Efficient symbolic computation of approximated small-signal characteristics of analog integrated circuits

IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 30, NO. 3, MARCH 1995 327 E icien Symbolic Compu a ion o App oxima ed Small-Signal Cha ac e is ics o Analog In eg a ed Ci cui s Pie Wambacq, F ancisco V. Fe nBndez, Geo ges Gielen, Willy Sansen, and Angel Rodiguez-VBzquez Abs ac -A symbolic analysis ool is p esen ed ha gene a es simpli ied symbolic exp essions o he small-signal cha ac e - is ics o la ge analog in eg a ed ci cui s. The exp essions a e app oxima ed while hey a e compu ed, so ha only hose e ms a e gene a ed which emain in he inal exp ession. This p inciple causes d as ic sa ings in CPU ime and memo y, compa ed wi h p e ious symbolic analysis ools. In his way, he maximum size o ci cui s ha can be analyzed, is la gely inc eased. By aking in o accoun a ange o he alue o a ci cui pa ame e a he han one single numbe , he gene a ed exp essions a e also mo e gene ally alid. Misma ch handling is explici ly aken in o accoun in he algo i hm. The capabili ies o he new ool a e illus a ed wi h se e al expe imen al esul s. I. INTRODUCTION URRENT ools o small-signal symbolic analysis o C analog in eg a ed ci cui s, like o ins ance ISAAC [ 11 and ASAP [2], a e able o e alua e ne wo k unc ions in he s-domain wi h he complex equency a iable and he ci cui pa ame e s (capaci ances, esis ances, ansconduc ances, e c.) kep as symbols. These unc ions a e ypically gi en as an expanded cancella ion- ee sum o p oduc s, in which xT = (21, ZZ,...,XQ} is he ec o o symbolic ci cui pa ame e s and he i and gj a e sums o p oduc s. Since hese exp essions a e calcula ed au oma ically, analog designe s a e eleased om he in ol ed calcula ions needed o ge insigh in o he ac beha io o ci cui s. Also, analog cells can be au oma ically sized o gi en ac speci ica ions h ough he i e a i e op imiza ion o he symbolic equa ions gene a ed o hei gain, poles, ze os, e minal impedances, PSRR, CMRR, e c. O he po en ial applica ions o symbolic analyze s, o syn hesis, s a is ical op imiza ion, es abili y, e c., exploi also he compu a ional ad an ages o pe o m epe i i e e alua ions o p ecalcula ed models [3]. Howe e , hese applica ions can be ealized a ully only i he au- oma ic gene a ion o symbolic exp essions uns pa allel o he au oma ic p uning o insigni ican e ms in hese exp es- Manusc ip ecei ed July 13, 1994; e ised No embe 4, 1994 P. Wambacq, G. Gielen, and W. Sansen a e wi h Ka holieke Uni e si ei Lue en, Dep. Elek o echniek, ESAT-MICAS, B-3001 He e lee, Belgium. F. V. Fe nandez and A. Rodnguez-Vhquez a e wi h Depa men o Analog Ci cui Design, Cen o Nacional de Mic oelec 6nica. Edi . CICA, E-4101 2 Se illa, Spain. IEEE Log Numbe 9408739. sions-simila o wha expe analog designe s do when hey analyze ci cui s by hand. Al hough exis ing analyze s like ISAAC and ASAP in- co po a e such simpli ica ion ea u e, hei algo i hms ha e wo impo an d awbacks: a) simpli ica ions a e pe o med only a e he exac symbolic exp ession is gene a ed in an expanded sum-o -p oduc o ma ; and b) he signi icance o each e m in he sums-o -p oduc s is assessed on he basis o nume ical e alua ions using ypical alues o he ci cui pa ame e s, a a single poin o he design pa ame e space. Since he size o he exac symbol exp essions inc eases exponen ially wi h he numbe o nodes and elemen s in he ci cui , he i s d awback pu s an uppe limi on he complexi y o analyzable ci cui s; a ound en ansis o s i each ansis o is ep esen ed by a high- equency model con aining abou nine ci cui elemen s. On he o he hand, app oxima ing symbolic exp essions by conside ing jus a single poin o he pa ame e space does no seem o be consis en wi h he e y na u e o he symbolic analysis p ocedu e, whe e he exac nume ical alue o he pa ame e s is, by