ECEN 5004, Spring 2018 Active Microwave Circuits Zoya Popovic, University of Colorado, Boulder LECTURE 2 SOME PASSIVE CIRCUITS

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1 ECEN 54, pring 18 Activ Microwav Circuits Zoya Popovic, Univrsity of Colorado, Bouldr LECTURE OME PAIVE CIRCUIT W hav alrady rviwd atching circuits, which ar -port ntworks. Thy ar passiv and can b rciprocal and losslss, but ar gnrally not atchd (othrwis you would not nd th). As you hav studid in prvious classs, a 3-port atchd, losslss and rciprocal circuit is not possibl, but usful 3-port ntworks includ unatchd T dividrs (.g. for antnna fds), lossy Wilkinson dividrs, and non-rciprocal circulators. Anothr usful 3-port ntwork is a bias-t, which w will nd to provid DC powr to a transistor so that it can ithr produc RF powr (in an oscillator) or aplify RF powr at th xpns of th DC input (in any aplifir)..1. BIA NETWORK Th activ dvic in an aplifir, oscillator, ixr, tc. nd to b connctd to on or or bias supplis, which idally not affct th RF prforanc of th circuit. Dsigning good biasing circuits is a larg part of aplifir dsign, and th following ar critical dsign paratrs: 1) th bias ntwork nds to b invisibl to th RF wavs, i.. as clos to an opn circuit as possibl. Th rason is that w cannot afford any of th RF powr to b lost in th biasing circuit and powr supply; ) th DC bias nds to b isolatd fro th RF circuit, i.. w do not want th DC voltag to b prsnt at th RF input (.g. w ight b daling with a -stag aplifir, and th prvious stag rquirs a diffrnt voltag); 3) finally, th DC bias circuit should b dsignd ovr th ntir frquncy rang whr th dvic has gain, so as not to caus instabilitis. In ordr to satisfy th first critrion, th DC bias lins could consist of an inductor with a valu chosn to prsnt a high ipdanc at th RF. It is difficult to ak a high-valud inductor at icrowav frquncis du to parasitic capacitanc. Anothr option is that th bias lins hav th charactristics of a low-pass filtr (rviw basic low-pass filtrs if ndd). In ordr to satisfy th scond rquirnt, a DC blocking capacitor nds to b addd to th circuit, and nds to b takn into account in th dsign. Capacitors ar not idal shorts at icrowav frquncis (thy hav parasitic sris inductanc and shunt conductanc). Th bias circuit can b intgratd with th aplifir atching circuit, or altrnativly, an xtrnal biasing circuitry can b usd. Extrnal bias circuits ar oftn rfrrd to as Bias Ts, a block diagra is shown in Figur L.1. Ths dvics ar xpnsiv if thy covr a broad bandwidth, and usually hav currnt liitations. Th rason is th inductor in th DC path, which nds to b ad of thin wir so as not to hav apprciabl parasitic capacitanc. 1

2 Corcial bias Ts ar fairly larg and hav typically MA connctors at th two RF ports. Oftn, biasing circuits ar part of aplifir dsign, and so xapls ar shown in Figur L.. DC in RF chok RF in Blocking capacitor DC in RF out Figur L.1. A bias-t idal quivalnt circuit. Th DC biasing circuit should b takn into account whn analyzing stability, i.. it is part of th input and output ntwork. Evn though it is dsignd to prsnt a high ipdanc to th RF signal at th dsign frquncy (convinc yourslf why this is so), it is not a ral opn circuit. For xapl, at a frquncy othr than th dsign frquncy, th quartr-wav shortd lin is not a quartr-wavlngth long, and thrfor is not an opn circuit to th RF signal. Thr is also so loss in th blocking capacitor th DC blocking capacitor has lad inductanc and so rsistanc, and it will not b prfctly atchd to th input RF 5- lin. In th groundd capacitor iplntation (righthand sid of Figur L.), th capacitor and via hol hav inductanc that is usually not wll charactrizd, and this also dtrins th quality of th opn circuit prsntd to th RF signal. Frrit RF chok Quartr-wav opn radial stub Frrit RF chok Frrit RF chok Groundd lupd capacitor Quartr-wav high-ipdanc lin Quartr-wav high-ipdanc lin Quartr-wav opn radial stubs Quartr-wav high-ipdanc lin DC blocking RF capacitor DC blocking RF capacitor DC blocking RF capacitor Figur L.. Exapls of icrostrip biasing circuits. Th frrit chok is an inductor at lowr frquncis and is ffctiv at choking frquncis up to a fw hundrd MHz. This is iportant, sinc th bias lins can b good antnnas for broadcast signals. At icrowav frquncis, howvr, th frrit is just a larg rsistor (th atrial is vry lossy), so th RF currnts will b vry attnuatd and will not rflct back into th circuit. Howvr, th powr is lost and any powr flow into th frrit lins should b iniizd.

