Context-dependent transformation of Pareto-optimal performance fronts of operational amplifiers

1 Con ex -dependen ans o ma ion o Pa e o- op imal pe o mance on s o ope a ional ampli ie s Elisenda Roca, Manuel Velasco-Jiménez, Ra ael Cas o-López and F ancisco V. Fe nández Ins i u e o Mic oelec onics o Se ille, IMSE-CNM, CSIC and Uni e si y o Se ille. A da. Ame ico Vespucio s/n, Isla de la Ca uja, 41092 Se ille, Spain. Phone: +34 954 46 66 66 FAX: +34 954 46 66 00 Co esponding Au ho : Elisenda Roca, [email protected] Abs ac : The use o Pa e o-op imal pe o mance on s in eme ging design me hodologies o analog in eg a ed ci cui s is a keys one o o e come he limi a ions o adi ional design me hodologies. Howe e , mos echniques o gene a e he on s epo ed so a neglec he e ec ha he su ounding ci cui y (such as he load impedance) has on he Pa e o- on , he eby making i only ealis ic o he con ex whe e he on was gene a ed. This s ongly limi s he use o he Pa e o on because o he s ong dependence be ween he key pe o mances o an analog ci cui and i s su ounding ci cui y, bu , mo e impo an ly, because his ci cui y emains unknown un il he Pa e o-op imal on is being used. Since pe o mance on gene a ion is a cos ly p ocess, his pape p oposes ha pe o mance on s o a new con ex o use o a gi en ci cui can be ob ained om on s ha we e p e iously gene a ed unde some di e en condi ions. Towa ds his goal, a ans o ma ion me hodology o pe o mance objec i es o ope a ional ampli ie s has been de eloped. Expe imen al esul s o a olded-cascode and a Mille -compensa ed ope a ional ampli ie s show ha his is a p omising app oach o euse he on s in mul iple con ex s. Keywo ds: analog in eg a ed ci cui s, design au oma ion, ci cui sizing, simula ion-based op imiza ion, mul i-objec i e op imiza ion 2 1 In oduc ion Elec onic design au oma ion o analog in eg a ed ci cui s s ill lags behind i s digi al coun e pa . Beyond any single eason, he inhe en complexi y o designing he simples o he analog sys ems ( he many non-ideal e ec s, he la ge sensi i i y o noise, e c.) has hinde ed he e olu ion o sys ema ic design me hodologies. Key asks in any analog design au oma ion low a e opology selec ion and opology sizing [1],[2]. Gi en a se o design speci ica ions, he i s ask e e s o he selec ion o an app op ia e opology, among a se o unc ionally equi alen ones, ha can mee he speci ica ions. Once a opology has been selec ed, gi en as an in e connec ion o sub-blocks, he second ask e e s o he de e mina ion o sub-block speci ica ions o which he highe -le el speci ica ions can be achie ed. A he ansis o le el, opology sizing educes o se ing he de ice sizes and bias ol ages and cu en s. In he pas couple o decades mos app oaches o analog sizing ha e been buil a ound op imiza ion-based app oaches: some i e a i e op imiza ion algo i hm wi h a pe o mance e alua o in he loop [3]-[9]. The sizing p oblem has usually been o mula ed as: min ( ) subjec o () 0 L H  x x XxX gx (1) In his equa ion, he ec o () x is he se o objec i e unc ions o be minimized. Vec o xco esponds o he design a iables, and L X and H X a e hei lowe and uppe bounds, espec i ely. The ec o () 0gx co esponds o he design cons ain s and delimi s he easible pe o mance egion. I ec o () x con ains only one objec i e, he design p oblem can be sol ed by using any single- objec i e op imiza ion algo i hm, e.g., simula ed annealing [10], gene ic algo i hms [11], di e en ial e olu ion [12], e c. I ec o () xcon ains mo e han one objec i e, he e a e usually mul iple solu ions o (1), each one co esponding o a di e en se o alues o he objec i es. In he majo i y o epo ed app oaches, he sizing p oblem has been sol ed by applying a single- objec i e op imiza ion algo i hm. Fo he case o se e al objec i e unc ions, his en ails he ans o ma ion o he mul i-objec i e op imiza ion p oblem in o a single-objec i e op imiza ion p oblem, usually by applying a weigh ed sum o he objec i e unc ions. This equi es selec ing a p io i one solu ion among he mul iple ones men ioned abo e. Mo e ecen ly, a wa e o inno a ion has a isen wi h he de elopmen and applica ion o mul i- objec i e op imiza ion algo i hms o his kind o p oblems. Se e al algo i hms ha e been epo ed (e.g., NSGAII [13], SPEA2 [14], MOEA/D [15], e c.) and success ully applied o design au oma ion p oblems [16]-[22]. The ou come