.Themagnitudeof theseeffects herealongwith a compensation atthelower microwavefrequencies[2] will beconsidered technique[3]. designsareoftentransformedinto distributeddesigns Prototypelumped-element 321
322
Design of RF and Microwave Amplifiers and Oscillators
by replacingthe inductorsandcapacitorsin thenetworkwith shortedstubs,open-circuited stubs,andcascadesectionsoftransmissionline. Therangeofseriesandshuntreactances, which can be transformedwith negligibleerror, will be examinedhere.It will also be shownthatsignificantlybetterresultscanbeobtainedby replacinglow-passT-sectionsand Pl-sectionswith sectionsof seriestransmissionlines.
9.2
MICROWAVERESISTORS
Thin-film techniquesareoften usedto manufactureresistorsat microwavefrequencies.By keepingthe dimensionsof a resistorsmall,the associated capacitance andinductancecan be minimized.Thecapacitance canbereducedfurtherby depositingthethin film on a low dielectric-constant substrate. A thin-film resistorcanbe characterized asa lossytransmissionline. Therelevant equationswereconsideredin Chapter7. Thin films with resistances of lOQto 1000Oper squareareavailable. Adjustrnentof resistance valuesby lasertrimming is only an optionat microwave frequenciesifa broadgapis used.
9.3
THE LIMITATIONS OF A SERIESTRANSMISSION LINE USEDTO REPLACEA LUMPED ELEMENT
All lumpedinductorsof finite dimensionhavesomecapacitanceto groundandassuchcan be consideredtransmissionlines of high characteristicimpedance.The characteristic impedancewill not be uniformwhenbondingwire inductorsor squareor spiralinductors are used. In order to get an idea of the boundson the indtctance that can be realizedwith lumped inductors,as well asthe limits on the inductancethat can be replacedwith series transmissionlines with negligibleerror, it is necessaryto considerthe transformation propertiesof a seriestransmissionline. Assuminga loadimpedanceof 2,.= Rt+jQ. Rr-,theinputresistance andreactance of a losslessseriestransmissionline havinga characteristic impedanceof Zois given by
&, = ftr[ +tanzo]/ z
(e.l)
and Xi^ = jfQRL(l - tan20) + Zotano- R; tano(t+ 82) | Zoll Z where Z =ll-
QRrtanl I Zol2+lRLtan} / Zol2
(e.3)
Microwave Lumped Elements,Distributed Equivalents, and Microstrip Parasitics
323
and
(9.4)
0 = 0/
In order to exhibit truly lumpedbehavior,the line lengthand characteristic impedance, respectively, mustbeshortenough andhighenoughfor theinputimpedance to be approximately Za = Rr + iQ Rr + jZotan9
(e.s)
For (9.5) to apply,the following inequalitiesmustbe satisfied: R,n= Rr: tanz0 << I Z o/ R L > > 2 Q t a n 1 (Zo I R)2 >> tan2e
(e.6)
Xin= jQ R, + jZotan$: tan2e<<1
Zo/ RL >> 2Qtan9 (zo I R)2 >> tan20 (Zo/ Z)2 >rI+ Q'
(e.7)
It follows from (9.1) and (9.3) that, evenif the characteristic impedanceof the line was equalto infinity, the resistance would still be transformedto R6 = Rl[ + tant 01
(e.8)
that is, the influence of the phaseshift does not becomenegligible with increasing characteristic impedance. With Z0approachinginfinity, the input reactanceof the line is given by Xi^ = jQ RtU - tan20f + jZotan\
(e.e)
Equations(9.8)and(9.9)canbeusedto provideupperboundson theline lengthfor which
324
Design of RF and Microwave Amplifiers and Oscillators
-j(Xr+ Zotan0)
RL
,r']t-i
.:
t
p
tlrrc
9.f
The equivalentcircuit usedto derive (9.10).
input impedanceof theline will beapproximatelyequalto thatgivenby (9.5).Oneway to do this is to evaluatethe reflectioncoefficientof the circuit in Figure9.1 for an infinite valueof the characteristicimpedanceof the line. The input reflectioncoefficientof this circuit is thengiven by "_t t -_
_
Rr(l + tan20) - R, + jlx
LQ- tan20) - Xrl
1- iQ 1 2/ t a n 2 e + \ -
(e.10) jQ
It follows from (9.10)thatthedeviationfrom lumpedbehavioris a strongfunction of the length of the line and the quality factorQ of the load impedance.This is clearly illustratedby the following results,which correspondto an insertionloss of 0.25 dB (Gr=l - l""lt):
Q =o : Q: I: Q:2: Q:3: Q:4:
0 :38" 0:33" 0:26 A= 2 2 0: 19"
Because of the finite characteristic impedance of any physical line, the exact deviation will always be greater than that predicted by (9.10). The exact reflection parameterfor any particular casecan be calculated by substituting R6 and{n, as given by (9. I ) to (9.4), into the equation
Jtt =
[R- - Rr] + j[Xi^- XL - Zotan9l [R- + R,l+ j[Xi" - XL - Zotan9l
(e.1l)
As an illustrationof thecombinedinfluenceof a reactiveloadandfinite valuesfor the characteristic impedance, the line lengths corresponding to an insertion loss of approximately 0.25 dB in the circuit shown in Figure 9.1 are tabulated in Table 9.1 as a
325
Microwave Lumped Elements,Distributed Equivalents, and Microstrip Parasitics
functionof the ratio Zo/R,.andtheline length. It is clearfrom the resultsin Table9.I thatthe rangeof characteristic impedances and line lengthsover which a seriestransmissionline canbe consideredto be a lumped inductor (and, inversely,over which the distributednatureof a seriesinductor can be ignored)is very limited, especiallywhenthe load Q is high. ' Table 9.1 The line tengthscorrespondingto an insertionlossof 0.25 dB in Figure 9.1 as a function of the characteristicimpedanceofthe line andthe p-factor ofthe load Line Length (')
4/Rt
Q=0
Q=1
26
t3 20
1.0 2.0 3.0 4.0 5.0
J)
Q=2
t0 l3 l5 l7 20 2l 23 24
' I
26 27 29 30 3l 3l
J I
38 38 38 38 38
I.J
10.0 15.0 20.0
Q=3
Q=4
3
2
5
J
9 ll t4 l5 t'7 l8
4 6 7 9 ll l3 l5
Q=5 I
) 4 7 9 l0 12
9.4 LUMPED MICROWAVE INDUCTORS Lumpedmicrowaveinductorscan be fabricatedin differentforms. For low inductance values,strip inductorsor bondingwire is frequentlyused,while largerinductancevalues arerealizablewith spiralor solenoidalinductors.The basicequationsrequiredto design theseinductorswill be consideredhere. Strip Inductors Theinductanceofan isolated(no groundplane),flat, ribboninductor(or strip inductor)is given approximatelyby [4]
L (r*l I mm) = 0.2{lnll / (w + r)l + 1.193+ 0.2235(w+ t) / l}
(e.r2)
wherew is thewidthof theribbon,t itsthickness, and/ its length. An approximate expressionfor the Q of a ribbon inductor is [5]
'o =2.r5,,sr
r(nH)' fe(cE)u'(f G]e'4\v' p \ K
/ \
)
z
)
(e.13)
326
Design of RF and Microwave Amplifiers and Oscillators
where p is the resistivity of the material used,and K is a correctionfactor for the current crowding occurring at the cornersof the strip [4]. K is given approximatelyby the following expression:
- 0.2319lnlw lrl + 0.2386fln(w I t)12 K = 1.3565 - 0.0536lln(w lr)13+ 0.0043 fln(wI t)la
(9.14)
Theinductance ofa stripinductoris decreased by thepresence ofa groundplane.The effectiveinductance for thiscaseis givenin termsof thefree-space valueby [6, 7] L"s =10.570- 0.145ln(W / h)l.L
(e.15)
Single'Turn Circular Loop Equations(9.12)and(9.15)canalsobe usedto calculatethe inductance ofa single-turn circularloop in thosecaseswherethewidth of thestrip is muchsmallerthanthe diameter. Whenthe groundplanecanbe ignored,the following expression[8] canalsobe used: L(nLI I mm) = 0.2|n(l / w + t) - 1.761
(e.16)
For (9.16)to apply,the inequalityI >> 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
Design of RF and Microwave Amplifiers and Oscillators
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
Design of RF and Microwave Amplifiers and Oscillators
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 <
3.0 4.0 5.0
/.) 10.0
,i._
15.0 20.0 25.0
0.8 1.4 2.0 2.6 3.2 4.5 5.8 9.3 13.0 21.0 29.0 37.O
Bonding wire d=25 pm
0.8 t.3 1.9 2.5 3.1 4.4 >.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
Microwave Lumped Elements,DistributedEquivalents,and Microstrip Parasitics
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.
Microwave Lumped Elements,Distributed Equivalents, and Microstrip Parasitics
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
Designof RF and Microwave Amplifiers and Oscillators
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 ^u _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:
Microwave Lumped Elements,Distributed Equivalents, and Microstrip Parasitics
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
Design of RF and Microwave Amplifiers and Oscillators
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'=
Design of RF and Microwave Amplifiers and Oscillators
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
Microwave Lumped Elements,DistributedEquivalents,and Microstrip Parasitics
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
Design of RF and Microwave Amplifiers and Oscillators
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<
3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.15 6.00
- I l.13 53.1 10.3748.9 -9.75 44.9 -9.23 41.1 -8.81 37.4 -E.48 33.9 -8.22 30.5 -8.02 27.4 -7.89 24.6 -7.81 22.0 -7.78 19.8 -7.79 17.9 -7.84 t6.5 -7.90 r5.7 -7.96 15.5 -7.99 16.0 -7.94 l7.l
-8.10 -7.51 -7.06 -6.73 -6.51 -6.38 -6.35 -6.41 -6.56 -6.81 -7.15 -7.57 -8.02 -8.40 - 8.53 -8.22 -7.45
41.5 36.3 31.4 26.8 22.4 18.2 14.3 10.8 7.7 5.1 3.4 2.7 3.6 6.5 I 1.3 17.0 2r.9
I\ T C
fi t(
c standing-waveratio (VSWR) of the two-stageamplifier in which this networkwas useddecreased from 1.72to I .65 with the transformation. The error in the input reflectioncoefficientof thenetworkitself (s,,(b))is, however,not insignificant,as
T is
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li. c( h ca hi
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Comparisonof theS-parameters ofthe distributedequivalentshownin Figure9.8(c)and the original lumped-elementnetwork (Figure 9.8(a)).
o1
Microwave Lumped Elements,DistributedEquivalents,and Microstrip Parasitics
345
is illustratedin Table9.9. Thereflectioncoefficientofthe originallumped-element networkis listedassr,(a)in this table. Better results can be obtainedby replacingthe seriesinductors,andthe remainingafterthis wasdone,exactlyat the highestfrequencyin the capacitance passband(referto Example9.2). The distributednetworkobtainedby doing this is shownin Figure 9.8(c).The input reflectioncoefficientof this networkis also ofthis networkarecomparedwiththoseofthe listedin Table9.9.TheS-parameters networkin Figure9.9.Notethatthehighestfrequencyon originallumped-element the differenttracesis not marked.
9.8 MICROSTRIPDISCONTINUITYEFFECTSAT THE LOWER MICROWAVE FREQUENCIES with bends,curves,changesin the line width, Microstrip discontinuityeffectsassociated inductanceand stubsaddundesirable T-junctions,crosses, andtheopenendofopen-ended to designedcircuits.Themagnitudeof theseparasiticsatthelowermicrowave capacitance frequenciesftelow X-band) [2] will be consideredin this section.A compensation techniquethat canbe usedto reducetheseeffects[3] will alsobe considered. Open-EndedStubs The effect ofthe fringing capacitanceassociatedwith the openendofan open-endedstub is similar to extendingthe lengthof the line slightly. The equivalentadditionalline lengthis givenempiricallyby [3]. Theexpression for the phaseshift (in degrees)is
+ 0'300W l h + 0 . 2 6 4 A0o"= 4.944xtor n7 ^[-e*t"-"rr v"r-etl e, - 0 . 2 5 8 Wl h + 0 . 8 0 0 .r
(e.6r)
where ft is the thickness of the substrate (in meters) and/ the frequency (in Hertz). The maximum relative error in (9.61) as comparedto the more accurateexpression of Silvester and Benedek [ 5] is less than 4o/ofor Wh > 0.2 and 2 < e, < 50 [ 3]. As an illustration ofthe magnitude ofthe open-endparasitic,the parasitic electrical line length (at 10 GHz) associatedwith different width+o-height ratios and dielectric constants (e,= 2.5 and e, = 10.2) are tabulated in Table 9.10 for a substrate with h : 0.635 mm. It is clear from these results that the parasitic influence of an open end cannot be neglected at the higher frequenciesand that this effect is more pronounced with higher dielectric constants and low impedance lines. The simplest way to compensate for the increase in line length is to reduce the length of the designed line by the correct amount. A distance of at least the equivalent line length should be allowed betweenthe end ofan open-ended stub and the substrateedge.
Design of RF and Microwave Amplifiers and Oscillators
Table9.10 The increasein electricalline length causedby an openend as a function ofthe dietectricconstant andthe width-to-heightratio of the line (/: l0 GHz; l:0.635 mm) el
z" (o)
wh
25 {n
7.20 2.80 1.35 0.70 0.38
5.6 5.0 4.4
6.90 3.35 0.90 0.30
9.3 8.4 6.2 4.5
0 :
t)
t00 125
l5 25 50 It
J. t
3.2
Stepsin Width The parasiticeffect of a stepjunction is similarto that of an openend.The effectof the fringing capacitance associated with thewiderline of thestepdiscontinuityis similarto an increasein the lengthofthat line. The changein the electricallength(in degrees)canbe estimatedby usingthe equation[16]
= Ae,"[1-Wz IWJ A0sep
(e.62)
whereA0o"can becalculated by using(9.61). An altemativeandmoreaccurateapproachto characterizinga stepdiscontinuity is to usethe equivalentcircuit shownin Figure9.10(b).An approximateexpressionfor the inductanceL": L, + Z, is (+5%for Wt/ W2< 5.0and Wzl h = 1.0)[3]. .t
l T ' t
Tr
l
L2
L r
T-" T-
l*'
Figure 9.10
The equivalentcircuit ofa stepdiscontinuity.
Tl
4-
Microwave Lumped Elements,Distributed Equivalents, and Microstrip Parasitics
/ m)=*t(# -lo)- zsr"r( z"(,,H
347
(e.63)
The individual inductancesare given by [3] Lt = L*r lfL,, + Lnzf .L,
(e.64)
and Lz = L*z lfL*t + L*zl. L,
(e.65)
impedances of the with the characteristic associated whereI,, andL*are the inductances two lines. C"in Figure9.10(b)is for thecapacitance An approximateclosedform expression (+10%for €, < 10and1.5< Wz/ Iryt< 3.5)|31
ftrrrorl
-3.r7 +ml#-rz6toge, m)=[10.11oge,
(e.66)
with stepdiscontinuities associated An ideaofthe magnitudeofthe parasiticefFects line as can be obtainedfrom the extensionsin line lengthresultingfrom an open-ended givenin Table9.10 and(9.62). techniquefor a stepdiscontinuitywouldbeto decrease A first-ordercompensation with a the lengthof the wider line by the appropriateamount.Thephaseshift associated with the in a line an open end stepdiscontinuitywill alwaysbe lessthanthat causedby lower characteristicimpedance.