3 A difficult probl is a broadband bias lin. In principl, a good inductor with svral hundrd nh inductanc would solv th probl, but icrowav inductors typically do not work abov a fw GHz du to parasitic capacitanc. If loss is not an issu, howvr, th Q factor of th inductor can b rducd by adding rsistors or frrits and vry broadband bias ntworks can b ad. A vry broadband aplifir fro Agilnt (-4GHz) nds vry broadband bias lins, as shown in Figur L.3. Th tiny con-shapd inductors with frrit loading ar fro Piconics, but Coilcraft aks th as wll. Th ida is that th Q is gratly rducd at high frquncis, so th rsonanc du to th parasitic capacitanc is not rlvant. Ths broadband inductors ar conical, so ffctivly thy look lik a continuous st of sris inductors with progrssivly highr inductanc valus and, corrspondingly, progrssivly lowr slf-rsonanc frquncis. Effctivly, this is a distributd sris of bandpass filtrs, shown in Fig.L.3. You can ak a broadband bias lin using svral sris inductors in this fashion. RF L 1 L >L 1 L 3 >L DC C 1 C >C 1 C 3 >C Figur L.3. Top: Photographs of iniatur conical coil with frrit loading ( Botto: discrt sris inductors which in th continuous liit bhav lik th conical inductor. If a frrit is addd, it incrass inductanc at th lowr frquncis, and adds loss at th highr frquncis... COUPLER 4-port ntworks can b ad losslss, rciprocal and atchd, and thr ar svral usful xapls, such as various 9-dgr and 18-dgr couplrs. Considr a 4-port ntwork that is atchd, rciprocal and losslss, with two-plan sytry (such as th coupld-lin couplr in Fig.L.6, whr th sytry plans ar shown in dashd lin). Th scattring atrix can b writtn in th following for: 3

4 , whr w hav usd th rciprocal condition and sytry. Nxt, w can us th losslss condition for th innr products of th various coluns of th * * * atrix, which givs, which in turn iplis that. This xprssion can b xpandd to: ( x jy )( x jy ) x x y y By looking at this xprssion, which is that of an innr product, w can conclud that and ar orthogonal, or 9 dgrs out of phas, or altrnativly that on of th is zro. Th sa analysis for th losslss condition of othr coluns of rsult in and, as wll as and bing orthogonal. inc this is ipossibl for 3 two-dinsional vctors to by utually orthogonal, w conclud that on of th has to b zro. If =, this ans that port 4 is isolatd. Furthror, th othr two ports ar in quadratur. inc th losslss condition also givs 1 this iplis that th powr is dividd btwn ports and 3 (not ncssarily qually)...1. Branch-lin couplr A branch-lin couplr is shown in Fig.L.4, and th usual odd and vn od analysis can b usd to find th -paratrs. Howvr, if w know that port is isolatd, it ans thr is no currnt flowing into it, and thrfor w can short this port. Latr, w will nd to prov that th solution is consistnt with thr bing no currnt in th short. Figur L.4. Branch-lin couplr, port is isolatd, and ports 3 and 4 ar in quadratur. On th right is th quivalnt circuit whn port is short-circuitd. First, by shorting port, w find th quivalnt circuit on th right in Fig.L.4. Th load at port 3 is Z/, so th ipdanc at port 1 aftr th quartr-wav sction is (Z/) /(Z/)= Z. Thrfor, port 1 is atchd. inc port 3 ss two Z loads in paralll, th currnt through th 4