de ini ion, unknown a p io i. E en in he case symbolic analysis is used o s udy c i ical pa ame e a ia ions in an al eady sized schema ic, simpli ying by using jus in o ma ion abou a nominal poin may lead o impo an inaccu acies in misma ch-sensi i e cha ac e is ics, as o ins ance PSRR o CMRR o ope a ional ampli ie s. This pape p esen s a simpli ica ion algo i hm o o e come bo h d awbacks abo e. Fi s o all, he complexi y limi s o analyzable ci cui s a e ex ended by gene a ing only he dominan e ms, wi hou i s compu ing he comple e exac exp ession. In his app oach, which is deno ed as simplijica ion du ing gene a ion, he dominan e ms a e gene a ed un il he accu acy alls wi hin a gi en use -supplied accu acy. As shown in Fig. 1, his is a much mo e e icien app oach, bo h in e ms o memo y usage and CPU ime, han he classical app oach ollowed in [l], [2]. The idea o simpli- ica ion du ing gene a ion was i s men ioned in [4], and la e ound also in [5] and [6]. Howe e , simpli ica ions in [4] and [6] a e pe o med a a nominal poin o he design space, and, consequen ly, any e alua ion o he exp essions in ano he ope a ing poin migh cause la ge e o s. To educe hese e o s, his pape u he elabo a es he app oach in [5] o combine he concep o simpli ica ion du ing gene a ion wi h he use o anges [7], ins ead o single alues, o he ci cui design pa ame e s. This inc eases he compliance o gene a ed exp essions, while keeping he compu a ion ime and he memo y esou ces needed o symbolic analysis o la ge analog ci cui s bounded. Also, he combina ion o bo h 0018-9200/95$04.00 0 1995 IEEE Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ma ch 20,2020 a 15:22:04 UTC om IEEE Xplo e. Res ic ions apply. 328 IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 30, NO. 3, MARCH 1995 ..... . . .,......... . .... . .. ..... .. ... ....... . ., nes ed - expanded app oxima ed (a) (b) Fig. 1. Schema ic ep esen a ion o he memo y usage du ing simula ion ime wi h he classical app oach (a) and wi h he p oposed app oach o simpli ica ion du ing gene a ion (b). Classically, a symbolic exp ession is i s gene a ed in a nes ed o ma . Fo a eliable app oxima ion, he exp ession is hen expanded and he cancelling e ms a e elabo a ed. This expansion can lead o a huge numbe o e ms whose s o age exceeds he memo y limi s (dashed line), while only a ew e ms a e e ained a e app oxima ion. This p oblem is ci cum en ed i he exp ession is simpli ied when i is gene a ed: he memo y usage inc eases wi h he equi ed accu acy o he numbe o e ms o he symbolic exp ession. echniques, simpli ica ion du ing gene a ion and he use o anges, demons a e be e esul s han p e ious app oaches o he handling o ma ching be ween symbolic pa ame e s and misma ches. Sec ion I1 p esen s he concep and ou lines he algo i hms he IC- h powe , spanning ees in dec easing o de mus be gene a ed con aining exac ly IC capaci ance b anches. This can be o mula ed as he ollowing g aph- heo e ical p oblem: gi en a g aph wi h n nodes and wi h ed (co esponding o ( ans)conduc ances) and blue (co esponding o capaci ances) weigh ed b anches, enume a e in dec easing o de he spanning ees ha con ain exac ly IC blue b anches and n - k - 1 ed b anches, in which IC can ha e a alue be ween ze o and n - 1. Fo his p oblem, an algo i hm [9] has been de eloped whose ime complexi y and memo y equi emen s inc ease linea ly wi h he numbe o gene a ed spanning ees. 111. GENERATION OF THE NUMERICAL REFERENCE AND APPROXIMATION OVER RANGES The ee enume a ion p ocedu e desc ibed abo e ob iously needs a s opping c i e ion o know when enough e ms ha e been gene a ed. The gene a ion o e ms o a ce ain powe IC o s in he enume a o o denomina o can s op when ICnum. e alua ion o gene a ed e ms1 (num. alue o coe icien o sk I > (1 -Ek). (2) - used o simpli ica ion du ing gene a