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 > 2 Q t a n 1 (Zo I R)2 >> tan2e
(e.6)
Xin= jQ R, + jZotan$: tan2e 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
.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
