Symbolic analysis tools-the state of the art

SYMBOLIC ANALYSIS TOOLS - THE STATE-OF-THE-ART F ancisco V. Fe n Andez and Angel Rod iguez-Vkzquez Dep . o Analog Ci cui Design, Cen o Nacional de Mic oelec cjnica Edi . CICA, A da. Reina Me cedes s/n, E-41012 Se illa, SPAIN Phone: +34 5 4239923, FAX: +34 5 4231832, E-mail: [email p o ec ed] ABSTRACT This pape e iews he main las gene a ion symbolic ana- lyze s, compa ing hem in e ms o unc ionali y, poin ing ou also hei sho comings. The s a e-o - he-a in his ield is also s udied, poin ing ou di ec ions o u u e esea ch. 1. INTRODUCTION Ci cui analysis is a basic miles one o e icien design o in eg a ed ci cui s. E e since powe ul compu e s ha e been a ailable, designe s ha e de eloped p og ams o analyze ci - cui s au oma ically. Today, all elec ical enginee ing p o es- sionals and s uden s use elec ical simula o s (such as HSPICE o ELDO). Howe e , elec ical simula o s do no co e all he analysis asks equi ed o in eg a ed ci cui design. Essen ially, hey only se e o e z y he pe o mance o p e iously sized ci cui s. Among o he hings, designe s mus be able o p edic he beha io o unsized ci cui s by acing ela ionships among pe o mance igu es and design pa ame e s. These ela ionships may be in he o m o ans- e unc ions, poles and ze os, oo loci, pa ame ic Bode plo s, ha monic dis o ion coe icien s, e c. Tools used o de i e hem au oma ically a e called symbolic analyze s. The i s gene a ion o symbolic analyze s appea ed in he la e 60’s [l], whe e he basic analysis echniques we e p o- posed. Mos o hese echniques a e compiled in a ecen book by P.M. Lin [2]. These ools did no sp ead on he designe communi y, pa ially due o hei compu a ional cos and because hey did no ully i he designe equi emen s. On he o he hand, he lack o in e es on cus om analog ICs du - ing he 70’s ende ed symbolic analyze s seconda y as com- pa ed o elec ical simula ions. This scena io changed du ing he 80’s wi h he inc eased economical in e es on cus om analog and mixed-signal ASICs. Symbolic analyze s we e p omp ly ecognized c ucial ools o: p o ide insigh and knowledge abou analog ci cui s; suppo au oma ic beha - io al modeling o analog in eg a ed ci cui s; p o ide com- piled models o au oma ic epe i i e e alua ions equi ed o analog syn hesis, among o he applica ions [3], [4]. I esul ed in he eme gence o a new gene a ion o ools du ing he la e 80’s. Main ea u es o his new gene a ion ha e been app oxima ion, pos -p ocessing and he ex ension o weakly non-linea ci cui analysis. This pape summa izes some main con ibu ions in hese second gene a ion ools, and compa es he cha ac e is ics o many o hem. 2. SYMBOLIC EXPRESSION APPROXIMATION Simpli ica ion in symbolic analysis is de ined as he educ- ion o o mula complexi y by elimina ing insigni ican e ms o sub-exp essions, based on nume ical es ima es o he sym- bolic pa ame e s. Simpli ica ion (o app oxima ion) is equi ed e en o small ci cui s gi en he exponen ial inc ease o he exp ession leng h wi h he ci cui size. Con- 0-7803-3073-0/96/$5 .OO ‘1996 IEEE 798 en ional simpli ica ion app oaches i s calcula e he com- ple e symbolic exp ession, and hen simpli ies i -- Simpli ica ion A e Gene a ion (SAG). Di e en e sions o SAG ha e been implemen ed in mode n symbolic analyze s: ISAAC [5], SSPICE [6] and ASAP [7] o expanded o ma exp essions, and SYNAP [8] and ASAP [7] o nes ed o ma exp essions. Di e ences a ise in he con ol o e o s and in he use o ei he a nominal