o he applica ion o hese algo i hms o p oblems wi h se e al mu ually con lic ing objec i es is a Pa e o-op imal pe o mance on , which shows he bes ade-o s among he objec i e unc ions. In a Pa e o-op imal on , one pe o mance objec i e canno be imp o ed wi hou wo sening a leas one o he o he objec i es. Pa e o on s allow, o ins ance, he selec ion o a design solu ion a pos e io i, once he comple e se o solu ions has been gene a ed. Besides, new possibili ies a ise. Fo ins ance, hey can be used as a esou ce o opology selec ion. Since he bes ade-o s among solu ions a e a ailable, i can be immedia ely known i a gi en opology can mee a gi en se o speci ica ions. Pe o mance on s ha e been e en p oposed as a basic ing edien o hie a chical mul i-objec i e bo om-up syn hesis me hodologies [21],[22]. 3 Mos popula algo i hms o pe o mance on gene a ion a e mul i-objec i e e olu iona y algo i hms [23],[24], whe e a popula ion o candida e solu ions ( ypically in he o de o some hund eds o a ew housands) e ol es along a numbe o gene a ions ( ypically in he o de o some hund eds). The e o e, he op imiza ion algo i hm equi es a numbe o execu ions o he pe o mance e alua o in he o de o housands o some hund ed housands. High accu acy is usually wished and, hence, he pe o mance e alua o is ela i ely cos ly: elec ical simula ion (e.g., HSPICE [25]) o cell-le el sizing o de ailed beha io al simula ion a highe le els [26]. I is easy o conclude ha he gene a ion o a pe o mance on is a cos ly p ocess ha may ake se e al hou s o compu a ion ime. The eason such a high compu a ional ime may be conside ed accep able is ha , unlike single-objec i e design p oblems, he ou come o mul i-objec i e op imiza ion algo i hms can be used o a wide ange o p oblems. Howe e , in his easoning a undamen al issue has been igno ed in he li e a u e. The objec i e unc ions canno be selec ed a bi a ily. They mus co espond o he ypical pe o mances ha he designe is in e es ed in, and he ul ima e alue o many commonly used pe o mance cha ac e is ics o analog ci cui s do no only depend on he block i sel , bu , also, on i s su ounding ci cui y, i.e., he gene a ed Pa e o- on depends on he con ex whe e he analog building block is being used. To add ess his con ex dependency o he on s, a ans o ma ion p ocedu e is p oposed in his pape 1. The pape is s uc u ed as ollows. Basic concep s o mul i-objec i e op imiza ion and gene a ion o Pa e o-op imal pe o mance on s a e in oduced in Sec ion 2. Sec ion 3 de ails he p ac ical p oblems ha mo i a e his wo k and Sec ion 4 p oposes a ans o ma ion p ocedu e ha sol es he p e ious p oblems. Sec ion 5 illus a es his me hodology wi h wo examples, a olded- cascode ope a ional ampli ie and a Mille ope a ional ampli ie , and di e en numbe o objec i es. Finally, conclusions a e p esen ed in Sec ion 6. 2 Gene a ion o Pa e o-op imal pe o mance on s As s a ed in equa ion (1), he sizing p oblem is o mula ed as he maximiza ion o minimiza ion o a se o n design objec i es,   () (), (), , () 12 n  xxx x , whe e x is he ec o o design a iables (e.g., de ice sizes, pe o mance cha ac e is ics o sub-blocks a in e media e hie a chical le els), and each () i xis a pe o mance cha ac e is ic o he block (such as dc gain), subjec o some cons ain s (e.g., slew a e la ge han a ce ain alue o an ope a ional ampli ie ). In he single-objec i e case, design candida es, each one ep esen ed by a ec o o design a iables x, can be easily o de ed by conside ing he objec i e unc ion alue () x. Howe e , in he mul i-objec i e case, a di e en o de ing me hod is equi ed: Pa e o dominance. A design poin , aX, is said o domina e ano he design poin ,  bX, (no ed as ab) i () ()  a b and ()< () ii ab o a leas one unc ion i 2. The design poin a is said o be non-domina ed i he e is no o he design poin ha domina es i . The non-domina ed se o he en i e easible sea ch space is known as he Pa e o-op imal on . All hese concep s a e illus a ed in Figu e 1 o a 2- dimensional objec i e space. 1 A p elimina y e sion o his pape appea ed a he XI In . Wo kshop on Symbolic and Nume ical Me hods, Modeling and Applica ions o Ci cui Design [27]. 