Microstrip Bends The equivalentcircuit for a microstrip bendwith lines of equalwidth is shownin Figure 9.11. and for the right-angledbenddiscontinuitycapacitance Closed-formexpressions inductanceare[2]
(14e"+ r25)Wt h-(r.83e,-2.25) 0j=91:, (W
= ftro,/m) (9.5e, + 1.25)W /h+5.2e,+7.0 (|rylh>l)
(e.67)
348
Designof RF and Microwave Amplifiers and Oscillators
Tl T2
Figure 9.ll
Tr
Lh
Equivalentcircuit for a microsnip bend.
Lb/ h(r*l/ m)= l001^m h - 4.211
(e.68)
Equation(9.67)is accurateto within S%ofor2.5< e, < 15 and0.1 < Wh < 5. The accnracyof (9.68)is about3o/ofor 0.5 < Wh < 2.0 [3]. Table9.Il The VSWR (theoretical)associatedwith an unchamfered90' bend in a 75 Q (e,=2.5) and a 50O (e,: 10.2)line asa functionof the frequency(/l = 0.508mm) er
zo
(c))
f
vswR
(GHz)
') A
8 t0
n 4 8 l0
1.03 1.06 t.t2 l.l5 1.06 l.l3 1.28 1.36
An ideaofthe magnitudeofthe parasiticsassociated with a benddiscontinuitycan beobtainedfrom Table9.I I . ThetheoreticalVSWRsassociated with two bendsareshown in this table as a function of the frequency. Although the effect of a single bend may be small at the lower microwave frequencies,it shouldbe kept in mind that it will increasewith frequency,the numberof bendsusedin cascade,andthe line width. The parasiticeffectsof benddiscontinuitiesareusuallyreducedby miteringthe bendasshownin Figvre9.l2. TheoptimumvalueofZ in this figureis aboutl.8W for 50Q lineson aluminaandrexolitesubstrates, andit seemsto be independent of the bendangle
l2l.
Microwave Lumped Elements,DistributedEquivalents,and Microstrip Parasitics
Figure 9.12
349
Compensationof a microstripbend.
Whenfi|lh > 0.25ande, < 25,thelengthZ canbecalculatedby usingthefollowing equation: th L I W = JT tt.o+ + | 3 e-L3sty 7
(e.6e)
The equivalentelectricalline lengthof the mitre (t) canbe estimatedby usingthe equation
l= L /.li
(e.70)
Whenthe line is too thin(lVlh < 0.25)the optimummiter cannotbe used.
Figure 9.13
The effectivecurving radiuswhen a line is curved.
350
,
,
Designof RF and Microwave Amplifiers and Oscillators
Curving a line is frequently a better option than mitering it. When the curving radiusis largerthantwice the width of the line, themainparasiticeffectis a changein the effectiveline length.The effectivelengthof the curve(3 < RIW< 7) canbe estimatedby assumingthe effectiveradiusto be [7]
(e.7r)
R r f f = R i n n " .+ 0 . 3 W
This is illustratedin Figure9.13. I thatthe directionofthe line canbe changed Curvinga line alsohasthe advantage with any arbitrary angle. T.|unctionr Hammerstad'sapproach[6] to characterizing the parasiticeffectsof a T-junctionwith constantmainJinewidth is illustratedin Figure9.14.Thedifferentparameters aredefined by the following equations:
T2
*'l
24-:
Figure 9.14
The equivalentcircuit for a microstripT-junction.
Dl - l 120n h / Zor@ir)
d;
)Dr - d 2
I =l
(e.72) (e.73)
2
Itl s r."(;
n2
Dz =l20nh / Zor@ir)
zDt Zu\ ?t^ hr)
" AZIL 2 )r^ Z@
[r-(-rye.\'I \ ?'', Dt) |
)
(e.74)
Microwave Lumped Elements,Distibuted Equivalents,and Microstrip Parasitics
351
dt I Dz = 0.05n2Zot I Zoz
(e.7s)
d; I Dt = {0.076+ 0.2(2Dt I t",)' + 0.663e-t'ttzottzoz - 0.I72ln(Zo, I Zo)| Zot / Zo2
(e.76)
I-zDr /X^)Zu I Zo2 Zu I Zo2<0.5
B,x^-l-t
ffi-l'' -2Dt I }\^ll3zu I Zo2-21 L^=Lo/J",_*
(e.77) Zu I Zo2>0.5 (e.78)
When261| Zo2> 2, the calculatedvalueof d, / D, is too high. In this range,a bettervalue for drcanbe obtainedby replacingZot/ Zo2in (9.76)with its inverse[16]. T-junctionscan be compensated easilyfor the referenceplaneoffsetsby simply adjustingthe lengthsofthe differentlines.Theoffsetin themainline is usuallyvery small, andthe main effectis on the lengthof the stub. The best solution to the transformereffect is to keepthe width of the stub nuurow enoughfor the transformingeffectto be negligible(r shouldbe closeto unity). Becauseof the approximatenatureof the equationfor the loadingsusceptance at thejunction, no compensation for this effectis recommended.As with the transformer effect, the best option is againto limit the stub width to valuesfor which the loading susceptance will benegligible.Ifthiscannotbedone,abettermodelforthejunctionshould be obtainedorphysicalcompensation ofthejunctionshouldbeconsidered U8]. Crosses As a first orderapproximation,a crosscanbe considered to betwo T-junctionsin parallel. Via Holes Thebestoptionwhenshortcircuits(connections to thegroundplane)arerequiredis to use via holes.Theparasiticeffectof a via holeis usuallynot severeandthe sameperformance canbe expectedeverytime (repeatability). Via holesaremadeby drilling holesin the substrateat the appropriatepositions beforethetracksareetched(a drill file is usuallycreatedfor this purpose),afterwhich the substrateis treatedchemicallyandmetalis depositedelectrolyticallyor by sputteringon the cylindricalsurfaceof theseholes.Manufacturingreliablevia holesin teflonsubstrates is not a simplematterandis bestleft to experts. The main parasiticassociatedwith the hole itself is the inductanceto ground. Dependingon the diameterand the substratethickness,this inductanceis usually very
352
Design of RF and Microwave Amplifiers and Oscillators
small. The via hole inductance (cylindrical via hole) can be estimatedby using the following equation[19]:
L"iu=*[,
r'
h+
-l *)<,- ,lr2+n2>f
(e.7s)
wherelr is the substrateheieht and r is the radiusof the hole.
TI *f
I
T: Tl
Ts To
o) Figure 9.15
(a) A singlevia hole and(b) an equivalentcircuitfor it.
with the with the via hole pad andany stepassociated The parasiticsassociated feedingline will oftendominatetheeffectof thevia andwill usuallyincreasetheeffective inductanceof the via significantly.An equivalentcircuit for the via hole is shown in Figure9.15(b).Note thatthe lengthof theopenstubusedis determinedby thepadsection to the right of the via hole (T5to T6).
EXAMPLE 9.5
An artworkexample.
Applicationof someof thematerialin this sectionis illustratedinFigure 9.16.The artwork of the synthesized[20] GSM (globalmobile system)amplifier shownin Figure9.l6(b) wasmodifiedto thatshownin Figure9.16(c)by addingvia holesto provide the groundconnectionsrequiredand by curving and bending(optimal mitre) someof the lines to reducethe size of the artwork. The lengthsof the relevantlineswereadjustedto compensate for the effectof the changesmade.
Microwave Lumped Elements,DistributedEquivalents,and Microstrip Parasitics
353
r-;-
il
F
hll n{r l\-/ | ll - h r o l t.l Fl
tl
u E
[-iT ll
f
a -5
n
u'J
tl (b)
Figure 9.16
t
f'l
flTf
lU
T
l
-flfll---$r
-l_rl_rJr ll I J YLIO
t--1-
ll
u
-
n n n
(c)
(c) Modification of the artwork of a GSM amplifier (a, b) by addingvia holes to provide the ground connectionsrequiredand by bendingand curving somelines to reduceits size (Courtesyof GrinakerAvitronics, Highveld Technopark,SouthAfrica).
The performanceof the amplifier before and after the modifications is essentially the same(Gr: 14dB; F: 0.5dB) .
9.9
A COMPENSATION TECHNIQUE FOR MICROSTRIP DISCONTINUITIES
It is possibleto reducetheparasiticeffectsassociated with openends,steps,bends,andTjunctionsby usingconstantimpedancetapers.Thebasicideais illustratedin Figure9.17. The characteristicimpedanceof an incrementalsectionof the line is a function of the effectivedielectricconstantand the capacitance to ground.The capacitance consistsof plate capacitance and fringing capacitance associated with the edge.If the line width is reducedsmoothly, the plate capacitanceis reduced,but the fringing capacitanceis increased.If the line width is reducedfast enough,the total capacitancecan be kept constant,while the line width is reduced. In the caseof a step,the width canbe reduceduntil it is the sameas that of the narower line. Thewidth shouldbereducedto approximate a pointjunctionwhenanopenendor a T-junctionis compensated (i.e.,the width shouldbe reducedto a predetermined limit that is mainly a function of the currentdensity).
354
Designof RF and Microwave Amplifiers and Oscillators
This compensationtechnique wasintroducedfor sriplinesby MalherbeandSteyn [3]. The sameapproachcanbe followedwith microstriplinesandthe relevantequations will be derivedhere.
(a)
Figure 9.17
(b)
(a) An asymmetricaland (b) a symmetricalconstantimpedancetaper.
The characteristicimpedanceof an incrementalsectionof the line is given by Zo=l/fvoC)
=Ji rvcl
(e.80) (e.81)
wheree, is the relativeeffectivedielectricconstantof the line, c the speedof light, andC the capacitanceper unit length.If the characteristicimpgdanceis to remainconstantwith the narrowingwidth-to-heightratio,the quotientC I Je , in(9.81) mustremainconstant. The capacitancecorrespondingto a sectionwith incrementallength dy and the untaperedwidth l/is givenby
Cnb = eoe,(w/ h)dy + El(w / Hr)dy +2c tody
(e.82)
wherc e, is the effective dielectric constant of the material used, C7, the fringing capacitanceper unit length correspondingto the width ll', H, the distancefrom the conductorto the cover,and/r the substrate thickness. The capacitance corresponding to an incrementalsectionof an asymmetricalline with taperedwidth x is given by C " d y = [ e o e , ( xl h ) + e o @ I H r ) + C o + C r s e c l l d y
(e.83)
(C2)is a functionof thecharacteristic Thefringing capacitance impedanceandthe effectiverelativedielectricconstantof anuntaperedline with width x. It canbe calculated asfollows: C* = 0.5[r/e* / (il o) - ele,x / h- eoxI Hr]
(e.84)
Microwave Lumped Elements,DistributedEquivalents,and Microstrip Parasitics
e,"is the effective dielectric constantofthe untaperedline (widthx). This equationcanbe derivedby combining(9.81)and(9.82)with I/ setequalto x in (9.82). By equatingthe characteristicimpedancesat x and lI/, the following result is obtained:
,+sec 0= rc,f-ry - - co}rco Y. nw l**. {rH
(e.8s)
With 0 known asa function ofx, it is a simplematterto constructthe taper.Altematively, theexactpositiony wherethe taperwidth is equalto Xcan be determinednumericallyby evaluatingthe integral
y =-
(e.86)
fcot}dx
Thelengthofthe tapercanbe determinedby settingtheboundaryXin(9.86)equal to the final valueof x. The electricallength of the taperedsectioncan be obtainedwith the following equation:
=306(.f t Q AO(degrees) [--,ta
cot1dx
(e.87)
where/is the frequencyof interestandXis the final valueofx. Whena symmetricaltaperis required,(9.86)and(9.87)canbeusedin conjunction with the following equations: Ct = 0.5[t!e^ t (il o) - eoe,2(xI h) - er2(x I Hr)]
(e.88)
(e.8e) Application of the taperingtechniqueto reducethe effect of stepjunctions is illustratedin Figure9.18.
Flgure 9.18
The taperingtechniqueappliedto reducethe discontinuityeffectsassociatedwith a stcp iunction.
356
Designof RF and Microwave Amplifiers and Oscillators
March, S., 1981.
l.
"Microstrip
Packaging:WatchThe Last Step,"Microwqyes,December
Gupta,K. C., R. Garg,andI. J. Bahl,Microstrip Linesqnd Sloflfnes,Norwood,MA: ArtechHouse,1979. "The Malherbe,J. A. G., and A. F. Steyn, Compensation of StepDiscontinuitiesin TEM-ModeTransmission L ines,"I EEE Trqns.Mi crowave Theory Tech.,Y oI.MTT 26, No. I l, November1978. 4. Terman,F.E., RadioEngineersHandbook,NewYork: McGraw-Hill,l943. 5. Young,L., Advancesin Micrawaves,New York: AcademicPress,1977. 6. Gupta K. C., R. Garg,andR. Chadha,Computer-AidedDesignof MicrowaveCircuits, Norwood,MA: ArtechHouse,1981. "The 7. Chaddock,R. E., Applicationoflumped ElementTechniques to High Frequency Hybrid IntegratedCircuits,"Radio and ElectronicsEngg.(GB),Yol. 44,1974, pp.414-420. 8. Dukes,J.M. C.,PrintedCircuits,TheirDesignandApplicatlon,London:MacDonald, 1961,pp. 120-135. g. March, S. L., "Simple EquationsCharacteizeBond Wires," Microwayes& RF, November1991,pp. 105-110. "Designing 10. Dill, H. G., Inductorsfor Thin-Film Applications," Electron, Des., February17, 1967,pp. 52-59. "Hybrid 11. Caulton,M., S. P. Knight, and D. A. Daly, IntegratedLumped-Element MicrowaveAmplifiers," IEEE J. Solid-StateCirc.,Yol. SC-3,No. 2, June1968. 12.
"Rf
andMicrowavePorcelainCapacitors,"Cazenovia,NY: DielectricLaboratories, Inc.,1998.
13. Garg,R., andL J. Bahl, 1978.
"Microstrip
Discontinuities ," Int. J. Electronics,Vol.45. July
"Interdigital 14. Alley, G. D., Capacitorsand Their Application to Lumped-Element MicrowaveIntegratedCircuits,"IEEE Trons.MicrowaveTheoryTech.,Yol.MTT18,No. 12,December1970.
Microwave Lumped Elements,Distributed Equivalents, and Microstrip parasitics
357
15. Silvester,P., andP. Benedek,"EquivalentCapacitance of MicrostripOpenCircuits," IEEE Trans.MicrowaveTheoryTech.,Yol.MTT-20,1972,pp.5ll-516. 16. Hammerstad,E. O., "Equations for Microstrip Circuit Design," Conferenci Proceedings,5frEuropeanMicrowaveConference, September1975,Hamburg. 17. FoundryManual,GaAsFoundryServices,TexasInstruments,Inc., January1991. 18. Chadha, R., and K. C. Gupta, "Compensationof Discontinuities in Planar TransmissionLines,"IEEE Trans.Microwwe TheoryTech.,Yol. MTT-30,No. 12, December1982,pp. 215l-2156. 19. Goldfarb,M. E., andR. A. Pucel,"ModelingVia Hole Groundsin Microstrip,"IEEE Microwaveand GuidedWaveLetters,Vol. l, No. 6, June1991. 20. MultiMatch RF and Microwave Impedance-Matching,Amplifier and oscillator Synthesis Sortware,Somerset West,RSA:Ampsa(PTY)Ltd; http://www.ampsa.com.