5 is th sa, so th powr divids qually. Thrfor, this is a 3-dB couplr. Furthr, sinc th ipdanc of th quartr-wav sction of lin btwn ports 3 and 4 is Z, th voltag at port 4 is th sa in agnitud as that at port 3, but 9 dgrs out of phas. o, th couplr is a 9- dgr 3-dB couplr. Th currnt at th input of port 3 lags th voltag at port 1 by 9 dgrs, I=-jV1/ (Z/). inc th currnt divids qually at port 3, V3=I Z/=-jV1/. Thrfor, voltag at port 3 lags voltag at th input by 9 dgrs. This happns bcaus th ipdanc that th currnt I ss is ral. Now w nd to show that th currnt in port is indd zro, and thrfor that roving th short dos not affct th circuit. W can copar V1 and V4 as follows: V1 = V4 sinc half of th powr gos into port 4. Also, V1 = V4-18 bcaus of two quartr-wav sctions. inc th ipdanc of th lin btwn ports and 4 is lowr than th ipdanc btwn 1 and 4 by a factor of, this copnsats for th lowr voltag at port 4, and th two currnt agnituds at port ar qual but of opposit sign. Thrfor, no currnt is flowing into th short at port. Can you apply a siilar rasoning to quickly solv th rat-rac couplr circuit?... Coupld lins Coupld lins ar paralll transission lins that ar clos togthr, so that th lctric and agntic filds on th lins ar not indpndnt and coupl fro on lin to th othr. Thy ar usful as coponnts in broadband dirctional couplrs (both 9 and 18) and bandpass filtrs. Th bandwidth of coupld lin couplrs can b uch largr (dcads) than in th cas of a branch lin or rat-rac (-3%). Figur L.5 shows lctric and agntic fild lins for two coupld icrostrip lins. If th lft lin is connctd to a voltag sourc, thr will b lctric coupling to th right lin through lctrostatic induction. If currnt is allowd to flow on th lft lin, thr will b agntic coupling btwn th two lins through Faraday s law. Ths two induction chaniss ar rsponsibl for coupld-lin bhavior at all frquncis. Th scond lin can also b connctd to a voltag sourc, and thr ar two possibl situations, shown in th figur and calld th odd and vn od, rfrring to th sytry of th lctric and agntic filds. Figur L.5. ktch of coupld icrostrip lins showing th lctric and agntic filds for th odd and vn ods. 5

6 Now lt us go through a littl ntal xprint. Assu you hav two lins, and you quickly bring qual lngth parts of th clos togthr, as shown in Figur L.6. If a wav is incidnt at port 1, you can iagin it coupling capacitivly to port 3 as soon as th wav rachs th coupld sction. Thn a wav is stablishd on th coupld sction and propagats until th lin splits again, with so phas dlay along th coupld part. It is not obvious if thr will b so rflctd wav at th input port 1. By adding a scond lin clos to lin 1, w hav changd th ipdanc of that part of lin 1, so on would xpct a rflction. In fact, lin 1 looks thickr in th coupld part, so it ss as if th ipdanc wnt down and thr will b a rflction. If w want to liinat th rflction, w can narrow both lins in th coupld rgion until w atch port 1. This will rais th inductanc and lowr th capacitanc of th lins in th coupld rgion. Figur L.6. Voltag wavs for a coupld-lin couplr. Not that optical fibr couplrs look siilar to ths, but oprat quit diffrntly. In th cas of fibr couplrs, th vanscnt od (xponntially dcaying away fro th cor of th fibr) coupl, and th aount of coupling dpnds on th lngth of th lin and th isolatd port is port 3. In our cas, th isolatd port is port 4. Th fact that th circuit in Fig.L.6 has a port isolatd and th othr two 9-dgrs out of phas is indpndnt on th lngth of th coupld sction. Port 1 is atchd, port 4 isolatd and ports and 3 in quadratur, and this will not chang with th lngth of th coupld lin sction lngth. Th only thing that changs is th coupling cofficint. Not that this is quit diffrnt than in th cas of th branch-lin couplr, which dpnds on phas-dpndnt additions and cancllations at th ports, rsulting in frquncy dpndnc...3. Quasi-static (long wavlngth) analysis Considr a short dg couplr, with all ports proprly trinatd, and an incidnt wav V1 at port 1, as in Figur L.7. W will ak a fw assuptions: - Wavlngth is uch longr than th lins - Coupling is wak, which ans that th coupld lin dos not affct th voltag and currnt on lin 1 - Bcaus th couplr is short, V1 and I1 ar th sa all along lin 1. 6