ion. Sec ion 111 explains how in e als a e inco po a ed in he s opping c i e ion ha con ols he gene a ion o e ms. In Sec ion IV i is explained how ma ching elemen s and co esponding misma ches a e handled in he new app oach. Finally, Sec ion V p esen s examples ha demons a e he sui abili y o he echniques p esen ed o analog cells con aining mo e han 20 ansis o s, which app oaches he size o p ac ical ci cui s used in odays IC designs. In his equa ion, he nume ical e alua ion is pe o med in a nominal ope a ing poin o he ci cui . The denomina o in (2) ep esen s he nume ical alue o he coe icien o sk in ei he he nume a o o denomina o o he ne wo k unc ion. The comple e coe icien s a e ne e gene a ed and, hence, hei nume ical alue mus be calcula ed in ad ance (wi hou knowing he symbolic exp essions). This is e icien ly pe o med using he polynomial in e pola ion me hod [lo]. Fo he ex ension o he s opping c i e ion o (2) o in e als, 11. SIMPLIFICATION DURING GENERATION The idea o simpli ica ion du ing gene a ion needs a e m by e m gene a ion mechanism, which inds he e ms in dec eas- ing o de o magni ude, wi hou skipping any e m. This can be achie ed wi h he undi ec ed ee enume a ion me hod [8]. This is a opological me hod ha ope a es on wo weigh ed g aphs, he ol age g aph and he cu en g aph, which a e easily de i ed om he gi en (small-signal) ne wo k. A e m is alid only i i s co esponding b anches cons i u e a spanning ee in bo h g aphs. The symbolic e m is gi en by he p oduc o he b anch weigh s (admi ances) in any o he g aphs. The sign o a e m is de e mined sepa a ely, using opological in o ma ion o bo h g aphs. By augmen ing he ne wo k in a special way wi h ic i ious elemen s, i is possible o gene a e he e ms o bo h nume a o and denomina o a he same ime [8]. The numbe o ees inc eases exponen ially wi h he ci cui size. Since we a e in e es ed only in he dominan e ms and he e o e no in all ees, he new algo i hm enume a es spanning ees in he ol age g aph in dec easing o de . Fo e e y spanning ee, i is checked whe he he co esponding b anches in he cu en g aph cons i u e a spanning ee as well. I so, a alid e m is ound and i s sign is de e mined. This echnique is pe o med o e e y powe o he e- quency a iable s in bo h he nume a o and denomina o o he ne wo k unc ion. Fo a nonze o powe o s, say i is assumed ha a symbolic pa ame e z can ake a alue inside a gi en in e al de e mined by i s lowe bound XL and i s uppe bound XH. The in oduc ion o in e als o he symbolic ci cui pa- ame e s gi es ise o mul idimensional in e als o he alue o he coe icien s i and gj om (1). These a e compu ed by an in e al ex ension o he polynomial in e pola ion me hod. The esul ing in e al o a coe icien is usually a pessimis ic o e es ima e. The e o e, in e als a e na owed using he algo i hm desc ibed in [ 1 I]. In e als o he small-signal ci cui pa ame e s a e ei he de e mined by speci ying a ela i e a ia ion a ound a gi en nominal alue, o hey can be de i ed om in e als o he bias alues and echnological pa ame e s. In e als o symbolic e ms o sums o p oduc s a e hen de e mined using he di ec in e al ex ension [ll], o , mo e accu a ely, wi h he mean alue o m [ll]. The s opping c i e ion gi en in (2) can now be e o mula ed as: (3) In his equa ion (SL, SH] ep esen s he in e al o he coe i- cien o sk ob ained as desc ibed abo e. The in e al [GL, GH] deno es he in e al o he sum o he signi ican e ms ha ha e al eady been gene a ed. The symbols L and U deno e he lowe and uppe bound o an in e al, espec i ely. L(l[GL, GHll) > - k), U(I[SL SHII) Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ma ch 20,2020 a 15:22:04 UTC om IEEE Xplo e. Res ic ions apply. IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 30, NO. 3, MARCH 1995 329 Fig. 2. CPU ime on a SUN SPARC 10 o he symbolic compu a ion wi h a 25% e