SELECTEDBIBLIOGRAPITY De Brecht,R., andM. Caulton," LumpedElementsin MicrowaveIntegratedCircuitsin the l-12 GHz Range,"IEEE G-MTTInternat.MicrowaveSymp.,May 1970,p. 14. Hammerstad,E.O., and F. Bekkadal,Microstrip Handbook,ELAB ReportSTF 44/A74 169,Universityof Trondheim,NorwegianInstituteof Technology,1975. Hammerstad,E. O., and O. Jensen,IEEE MTT-S SymposiumDigest, June 1980,pp. 407-409. PettenpaulE., H. Kapusta,A. Weisgerber, H. Mampe,J. Luginsland,andI. Wolff, "CAD Models of Lumped Elementson GaAs up to l8GHz," IEEE Trans.Microwave TheoryTech.,Yol.36, No. 2, February 1988. "Application Sobol, H., of IntegratedCircuit Technologyto MicrowaveFrequencies," Proc.IEEE,Vol. 59,Augustl97l,pp.1200-l2ll.
CHAPTER 10 THE DESIGN OF RADIO.FRBQUENCY AND MICROWAVE AMPLIFIERS AND OSCILLATORS 10.1 INTRODUCTION In thischapterthedesignof radio-frequency andmicrowaveamplifiersandoscillatorswill be considered.Many ofthe considerations applyingto amplifiersalsoapplyto oscillators and vice versa. Basic considerationssuch as stability, tunability, and unilateralnesswill be introducedfirst. The dynamicrangeof an amplifierwasconsideredin chapter2. Stabilityis evaluatedin termsof stabilitycirclesor in termsof theRolletteandthe Stemestability factors.The Linville stabilityfactorwill alsobe considered. Experiencehasshownthattheloopgainshouldalsobeevaluatedwhenthe stability of an amplifier is considered.The gainandphasemarginsassociated with the loop gain shouldbe calculatedfor eachstagein which feedbackis used. It is importantto understand thatthestabilityfactorsnormallyusedare"black-box" parameters andprovideno informationontheeffectthatcomponent tolerances (sometimes very small changes)can have on the stabilityof the circuit. In contrast,the loop gain is appliedat the sourceof the instability(thevariousfeedbackloops) andprovidesa much clearerpictureofthe stabilitysituation. By using andextendingthe materialprovidedin ChaptersI and2,a wide rangeof amplifierscanbe designed. The basic procedureproposedhere for the design of cascadeamplifiers is summarized inthe flow diagramshownin Figurel0.l [ ]. Theprocedureconsistsbasically of designingthe stagesin theamplifierchainsequentiallyfrom onesideto the other.Each stageis designedby first adding feedbackand/or loading networksto the transistor selected,after which the performancetargetedfor that stageis realizedwith a lossless impedance-matchingnetwork. The two amplifiers shown in Figure 10.2 areexamplesof cascadeamplifiers designedby following this approach.The first amplifier is a switchedamplifier module covering the frequency range 2-18 GHz, while the second is a GSM amplifier (905-915MHz). Couplerswereusedin both amplifiersto improvethe VSWRs.
359
I
il
Designof RF and Microwave Amplifiers and Oscillators
thenumberofstasesto beusedin
Evaluatetheresultsobtainable with the differentmodificationontions.
;
Modifloptimize the transistor for improved stability/VSWRV gain slope (G^ C-_,fl, MAG, MSc) and power perfomance. Evaluate the influence oftolerances in the values ofthe passtvecomponents.
gain/NF/power lossless control
Specif the operating power gain/available power gain/ transducerpower gain/noise figure/output power required from stage i. Synthesizesolutions to th€ defined impedance-matchingproblem. Evaluate the influence ofcomponent toleranceson the different solutions synthesized.Select on€ ofthe solutions.
Synthesize a lossless networkto control thegainrippleor theVSWRof thefinal impedanc€-matching network.
Analyzeandoptimizethesynthesized amplifier.Transform theamplifierto microstdpor striplineform,if required.Changetheorientationof therelevantstubsmd curveor meander anylonglinesin theartworkcreat€d.
Ittrl
l0.l
A flow diagramof a typical amplifier synthesiscycle Il].
The Desip of Radio-Frequencyand Microwave Amplifiers and Oscillators
361
(b) Figure 10.2
(a) A switched amplifiermodule (2-18 GHz; the switching is doneonthe output sidesof the different modules) and (b) a GSM amplifier (Coudesy of Grinaker Avitronics, Highveld Technopark, South Africa).
362
Design of RF and Microwave Amplifiers and Oscillators
It will be shown here that when wideband amplifiers are designed, devicemodification(addingresistivefeedbackand/orloadingsectionsto the transistorsused)is probablythemostcriticalstepin thedesigncycle.Device-modification servesto exchange the excessin gain and noise capabilitiesat the lower frequenciesfor more desirable (improvedstability,gain leveling,lower VSWRs).It can also be usedto characteristics bring the optimumnoiseor powermatchcloserto a conjugatematch. Amplifiers with excellentnoise performanceand low input VSWRs are often required.Ifhybrid couplersor isolatorscannotbe used(losses,costfactor,etc.),the first stepin realizingsuchanamplifieris device-modification. Ifthe VSWR associated with the optimumnoisematchis still not satisfactory,the nonzeros,2of the modifiedtransistoris often usedto improvedthe input VSWR. The designof a high power stagewith a good outputmatchproceedsalongsimilar lines. The gain of each amplifier stageis controlledby its operating,available,or transducerpowergain.In doingthis,it is not necessary to ignorethe effectofthe reverse transfergain (s,r) asis often done. Thedesignofmatching networksfor passiveproblemswasconsideredin Chapter8. It will be shownherethat, if the (modified)transistoris inherentlystable,the activegain or noisefigurecontrolproblemscanbetransformed exactlyto equivalentpassiveproblems. The impedance-matching techniquesoutlinedcanthereforealsobe usedto solveactive problems. Note that inherentstabilityis not requiredwhenthe noisefigure is controlled. It is importantto realizethat the correctchoicefor the transistorto be usedin a particularstagecanhavea dramaticeffecton boththeperformance andthesensitivity.The samecanbe saidfor using a suitablenetworkto controlthe gain,the noisefigure,or the outputpowerof eachstage.In orderto find theright network,thecapabilityof synthesizing over differenttopologiesis essential. When a high dynamicrangeamplifier is requiredat RF frequencies,a lossless feedbackamplifiershouldbeconsidered. Theperformance obtainablewiththeseamplifiers is excellent.It shouldbe noted,however,that with careful designand the right choice of thetransistor,similarperfonnancecanoftenbeobtainedwith a cascade amplifier.Lossless feedbackamplifierswill be consideredin Section10.10. The designof reflectionand balancedamplifierswill be also considered.For an excellenttreatmenton the designof distributedamplifiersreferto [2]. Oscillatordesignwill be considered in section10.13.
IO.2 STABILITY In orderto designan oscillatoror to preventan amplifierfrom oscillating,it is necessary to know more aboutthe conditionsunderwhich oscillationscanoccur.Theseconditions will be establishedhere. It will be shownthat steady-stateoscillationswill occurwhenthe input andoutput admittanceof an activecircuit is equalto zero.Oscillationis not possibleat anyfrequency at which the input conductanceis positive or, equivalently,the magnitudeof the input and
The Design of Radio-Frequencyand Microwave Amplifiers and Oscillators
363
outputreflection coefficientsaresmallerthan unlty. The boundarf condition at which the is equalto zeroor themagnitudeof thereflectioncoefficientis equalto input conductance unity will be consideredon the admittanceplaneandon the SmithChart,respectively.It will be shownthat the stableandpotentially unstableareasareseparatedby circlesin both planes.The areainsidethe relevantcircle is usuallythe areaofpotential instability. Whenoscillationscannotoccurwith anypassivetermination,a transistoris saidto be inherentlystable. The stabilityof a two-portcanbe consideredby establishingthe positionsof the stabilitycircleson the planeof interestor by calculatingvariousstabilityfactorsdefined for this purpose.The Rollette,the Sterne,andthe Linville stabilityfactorsarefrequently used. Although essential,the stability factorsby themselvesdo not provide sufficient informationon the stabilityof an amplifier.Themain problemis that the stability of the two-portasa wholeis consideredwithoutreferenceto the specificcauseof the instability (theintrinsicand/orexternalfeedbackloops).Smallchangesin thecomponentvaluescan sometimeshavea very pronouncedeffecton the stabilityfactors.In orderto get a clearer pictureof the situation,the following shouldalsobe done: l.
" 2.
Figurc 10.3
requiredon eitheror both sidesof Theseriesor shuntstabilizingresistance the two-port shouldbe calculated(seeFigure 10.3).In the well-behaved with increasingfrequency. requiredwill decrease case,the seriesresistance will increasewith increasingfrequency. Similarly,the shuntresistance Note that the use of stabilizing componentsis not necessarily required,however,providesa goodideaofhow intended.The resistance severethe potentialinstabilityis. Theloop gainfor eachtransistorstageto whichexternalfeedbackis applied gain andphasemarginsshouldbe shouldbe calculatedandthe associated If morethanoneloop is used,the gainfor eachloop shouldbe established. calculated. It is importantto realizethat the loop-gainis dependenton the ter-
A two-port network augmentedwith its terminations.
36
Design of RF and Microwave Amplifiers and Oscillators
s0o). Thereflectiongainat theinputandtheoutputof each(modified)transistor shouldbe calculated. The reflection gain is also dependenton the terminationsused.
3.
It should be notedthat oscillationswill startup becauseof favorableloop gain conditionsand may also start up becauseof favorablereflectionconditions(negative theseconditionswill apply simultaneously. At steady-state resistance). Calculationofthe stabilitycircles,the stabilityfactors,the stabilizingresistance, andthe loop andreflectiongainwill be consideredin this section.
10.2.1 StabitityCircleson the AdmittancePlane Consider the two-port network in Figure 10.3. The terminal crrrents of the two-port augmentedwith its terminationscanbe calculatedby usingthe equation
[t'l=lhr+Y, Vr)
| t^
ln l[t'] Y22+YL)lvz)
(lo.l)
If the circuit is oscillating,theterminalvoltageswill not be equalto zero,eventhoughthe terminal currentsare equalto zero. In orderforthe voltagesin (10,1)notto beequalto zerowhile thecurrentsareequal to zero, the determinantof the extendedl-parametermatrix must be equalto zero. Were this not the case,the l-parametermatrixwouldhavehadaninverse,andthe only solution corresponding ro zerovaluesfor I and12would havebeenzerovaluesfor both V1and V2 aswell. A zerovalue for the determinantimplies that
lyu +Y,llyn + Yr7- lnlzt = 0
(10.2)
leadingto !rr*Y,-
!o! Ll =g lzz*rt
(10.3)
and y"" iY, " - lnln h*Y,
- g
(10.4)
The lasttwo equationsareeasilyrecognizedasthe equationsfor the input andoutput
The Design of Radio-Frequencyand Microwave Amplifiers and Oscillators
365
admittancesof the augmentedtwo-port, respectively. A necessaryconditionfor any oscillationto occur,therefore,is that the input and output admittanceof the augmentedtwo-port must be equalto zero. This condition will apply at steady-state. It is clearfrom (10.2)to (10.4)that if the input admittanceof a networkis equalto zero,the samewill applyto its outputadmittance,that is, aslong asthe produc'tyr2y21is not equalto zero. It is alsoclearfrom (10.3)and(10.4)thatwhenevertheresistivepartof the input or outputadmittanceis greaterthanzeroat anyparticularfrequency,oscillationscannotoccur at that frequency.Therefore,ifcare is takento ensurethatthe input (or output)conductance of an amplifier is alwaysgreaterthanzero,oscillationswill not be possible.The locusof load admittances(sourceadmifiances)for which the input (output) admittancewill be is thereforeofinterest. purelyreactive(zeroconductance) the input admittancefor the moment, the locus of the load Considering only admittancefor which the input conductancewill be equalto zerocanbe derivedeasily by settingREALUI" (Ir)l = 0:
s(ri,)=4= r,-n(frfi=o where 8rr* i\t
(10.s)
= ln
with Snrjbzz=lzz
(10.6)
lnlzt=P+iQ
(10.7)
and (10.8)
Gr+jBr=Y, it followsthat
--t - s ' 6s,, r r -\r{e G.*
+ iQ)tkrr!cr)-i(trrta)ll-, (Lzz+Gr)'+(brr+Br)2 )
This equationcanbe manipulatedinto the following form:
'# ,*. Design of RF and Microwave Amplifiers and Oscillators
366
lo'*r"
G 2(G, + grt) (10.e)
R
plane.Thecenter of a circlein thelinearadmittance (10.9)is theequation Equation of the circle is given bY
G,+iB,= [*h
- s,,f.tlT#
b,,) rr-
^i
:j I I
while the radius is given bY
Y -" l n = l Y +" g" l2(G,
IS
(10.10)
(
I
(10.1)
)l
input conductance For all load admittanceson the circumferenceof this circle, the is alsoequalto zero,the amplifierwill will be equalto zero,andif the input susceptance falling inside is usuallynegativefor all loadadmittances oscillate.The input conductance < the stabilitycircle.The exceptionoccluswhengtt 0' conditionwill clearlybe when worst-case the considered, are If only passiveloads G": 0. Underthis condition(10.10)and(10'11)simplifr to
ci+1ai=l*-",,]*,1*- r,)
(10.12)
f a
and
p', " -lyrryt,l l 2 s "I
a
(10.13)
axis of the admittance whenever this stability circle lies to the left of the imaginary particularfrequencyaslong plane,it will notbe possibleior anamplifierto oscillateatthat inherentlystable' asits terminationsarepassive.Suchan amplifieris saidto be in thesourceplanecanbe circle stability of the parameters Proceeding* uuou.,ttr" determinedeasily.The resultingequationsare
The Design of Radio-Frequencyand Microwave Amplifien and Oscillators
G'+iB,=l;,r-
s',].,1*%- 4,]
p_l y,ryr,I + grr)l lZ(G,
367
(10.14)
(r0.1s)
The stableareawill beoutsidethis circle aslong asg22> 0. Theworst-casecondition is againassociated with Gr:0. TheLinville stabilityfactorcanbedefinedin termsofthe parameters of thestability circle in the following way: p' /a _
Gi
(10.16)
lv,rvr,l
=- l'r" I 2gu
(10.17)
8zz
Whenever0 < C < l, the stabilitycirclewill lie to the left of the imaginaryaxis of the admittanceplane.If 91 is positive,the insideof the circlewill representthe unstable area(Ij" =y11when Ir--) andthe deviceunderconsideration will be inherentlystable. The stabilitycircle is plottedfor differentvaluesof C in Figure 10.4. TheLinville stabilityfactoris independent of thetwo-portterminationsandis only a functionof the parameters of the deviceused. Anotherusefulmeasureof stabilityisthe Sternestabilityfactor.TheStemestability factor takesthe influence of the resistiveloading by the load and sourceadmittanceinto account:
K*
gzz+ Gt
P lnlzr 2(gu+G") 2(grr+G")
(r0.18)
An amplifierwill be stableat any frequencyfor which K > l, that is, aslong asthb terminationsusedarein place. Equation( I 0.I 8) is basedon the fact that for inherentstability,it is requiredthat
ni+ci<0 thatis, if g,, > 0.
(10.1e)
Design of RF and Microwave Amplifiers and Oscillators
Whenevereither or both of the terminationscan change,it is advisableto set G1 and/orG" in (10.18)equalto zero.
Figure 10.4
;
I F b
.
10.2.2
Stability Circles on the Smith Chart
The locusof load and sourcereflectioncoefficientsfor which the input or output or resistanceof an amplifierwill be equalto zeroarealsocircleson a Smith conductance Chart. The equationsfor the stabilitycircleson the SmithChartcanbe derivedby using in ChapterI (1.86) and (1.87)).For inherent derivedfor,s,,, and s22o the expressions stabilityit is requiredthat
1",,. |. I and
;
The relationshipbetweentheLinville stabilityfactorandthepositionof a stabilitycircle relative to the imaginaryaxis ofthe admittanceplane.(The stableareais on the outsideofeach circle as long asB' ' > 0.)
