A Methodology for Analog Circuit Macromodeling

Transcription

1 A Methooloy for Aalo Crcut Macromoel Roha Batra, Pe L a Lawrece. Ple Departmet of Electrcal a Computer Eeer Caree Mello Uerty, Pttburh, PA 5 {rbatra, pl, ple}@ece.cmu.eu Yu-u Che SoC echoloy Ceter Iutral ech. Reearch Ittute, awa mozart@tr.or.tw Abtract h paper ecrbe a complete framework for eerato of compact aalo crcut macromoel whch fcatly reuce the moel complety whle tll captur the omat lear a olear repoe of the crcut. he techque applcable to a broa cla of crcut that ehbt weakly olear behaor uch a mer, RF power amplfer a wtche-capactor crcut. he Volterra-bae crcut moel are frt characterze u a combato of SPICE mulato a effcet umercal ftt techque. he complety of the etracte crcut moel further reuce by moel orer reucto techque whle mata a hh eree of accuracy. he effcacy of our macromoel methooloy erfe by comparo wth SPICE mulato. he effcecy of our macromoel make them utable for whole-ytem erfcato a hh-leel e aaly.. INRODUCION Verfy a complete aalo ytem a trator-leel mulato a etremely ffcult proce a ca ofte become feable ue to the lmtato of mulato capacty. A mlar ffculty ecoutere whe hhleel e aaly performe for the whole ytem. For thee reao, compact macromoel of aalo block are ere whch ca be ubttute place of the actual trator-leel etlt to peeup the mulato wthout acrfc ay of the requre accuracy. he NORM alorthm that wa recetly propoe [] utlze the Volterra-Sere to repreet olear trafer fucto a employ projecto-bae techque to fcatly reuce the ze of the olear ytem equato, thereby eerat compact repreetato for aalo a RF crcut. h alorthm ca be apple to tme-arat a well a tme-ary weakly olear crcut. Some eteo hae bee ecrbe [7][].o eerate aalo macromoel that ca be ue commercal mulato eromet, the crcut uer coerato mut be characterze a the moele bae o utry-taar ece moel thee eromet. I th paper, we eelop a complete methooloy whch ete the NORM alorthm to eerate olear reuce-orer macromoel rectly from trator-leel etlt. he reuce olear macromoel wll capture A of Auut 00, the author ha joe the Dept. of Electrcal Eeer, ea A&M Uerty, Collee Stato, X 77 the olear charactertc of correpo crcut block, uch a IIP, HD a a compreo, a compact form whle mata a accuracy comparable to commercal mulator uch a SpectreRF a HSPICE. he purpoe of eelop compact olear aalo macromoel two-fol. Frtly, macromoel ca facltate effcet ytem-leel e eplorato by allow eer to effectely re-ue the macromoel from ther preou e to prect the ytem-leel behaor. Hh-leel eco a traeoff aalye ca be mae effcetly by ealuat ytem pecfcato throuh the ue of a lbrary of reuceorer macromoel correpo to a arety of crcut topoloe a cofurato. Secoly, compact compoet macromoel alo facltate the whole-ytem erfcato whch otherwe tractable.. BACKGROUND Volterra Sere proe a eleat way to characterze weakly olear ytem term of olear trafer fucto. For a crcut wth put u(t) the repoe (t) ca be epree a the um of repoe at fferet orer = = ( t ) ( t ) () where, (t) the th orer repoe. More eerally, we ca ue Volterra Kerel to capture both olearte a yamc by cooluto [] ( t) =... h ( τ,..., τ) u( t τ)...( u t τ ) τ h ( τ,.., τ.. τ where, ) the th orer Volterra kerel. he frequecy oma traform of the th orer Volterra kerel eote by H,.., ) eerally referre to a ( the th orer olear trafer fucto. hee olear trafer fucto are epeet of the put a fully ecrbe the weakly olear behaor of the crcut. I orer to the Volterra olear trafer fucto for a SIMO weakly olear ytem we ca coer t taar MNA formulato ()