poin o a nominal in e al o es i- ma e he magni ude o he symbolic e ms. Due o he necessi y o gene a ing comple e exp essions, SAG echniques a e cons ained o small and medium com- plexi y ci cui s (below abou 100 symbols). La ge ci cui s can be analyzed by using he newes app oaches o simpli y- ing du ing, o be o e o, he analysis s age. They a e called Simpli ica ion Qu ing Gene a ion (SDG) and Simpli ica ion Be o e Gene a ion (SBG) echniques espec i ely. The a ionale behind SDG echniques is ha e ms can be gene a ed s ic ly in dec easing o de o magni ude s a ing wi h he la ges . The classical undi ec ed ee enume a ion me hod has demons a ed o be he mos e icien o sol e he unde lying ma oid in e sec ion p oblem. This app oach has been implemen ed wi h sligh di e ences in RAINIER [9],[10] and ADAGIO [11],[12], he new ool e ol ed om ASAP and ISAAC. SBG can ake place in he ne wo k unde analysis, eplac- ing hose elemen s (o subci cui s), whose con ibu ion (app op ia ely measu ed) o he ne wo k unc ion is negligi- ble, wi h a ze o-admi ance o ze o-impedance elemen . These app oxima ions a e usually done when de i ing ne - wo k equa ions manually. Howe e , excep o he mos i - ial eplacemen s, no au oma ic p ocedu e has been epo ed ye . Repo ed app oaches pe o m app oxima ions di ec ly on he ne wo k equa ions, ei he in g aph [SI o ma ix o m [13], ~41. 3. POSTPROCESSING TOOLS These a e basically in ended o easie in e p e a ion and manipula ions o symbolic o mulae, o o ex ac in o ma- ion om hem. 3.1. Symbolic pole-ze o ex ac ion Poles and ze os a e commonly used by analog designe s. [ 151 in oduced a echnique o symbolic ex ac ion o poles and ze os, based on heu is ic app oxima ions, simila o hose pe o med by expe analog designe s, e.g.: he oo -spli ing echnique. This app oach is also used in [16] which subse- quen ly applies a symbolic New on i e a ion o e ine he polelze o loca ions ob ained wi h he oo -spli ing echnique. Conside , o example, he ac i e eed o wa d compen- sa ed ampli ie o Fig. 1. A symbolic ex ac ion o he i s pole/ze o pai gi es [ 151 : 3.4. Fac o iza ion and complexi y educ ion E en app oxima ed exp essions can be di icul o in e p e due o he conside able size hey ha e when calcula ed wi h enough accu acy in medium o la ge ci cui s. Fac o iza ion o common ac o s and cancella ions o common ac o s in nume a o and denomina o a e some o he needed capabili- ies o imp o e he in e p e abili y wi hou dec easing he accu acy. Many mode n ools inco po a e some p imi i e o m o ac o iza ion. Bu mo e sophis ica ed mechanisms a e s ill needed, o ins ance o elimina e he con ibu ion o bias ci cui y which usually a ec equally nume a o and denom- ina o o mos ans e unc ions. ? Figu e 1 : Ac i e eed o wa d compensa ed ampli ie The in e es o symbolic poles and ze os can be seen h ough his example. F om (1) one can in e s he alue o he com- pensa ion capaci o ha annuls he pole/ze o misma ch and hence imp o es he high equency beha io o he ampli ie , Ano he ex ac ion echnique, epo ed in [ 171, ex ends he oo spli ing echnique o clus e o poles o ze os which a e qui e apa om o he clus e s and calcula es app oxima e exp essions o he ne wo k unc ion o a equency ange app op ia e o he oo s being calcula ed. 3.2. G aphical in e aces Some ools inco po a e g aphical in e aces o pa ame ic ep esen a ions o Bode diag ams o oo locus. Fo illus a- ion, Fig. 2 shows he g aphical ep esen a ion o he eal pa o poles and ze os o he ol age gain o