(10.20)
The Design of Radio-Frequencyand Microwave Amplifiers and Oscillators
l t n , l .t
369
( r 0.21)
for anypassive (i.e.,forlf, | < t analf" | < t 1. termination It followsfromtheseequations that[3]
-Asrl< lt-s,rsrl ls,,
(r0.22)
:
and
l"r-AS"l.ll-",,S"1
0o.rrl
where A = , s , , . l r ,- s t z s z r
(r0.24)
Theparametersfor the load stability circle (thevaluesof S, for which I,r11, | = 1,that is,theloadterminationsforwhichthe inputimpedance will bepurelyreactive)followfrom (10.22)andaregiven by
n "'--(s z z - A t i r ) .
(10.2s)
lsrrl' -l^f
D
l\I
"
--
-----------;
llt,,l'-ld'l --' ' ' I l'
(10.26)
where C1is the centerof the circle andR, its radius. Similarly,the parameters for the sowcestabilitycircle (thevaluesof & for which = that is, the source terminations for which the outputimpedancewill be purely | l"rr. | reactive)aregiven by
"'-
-Aq'r)' ,- - (srr
P,,;'-^t
(r0.27)
Designof RF and Microwave Amplifiers and Oscillators
p,,l'- lol'l
}
(10.28)
The stableareacould be outsideor insidethe circle. The specificcasecan be respectively'If, for example, establishedby observingthe magnitudesof s' and .s22, positivewith a 50Oload;andif the 50Oloadfalls outside is resistance . input l, the lr,, I ihe ioad stabilitycircle,the areainsidethe circleis the unstablearea. It follows that the following conditionsmust be satisfiedfor a transistorto be inherentlystable:
F L E F
llcl-lR"ll'l
(10.2e)
llc,l-ln,ll>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
Design of RF and Microwave Amplifiers and Oscillators
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.
The Design of Radio-Frequencyand Microwave Amplifiers and Oscillators
373
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(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
Designof RF and Microwave Amplifiers and Oscillators
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
The Design of Radio-Frequencyand Microwave Amplifiers and Oscillators
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
Design of RF and Microwave Amplifiers and Oscillators
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
,.*
The Design of Radio-Frequencyand Microwave Amplifiers and Oscillators
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7: l:lm t0:S:2
dB
t.m
s.0
- 5.3C1
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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).
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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
Designof RF and Microwave Amplifiers and Oscillators
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
Design of RF and Microwave Amplifiers and Oscillators
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.
The Design of Radio-Frequencyand Microwave Amplifiers and Oscillators
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)
Design of RF and Microwave Amplifiers and Oscillators
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
The Design of Radio-Frequencyand Microwave Amplifiers and Oscillators
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
Design of RF and Microwave Amplifiers and Oscillators
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
The Design of Radio-Frequencyand Microwave Amplifiers and Oscillators
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)
Desigr of RF and Microwave Amplifiers and Oscillators
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
Design of RF and Microwave Amplifiers and Oscillators
(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
The Designof Radio-Frequencyand Microwave Amplifiers and Oscillators
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)
The Designof Radio-Frequencyand Microwave Amplifiers and Oscillators
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
Design of RF and Microwave Amplifiers and Oscillators
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
The Design of Radio-Frequencyand Microwave Amplifiers and Oscillators
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.
The Designof Radio-Frequencyand Microwave Amplifiers and Oscillators
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
Design of RF and Microwave Amplifien and Oscillaton
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
Design of RF and Microwave Amplifiers and Oscillators
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)
The Design of Radio-Frequencyand Microwave Amplifiers and Oscillators
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
A^u=
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)
The Designof Radio-Frequencyand Microwave Amplifiers and Oscillators
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
Desigr of RF and Microwave Amplifiers and Oscillators
?: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
The Design of Radio-Frequencyand Microwave Amplifiers and Oscillators
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
Designof RF and Microwave Amplifiers and Oscillators
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
The Design of Radio-Frequencyand Microwave Amplifiers and Oscillators
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
Design of RF and Microwave Amplifiers and Oscillators
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
The Design of Radio-Frequencyand Microwave Amplifiers and Oscillators
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
Design of RF and Microwave Amplifiers and Oscillators
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
The Design of Radio-Frequencyand Microwave Amplifiers and Oscillators
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).
Design of RF and Microwave Amplifiers and Oscillators
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
The Design of Radio-Frequencyand Microwave Amplifiers and Oscillators
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
Design of RF and Microwave Amplifiers and Oscillators
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
Design of RF and Microwave Amplifiers and Oscillators
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].
The Design of Radio-Frequencyand Microwave Amplifiers and Oscillators
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-
Designof RF and Microwave Amplifiers and Oscillators
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.
The Design of Radio-Frequencyand Microwave Amplifiers and Oscillators
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
Design of RF and Microwave Amplifiers and Oscillators
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
The Design of Radio-Frequencyand Microwave Amplifiers and Oscillators
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
Desigr of RF and Microwave Amplifiers and Oscillators
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
Transducerpower gain
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
Transducerpower gain
(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.
The Designof Radio-Frequencyand Microwave Amplifiers and Oscillators
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
Desigr of RF and Microwave Amplifiers and Oscillators
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
233.2 201.4 170.7 140.6 111.4 83.0 55.4 28.4 1.8 33s.2 308.3 280.5 25r.4 220.8
7' 1<
33.05 33.23 33.29 33.28 33.22 33.t5 33.08 33.05 33.06 33.07 33.03 32.84
(dB)
(")
-21.17 -22.32 -23.62 -25.04 -26.51 -27.88 -28.86 -29.13 -28.53 -27.06 -24.96 -22.50 - 19.91 -17.37
122.3 105.3 86.9 67.2 46.3 24.1 0.7 336.8 314.5 294.7 277.6 262.2 247.5 232.3
The Design of Radio-Frequencyand Microwave Amplifiers and Oscillators
431
10.10 LOSSLESSFEEDBACKAMPLIFIERS Losslessfeedbacknetworksimplementedwith transformers or directionalcouplers[2] can be usedto removethe gainslopeof a transistorin theplaceof usingresistivefeedbackor loadingnetworks.Whenthis is done,no (or very little) poweris dissipatedin thefeedback by the transistorendsup in the load.At the same networks;most of the powergenerated in the output voltage and/orthe output current is reducedby the time, anydistortion feedback. Low input andoutputVSWRscanbeobtainedby choosingthecorrectturnsratios Most losslessfeedbackcircuitsaredesignedto minimizetheVSWRs. forthe transformers. It shouldbe notedthat the noisefigure andthe outputpower,and,therefore,the dynamic range,arenot necessarilyoptimizedwhenthis is done. A circuit usingthe principleof losslessfeedback(Figure 10.45)was patentedin Novemberl97l by D. Norton (AnzacCorporation)in the United Statesof America[8]. Two transformersare usedand configuredso that the load voltageis sampledby one and fed backasa voltagein serieswith the input of the amplifier,while the otheronesamples the loadcurrentandfeedsit backasa currentto the input ofthe amplifier(voltage-series and current-shuntfeedback).The turns ratios ofthe transfonnerswere chosento createa directionalcoupler arrangement,the main purposeof which was to provide excellent anypower VSWRsandto controlthegain.Because ofthe directionalcouplerarrangement, enteringthe input port is directedto the input ofthe transistorandto the outputport, and any power generatedby the transistoris directedback to its own input as (negative) feedbackandto the load asexternalpower. Themain advantage of this circuit is that gainlevelingat very low gainvaluescan be obtainedwithout a degradationin performancerelativeto that associatedwith the that any power highestgain obtainablewith this configuration.It hasthe disadvantage incidenton the outputport (s,2)will be directedtowardthe inputport andthe outputport
o
l
o
N2
/vl
Pl
Figure 10.45
t
o
o
P2
The losslessfeedbackamplifier patentedby David E. Norton in l97l [8].
432
Designof RF and Microwave Amplifiers and Oscillators
P2
: l ry'r
"l
II
/y, P3
:
P4 o
Figure 10.46
P2
Pl
Ll p-
HJ,-^-^-p
l
The losslessfeedbackcircuit @owerFeedbackTechnologyru)patentedby C-bit I9].
ofthe transistor.Theisolationof this amplifier,therefore,is usuallypoor,especiallywhen an amplifierwith low gain is designed. Theisolationproblemwassolvedby Q-bit [9] by usingtwo couplersinsteadof only one(PowerFeedbackTechnologytt).In thisanangement(referto power Figure10.46),the incidenton the output side is directedat the transistorand the terminationof the input coupler,insteadofthe inputport. Theisolation,therefore,tendsto be thatofthe transistor only plus the throughlossesof thetwo couplers(whichshouldbe small).However,some (most)of the power fed back is dissipatedin the terminationof the input coupler.This actuallyviolatesthe principleof losslessfeedback. Theinputpowerin this Q-bit circuitis directedat theinputof thetransistorandthe terminationof the input coupler,which degrades thenoisefigure (slightly)anddissipates power. someof the input An altemativecircuit (Figure 10.47)was introducedin [0]. The outputvoltage acrossthe load and the output current are sampledwith two of the windings of an impedance-matching transformerwith threewindings,andboththe currentandthevoltage arefedbackto thethird winding,which is in serieswith theinputterminalof thetransistor (thewindingthatsamplesthecurrentdetermines thecurrentthroughtheinputwinding,and thewindingthat samplesthevoltagedetermines thevoltageacrosstheinputwinding).The impedanceassociated with the third winding,therefore,is completelydeterminedby the voltageandcurrentsampled(altematively,theinputimpedance requiredwoulddetermine the ratio betweenthe voltageandthe currentfeedback). Ideally,theinputimpedance ofthe transistorusedshouldapproximate a short-circuit in this arrangement, while its outputimpedanceshouldlook like an open-circuit(a bipolar transistorusedin the common-base configurationcan usually be usedto presentsuch impedances). It shouldbenotedthatwhile theoutputcurrentof thetransistoris actuallysampled in this circuit, the transfomer arrangementandthe fact that the input currentis very low comparedto theoutputcunent(i6: i"/ B) forcetheloadcurrentto be directlyproportional to the transistorcurrent. With a correctchoiceof the tums ratio of the transformer,a two-wayimpedance matchcanbe obtainedeasilywith this arrangement. This circuit is frequentlyused.It hasthe advantages thatno poweris dissipatedin
The Design of Radio-Frequencyand Microwave Amplifiers and Oscillators
433
RL
Rr=R"; Y= 742-Y-1 Zr,=(N+M)R, Z"_,:2\" Gr:M'' M=4' Gr= 12dB
Figure 10.47
The impedance-matchingtransformer configuration patentedby Norton and Podell [0}
presentedby the circuit the terminationsof directionalcouplersand that the impedances to the transistor tend to approximatethose required for optimum output power in a common-emitter or a common-baseconfiguration at low currents and at the lower frequencies. Thehigh loadimpedance atthecollector(l5Rrwhentheterminationsareequaland when the transformeris designedfor an amplifier gain of 12 dB) tendsto limit the bandwidthobtainablewith this arrangement. The isolationis also not as good as that obtainablewith the Q-bit circuit,but a two-stagedesigncanbe usedto improvethis (refer to Figure10.48). A highergain cascadeversionof this type of amplifieris shownin Figure 10.48. The gain claimedfor this amplifieris l9 dB overthe bandwidth70-570MH2. The singletransistorversionof this amplifier(seeFigure10.49)canalsobe used with the transistor in a common-emitterconfigurationinstead of a common-base is obtainedby simplyrotatingtheemitter/winding configuration( thecorrectconfiguration combinationto be the commonbranch).A transformerwill be requiredon the input to transformthe high input impedancedownwardasshownby Rohdein [11]. However,the (rt", : rt t' + 9 g^) input impedancewill be a strongerfunctionof thetransistorparameters (negligible) with configuration with its low input than was the case the common-base impedance.
Flgure 10.48
A cascadeexampleofa high dynamicrangeamplifier using losslessfeedbackbasedon the impedance-matching transformerprinciple [2].
I
t
434
Designof RF and Microwave Amplifiers and Oscillators
r I
l
1rl Figure 10.,19
A different variation of the impedance-matching transformerlosslessfeedbackamplifier
t8t.
;
A better alternativewould probably be to use the original configwation with an input transformerto providean additionaldegreeof freedomon the designparameters, if required. An interestingvariationon losslessfeedbackwith transformers wasalsointroduced by Rohdein [ 2]. In thisvariationtheloadcurrentis sampledandfedbackasa current,and the outputvoltage(actuallythe voltageacrossthetransistor)is sampledandfed backasa voltagein serieswith the input (current-shunt andvoltage-series feedback).The circuit is shownin Figure10.50. If widebandperformance is required,thebestchoiceseemsto bea modifiedversion of theNorton couplercircuit (seeFigureI 0.5I ). TheNortoncouplercircuitwasoriginally investigatedfor bipolartransistorsonly andwasconsideredto be a goodsolutiononly if thetransistorto be usedhadinput andoutputimpedances thatwerecloselymatchedto the terminationspresentedto the couplercircuit. However,excellentresultscanbe obtained by using FETs (capacitiveinput impedance;resistiveoutputimpedance)in this circuit. While the original couplercircuit usedtwo identicaltransformers,it was found that better resultscould be obtainedby increasingthe turns ratio for the input transformer,that is, whena FET is used(analtemativeis to increasethesourceimpedance). This circuitis also not very sensitiveto reductionofthe couplingfactorby leakageflux, anda simpleshunt capacitorcanbe usedto compensate for the effectas long asthe couplingfactorremains
Figure 10.50
Another losslessfeedbackamplifier configurationintroducedby Rohde[2].
The Designof Radio-Frequencyand Microwave Amplifiers and Oscillators
435
4
ouT eT
T
RFC N2
r'{l
-. |
P " *
Pl
_
Figure 10.51
1"-
|
A modified versionof tle Norton coupleramplifier [],
fairly good (& > 0.9). The isolationof the modifiedNorton couplercircuit alsoturns out to be muchbetterthanexpected. It shouldbe notedthat the magnetizinginductancerequiredin a modifiedNorton coupleramplifier is not only a functionof the terminationsand the lowestfrequencyat which acceptableperformanceis required,but is alsoa functionof the transconductance also ofthe transistorto beused.Thetransconductance andtheinput andoutputimpedances (reverse gain) values are Lowertransconductance ofthe amplifier. the isolation determines isolation. associated with better obtainablewith themodifiedNortoncouplercircuitis impressive. Theperformance Assuming a coupling factor of unity and no interwinding capacitance,an amplifier was designed over the passband10 MHz to 1 GHz. The expectedpower gain was approximately10.49 dB, the l-dB compressionpoint was close to 23 dBm, and the isolationwasbetterthan 19.7dB.Theexpectedefficiencywasaround39Vo.Theexpected input VSWR wassmallerthan I .81overthewholebandandlessthan I .5 up to 625MHz. The expectedoutputVSWR was smallerthan 1.5overthe wholeband. The performanceof a manufacturedprototype tumed out to be close to that whichwasreducedto around500MHz. predictedexceptfor theupperendofthe passband, A modificationto the circuit wasalsorequiredto eliminateoscillationsabove2GHz. The transistorwasbiasedwith an activebiasingcircuit. An activebiasingcircuit suitablefor FETs is shownin Fisure 10.52. +vn
_T_-.-T--
tI < l _
ZTFF
- . t i + | +
, ,
t
- l _ \ -
t Figure10.52
l l
l H
I'r l
-vs
An activebiasingcircuitsuitablefor FETs.