7 Capacitiv coupling Th voltag on th lin 1 will rais th voltag on lin, so currnt will flow towards both of th trinating rsistors. inc thr is a capacitanc btwn lins 1 and, th currnts in lin will lad V1 by 9, which ans that th currnt will flow in th +z dirction at port 4, and th z dirction at port 3. Now w can writ a transission-lin quation for th currnt on th scond strip: di ' I jcv1, dz whr C is th distributd utual capacitanc, and th pri rfrs to diffrntiation with rspct to z. By sytry, thr is no currnt in th iddl of th scond strip, and this givs a rfrnc point that can b usful to intgrat th quation which w assu is linar in z sinc th wavlngth is long copard to th coupld lin lngth: I ( z jc V z C ) 1 Figur L.7. Capacitiv (a) and inductiv (b) coupling on a coupld lin. Th currnt along th strip varis linarly with distanc, bcaus w can think of it as physically coing fro a row of tiny capacitors btwn th strips. Mor currnt is accuulatd as w approach th nd of th lin. Th utual capacitanc has to b ngativ in ordr for th currnt to hav th right phas. Th currnt agnitud at th nd is: I C ( l / ) CV1 l / 7

8 Inductiv coupling Th lins ar a wakly-coupld transforr. Whn a currnt flows in lin 1, thr is a agntic fild around it and flux btwn th lins, rsulting in utual inductanc. This causs, through Faraday s law, a circulating currnt in lin and th trinations. Th currnt will oppos th flux. Thr is a voltag now across th rsistor in port 3, and an qual but opposit voltag at port 4. By sytry, th voltag in th iddl is zro, and again this is th rfrnc point for intgration of th transission-lin quation: dvz V ' jl I1 dz V jli1z whr L is th distributd utual inductanc. Th inductivly inducd voltag should lag I1, and this ans that th utual inductanc is positiv. Th voltag agnitud at th trination is: V ( l / ) LI1 l Th capacitivly and inductivly coupld voltags at port 3 ar in phas, but at port 4 thy ar out of phas, and tnd to cancl. Thr is coplt cancllation undr th following condition btwn th trinating ipdanc and th utual capacitanc and inductanc (kping in ind that th utual capacitanc is ngativ): LI1 Z C V For a short couplr, V1/I1=Z, rsulting in 1 L Z. C This ans that port 4 is isolatd if w dsign th coupld lins to hav this typ of utual inductanc and capacitanc. Notic that th rsult dos not dpnd on th lngth of th lin undr ths assuptions...4. Transission-lin (distributd, high-frquncy) coupld-lin couplr analysis What typs of wavs propagat on a lin that is not vry short copard to a wavlngth? W can writ th transission lin quations for th lins, but with two lins, thr will b two voltags, two currnts, and distributd utual and slf-capacitancs and inductancs. W can writ th transission lin quations in vctor for as: / whr th V, I, C and L ar givn by: I' jcv V ' jli V1 V, V I I1 I, L L L L L, C C C. C C 8

