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*"