Design of RF and Microwave Amplifiers and Oscillators
10.1T REF'LECTIONAMPLIFIERS
t
lF t 6
t
ir
rr
At very high frequencies,Impaff, Gunn, and tunnel diodes m also used to provide arnplification. These negative resistancesingle-port devices are usually used in combinationwith circulatorsand occasionallywith 3-dB hybrid couplers.Only the circulator-typewill be consideredhere. The S-parametermatrix of an ideal circulator is given by
"=f? :I L 0 r 0 l
(l 0.r33) t.
!-
This impliesthat
F
t l
b I
h
I I
tI
r I
it .
r
LI] [;]
(10.134)
and,therefore,the energyincident at port I is alwaysdeliveredto the loadconnectedto port2,the energyincidentat port 2 to the loadconnected to port 3, andtheenergyincident at port 3 to the load connectedto port L Consequently,the energy is propagatedin a circularfashionaroundthe circulator;hencethe namecirculator.Theserelationshipsare illustratedin Figure 10.53.The configurationof a circulator-typereflectionamplifier is shownin Figure10.54.
I
tl I
I a
iI
I
F Ftgure 10.53
The relationships betrveenthi normalized incident and rrflected componentsofan ideal circulator.
437
The Design of Radio-Frequencyand Microwave Amplifiers and Oscillators
* 6 ,= S -
4
ar=fyb2
am=bz-
bz: at
but= az -
-
4@
Matching -
Network
7
zD
I 1,,=o =42
I
Figure 10.54
The configurationofa circulator-typereflectionamplifier.
The transducerpower gain of the amplifier is definedby
G'- =
Pt
( 10.13s)
P*-t
whereP"u-r,isthe power availablefrom the source.By usingthe relationshipsshownin Figure 10.54,it follows that
r. t2
G', = l b t l -
lo'l' -lo'l'
lu'l' -lb''l' lou'l' -lb"l' t
t
lauzl
Z
--lz"*-zL l4"J z,
(l 0.136)
438
Design of RF and Microwave Amplifien and Oscillators
vrhereZo = - RD+ jxDis the impedanceof thenegativeresistance diode,andZr' indicates its conjugate.
Gru= | - llGr
Figure 10.55
Zn= Ro+jXo
The matchingproblemto be solvedwhen the amplifier in Figure 10.54is designed.
Equation(10.136)canbe manipulatedin the following way:
'12 = 1 1 1 2 . * - ( nf io) + lZour+(Rr+fi)l
= t / lfr*l'
(10.137)
whereIr* is the reflectionparameterofthe networkshownin Figure 10.55with the source Theproblemof maximizingthe andload impedances shownasnormalizingimpedances. gain of a circulator-typereflectionamplifier,therefore,is equivalentto minimizing the mismatchbetweenthe sourceandthe load shownin Figure 10.55. Whenthe amplifieris designedto havea specifiedgainversusfrequencyresponse, the gain of the equivalentmatchingnetwork shouldbe Gru=l-I/Gr where G. is the transducerpower gain specificationfor the reflection amplifier.
(l 0.l 38)
The Design of Radio-Frequencyand MicrowaveAmplifiers and Oscillators
439
Table l0.lE The specificationsfor the outputmatchingnetwork of the reflectionamplifier Frequency
Sourceimpedance
Load impedance
(GHz)
(o)
(o)
7.0
50.0+J'0.00 50.0+70.00 50.0+70.00 50.0+J0.00 50.0+/0.00
10.0+ j3.0 12.0+j7.0 15.0+j10.0 19.0+jI3.0 25.0+jI5.0
t.)
8.0 8.5 9.0
EXAMPLE 10.7
Transducerpower gain
U.9UO 0.900 0.900 0.900 0.900
Designinga matchingnetwork for a reflection amplifier.
As an exampleof designingthe matchingnetwork of a reflectionamplifier, a matchingnetworkwill bedesignedfor a Gunndiode(IWA-COM,MA-491l0) with input impedancecorrespondingto the load impedanceof the corresponding equivalentmatchingproblem(asgivenin Table 10.I 8) anda gainof l0 dB across the passband7-9 GHz. With the requiredtransducerpower gain equalto l0 dB, the transducer power gain of the equivalentmatchingproblemis foundto be Gru =l-l/Gr = 1- 1/10.0 = 0.90
0.67nH
Figure 10.56
The designedmatchingnetwork for tle reflection amplifier of Example 10.7.
The specificationsof the equivalentmatchingproblemis shownin Table 10.18. Thedesignedmatchingnetworkis shownin Figure10.56.Themaximumdeviation from the specified gain responseis 0.16 dB, and the transformationQ-factors corresponding to the solutionare I . I 83, 1.506,and0.5I 1, respectively.
l
tn
Designof RF and Microwave Amplifiers and Oscillators
r0.12
BALANCED AMPLIFIERS
Srln a balancedamplifier, the input signalis split into two or moreamplifiers,andthe output :nals of theseamplifiers are combinedto a single load, with isolation betweenthe Ddividual amplifierportsin bothcases.Themostcommonlyusedconfigurationis shown b Figure10.57. The S-parameter matrix of a 3-dB, 90" hybrid divider is given by [13], with the prts numberedas in Figure 10.57. For the divider, the energyincidentat port l, therefore,is deliveredto the loads coorcted to ports2 and3 with a 90' phaseshiftbetweenthetwo components. Theenergy :-cident at ports2 and3 in the combineris routedto port l, againwith a 90' phaseshift -:iseenthe two components.
l Su
outpl refle indiv ampl ampl
Gr=
j '-l [o =0.107 li o ol
(10.139)
L l 0 o J
Gr=
I hd ftd for a 3-dB, 90'hybrid combinerby
whicl
[t o o- i 1l
sr=0.707 0 rl 10
(lo.t4o)
li 1oJ
t -
The S-parametermatrix of the amplifier is given in termsof theS-parameters of the . nro individual amplifiersby [13]
I
indiv only r
be re< (10. decid
10.
3-dB 90" hybrid coupler
3-dB 90" hybrid
Oscil or the gain. at sta to ens
feedb 10.5
ftrrc
10.57
The most commonly usedbalancedamplifier configuration.
to the first n the se
The Designof Radio-Frequencyand Microwave Amplifiers and Oscillators
S r = 0.,I
srr,r- s,,.2 ,l(srz,r+ s r r r ) l
[f(szr,r+ szr,z) sz2,l*
Snl
I
441
(l0.l4l)
I
It is clearfrom this equationthat if amplifiersI and2 areidentical,the input and outputreflectionparameters of thebalancedamplifierwill beequalto zero,evenwhenthe reflectionpararneters of the individual amplifiersare not equalto zero.As long as the individual amplifiersare almost identical,the input and output VSWRs of a balanced amplifier will thereforebe very low, independentof the vSWRs of the individual amplifiers. The tansducer power gain of the balancedamplifier is given by
Gr = 0.25''s rr,,* r rr,rl'
(r0.r42\
When the individual amplifiers are identical, this reducesto t P ur = lszr,rl
(10.143)
which is identicalto the gain of a singleamplifier. Although the gain of the balancedamplifier is thereforeidenticalto that of each individual amplifier in the idealcase,the outputpower is twice that obtainableby using only a single-endedstage. Shouldoneof the amplifierscomprisingthe balancedamplifierfail, the gainwill reduced be to one-fourthofits originalvalue.This canbe provedeasilyby settingszr,rin (10.142)equalto zero.Insomeapplications thisadvantage canbeanimportantfactorwhen decidingwhethera balancedor single-ended amplifiershouldbe used.
10.13 OSCILLATORDESIGN Oscillatorscanbe designedby controllingthe reflectioncoefficient(negativeresistance) or theloop gainof thetransistor[1- 4]. Thebetteraltemativeusuallyis to controlthe loop gain.At steady-state,both conditionswill be satisfied,but this doesnot necessarilyfollow at start-up.Independentofhow the designwasdone,both conditionsshouldbe checked to ensurethat spuriousoscillationswill not occur. The two basicoscillatorconfigurationsareshownin Figure 10.58.Voltage-shunt feedbackis used in Figure 10.58(a),while current-seriesfeedbackis used in Figure 10.58(b). In orderto control the outputpowerof an oscillator,the loadterminationpresented to thetransistorshouldbe controlledtoo. Theloadterminationcanbe controlledeasilyby fust modifring thebasicconfigurationsto thoseshownin Figure10.59[4]. In thecaseof the seriesfeedback,the original groundconnectionwasfloatedanda virtual groundwas
442
Designof RF and Microwave Amplifiers and Oscillators
(b) Figure 10.58
The basic configurationsfor oscillatorswith (a) shuntfeedbackand (b) seriesfeedback.
introduced.No physical changeis requiredin the shuntfeedbackcircuit. In the seriesfeedbackcase,any transmissionlinesusedshouldfirst be converted to lumpedT- or Pl-sectionequivalentsbeforethe groundconnectionis changed. Any extensionlines shouldbe kept as short as possible.The extra phaseshift aroundthe loop will reducethefrequencyrangeoverwhich oscillationis possibleandwill alsoincreasethe start-uptime. A simplifiedflow diagramofthe oscillatordesignprocessis shownin Figure10.60. In order to control the output power,power contourscan be generatedfor the transistorby using the power parameterapproachdescribedin Chapter2 or by using a nonlinearsimulator.A suitableload line can then be selected,after which a feedback networkcanbedesignedto providethis loadterminationto thetransistorwith theloop gain shouldbecontrolled,butexcellentresultscan required.Ideally,theloadline at steady-state if the loop gainrequiredis low. alsobe obtainedwith the small-signalparameters
The Design of Radio-Frequencyand Microwave Amplifiers and Oscillators
443
(a)
(b) *
ZLr
Zn= Zt r
(c) Figure 10.59
(a), (b) The two oscillatortopologiesshownin Figure 10.58modified for the purposeof calculating the transistorload-terminationand the loop gain [l]. (c) The voltage and impedanceat steady-state.
The output power of an oscillator will increaseinitially as it is driven harder into compression, after which it will decrease. The transistor will be driven harder into compression as the loop gain or the negative resistancein the input loop (series feedback case) is increased. The gain compressionassociatedwith the maximum effective output power can be estimatedby assumingthe power saturationcharacteristicto be governedby an exponential law function [5]. Under the assumptionsmade, this point is only a function of the smallsignal gain associatedwith the load termination chosen.The relevant equationsare derived i n S e c t i o n1 0 . 1 3 . 1 .
F
F
444
Design of RF and Microwave Amplifiers and Oscillators
Selecta transistor.
Find a suitableload termination for the transistor.
Decide on the constraintsto be imposedon the three T- or PIsection impedances(only four ofthe six parametersare requiredto control the loop gain and the load terminationofthe transistor).
Determinethe valuesofthe three impedancesin the T- or Pl-sectionfeedbacknetwork.
Design the resonatorcircuit to be used(if any).
Synthesizenetworksto realizethe impedancesrequired.
Analyze the oscillator and check for and eliminate any spuriousoscillations.
Veri! the performancewith a nonlinear simulator and optimize the performanceif an accuratenonlinear model for the transistoris available.
Figure 10.60
A simplified flow diagramof the oscillatordesignprocessoutlined [].
If the compressionrequiredis relativelylow (a few decibels),the compressionat steady-statewill be approximatelythe sameas the loop gain at start-up.In this casethe loop gain at start-upcanbe useddirectlyto forcethe transistorto its peakpowerpoint. Substantialcompressionis frequently required to extract the maximum output powerfrom an oscillator.It is importantto realizethat in thesecasesthe loadtermination presentedto the transistorand the oscillatorfrequencywill changeas the transistoris driven into compression.In order for this changeand the changein the oscillation
us
The Design of Radio-Frequencyand Microwave Amplifien and Oscillators
frequencyto be small,the conditionslistedin Section10.13.1mustapply. If theseconditionsdo not apply,a betterapproachwouldbeto makeuseofthe fact that, with a well-behavedload line, the main nonlineareffect in the transistorwould be the (G,). Thetransconductance in thesmall-signalmodel compression ofthe transconductance asrequired. canthereforebe reduceduntil the large-signaloperatinggain is compressed instead setof S-parameters Thefeedbacknetworkcanthenbedesignedwith theassociated is controlled instead load line parameters. In case the steady-state this small-signal of the of the load line at start-up. When the goal is low phase-noiseand not power, the steady-statecompression shouldbe kept low. If this is done,theconversioneffrciency(mixing effects)will be low, of theflickernoise.A well-behavedload efflecton theup-conversion with a corresponding line for the transistoris still desirableas it will preventrunning into nonlineareffects with a poor choiceof the loadline. associated is required,extracareshouldbe takento maximizethe loaded If low phase-noise ofthe oscillator.This (or the slope in thephaseofthe loop gainresponse) equivalently Q will reflecton thechoiceof theresonatorto beused,aswell astheloadline chosen(higher with higherQs).In simplecasesthe will be associated parallelor lower seriesresistance by using( I 0.44).Instead loadedQ at start-upcanbeestimatedfrom theloopgainresponse of trying to estimatethe loadedQ, abetteroption seemsto be to controlthe slopein the loop phasedirectly. The feedbacknetwork (refer to Figure 10.58)must be designedto provide the (or an approximationof requiredload line and loop gain at start-upor at steady-state
Table 10.19 An exampleof a table of the T-sectionimpedancesrequiredat a specificfrequency(3.5 GHz) as a function of the loop gain [] Loop gain
(dB)
RL
(o)
XL
Ln Cr
Lo C"
v
(o)
(o)
(nH, pF)
(o)
(nH, pF)
-1.952 -2.t90 -2.457 -2.757 -3.093 -3.471 - 3.894
4.651nH 4.665nH 4.677nH 4.691nH 4.706nH 4.723nH 4.743nH
t02.356 t02.594 I 02.861 t 0 3 l.6 l 103.494 103.875 104.298
-4.902 -5.500 -6.172 -6.925 -7.770 - 8.718
4.81n 6H 4.846nH 4.881nH 4.919nH 4.962nH
105.905 106.576 r0'1.329 108.t74 109.122
-0.0927 0.9073 1.9073 2.9073 3.9073 4.9073 5.9073
49.620 49.5'73 49.521 49.463 49.39'l 49.324 49.24r
l.808 2.029 2.277 2.555 2.866 3.216 3.608
23.300pF 20.765pF pF 18.508 pF 16.495 pF 14.701 1 3 . 1 0p3F
7.9073 8.9073 9.9013 10.9073 I 1.9073 t2.9073
49.015 48.928 48.798 48.651 48.486 48.30t
5.097 5.719 6.417 7.200 8.078
8.267pF 7.368pF 6.567pF 5.853pF 5.216pF
@trc
xF
11 . 6 7 8
Note: The highlighted loop gain is equal to the estimatedcompressionrequiredto maximize the output power.