2 f ( ( t)) ( q( ( t)) = bu( t), y( t) = ( t) () t For a crcut wth a tme-arat operat coto e by = 0, the frt orer lear trafer fucto e by ( G C) H( ) = b () he ymmetrze eco orer olear trafer fucto eterme by [] [ G C ] H(, ) = [ G C ]. H( ) H( ), (5) where = a for equato () a (5) G = ( f ) C = ( q),!! = 0 = 0 H( ) H( ) = ( H( ) H( ) H( ) H( ) Whe a olear crcut ha a lare put ectato, howeer, t caue the operat pot of the acte ece to chae wth tme. For eample, for a mer crcut, the operat coto wth repect to RF al eterme by a lare LO al rather tha a fe operat pot. h requre the aaly of a mallal ectato oer a lare peroc operat coto. herefore, we ca apply a tme-ary formulato of the Volterra trafer fucto H ( t,,,... ) whch ca be formulate mlar to the tme-arat cae [][7] []. Volterra bae olear ecrpto, howeer, ofte creae ramatcally wth problem ze, thereby mak them effecte whe ue rectly. herefore, we tea apply the projecto bae olear reuce orer metho (NORM) propoe [] to reuce the moel ze. he alorthm compute a projecto matr by eplctly coer momet-match of olear trafer fucto. For eample, f we epa the frt-orer trafer fucto H ( ) at the or M, k = ( ) k=0 H k M, (), k where a kth orer momet for the frt-orer trafer fucto. Now, epa the eco orer olear trafer fucto H, ) at the or (0,0) where, M, k, l ( k l k l H (, ) M, k, l k = 0 l= 0 =, (7) a kth orer momet of the eco-orer M, k, l trafer fucto. he actual epreo for ca be obtae by frt ubttut () to (5) a epa w.r.t =. h proceure ca be alo apple to obta the momet of the thr orer trafer fucto. I NORM, a projecto matr bult uch that the reuce orer moel wll match certa umber of trafer fucto momet. It ha alo bee emotrate that mult-pot epao bae approach prouce much more compact moel tha the le-pot epao.. OVERALL MACROMODELING FLOW We outle the complete flow for the eerato of reuce-orer moel from trator-leel etlt F.. Frt, we mulate the trator-leel etlt a commercal mulator uch a SPICE to eterme a proper operat coto for the crcut. I the cae of a tme-arat crcut, a fe DC operat pot wll be compute. Otherwe, a lare-al tme-ary operat pot wll be compute for a tme-ary crcut uch a a mer. We moel the olearte for each trator the crcut a a thr-orer polyomal. We mulate each trator the crcut multple tme, ary the ba-oltae for t termal to eerate accurate ata-pot for ftt the polyomal. We the cotruct the full Volterra-bae moel of the crcut a eerate the reuce-orer moel of the crcut u NORM[]. - - Reuce Orer Moel = a a a Spce mulato Determe op pot Collect ata-pot Ft olearte Reuce-orer moel Fure.. Etracto of reuce-orer moel. EXRACION OF VOLERRA PARAMEERS he olear moel techque outle Secto epe o etract the parameter of the Volterra moel accurately. I Volterra ere, a olearty repreete a a power ere epao arou a ba pot. o llutrate, let u coer a olear ece charactertc f () epae about a ba pot 0 where, f ( ) = f ( ) a ( ) a ( )... () 0 0 0