o o he ol age gain Vou /K, o he esis i e ladde ne wo k. The do ed line co esponds o imes measu ed wi h ASAP (con en ional symbolic analyze ). The solid line co esponds o he new app oach. The use o in e als p o ides a e y good ade-o be ween accu acy and complexi y. Ob iously, mo e e ms appea in he inal esul han when using ixed alues, and i he in e als a e aken oo wide, hen he in e p e a ion o esul s can become complica ed again. IV. MATCHING ELEMENTS Ma ching elemen s play an impo an ole in analog and especially in di e en ial in eg a ed ci cui s. In symbolic cal- cula ions hey a e ep esen ed by he same nominal symbol. A e doing so, p oduc e ms can occu wi h exac ly he same symbols, so ha hey cancel o add, depending on hei sign. The de ec ion o ma ching e ms equi es a lo o o e head in CPU ime and memo y consump ion in con en ional symbolic analyze s [ 11, [2]. Wi h he new echnique, howe e , ma ching e ms a e easily de ec ed: since hey a e equal in magni ude, hey a e gene a ed one immedia ely a e he o he . Hence, he cancella ions can be elabo a ed by looking only a he las ew gene a ed e ms ha ha e he same magni ude as he las gene a ed e m. Misma ches a e modeled explici ly by adding a small sym- bolic misma ch elemen in pa allel wi h he nominal elemen . F om ha momen , bo h elemen s a e handled independen ly, and wi h hei own nume ical magni ude. Fo example, he ansconduc ances o wo ma ching ansis o s MIA and M~B a e w i en as gmM1 and gmM1 + ASmMleA, espec i ely. In his way, p oduc e ms con aining misma ch symbols a e gene a ed much la e han he co esponding nominal e ms and only when necessa y. In echniques p e iously used [ 11, [2] in symbolic analyze s, misma ch e ms we e always gi en a magni ude ( he maximum de ia ion) and a sign. This is no ealis ic, since hei sign is no known in ad ance. This p oblem is o e come he e by ep esen ing a misma ch e m by a symme ic in e al a ound ze o. V. EXAMPLES The new echnique no only exceeds he limi s o a con- en ional symbolic analyze , i can also- o smalle ci - cui s-gene a e an app oxima e exp ession in a CPU ime ha is up o se e al o de s o magni ude smalle han wi h con- en ional analysis. This is shown wi h he symbolic analysis o he esis i e ladde ne wo k shown in Fig. 2, which is o en 9- -1 mD .- . . GND .. Fig. 3. A illy di e en ial BiCMOS ope a ional ansconduc ance ampli ie wi h common-mode eedback. 1 0‘ !Io3 1 o2 / IO’ ela l eem 10’ IO-’ IO-* IO” loJ IO” # e ms 32 182 664 1898 1606 9334 #di e en e ms 7 36 120 339 788 1606 Fig. 4. CPU ime (in seconds on a SUN SPARC 10 wo ks a ion) e sus he ela i e e o o he gene a ed symbolic exp ession o he denomina o o he low- equency di e en ial-mode gain o he BiCMOS ampli ie (Fig. 3). The numbe o enns ha co esponds o he accu acy is indica ed as well. used as a benchma k ci cui o symbolic analyze s. The CPU ime is shown as a unc ion o he numbe o s ages o he symbolic compu a ion o he ol age gain wi h a 25% e o . The d ama ic inc ease in CPU ime wi h he numbe o s ages o con en ional symbolic analysis is due o he ac ha he exac exp ession mus be gene a ed. The e iciency o he simpli ica ion du ing gene a ion ech- nique in e ms o CPU ime is illus a ed wi h he symbolic compu a ion o he sys em de e minan o he BiCMOS am- pli ie o Fig. 3. This ci cui , con aining wen y ansis o s, is a oo complex o be analyzed wi h classical symbolic analyze s. Fig. 4 indica es how wi h he new echnique he CPU ime inc eases wi h he accu acy o he gene a ed sym- bolic exp ession and hence wi h he numbe o e ms, jus as wi h he p inciple idea shown in Fig. 1. In con as wi h he con en ional symbolic analysis app oaches, he less e ms a e gene a ed ( he la ge he e o ), he less CPU ime is equi ed, which is a e y “na u al” way o gene a ing e ms ha cons i u e a la ge exp ession. Fo la ge ci cui s complica ed