u6
b
r
Design of RF and Microwave Amplifiers and Oscillators
steady-state).Two of the three impedances(seriescase)or admittances(shunt feedback case)areusuallyassumedto bepurelyreactive(i.e.,at leastduringthe initial stagesof the design),while the outputpoweris extractedfrom the third impedanceor admittance. ofthe transistoris alsoknown, Becausetheloadline is known,theinputimpedance andit follows that the terminationsfor the T- or Pl-sectionfeedbackareknown.With the terminationsand the gain of the transistorknown, equationscan be derived for the componentsthat will provide the required loop gain, as well as the required load termination.This is donein Section10.13.2[1,. An exampleof a tableof the2,.,Zr, andZ valuesrequired(seriesfeedbackcase) GHz to realizedifferentvaluesof the loop gainanda specifiedload terminationis at 3.5 given in Table10.19.In this edsa,ZpandZ, were chosento be purely reactive.The highlightedloop gain is equalto the estimatedcompressionrequiredto maximizethe output power. for this oscillatorfrom 3.5to Table 10.20givesthe2,.,Zr, andZ valuesgenerated 4.5 GHzafterselectingthe loop gainestimatedfor peakpower.Therequiredterminations are displayedon a Smith Chart in Figure 10.61.Table 10.19showsthe T-section requiredat a specificfrequencyasa functionof the loop gain. impedances to be usedmustrotate Note thatthe tracefor at leastoneof the setsof impedances aroundthe Smith Chartin orderto ensurefrequencystability (i.e.,the counterclockwise oscillatormust lock at the frequencyof interestand not drift aroundin frequency).Such will be referredto asof varactortype. impedances The equivalentstatementin termsof the loop phaseversusfrequencyresponse (displayedon a rectangularplot) is thatthephasetracemustpassthroughzerowithoutany jiner and must not crossthe zero-degree line againbeforethe loop gain is too low for oscillation. With the T- or Pl-sectionimpedancesknown over the frequencyrangeof interest, networksmust be synthesizedto approximateeachof the impedancesover the frequency rangeof interest.Onewould generallyselecta combinationthatwouldresultin onefixedvalued component,a varactor,or a resonatorcircuit and a complex impedance(to be network). realizedwith an impedance-matching oscillator(VCO) is designed,betterresultscanusually Whena voltage-controlled network. be obtainedwith t'wovaractorsandoneimpedance-matching The impedanceassociatedwith the load terminationis often takento be the actual network load (50O),but this is clearlynot optimum.In general,an impedance-matching is requiredto realizethe impedancerequired. The reactancesrequired can be realizedwith capacitors,inductors,transmission ThedesignofhighQ dependingon therequirements. lines,varactordiodes,or resonators, resonatornetworksis consideredin Section10.13.3,while that of varactornetworksis in Section10.13.5. considered If a resonatoris used,the resonatorimpedancemustbe transformedto presentthe impedancerequiredat the relevantposition. This can often be doneby simply using a impedanceandlength.This is illustrated transmissionline with the correctcharacteristic in Section10.13.4. Onewould generallyuseseriestunedvaractornetworksin a seriesfeedbackoscil-
447
The Design of Radio-Frequencyand Microwave Amplifiers and Oscillators
Table 10.20 An exampleof a table of the T-sectionimpedancesrequiredto providethe specifiedload termination and the specifiedloop gain over the oscillationband (VCO with two varactors)[] Frequency (GHz)
3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50
L"'C'
x"
(o)
(nH, pF)
(o)
-4.369 -4.430 -4.497 -4.564 -4.639 - 4.715 -4.820 -4.935 -5.046 -5.166 -5.290
4.764nH 4.452nH 4.1't'tnH 3.916nH 3.684nH 3.464nH 3.273nH 3.095nH 2.925nH 2.772nH 2.630nH
104.773 100.707 97.O97 93.492 90.275 87.051 84.305 81.683 79.038 76.645 74.352
XL
L1,C1
XF
(cl)
(0)
(nH, pF)
49.149 49.114 49.078 49.041 49.004 48.965 48.923 48.878 48.832 48.786 48.738
4.049 4.085 4.126 4.165 4.212 4.257 4.327 4.403 4.475 4.552 4.632
pF 10.408 9.979pF 9.565pF 9.178pf 8.797pF 8.438pF 8.053pF 7.679pF 7.335pF 7.002pF 6.68spf
RL
lator and parallel tuned networks in a shunt feedbackoscillator. The particular choice would dependon the componentvaluesand the behavioroutsidethe oscillationband. Whena seriestunednetworkis usedin a shuntfeedbackoscillator,andvice versa,losses in the varactornetwork could havea seriousstabilizingeffect on the circuit. If sucha choicewas made,be sureto checkthe effect of suchlosseson the performanceof the circuit.
MEASSYreE lru.4.ffi
Figure 10.61
ml: ru2
A.@ $.6
The T-sectionimpedancesin Table 10.20displayedon a SmithChart[]. Note that at least one of the setsof impedancesshouldrotatecounterclockwisearoundthe Smith Chart to ensurefrequencystability (Zo in this case).
448
Design of RF and Microwave Amplifiers and Oscillators
Care shouldbe takenwhen decidingon the impedanceto be approximatedwith a fixed capacitoror inductor.Ideally,thechoicemadeshouldresultin a topologythatcannot sustainoscillationsat very low or very high frequencies. Whensuitablenetworkshavebeenfitted to the targetimpedances, the oscillator shouldbe analyzedto confirm its performanceandto checkfor anyspuriousoscillations. Becauseloopsmay be present,the analysisshouldbe donefairly densely.Both the loop gain andthe reflectiongainperformanceshouldbe checked. If an accuratenonlinearmodel for the transistorusedis available,the oscillator performanceshouldbe verified andoptimizedwith a nonlinearsimulator. An exampleof a dielectricresonatoroscillator(DRO) designedasdescribedhere is shown in Figure 10.62(Courtesyof PlesseyAvionics, Retreat,SouthAfrica). The topologyis shownin Figure 10.63.Theoscillatorwasdesignedto oscillateat 15.65GHz with theoutputpowerhigherthan l0 dBm (Biaspoint: 2V,20 mA). Theperformance was in the supplyvoltageandthepuck position. realizedwith slight adjustrnents Note that becausea nonlinearmodel for the transistorusedwas not available,a nonlinearsimulatorwasnot used. The loop gainperformance ofthe oscillatoris shownin Figure10.64.Oscillations seemto be possiblearound6 GHz too. However,a modificationwas madeto the basic oscillatorcircuit (a gap capacitorwas insertedbetweenthe transistorandthe resonator circuit) to delaythe changein the loop phasein this area,andthe gain margin in this case
Figure 10.62
AnexampleofaDROoscillator(CourtesyofPlesseyAvionics, Reheat,SouthAfrica). The oscillationfrequencyis 15.65GHz andthe outputpoweris aroundI1.6 dBm. The puck is coupled to a line connectedto the gate ofthe transistor.
The Design of Radio-Frequencyand Microwave Amplifiers and Oscillators
Thc currnt-senes feedbackapplied to thc two sources
The gap capacitor used to eliminate the spurious oscillations at 6GHz
Figure 10.63
The schematicof the oscillatorshownin Figure 10.62[l].
450
Design of RF and Microwave Amplifiers and Oscillators
DROOSC
8i l:1999 8:al:13
dE 3!.00
r95.0
22.20
165.0
n
11.40
135.0
6.607
105.0
- 1.190
75.00
v\,
Lsp_G
PHASE
- 16.70
15.0O
- 24.5t
- 15.00
- 3:L3E
- ,15.00
t
"l V
-,o.t8 - 47.96 0.100
9.mo
3.000
- 75.d!
V 12.250
- 105.0
20.000
FREO(GHz)
Fos15.6435GH2 Gloop-6.9€ t dB
66..Eg5.2tS5.rGHr
(a)
DROOSC
8: t:1999 8:4'.49
dB 25.O
,{0.0o
21.U
31.50
'tE.67
23.00
15.51
14.50
12_y
6.000
tfip_G
PHASE
6.010
- | 1.00
2.U5
- 19.50
---\
- 0.320
- 28.00
\
- 3.,184 - 6.6,19
'15.000
15.500
16.m0
16.500
>a \ 17.OOO
.,'
./
- 36.50 - 45.00
18.000
FREO(GHz) F6c=l 5.6,135GH2 GloP=5.96169
64..tg5.2t55'lGHz
(b) Figure 10.64
(a), (b) The theoreticalloop gain andphaseofthe oscillatorshownin Figure10.62[l].
The Design of Radio-Frequencyand Microwave Amplifrers and Oscillators
451
is actually quite large.Interestinglyenough,the circuit doesoscillate around6 GHz if the the fact that changeintroducedis not made.While the spuriousoscillationis undesirable, with relative ease servesas be it canbe predictedwith suchaccuracyandcan eliminated a validationfor the loop gain approach. The spurious oscillation can also be eliminated by using a different (more expensive)transistor.
Associatedwith the 10.13.1 Estimationof the Compression Maximum EffectiveOutput Power If the power gain of a transistoris consideredasa function of the drive level, it is clearthat the gain is equalto the small-signaloperatingpowergain (G^) whenthe input power is low andthe outputpowerwill approachthe saturationlimit whenthe input poweris high (seeFigure 10.65).Assumingthe transitionto be exponential,the outputpowercould be describedby the following equation[5]: /P* Pou,= Pr* [1 - e-G* 4" 1
(10.144)
Pin t
-l
Figure 10.65
Typical saturationcharacteristicsfor a transistor.
452
Design of RF and Microwave Amplifien and Oscillators
Themaximumeffectiveoutputpower(Pou,- P;")is deliveredby thetransistorwhen
l
l i
r l
0(P"*- 4") -,^, 0P^ that is, when dPou,-,
oPn aboveyields Applyingthisto theequation (10.r45)
4" = P."[n(G.")/G."] and Poot_-o = P.", (l - I / G.")
",
(10.146)
from which it follows that - p rDo s c m a - ' o u t
_ p ma< 'in
= P,",[1-l I G,, - ln(G.,) / G,"]
(10.147)
The correspondingvalue of the large-signaloperatingpower gain (G"y)at this maximum effectiveoutputpowerpoint is givenby G.t = Pou,_.o/P^ = (G," - l) / ln(G,")
(10.148)
The ratio of the small-signalandthe large-signaloperatingpowergainis therefore G,, I G,r = [G." I (G,, - l)]ln(G,")
(10.149)
a first order with a setof small-signalS-parameters, If an oscillatoris synthesized power is possible output gain in the maximum will result that the loop for approximation is, ratio, that root of this the square
tc-
=.1# G*r-oo, s<,ll U
453
The Designof Radio-Frequencyand Microwave Amplifiers and Oscillators
=m
(lo.lso)
This approximation would apply to the degreethat the following conditions [1] applv.
SeriesFeedbackCase
lz" +Zr + znlr,lzrl l,"_^.ol=
(l0.lsl)
and
lz, /G,^o_*lrrl"rrl
(10.l s2)
Shunt Feedback Case
lr"+ Yr+lnltt lrrl lr,_,*rl=
(10.I s3)
and
ll,rl lr" / G*o_"o,ltr 10.13.2
(l0.ls4)
Derivation of the Equations for the T- and Pl-Section FeedbackComponentsRequired
to be parameters in Figure10.66assumed of thetransistor Withthetransmission
ft A, ' 8,1l lC, D,)
it followsthat[1] Li-
"'
and
--
A,2,, + B, C,Zr, + D,
(10.155)
454
Designof RF and MicrowaveAmplifiers and Oscillators
Figure 10.66
/-
'
The T-sectionnetwork in an oscillatorwith seriesfeedback.
Z2lt
-
(10.l s6)
zrr, +Z u
tvhercZuis the specifiedloadterminationfor the transistor. Zuis givenin termsof Z,nandthe T-sectionimpedances by
Zo= Zr+
-- Ztt
Y,+zi-^ Z,+Z^
(10.157)
l+Yo(Z,+Zio)
while the loop gain (G1""0) is given by r r Z G,oorlr13= ZteJ
-
zJ
A,It l+ Yr(Z, + Z,n)
I,
( 1 0 .s18 )
from which it follows that z _ L F --
Zr+Zro
---7-
nt
+l Gtoop
(10. rse)
The Designof Radio-Frequencyand Microwave Amplifiers and Oscillators
455
This expressioncanbe written as Zin Z "= =A - ' a*Z "l A- ' * l Gtooo
(10.160)
G,ooo
Equation(10.159)canbe usedto simplifr (10.157): 7
zu=Zr+-*
t-7
t , _ lA , * l \. Gtooo
)
- z "L -A9 rt r , + z i n )
(10.161)
which can be reananged to give an expression for Zrin terms of Z:
z -Gro z r "| : - 2 . A,
* r , (t , I
,G,oo zJ,, -r ----Lin .ti.r \
or') /
(10.r62)
I
Equations(10.162)and(10.160)canbe combinedto eliminateZ,and givean expression for Z.interms of Zr,:
_ LF
,,-(r..Tt")
1
= -----
_
n'
G,ooo
+l
G,ooo
AI
Zin
A' *l G,-o
(10.163)
which canbe simplifiedto f
n
"
l
Z"r = - l l + - ' 9 o o lzr*zo L A 'l '
(10.164)
Equations (10.160), (10.162),and (10.164) can be used to solve for the required cornponentsonce the relevant constraints on 26 2,, or Z,,have been established.The real parts of two of these impedancesare usually assumedto be z,eroor to be fixed, after which these equations can be solved for the remaining values.
Design of RF and Microwave Amplifiers and Oscillators
l4# YLL
Y2
The Pl-sectionnetwork in an oscillatorwith parallel feedback.
Figure 10.67
The derivation of the equationsfor the Pl-sectioncomponentsproceedssimilarly to that for the T-sectioncomponents. With the transmissionparameters of the transistorin Figure10.67assumedto be
f ,q. 8.1 t ' l lC, D,) it followsthat 'u^ -- C , Z r r + D , 1,7ua4
-
r'i':
(10.16s)
and -' L =
(10.r66)
| A, + BTYLL
wfure Y2 is the specifiedload terminationfor the transistor. I'r, is given in termsof lln, andthe Pl-sectionimpedances by
Yu =Y, +
z'. +
| \+Y,^
v , v
-- W - t i - t
l
"
^S
-ln
ZF(I,+)'*)+l
(r0.r67)
The Design of Radio-Frequencyand Microwave Amplifiers and Oscillators
457
while the loop gain (G6) is given by
v2 rl-vl o o p z l -t r - - F
Yr+Yrn
-::--:* Zr
AuVt
l+ zr(r" +rr")
(10.l 68)
Y, +Yin
from which it follows that Yr*Yrn u -_ --Z'r "
v
(l0.l6e)
I
G,*o Equation(10.169)canbewrittenas
r
r
Ye=TY,+_zn u - r + - t G,ooo
(lo.l7o)
G,ooo
which providesa simpleexpressionfor I,, in termsof )/.. Equation(10.169)canalsobe usedto simpli$ (10.167):
v
- I /
"
Y+Y
' J ' ' i n
,
Zr(\ + I/,,)+ I
=yr*7-I:+ u'-rl+r I Gto.o \
)
n e-t-o 9ooo =Y, 'L * f- * -f*r n Au Au
(10.171)
which can be rcaranged to provide an exprcssion for 11 in terms of {:
,,=-Tx*(t--Tt^)
(r0.172)
458
Design of RF and Microwave Amplifien and Oscillators
Equations(10.170)and(10.172)canbe combinedto eliminateI", which will give an expressionfor I, in termsof Io too:
qlo -tlr, +ru r,- = | t's-
\
\ 1 "
) '
(10.173)
Equations(10.170),(10.172),and(10.173)canbe usedto solvefor the required The real componentsoncethe relevantconstraintson Yb Y",or Ir havebeenestablished. partsof two of theseimpedancesareusually assumedto be zeroor to be fixed, after which theseequationscanbe solvedfor the remainingvalues.