3 a =! f ( ) = 0 May fferet mall-al moel for MOS trator et, a mot ophtcate moel clue ubtrate coupl effect a tracapactace[][]. Spce moel lke BSIM ot oly repreet phycal effect but alo clue may umercal parameter whch further creae the complety of the moel equato. It feable to f the coeffcet of the equato e () by f the hher-orer erate from the moel equato BSIM a other moel. Itea, we employ leat-meaquare error (LMSE) ftt techque to f the coeffcet []. We wll how how the olear parameter for the ra curret of a MOS trator are etracte. We moel the ra curret I a a thr-eree polyomal wth repect to the ra, ource a ate oltae. For mplcty, we hae ue the boy termal a the referece oltae althouh other poblte ca be ealy accommoate. he equato clue ual oltae term a well a cro-term. Compare to orary haaaly equato we moel ot oly the frt-orer olearte but alo the eco a thr-orer olearte a mall-al quatte arou the ba pot I = I 0.. (9) where, I = ba curret alue at operat pot 0 y = mall-al oltae at termal = {,,} = frt-orer coeffcet for oltae at termal = eco-orer coeffcet for cro-prouct of oltae at termal a y = thr-orer coeffcet for cro-prouct of yz oltae at termal, y a z herefore, there are frt-orer term, eco-orer term a 0 thr-orer term the equato. It ot poble to et the eco a thr-orer term rectly from trator-leel mulato o we hae formulate a effcet way to et thee term a moel the olearty accurately. We etract the frt-orer moel parameter from Hpce mulato[5]. For a tmearat crcut, we perform a DC operat pot aaly to obta the ba curret alue a the frt-orer coeffcet. For a tme-ary crcut, we perform a le-toe traet aaly for a uffcet ettl tme a the ample a le tme-pero of the ettle repoe to obta tme-ary operat pot for the crcut. We the perform a DC operat-pot aaly at each of thee pot to et the frt orer coeffcet, a. We ca epre thee coeffcet term of the more commoly ue mall-al coeffcet G m, G a G mb = G, = G a = G G G ) () m ( m mb For both the tme-arat a tme-ary cae, the ba oltae for each trator are perturbe by mallamout to obta ata-pot for ftt the eco a thr-orer coeffcet the approprate ftt rae repreete by the bou bo how F.. he fure how the ra curret a a fucto of a. It poble to meaure the curret I by perturb the a lhtly arou each ba-pot to obta may fferet pot. From (9) ( I I 0 ) =.. (0) where, the ubcrpt eote the -th ata-pot a ( I I 0 ) calle the reue. o ole the coeffcet of the RHS equato (0) we wrte the power a cro-term of, a for ampl pot to matr Y, the correpo coeffcet to the ector p a the reue I ) to matr R Y = p = [ ( I 0... ] a [... ] 0, R = I I I I = I I () We hae to ft the coeffcet of (0) uch that the error for each of the ata-pot arou the operat pot mmze. he areate error for the th ata-pot eote by ε. We hae to mmze the error = [ ε ε... ε e ] Yp R = e () he LMSE alorthm etmate p by mmz the um of quare error F = e e = ( Yp R) ( Yp R) () h lea to the optmal oluto p = ( Y Y).( Y R) () I orer to uaratee a oo ft for the olearte we eure that the ftt rae for the ata correct [] (F. ). he rae mut be lare eouh to ft the olearte

4 accurately but t houl ot attempt to coer the effect oute the al-w rae. For eample, f the ate of a trator ha a epecte al w of ±0mV, the ftt rae for of th trator houl be lmte by the al w. It alo mperate to elect eouh ata-pot to ft the olear parameter accurately.. RESULS he methooloy preete the preou ecto ha bee emotrate o a ouble-balace mer a a opamp. he macromoel eerate u th approach are compare wth etale trator-leel mulato of the crcut wth HSPICE.. A Double-Balace Mer Vout - Vlo Vrf Fure.. Effecte ftt rae for Volterra parameter I ome cae, the ftte reult mht tll caue lare relate error for certa pot the ata. he ft may be mproe by u a wehte-leat quare metho tea of the coetoal metho. I th cae, t mportat to elect ual weht for each equato w ( I I0 ) = w w... w.. (5) uch that the effecte reue w ( I I 0 ) for each ata-pot the ame rae. h weht cheme e each ual ata-pot the ame mportace a far a the ftt proce cocere. h a reuc the error for each pot a e a better ft for the etre rae of ata-pot. Frt, we ca perform a LMSE ft o the ata to et a tal etmate of the pot where the error ε may be lare. We elect the approprate weht w for each ata-pot a cale the reue accorly. After perform the wehte leatquare ft u () a (7) we ca look at the error aa a mofy the weht f we are tll ot atfe wth the reult. h oe for a few terato tll we ca o loer mproe the reult. For the wehte leatquare metho we trouce aother matr W, whch the aoal matr of the ual weht. We hae to mmze the wehte leat quare error fucto F = ( Yp R) W( Yp R) () where Y,p a R hae bee efe earler. h ca be ole to et the olear parameter (9) p = ( Y WY).( Y WR) (7) ε Fure.. A Double-Balace Mer A ouble-balace mer (F. ) moele a a tmeary weakly olear ytem wth repect to the RF put. he LO frequecy et at Ghz, a we calculate the tme-ary operat pot of the crcut by ett the RF put oltae to zero a u traet aaly HSPICE to ample a le tme-pero of the ettle repoe. he thr orer olearte are moele arou th tme-ary operat pot u umercal ftt techque outle Secto. he ftte eco a thr orer coeffcet are ue to eerate a Volterrabae full moel for the crcut. A le-toe RF put ue to erfy the moel reult wth the traet mulato reult. he throrer harmoc of the RF put frequecy ow-coerte wth repect to the LO frequecy compare betwee the moel a the mulato reult. he eco orer olearte houl eally be zero ecept for umercal oe, by e. We performe traet aaly for the crcut Hpce followe by a accurate Fourer raform of the output tme-oma waeform to erfy the reult. he RF put frequecy are from 00Mhz to 00Mhz. he mamum error the full moel a compare to Hpce mulato for the frt-orer reult le tha % for all frequece. he mamum error the thr-orer reult le tha 0% for throrer for all frequece. he reult F. hae bee ormalze wth repect to the RF put ampltue.