exp essions may be gene - a ed. This is illus a ed in Fig. 5, which shows he symbolic exp ession o he low- equency di e en ial-mode gain o he ampli ie o Fig. 3. This exp ession has been gene a ed in 58 s on a SUN Spa c 10. The exp ession, howe e , can be u he simpli ied wi hou inc easing he e o by a symbolic pos p ocessing p ocedu e, ha ac o izes he exp essions and ha akes ad an age o he ac ha he e o on a a io o wo coe icien s o a ne wo k unc ion is o en much smalle han he e o on he coe icien s indi idually. Doing so, he Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ma ch 20,2020 a 15:22:04 UTC om IEEE Xplo e. Res ic ions apply. 330 IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 30, NO. 3, MARCH 1995 Fig. 5. App oxima ed exp ession (E = 20%) o he low- equency di e - en ial-mode gain o he ci cui o Fig. 3. The e ms a e so ed in dec easing o de . Due o ma ching, se e al p oduc e ms occu mo e han once, which explains he occu ance o in ege coe icien s 4 and 8. The elemen geq2 is a lumped elemen 111 consis ing o he pa allel conduc ances goM9 and gOQp. Elemen s om he bias ci cui y (like gmM6) o om he common-mode eedback ci cui y (like gm~4) ha don’ in luence he di e en ial-mode gain a all, disappea a e ac o iza ion. di e en ial-mode gain educes o (4) I is ound ha he ou pu conduc ance o he bipola cascode is oo small, e en wi h he inclusion o in e als, o con ibu e signi ican ly o he conduc ance seen a he ou pu node. An e en mo e complex ci cui is he comme cial pA741 opamp. This ci cui con ains 23 nodes, 22 ansis o s, and 13 esis o s. The gene a ion o a symbolic exp ession o he ampli ie ’s ans e unc ion wi h an e o o 0.1% (1 10 e ms) equi es 38 seconds on a SUN Spa c 10 wo ks a ion. VI. CONCLUSION A new p og am has been p esen ed ha gene a es app oxi- ma ed symbolic exp essions o small-signal cha ac e is ics o analog ci cui s. The app oxima ion is pe o med du ing he gene a ion o he exp ession. In his way, only he necessa y e ms o he simpli ied symbolic exp essions a e gene a ed, which con as s o app oaches o con en ional symbolic an- alyze s which equi e a lo o o e head o he gene a ion o he exac symbolic exp ession, which is hen p uned. The new app oxima ion echnique also akes in o accoun a ange o he alue o he symbolic ci cui pa ame e s a he han one single alue. This ex ends he ange o alidi y o he gene a ed symbolic exp essions. Mo eo e , he new echnique allows an accu a e con ol o he app oxima ion e o . The in e p e abili y o he exp essions can be enhanced by u he pos p ocessing. Se e al examples ha e demons a ed ha his app oach enables he symbolic analysis o la ge analog in e- g a ed ci cui s o he size o p ac ical analog cells, which we e impossible o analyze p ope ly be o e. ACKNOWLEDGMENT The au ho s wish o hank P. Eindho en, The Ne he lands, and he Human Capi al and Mobili y P og am o he CEC o hei suppo . REFERENCES G. Gielen and W. M. Sansen, Symbolic Analysis o Au oma ed Design o Analog In eg a ed Ci cuj s. No well, MA: Kluwe Academic, 1991. F. V. Femindez, A. Rod iguez-Vizquez, and J. L. Hue as, “In e ac- i e ac modeling and cha ac e iza ion o analog ci cui s ia symbolic analysis,” Kluwe Jou nal on Analog In eg a ed Ci cui s and Signal P ocessing, ol. 1. G. Gielen, P. Wambacq, and W. M. Sansen, “Symbolic analysis me hods and applica ions o analog ci cui s: A u o ial o e iew,” P oc. IEEE, ol. 82, pp. 287-304, Feb. 1994. P. Wambacq, G. Gielen, and W. M. Sansen, “A cancella ion- ee algo i hm o he symbolic analysis o la ge analog ci cui s,” in hoc. IEEE In . Symp. Ci cui s Sys ., 1992, pp. 1157-1 160. P. Wamhacq, F. V. Femindez, G. Gielen, and W. M. Sansen, “E icien symbolic compu a ion o app oxima ed small-signal cha ac e is ics,” in P oc. CICC 1994, 1994, pp. 21.5.1-21.5.4. Q. Yu and C. 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