10.13.3 lJigh QResonatorCircuits High Qresonatorcircuits canbe realiz-edbyusing dielectric resonators,cavity resonators or a magneticallybiasedyittrium iron garnet(YIG) sphere. TheYIG resonatoris a highQ, fernitesphereof yittrium iron gamet(Y2Fe2(FeOa)3) that can be tuned over a wide band by varying the biasing dc field. In a YlG-tuned oscillator,a YIG sphereis normallyusedto controltheinductanceof a coil in theresonant circuit. BecauseYIG is a ferri-magneticmaterial, its effective permeabilitycan be controlledwith anextemaldcmagneticfield,thuscontolling theoscillatorfrequency.YIG oscillatorscanbemadeto tuneovermorethana decadeof bandwidth,whilevaractor-tuned oscillatorsarelimited to a tuningrangeof aboutan octave[3]. areusuallyrealizedwith lowloss coaxialline or waveguide.The Cavityresonators simplestcoaxialcavity is a quarterwavelength(V4) shortedstub.The signalis coupled into the cavity with a shortedloop or an openprobe.The resonantfrequencyis usually adjustedwith a tuningscrewneartheopenend.It canbe shown[3] thattheimpedancenear the resonancefrequencyandthe Q ofthe resonatoraregivenby zin =
o*
,, _al+ I n _Arrl I^ Z o ' 2 a 0 "
n - P 4al
2a
(10.r74)
(10.175)
Equations(10.174)and(10.175)arederivedby startingwith theexpression
Za=Zotanh[(cr+jp)4
(10.176)
The Designof Radio-Frequencyand MicrowaveAmplifiers and Osciltators
459
Open-circuited)'/2 resonatorsare often usedon microstrip. The input impedancc andQ ,rnthis casearegiven by [3]
Za=
oo
al Arrl -, _ + Jn-Io Lo oo
T - B 2al 2a
(10.177)
(10.178)
The width and length of the smallestrectangularwaveguidecavity is )rlT (Tgrcl rnode).Therectangularcavityis a waveguideversionof a short-circuited .1./2 transmissionline resonator[3]. Becauseof the smallsizeandlow cost,dielectricresonators arefrequentlyusedat microwavefrequencies.The high dielectricconstantof the resonatorpuck ensuresthat mostof the fields arecontainedwithin the dielectric,but thereis somefringing from the sidesandendsof thepuck.The fringing fieldsprovidea convenientmeansof couplingto a microstripline. The spacingbetweenthe puck andthemicrostripconductordetermines the amountof coupling. Only dielectriclossesarepresentin the puck, and Qs ofseveral thousandcanbe realized.Metallicshieldingis requiredto minimizeradiationlosses.Thep canbeincreased by with a dielectricspacerunderthepuck. The resonantfrequencyofthe puck canbe adjustedby usingan adjustablemetal plateaboveit. Thelowestorderresonantmodefor a dielectricpuckis theTE616 mode.This mode coupleseasilyto a microstripline. Theresonantfrequencyfor a puck canbe estimatedby solvingthe following transcendental equationiteratively[3]:
BL - -cr
tan'
2
(10.l 7e)
p
where
2.40s\2 "
)
- \(2.+os\' " )
(l 0.l 80)
( 1 0l.8 l )
m
F
Design of RF and Microwave Amplifiers and Oscillators
Figure 10.68
,
Two commonly useddielectricresonatoroscillatorconfigurations.
2nf
( 1 0 .8 12 )
T o = -
c
In theseequationg/is the requiredresonantfrequency,e,is the dielectricconstant . of the puck material,l, the heightof the puck,andc its radius. The resonantfrequencymustlie in the interval[fr,fr],where [3]
2.405c ir=7-TzltiEra
(l 0.183)
and
, _2.405c J2 - --:zna
(10.184)
Equations(10.183)and (10.184)are necessary conditionsto ensurethat the roots in (10.180)and(l0.l8l) canbetaken. The unloadedQ of a dielectricpuck canbe estimatedas [3] I
Q=-
(10.l 85)
tand
Two commonlyusedDRO configurations are shownin Figure 10.68.The oquivalent circuitsassociated with thecoupledsections areshownin Figure10.69.
t
10.13.4
Transforming the ImpedancePresentedby a Resonator Network to That Required in the T- or Pl-section Feedback Network
Whena resonatoris used,the impedance presented by the resonatormustusrally be transformed to presentthetargetimpedance in theT- or Pl-sectionfeedback network.This
The Design of Radio-Frequenryand MicrowaveAmplifiers and Oscillators
461
(a)
f----------!.'€
(b) Figure 10.69
The equivalentcircuits associatedwith (a) a dielectricpuck coupledto a microstrip line and (b) a puck usedto coupletwo microstrip lines.
can often be done by simply using a transmissionline with the correct characteristic impedanceandlength. Theprincipleis illustratedin Figures10.70andI 0.71below.Thetraceon therighthand side of Figure 10.71(traceA; the smallerarc) is the measuredimpedanceof a dielectricresonatorpuck coupledto a microstripline (15.6-15.7 GHz).Theimpedanceat theoscillationfrequency(15.65GHz)is in theareaof a secondmarkershownon this trace (point a). The targetimpedancein the T-sectionfeedbacknetworkdesignedis in the area ofthe cursordisplayedon theleft-handsideof this figure(traceB, point b). Themeasured impedancewas transformedto this point by cascadinga transmissionline with suitable characteristicimpedanceand lengthto the resonatorcircuit asshownin Figure 10.70.
l":-l
l o l 24.9pF
(b) Figure 10.70
(a) The transmissionline used to transform the impedancepresentedby a dielectric resonatorpuck coupledto a microstripline asshownin Figure I 0.69(electricalline length specifiedat I 5.65 GHz). (b) The equivalentcircuit fitted to resonatorcircuit.
62
Desigr of RF and Microwave Amplifiers and Oscillators
DRO RES Soludd! I 2 3 6: l:1909 8:5lr5,l
B Slr
FREAUENCYMNGE ,|5.600- 15.700c1'lz
Figure 10.71
50.m 0.tm
An exampleof transformingthe impedancepresentedby a resonatorcircuit to the target impedancein the T-sectionfeedbacknetwork at the oscillationfrequency[l].
The equivalent circuit fitted to the resonatorcircuit is shown in Figure 10.700). If wideband measurementsare not available, an equivalent circuit can be used to check for spurious oscillations. The size ofthe modified resonator loop is a function ofthe characteristic impedance chosen. Note that if the characteristic impedance used was 50O, the original loop will simply be rotated around the origin as the line length is increased.This follows becausethe electrical length ofthe line added is basically constant over the n€rrow frequency range considered.
10.13.5
Designing Varactor Circuits to Realizethe Varactor-Type ReactanceRequired
on theSmithChart) (impedance rotatingcounterclockwise reactances Thevaractor-type networks. with varactor realized must be network feedback in theT- orPl-section required The designof the varactornetworkusuallyconsistsof finding a varactorwith a tuning range bigger than the minimum requiredand calculatingthe loading capacitanceor inductancerequired. The loading inductanceor capacitancerequired is usually realized with lumped elements. Theeffect ofthe parasiticinductanceofthe varactordiodeshouldbe includedwhen the tuning circuit is designed.
The Desigr of Radio'Frcquency and Microwave Amplifien and Oscillators
463
(a) Figure 10.72
Loading a varactor with (a) a parallel inductor and (b) a seriesinductor to obtain the series reactancerequiredat the passbandedges.
The basicvaractornetworksareshownin Figure 10.72. ratio of With a maximumachievablevaractorcapacitance (l 0.l 86)
Pr.o=cr,.,t*/cn^a
and specifiedlimiting valuesof the seriesreactancerequiredat the passbandedges(X(o") or inductancerequiredcanbe calculated. andX(rlr)), the loadingcapacitance loading circuit (Figure10.72(b), it follows that the series Considering
X(ron)=Xe(ao)+arLt o gCr-.in
(l 0.l 87)
and
X(a )=
Xn(a t)+a tlt
ozCr-.*
(10.188)
whereZu is the packageinductanceofthe varactor. Theseequationscan be manipulatedto give expressionsfor the minimum and in ( I 0.I 86) Substitutionof theseexpressions maximumvaluesof thevaractorcapacitance. yields an expressionthat canbe solvedto obtainthe valueofXr. It follows from this equationthat ,
_ c r X ( r ot ) - X ( a n )
LA:--u B
(J"r0-lDL-rJJH
with
|
(10.189)
&
Design of RF and Microwavo Amplifien and Oscillators
OrCu ro
J'=-=-Purno @gCu-rin
o) r
(l0.re0)
a H
If LAisfoundto be negative,a seriescapacitoris requiredinstead.Thevalueofthe cryacitor is given by
I 0JH
C,r =
cr, o)r
@nLn - d@LLB - X(a n) +aX(ror)
(l0.lel)
Thevalueof Cnor Lnrequiredin theparallelcasecanbe foundby startingwith the equations
- B(oro)= -1. a aLt
*
(l0.le2) a nln
orCr-.in
and
- B ( r)o= 4 *
I
a,Lo-
( r 0.le3)
orCr.o
Theseequationscan be manipulatedto give expressionsfor the minimum and na:
z.=-L>o
P ! , 1 P "- Q
(r0.le4)
qfrere @H - t] p=L[', + 2 l x ( a r ) X ( a n ) L u) rd
(r0.r95)
465
The Design of Radio-Frequencyand MicrowaveAmplifiers and Oscillaton
.r, a= X(ao,\X(a ,)
10.13.6
(a n I x(a r))-a rp,-^ l X(a r)) _ | (a , I a r,)- rotpu^*l a n LB
(l0.le6)
Considerations Applying to Oscillators with Low Phase Noise
The phasenoiseof an oscillatorcanbe minimizedby doingthe following: l.
Selecta transistorwith a low noisefigure andlow flicker noise;
2-
Bias the transistorcorrectly;
3.
Maximizethe loadedQ or therateat which the loop phasepassesthrough zero',
4.
Keepthe conversiongainof the oscillatorlow.
The conversiongain can be kept low by keepingthe outputpower well below with resistive saturation(limiting theloopgain)and/orby linearizingthetransconductance conversion (a the emitter). The can in the source or resistor be used feedback small series gain will alsobe low if the loop gainat start-upis kept low. If the amplitudeof the oscillationtendsto be unstablebecauseof thesemeasures, a linear automaticgain control (AGC) loop canbe designedto stabilizeit. A pin diode circuit is probablythe bestoption. flicker noiseby designing It is alsopossibleto reducethe levelofthe up-converted suitablelow-frequencycircuitry for the oscillator[6]. The aim of suchcircuitry is to reducethe level ofthe flicker noiseacrossthe nonlinearjunctions. When a fixed frequencyoscillator is designedand large-signalinformation is available,AM to PM conversioncan be minimizedif a networkis designedto ensure rectangularcrossingof the impedanceversusfrequencyandimpedanceversusamplitude tracesdisplayedon a Smith Chart.
REFERENCES l. MultiMatch RF and Microwave Impedance-Matching,Amplifer and Oscillator West,RSA:Ampsa(PTY)Ltd; http://www.ampsa.com., Software,Somerset Synthesis 1998. 2. Vendelin, G. D., A. M. Pavio, and U. L. Rohde,MicrowaveCircait Design Using New York: JohnWiley and Sons,1990' Linear and NonlinearTechniques,
rr 466
Designof RF and MicrowaveAmplifiers and Oscillators
3. Poza4D.M., MicrowaveEngineering,Reading,MA: Addison-wesleypublishing Company,1990. 4. Boyles,J. w., "The oscillator as a ReflectionAmplifier: An Intuitive Approachto OscillatorDesign,"MicrowaveJournal,JuneI 9g6. 5. carson,R. s., F/rE&FrequencyAmplifiers,Newyork: Johnwiley and sons, 1979. "A 6. Abrie, P. L. D., andP. Rademeyer, Methodfor Evaluatingandthe Evaluationof the Influenceof the ReverseTransferGainon theTransducerPowerGainof Some " MicrowaveTransistors, I EEE Trans.Mi crowave Theory Tech., Vol. MTT-33, No. 8, August1985,pp. 7ll-713. 7. Abrie, P. L. D., "A Seriesof cAD Techniquesfor DesigningMicrowaveFeedback Amplifiers and simpli$ing the Design of ReactivelyMatched Single-Ended Amplifiers," IEEE MTT-SDigest, 1990. 8. Norton,D.,u.s.PatentNo.3,426,29g,1969;"HighDynamicRangeAmplifier,"u.S. PatentNo. 3,624,536,November1971,AruacCorporation. 9. Mead,H. 8., andG. R. callaway:"BroadbandAmplifier," u.S. patentN o. 4,042,gg7 , August 1977, Q-bit Corporation. 10. Norton, D. E., and A. F. Podell, "TransistorAmprifier with Impedance-Matching Transformer,"US PatentNo. 3,891,934,June1975,AnzacCorporation. I l. Rohde,U. L., "WidebandAmplifier Summary,"Hqm Radio,November1979. 12.Rohde,u. L., "The Designof a wide-BandAmplifier with LargeDynamicRangeand Low NoiseFigureUsingcAD Tools,"IEEE LongIstondMTT symposiumDigest, April 28, 1987,pp.47-55. 13.Russel,K. J., "MicrowavePowercombining Techniques,"IEEE Trqns.Microwave TheoryTech.,MTT-27, No. 5,1979,pp.472-478. 14.Rauscher,c., "Large-Signalrechniquefor DesigningSingle-Frequency andvoltagecontrolledGaAsFET oscillators,"IEEE Trans.MicrowaveTheoryTech.,yol. MTT-29,No. 4, April 1981. , 15. Johnson,K. M., "Large Signal GaAs MESFET oscillator Design," IEEE Trans. MicrowaveTheoryTech.,Yol.lr'4TT-Z7,No. 3, March1979. 16. Prigent,M., and J. obregon, "PhaseNoise Reductionin FET oscillators by LowFrequencyLoadingandFeedbackcircuitryoptimization," IEEETrans.Microwave TheoryTech.,Yol.MTT-35,No. 3, March 1987.
The Desigr of Radio-Frequencyand Microwave Amplifiers and Oscillaton
467
SELECTEDBIBLIOGRAPITY "Two Port Power Flow Analysis Using GeneralizedScattering Bodway, G. E., MicrowaveJournql,May 1967,pp. 6 I -69. Parameters," Hq T. T., Solid-StateMicrowqveAmplifier Design,New York: John Wiley and Sons, 1981. Krauss,H. L., W. B. Bostian,andF. H. Raab,Solid-StateRadioEngineering,New York: JohnWiley andSons,1980. "Design Theory of BalancedTransistorAmplifiers," Bell Syst.Tech.J., Kurokawa,K., Vol.44,No.10,1965,pp. 1675-1698. Reston,VA: RestonPublishing Roddy,D., and J. Coolen,ElectronicCommunications, 1981. Company. "Design of Broad-BandPowerGaAsFetAmplifiers," IEEE Tajima,Y., andP. D. Miller, Trans.MicrowaveTheoryTech..Yol.MTT-32,No' 3, March1984.
/
APPENDIX A THE UNBALANCED TRANSMISSION LINE The basicequationsassociatedwith a transmissionline whenthe currentsareunbalanced [l] will be derivedhere.For the sakeof simplicity, it will be assumedthat thereis no magneticcoupling betweenthe two conductorsof the line. The equivalentcircuit shown in FigureA.l applies.
1r(0)
Figure A.1
tdx
Lrh
Inx
Ldx
The equivalentcircuit of an unbalancedtransmissionline.
The currentat positionx on the line will be consideredfirst. The currentcan be expressed in termsof balancedandunbalanced components asfollows: Ir(x)=Io@)* I"(x)
(A.l)
and I.r(x)=Jo1v)+1"(x)
(A.2)
(x) in these equationsis the balancedcomponentof the current,while 1"(x)is the unbalancedcomponent. It follows from thesetwo equationsthat Ir(x)=fl oQ) - I,(x))+ 21,(x)= l, (x) + 21"(x) 469
(A.3)
tf*
4l
F
*f
biCtofMandMicrowaveAmplifiersandOscillators
fl
= Ilx) + /o(:) - *:. ri1x.p is G fr*n"e
(A.4)
berween thennocurrents.