Fig. 1 e sus he alue o Cpz. The alue o which he pole/ze o pai cancel is clea ly seen. -io* -106 -104 -102 0 102 io4 io6 108 io10 Re(s) ( adls) (log) Figu e 2: Pole/ze o loca ion ( eal pa ) e sus Cpz. 3.3. Sensi i i y analysis Sensi i i y analysis also plays a undamen al ole in analog ci cui design, specially in he calcula ion o he in luence o componen misma ches. The sensi i i y o a ci cui cha ac e - ized by a symbolic sys em unc ion H(s, x) can be ob ained by simply de i ing he sys em unc ion wi h espec o he ci - cui pa ame e [ 181, [ 191, [20]. 4. AN OVERVIEW OF CURRENT TOOLS Many symbolic analyze s ha e e ol ed as membe s o his second gene a ion. Exac compa ison o hei pe o mance is di icul due o he lack o s anda d benchma ks o es ci - cui s and pla o ms. Some key cha ac e is ics o main epo ed symbolic analyze s a e compa ed in Table 1. A b ie desc ip ion o hose analyze s ollows. SCAPP [22] has been implemen ed a Iowa S a e Uni e - si y, in he amewo k o he de elopmen o a new me hod- ology o symbolic analysis o la ge ci cui s. The p og am pa i ions he ci cui s hie a chically and symbolically ana- lyzes each sub-ci cui sepa a ely. Once he e minal blocks a e analyzed, an in e se un o he pa i ion p ocess econ- s uc s he global sys em unc ion. Since his analyze uses an app oach o la ge ci cui s based on hie a chical analysis, he sys em unc ions a e gi en as a se ies o exp essions a he han only one exp ession. Thus, he numbe o symbolic e ms g ows linea ly, ins ead o exponen ially, wi h he ci cui size. SCAPP can be ob ained o ee ia anonymous p a cp e 1 .ee.ias a e.edu:/pub/scapp SCYMBAL and SYBILIN [23] a e wo simula o s de el- oped in CNET (Bagneux, F ance). A dis inc i e ea u e o SCYMBAL is ha i was de eloped speci ically o swi ched- capaci o ci cui s. I allows no only con inuous a iables such as conduc ances, capaci o s, e c., as symbolic pa ame- e s, bu also boolean a iables such as clock phases. On he o he hand, SYBILIN is a con inuous- ime analyze , specially concei ed o he analysis o mic owa e ci cui s. These ools can be ob ained om he au ho s on eques . The p ice depends on he ype o ins i u ion. SC [24] was de eloped a he Uni e si y o A izona, and concei ed o class oom use, and hus de eloped o pe sonal compu e s. I has a se o pos -p ocessing ools o ease in e - p e a ion o symbolic da a: in e ac i e nume ic e alua ion, nume ic calcula ion o poles and ze os, and g aphic ep esen- a ion. SC can be ob ained om he au ho s wi h a $6 han- dling cha ge. SAPEC [25] was implemen ed a he Uni e si y o Flo- ence, also o PCs. One o i s main cha ac e is ics is he g oup o applica ions de eloped a ound i , such as symbolic sensi i i y analysis, ansien analysis o elec onic powe ci - cui s, calcula ion o es abili y index and aul diagnosis. SAPEC can be ob ained o ee om he au ho (e-mail: mane i@ ing i 1 .ing.uni i.i ) SSPICE [6] was de eloped a Michigan S a e Uni e si y o bo h pe sonal compu e s and main ames. I inco po a es a as albei no e y accu a e SAG echnique. I is also capa- ble o pe o ming sensi i i y analysis [19]. SSPICE is a ail- able om Ins uc ional Media Cen e (Michigan S a e 799 Uni e si y) wi h a cos o $5OO/copy and a license ee o $250/yea . The e is a limi ed s uden e sion o $25. SYNAP [8] was de eloped a ETH in Zu ich (Swi ze - land). Toge he wi h ISAAC and ASAP, i was especially concei ed o he modelling and design o analog in eg a ed ci cui s and, mo e speci ically, o be included on CAD ame- wo ks o au