5 Relate Error Fure.. hr-orer harmoc (ow-coerte) for fferet put frequece Oce we hae the Volterra Sere bae full moel, t poble to meaure the thr-orer repoe at more ueful harmoc alo. For eample, F. 5 how the plot for the thr-orer trafer fucto H t, jπ f, jπ f, jπ ) ( f f 900 where, 00Mhz f, f 00Mhz a = Mhz. he full moel ha 50 tme-ample crcut ukow whch reuce to appromately crcut arable u the NORM metho, whle tll captur the omat repoe of the crcut. he relate moel error betwee the full a reuce-orer moel for the frt-orer reult le tha 0.0%. F. how that the relate percetae error betwee the full-moel a reuce-orer moel for the thr-orer reult le tha % for all cae Fure.. Relate moel error for thr-orer trafer fucto. A Operatoal Amplfer M M M M M5 V- M V M M7 Fure. 7. A two-tae opamp M0 M7 M V M9 M5 M M M M M 0 M9 M0 Vout /V Fure. 5. hr-orer trafer fucto for mer(full) A two-tae Operatoal Amplfer topoloy how F. 7. he cloe-loop AC repoe of th crcut how F.. U the etracto metho ecrbe Secto, t poble to match the AC (frt orer repoe) of the crcut accurately to about 99-00% compare wth Hpce mulato. For th crcut, eco orer olearte are more mportat tha the throrer olearte ce they are much hher matue. he opamp moele a a tme-arat ytem a learze at the DC ba pot to ft the eco a thr orer coeffcet for each trator the crcut. he eco-orer torto for a le-toe put how F. 9. We compare the Hpce mulato reult for put frequece ra from Mhz to 00Mhz wth our moel reult. he relate error betwee the full-moel a the mulato reult le tha 0% for all put frequece. he umber of tatearable for the crcut reuce from the full-moel

6 to about 5 the reuce orer moel. he comparo of the full a reuce orer moel reult how that there le tha 0.0% error for both frt a eco-orer repoe. aopt thee compact macromoel behaoral lauae uch a Verlo-A.. REFERENCES Oral H Matue Fure.. Cloe-loop AC repoe of the opamp Seco Orer Dtorto (V) Hpce Smulato Full Moel Reuce Orer Moel [] P. L a L. Ple, NORM compact moel orer reucto of weakly olear ytem, Procee of ACM/IEEE DAC, 00. [] J. Vuole a. Rahkoe, Dtorto RF Power Amplfer, Artech Houe, 00. [] W. Sae a P. Wambacq, Dtorto Aaly of Aalo Iterate Crcut, Kluwer Acaemc Publher, 99. [] Y., Operato a Moel of the MOS rator, McGraw-Hll, 999. [5] Hpce Uer Maual, P. -, Vero I, 99 [] J. Roychowhury, Reuce-orer moel of tmeary ytem, IEEE ra. Crcut a Sytem II Aalo a Dtal Sal Proce, ol., o. 0, Oct [7] P. L a L. Ple, Moel Nolear Commucato IC u a Multarate Formulato BMAS workhop 00. [] P. L, X. L, Y. Xu a L. Ple, A hybr approach to olear macromoel eerato for tme-ary aalo crcut, Procee of ACM/ICCAD, Frequecy (Mhz) Fure. 9. Seco orer torto a a fucto of frequecy 5. CONCLUSIONS I th paper, we hae preete a methooloy for eerat aalo crcut macromoel from the tratorleel etlt. h methooloy ca be apple to a broa rae of tme-arat a tme-ary weakly olear crcut. he macromoel eerate u th methooloy are characterze u effcet umercal ftt of mulato ata a moel orer reucto techque Our epermetal reult hae how that the macromoel offer fcat ecreae moel ze wth comparable accuracy to full trator- leel mulato Hpce. We woul lke to further eplore the poblty of