Becausc6ere is no groundpathon the line itself, the differencecurrent(/s(x) must bethesamearall positionsalongtheline (referto Figure6.17,if necessary), and(A.4) can thereforebe sinplified to
# F
(A.s)
Ir(x)= Ir(r) + Io With the qrrents defined,it follows that
l
D,' Ir(x)= -sCVn(x)
(A.6)
r
D,'.fr(r) =-sCVn(x)
(A.7)
I
D,'Yr(x) = -sL IJx)
(A.8)
I
D , 'Vr(x)= s[,I2(x)= s1 I,(x)+ sL I,
(A.e)
tl s
I ;
YnG )= r , @) - Y r ( t )
(A.10)
Differentiation of (4.6) yields that
o] .r,1x;=-sC[D, .rrr@)] = - sC ID, .Vr(x)-D,.Vr(x)] = - .rCf- sLI r(x) - sLI,(x) - sLI o'l = 2 s 2L C I l x l + s 2 L C I o thatis, (o1 -2r' LC)'1,(x)= szLC Io
(A.lr) :
Thesolutionto theequationis Ir(i= as-J'LC''"+ BeW"'
- Io / 2
(A.12)
Appendix A: The UnbalancedTransmissionLine
=-Iol2+Ae-r'+Be*t'
:
1
471
(A.13)
where
f =J?-LC.s= il,l'"[ztC
(A.14)
The equationfor 1dx) canbe derivedsimilarly andis givenby Ir(x) = Io /2+ Ae-r*+ Be*r'
(A.15)
An expressionfor I/1(r)cannow be obtainedby integrating(A.8) aftersubstitutionof 1'(r) andby using(A.13):
Vr(x)=rr(0)- z0I 2'(A - B) + z0I 2'l.Ae-"- B.t'I + s L x . l ol 2
(A.16)
where (A.17)
for Vr(x)followssimilarly: Therequiredexpression V2(x)= Vr(O)+ ZoI 2' (A - B) - Z0I 2'lAe-" - B"t'l + s L x - I lo2
(A.18)
by using(4.6): Vrr(x)canbeobtained An expressionfor Yrr(*)= T/r(x)- Vr(x)= 701tre-f'- Bet' l
(A.le)
below: derivedaresummarized Theequations Ir(t) = -Io l2+ Ae-rr*3"+fx
(A.13)
Ir(x) = Io / 2+ Ae-r' + Be*r'
(A.ls)
Vr(x)=Vr(O)- ZoI 2'(A - B) + z0 I 2'1.A"-" - Btt'l +sLx.Iol2
(A.16)
Design of RF and Microwave Amplifiers and Oscillators
vr(x )=z r ( 0 )+ z 0 1 2.(A -B )- zo1 2 .[l e -t' - B .t' l + s L x . I o/ 2
(A.18)
=vr(x) - vz@)= z of Ae-r'- 6.111 v12(x)
(A.le)
REFERENCES l. Abrie, P. L. D., Impedance-Matching Networluand BandwidthLimitationsof ClassB Power Amplifiers in the HF qnd vHF Ranges,Master's Thesis,University of Pretoria,1982. .J:
.'*
ai
.
!""
,ri
1,F
t,-
'rJ.r.
.*?
INDEX Center frequency, 127 Ceramics,96 Chamfer.348 Characteristicimpedance,I l7-l I 8, l2l-122, 179,I 85, t96-t97, 226-227,323 function,257 Chebyshev Chip capacitor, 95 coil, 100 Circle, 53, 366, 386-387, 390 CircuitQ, 135 Circular loop,326 spiral,328 Circulator,436 Class A,62-63,67,7E,2r2 4B,62 B, 62, 65-67,78, 185,212,215
I :4 transmission line transformer, 180 point, 62,67,78 l-dB compression Active biasing,435 Admittance. 161 Admittanceplane, 53, 366,381 Al-product, I 14-l 16 All-pass function,259 American wire gauge,100 Amplifier chzn.72 synthesis,360 Attenuation, I 17- | 18, 226 constant.12l Auto-transformer,179 Available power,5,20 gain,386,397 circle.403
c,2r2 Clipping,64,80 Coaxial cabfe,I 17-l 18, 179,197 line,I 17 C o i l , 9 9 ,1 0 7 , 1 0 9 - l l 0 Combiner,187 Common-base,7 Common-collector,8 90, 9l Common-drain, Common-emitter,8 Common-gate,88-90 88-89,9l Common-source, Compensation,203-207, 210, 354 Complex impedance,ul46 termination,144 Compression,444 point,73 Conductionangle,62 Conductor.l0l Configuration,88, 133,l8l, 183-184,186, 433,442,460 Conjugatematch,384
Balancedamplifier, 440-441 B a l u n ,l l 3 - 1 1 4 , 2 1 3 c o r e ,I 1 3 ,l 5 l Band-passidentities,272 Bandwidth,125, 127, 139,146, 149, 156,180, t96,208,266 Biaspoint,399 Bipolar transistor,78, 399 Bond wire, 223 Bonding-wire inductor, 326 Boundary,66 lines, 77 Branch,16,3l multipliers,3l Buildingblock, 188, l9l-192 Capacitance,99, 179, 190, 234 ratio,463 Capacitor,93-94,331 72,77,82 Cascade, network,11,59,82,84 representation,55, 57
475
476
Design of RF and Microwave Amplifiers and Oscillators
Connection,93 Constant impedancetaper,354 noisefigure circle, 396 operatingpower gain,389,392 circle,39l Constraints,21, 256, 286, 297-298 Contour.67 Conhol network.4lT Conversion,44 Copperlosses,122,154 Core,I 13, 179,189,196 dimensions,104,t l4-l l5 Correlationmatrix, 55, 57, 59 Coupledcoils, 175 Coupling, 179, 190,227, 233, 459 factor,154,156,158,161,166,173,175 Current,76, 180, 188-189 Current-representation, 55 Curvingradius,349 Cut-off frequency,| 57, 196, 205 Darlington synthesis,249, 258 DC component,69 dissipation,66 power, 64, 66 Decoupling,93,221 Device-modification,405 Diagram,l6 Dielectric constant, | 18, l2l resonator,458 puck,460,461 Diode,438 Discontinuity effect 345 Dispersion,l2l Dissipationfactor,95-96, 122 Distortion, 72 Double-tuned transformer, I 66 DRO,448 Dualism,13l Dynamic range,67-68, 386, 433 Edgecapacitance,223 Effective output power, 398 Efficiency,66,212 Electricalfield, l0l Equivalent circuit,2-3, 9, 4, 14,75, 94, 97, l5l, l ss- I 56, 162, 167-168, 198,223, 226,228-229,232,24r, 387,46|
noise soute,48 passiveproblem,386, 403 Extemal load,47,79 Feedback,75,442 network, 375 Fenite materials,104 FBT,78,399 Field,l0l Field strength,102 Film resistor,2l9 Fixed-valuedcomponent,446 Flicker noise,465 Flow diagram,360,444 graph,32 F I u x ,l 8 l coupling,152 density,105-106,l14, ll6,201,2w Fourierseries,65 Freespace,226,229 Frequencyresponse,126 Friiss' formula,60 Fringing capacitance,354 Fundamentaltone, 65, 66-67, 7 l Gain circle,371, 390,392,399, 401 compression,69-7 0, 443 leveling,43I slope,405 tapering,2l2 Gain-bandwidthconstraints,239, 405 Gapcapacitor,2lT Gap-capacitor,220, 236 Gm,78 Gradientvector,304 Ground plane,93, 223, 326, 327 Half-sinusoid,65 Harmonic components, 69 currents,TS High-passidentities,271 Hybrid combiner,440 divider,207,440 transformer,187 IIV constrants,6T IIV-plane,66,76 Ideal transformer,151 Imagereflection,327
l
477
lndex
Impedance,153, 193, 220 function.240-250 matrix, l5 Incidentcunent. l8 Incrementfrequency,276, 278 Indefinite admittancematrix. 7 S-matrix,35 Inductance, 94, 105,107,l14, l16, l8l, 190 Inductor,93,97, ll3,322,330 lnfinite chain,60, 73 Inherentlystable,390 Input admittance, 6, 292-293 , 365 impedance,270 Input loop,375 Insertionloss,60, 126, 147-148, 172,224-225 Interceptpoint, 70-7 3, 77 Interdigital capacitor,332 Intrinsic load, 47, 64,79 line.74 termination,66 Isolation, 187- 188, 432 L-section, 125, 133, 135,214 Ll enor, 399 Large-signal,452 LC transformers,253- 254 Leakage,155 flux. 153 inductance.15i Line segments, 64, 273 LltrJ,earity,62,67 Linville stability factor,367 Load impedance,154 line,64, 67,442 area,77 Load-pull contours,8l LoadedQ, 380 Loop gain,363,374-379,Ml,446,448, 450,457 product,3l Loops,33-34 Losstangent,95,122 Losses,75, 94, 98, 100,104, 106, l2l,152-153 Lossless feedback, 431-432, 434 network.12.386 two-port,43 Lossy transmission-line,322
Low-possidentities,27|
:'-"'
MAG,5,386 Magnetic core.| 13.180 f i e l d . 1 0 l .1 8 9 flux,l5l material,93, 98, I 13, 189,196 Magnetizinginductance, l5l, 153,l8l, 196, 198.200.210 Matchingnetwork, 214-215, 3 | 5 Matrix.2 Maximum relativedeviation.303 tunablegain,384 Microstrip, | 18, l2l, 218-220, 233-235, 345 bend,347 line,I 17 MIM capacitor,236 Minimum detectablesignal,67, 68 Minimum-admittancefunction, 24 I Minimum-impedancefunction, 240, 279 Miter,349 Model,48, 7E,| 53,218,235,236, 399, 445 Modification circuit.414 network,411 section,386 MSG.5 Multi-stage,4l7 Negativeresistance,375 N o d e . 3 l .1 3 l Noise figure, 6, 49, 53, 59-60, 386, 396, 404 circle,54, 387 floor, 68 match.6 measure,60,73 parameters, 47, 48 power,50 source,48,57 Normalizing impedance,16, 25-26, 29, 38 One-port,25-26 Open-ended stub,333,345 OpenJoopimpedance,375 Operatingpower gain, 384, 386, 451-452 Optimum termination,2l I Order,31 Oscillation,362,37l -372, 376 frequency,444 Oscillator,373, 378-379, 441, 456, 460
17t
Designof RF and Microwave Amplifiers and Oscillators
configuration,44l .:.,.:. li design,444 orrQut current,65, 67 impedance,252,260 power,47, 61, 65,210, 398, 441, 443,451 voltage,6T Packageparasitics,399 Faralleldouble-tunedtransformer,I 66 Parallel-platecapacitor,217-218, 226, 235 Parasitic inductance.462 absorption,248-249 capacitance,97, 107 Parasitics,93 Passive cascade.60 network, 59-60, 82 Pa$,33-34 Peakenvelopepower, 313 PEP.3 13 Phase noise,445,465 shift,442 Phasevelocity, 120 Pl-section,125, 138-139, 140-142, 144, 338-339,442, 446,456 Planartransmissionlines, 94 Pointjunction, 94 Polarity, l8l Poles,38, 241, 249-250, 252, 259 Potentiallyunstable,39I Power,19, 40, 147,152,398, 446 amplifier, 210-21 l, 213-215, 3 13 contour,67, 73-75, 80, 125,400 gain,4,194,385 Frameters,47, 62, 76, 80-81, 85-86, 88-90.442 splitter,187 tcrrnination,413,416 hfulry current,152 inductance,156 Pr,opagation constant,I 0 I hoximiry effect, 103, 123 hrch 448,459 hrsb-'pull, 185
g, r07 Q-tuo496, 98, I10, 127-128 Q-vatue,100 adiry frctor,95
Ratings,210 Ratios,l8l Reactive loading,463 load line,67 Real-frequency,273 Referencetemperature,50 Reflected component,l8 power,20 Reflection amplifrer, 437, 439 coeffrcient,12,22, 259, 324, 368, 438, 441 gain,364,372,374 parameter,20, 23, 25, 27, 29 Rejection,128-129, 166 Representation, 48 Residue.24 I Resistance, 98, 102, l8l level.294 Resistiveload line,67 Resistor.311.322 Resonance.93 Resonant circuit, 78, 126, 132, 458 frequency,93, 95, 98-99, 109, 221-222, 225-226 Resonatingsection,296 Resonator,446, 461 Ribbon,223 inductor.325 Richards'transformation,267 Ripple,166,168 RLC network,3ll matchingnetwork, 212 Rod core,I 13 Rollettestability factor, 370, 376 Route,33 S-parameters, I l, 14, 22, 30, 42, 44, 175-176, 284,440 Saturation, 451 resistance,62, 2l I voltage,62,2ll Scalefactor, 286,304 Scaftering matrix,20,40 parameters,27 Search,291 Secondary inductance,156 winding, 175
=G
;,ffi lndcr
99, ll | -l 12 Self-capacitance, Self-resonantfrequencY,332 Semi-infinitefunction, 277-27 t Sensitivity, 417-419 factor,4l9 Series double-tuned ransformer, I 72 feedback,86, 3'l 5, 454-456 resistance,96 resonantcircuit. l3l Shunt feedback,376 inductor, 333 Signalflow graph,3l Single-tuned transformer, I 58 Size, ll4,22l Skin depth,l0l-102 effect,117, 123,220 Small-signalmodel,78, 80, 399 SmittrChart,54, 368, 370-371,373,379,3t7' 389, 391, 394,397, 403,447 Solenoidalcoil, 99, 107,329 Spiral inductor, 328 Spuriousfree dynamicrange,68 Squarespiral, 327 Stability, 362,405 circle,368-369,371 factor, 363 Stabilizingresistance,363 Stackedcore,lL3,20l l13 Stackedtoroids, Standardwire gauge,100 Step,355 discontinuity,346 junction, 346 Sternestability factor, 367
stub,224 Superposition,50 T-junction,350 T-section, 138, 142-145, 338, 441, 454 Tappedcoil, 159-160,162 Temperature,106 Termination.23 Thermalrunaway,96 Thin film, 322 resistor.94 TOI.67 Topology,33,138,443 Toroid,l13, 15l Toroidalcore.I 14-l 16, 156,201 445 Transconductance.
{ *o
.#
4Vt
r1it'
s..
Tt&
.q;
porq frA, 6t. l4t. n9'?!0..2?l. 3l l' 371, 3t7-ltt"l37 43t' 4l circle.3!X 133,135,l3t, 140'239,247 Transformation. diagram,137,145 distance.255 factor, 163 matrix, 56 8, t34-t35,274,291 ratio, I 82 step, I 34 Transformer,| 51, 179, 182, 19l-192, I 95- I 96' 248 Transistor.399 Transmission line. 13, ll7, 122,179, 188-189,197,219' 3 3 3 .3 3 8 matrix, 10,57,220 10,23 parameters, zero,284 Transmission-linetransformer,179' 182' 212 Tunabilityfactor,383 Turns.109,l14 ratio, I 54 Twistedpairs,ll7,122 Two-port, 3, 9, ll, 25-26, 3 | -32, 42, 48, 53 Two-tone intercept Point, 62 products,6T signal,70 Unbalancedcurrent,180 Unit element,268-269, 272 Unitary matrix,40 UnloadedQ, I 13-l 14, 132,147,164,460 Varactor,446,463 network.462 Varactor-tYPe,446, 462 Variable,3l Via hole.351 Voltage, 76, 97, l3l, IEO gain,176 Voltage-controlledoscillator,446 55 Voltage-representation, Waveforms,62 Weight factor,277 Winding,179 Wire thickness,109 I-parameter matrix, 364
"=
480
Designof RF and Microwave Amplifiers and Oscillators
I/-parameters,l, 44, 23 | -232, 389 Z-pararneters, 8, 154, 175-176 Zero, 38,244, 250,252