oma ic design o analog in eg a ed ci cui s. I inco po a es encapsula ed models o semiconduc o de ices and analog unc ional blocks, and can handle misma ches. I s newes e sion is capable o analyzing linea con inuous- ime and sample-da a ci cui s, and also inco po a es a p imi i e DC analysis capabili y. SAG echniques implemen ed in SYNAP use he concep o "lazy expansions", which pe - o ms app oxima ions di ec ly on he nes ed exp essions. SYNAP is a ailable om he au ho only o uni e si ies and non-p o i esea ch ins i u ions. The i s e sion o ISAAC [5] was de eloped in he Ka holieke Uni e si ei Leu en. I was he i s ool o inco po a e he analysis o weakly nonlinea cha ac e is ics [26]. Like ASAP, SYNAP, and SSPICE i ob ained simpli ied exp essions using SAG echniques. I uses exac exp essions in expanded o ma and elimina es hose leas signi ican e ms whose accumula ed sum emains small. ISAAC o ms a undamen al pa o he au oma ic design sys em ARIADNE [27]. ISAAC is a ailable om he au ho s o a $400 handling cha ge. A Lucid Common Lisp license is equi ed. Dis ibu ion o companies is done on a case by case basis. Cu en ly i is being imp o ed o include SDG and SBG, among o he new ea u es -- a coope a i e p ojec sha ed wi h ASAP. ASAP [28] was de eloped a he Cen o Nacional de Mic oelec hica in Se illa (Spain). Like SSPICE, i uns on wo ks a ions and PCs. I sha es wi h ISAAC he capabili y o pe o ming g oupings o elemen s and oge he wi h SYNAP, he explici exp ession o misma ching. I calcula es app ox- ima e poles and ze os symbolically. ASAP also inco po a es a new SAG c i e ia which educe e o s by conside ing in e - als o he symbolic pa ame e s ins ead o ixed nume ic al- ues. App oxima ion wi h a ia ion anges has also been used o de elop a eliable and p ecise c i e ion o simpli y nes ed exp essions esul ing om hie a chical analysis. ASAP has been comme cialized wi hin he ELDO amewo k and is a ailable h ough ANACAD-EES. SIFTER [ 141 and RAINIER [9] ha e been de eloped a he Uni e si y o Washing on a Sea le, and ha e mainly add essed he implemen a ion o SBG and SDG echniques, as discussed in Sec ion 2. RAINIER is a ailable o ee ia anonymous p a wol 8.ee.washing on.edu. 5. FUTURE DEVELOPMENTS One o he majo challenges o u u e symbolic analyze s is inc easing he complexi y o ci cui s, while keeping he in e p e abili y o esul s. A possible solu ion has been shown o be he applica ion o hie a chical analysis echniques. These echniques a e specially adequa e o la ge ci cui s consis ing o loosely coupled sub-blocks. Fo s ongly con- nec ed ci cui s new SBG and SDG echniques a e a ising. The symbolic analysis o la ge ne wo ks has been he ocus o in ense a en ion in he las yea s, is s ill o in e es oday, and doub less will con inue o be o in e es in he u u e. Table 1 shows ha mos ools a e limi ed o he analysis o linea (ized) ci cui s. The i s a emp o nonlinea analysis has been epo ed in [26] al hough limi ed o weak non-linea - i ies. Reducing he ac ual symbolic analysis echniques o he s udy o linea o weakly nonlinea cha ac e is ics limi s i s p ac ical applica ions. I can be assu ed ha he possible u u e a ailabili y o la ge signal symbolic analysis ech- niques will suppose he ull in eg a ion o hese ools in au o- ma ic design and es sys ems, and in gene al, wi hin any CAD en i onmen . Rega ding he ime domain, he i s a emp s a ime-domain symbolic simula ion ha e been epo ed in [291. 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