9 Th subscript stands for slf and M for utual. Th currnt can b liinatd, just as in th singl lin cas, to giv a vctor wav diffrntial quation: V '' CLV whr V CLV is th charactristic quation. Th solutions ar, just in th cas of a singl lin, or th for Vxp(z). This ignvalus of this quation giv th propagation constants, and th ignvctors ar th voltag ods. W hav assud that L and C ar sytric atrics, so thir product is also a sytric atrix. Going back to your linar algbra class, you can conclud that th ignvalus ar ral and th ignvctors orthogonal. Evn and odd ods ar orthogonal, and this can also b concludd fro physical rasoning. (a) (b) Figur L.8. Evn and odd ods (a). Equivalnt boundary conditions for odd od (b). Evn and odd ods ar orthogonal, and this can also b concludd fro physical rasoning, rfrring to Figur L.8. Assu that th two ignvalus ar diffrnt (propagation constants not th sa). This is tru in icrostrip, but not in coax. Now lt us invrt th x-axis. This is th sa as rplacing indx 1 by indx and vic vrsa. By sytry, th lin has not changd, so this nw pair of voltags ust also b a solution. In fact, it has to b th sa as bfor bcaus th propagation constants stay th sa. This ans th od dos not chang whn th indics ar intrchangd. This in turn is only possibl for odd and vn voltags: V1=V or V1= - V If th od is odd, thr is a sign chang, but this dos not chang th od, sinc th ods ar dfind only to an arbitrary scalar constant. inc th ignvalus ar odd and vn, this ans that th voltag and currnt ignvctors will b odd and vn as wll, and writtn as follows: V I Z 1 xp 1 V / Z ( L L C L L C j z )( C C ) and V I Z 1 xp j z 1 V / Z ( L L C L L C )( C C ) 9

10 Th Ls and Cs can b solvd using quasi-static analysis, which w will not do. Howvr, considr Figur L.8(b) which shows th quivalnt boundary conditions for th odd od. Thr is no y-orintd E-fild coponnt in th sytry plan, so a tal wall can b placd thr. This rsults in incrasd capacitanc pr unit lngth copard to a singl lin. This ans that th utual capacitanc has to b ngativ, rfrring to th xprssions abov. Likwis, inductanc pr unit lngth dcrass, so th utual inductanc is positiv. Analyzing a couplr in trs of odd and vn ods aks things siplr, sinc w can writ th voltags and currnts on th uncoupld lins in trs of th vn and odd ods as wll and in this cas th vn and odd od ipdancs ar th sa and qual to th charactristic ipdanc of th transission lin. This suggsts splitting th circuit into two indpndnt probls: on for th odd od and on for th vn od, and trat th sparatly as long as nothing prturbs th sytry. Th siplst couplrs ar quartr-wav long at th dsign frquncy, Figur L.9. Lt us assu odd and vn od propagation constants ar th sa (so that both ar quartr-wav long). Th lins act as quartr-wav transforrs with input ipdancs as indicatd in th figur (noralizd to Z). Figur L.9. Equivalnt odd-od (top) and vn-od (botto) lins with noralizd ipdancs. Th rflction and transission cofficints for th two circuits ar shown blow, along with th total rflction and transission cofficints: z 1 z 1 s s s s jz ( z 1) 1

11 for z =1/zo. Notic th 9-dgr phas lag that w xpct. Th wav at port 4 is th diffrnc of th two qual transission cofficints, so this port will b isolatd as long as th vn and odd- od ipdancs ar invrss vrywhr along th couplr (s41=). This analysis ffctivly rducs th dsign of a 9-dgr couplr to dsigning a quartr-wav transforr. Th coupld wav is givn by th rflction cofficint of th transforr, and th through-wav by th transission cofficint. A quartr-wav 9-dgr couplr has a bandwidth ratio of about on octav (:1), uch highr than a branch-lin couplr. Mor sctions can b addd to incras th couplr bandwidth. If th nubr of sctions is odd, th couplr is a 9-dgr couplr, and if thy ar vn, it is a 18-dgr couplr. 11