2 This o mula ion is alid o minimiza ion p oblems. A simple change o sign applies o maximiza ion. 4 The compu a ion o an app oxima ion o he ideal Pa e o-op imal on is ypically pe o med by using mul i-objec i e e olu iona y algo i hms [23]. O de ing o solu ions acco ding o he Pa e o dominance concep equi es a pe o mance e alua o , which assesses objec i e and cons ain alues o each design solu ion, ypically named indi idual in he li e a u e on e olu iona y algo i hms. These mul i-objec i e e olu iona y algo i hms ypically s a wi h a andom popula ion o indi iduals ha , a e being e alua ed by he pe o mance e alua o , is modi ied in such a way ha a e n i e a ions (called gene a ions) a popula ion o non-domina ed indi iduals is ob ained: he Pa e o-op imal on . Being o s ochas ic na u e, he compu a ional cos due o he high numbe o equi ed pe o mance e alua ions is he main d awback o hese algo i hms. E iciency, con e gence o he ideal Pa e o-op imal on , and di e si y o solu ions a e a eas o in ense and cu en esea ch. An illus a i e example o ecen esea ch in hese a eas is he de ini ion o quali y me ics ha , p ope ly acked, can con ol when he op imiza ion algo i hm should be s opped [28]. In o de o illus a e he po en ial o Pa e o-op imal on s, le us conside he olded-cascode ope a ional ampli ie o Figu e 2, whe e a capaci i e load o 1pF has been conside ed a he ou pu . The ci cui is op imized o ob ain he maximum alues o wo con lic ing objec i es: he dc gain, 0 A , and he uni y-gain equency, u . The design a iables a e he wid h and leng h o he ansis o s, as well as he bias cu en s. Di e en cons ain s a e imposed in o de o ob ain co ec and use ul sized ci cui s; o example, dc gain is se o be la ge han 20dB, phase ma gin is se o lie be ween 10º and 90º, and ansis o s a e en o ced o ope a e in he sa u a ion egion. The Pa e o-op imal on was gene a ed by coupling he elec ical simula o HSPICE o he mul i- objec i e e olu iona y op imiza ion algo i hm NSGAII. The popula ion size and numbe o gene a ions we e se a 1500 and 100, espec i ely. The gene a ion o his Pa e o-op imal pe o mance on ook 1 hou o CPU ime on a 2.2-GHz p ocesso . The esul is a 1500-poin s app oxima ion o he 2-dimensional Pa e o on , which can be seen as black poin s in Figu e 3. Each poin ep esen s a sized ci cui showing a bes ade-o among he wo pe o mances being conside ed. To illus a e he e olu ion o he popula ion o design solu ions owa ds he Pa e o- op imal on , Figu e 3 shows he popula ion a e 8 gene a ions ( ed poin s), 25 gene a ions (blue poin s) and 100 gene a ions (black poin s). Figu e 1. Illus a ing Pa e o dominance and Pa e o-op imal on concep s o a 2- dimensional pe o mance space. 5 3 Po en ial p oblems in using pe o mance on s As discussed in p e ious sec ions, he gene a ion o Pa e o-op imal pe o mance on s may be a compu a ionally expensi e p ocess. Ye , hei g ea po en ial bene i is ha mul iple pe o mance ade-o s a e a ailable. Ei he i hey a e used in e ac i ely o wi hin some design au oma ion low, once hey a e gene a ed hey can be eused many imes by selec ing he design solu ion wi h he app op ia e pe o mance ade-o in each case. Howe e , he e is a undamen al law in his easoning: some ci cui pe o mances do no depend only on he ci cui i sel bu also on he o he ci cui y o which i is connec ed. Conside , o ins ance, he load impedance ha a ypical analog block such as he Mille -compensa ed ope a ional ampli ie in Figu e 4 has o d i e. Al hough, M2 M1 M 0 Vip Mbn Ibn Vdd Vin M3M4 M5M6 M8 M7 M10 M9 Mbp Ibp Vss Vdd Vdd Vss Vss Vss Vcp Vcn Vou ZL Figu e 2. Folded-cascode ampli ie . Figu e 3. Illus a ing he Pa e o-op imal on gene a ion o he olded-cascode ampli ie o Figu e 2 whe e dc gain and uni y gain equency ha e been maximized. 6 usually, a highe numbe o pe o mance objec i es has o be conside ed, o illus a ion’s sake we will conside only wo: dc gain and uni y-gain equency. The plo wi h black ci cles in Figu e 5 shows he pe o mance on o he Mille opamp exhibi ing he dc gain s. uni y-gain equency ade-o o a phase ma gin cons ain PM>60º and a capaci i e load o 1pF. I he load impedance is changed o he pa allel connec ion o a 2-pF capaci ance and a 10-k esis ance, he pe o mance on ob ained is he one shown in he same igu e wi h g ey ci cles. The e o e, i becomes clea ha he in o ma ion ha he pe o mance on o e s abou he ade-o s ha a opology can achie e, depends on he con ex whe e he ci cui is going o be used. The conclusions d awn om he pe o mance on ob ained o some gi en condi ions canno be hus di ec ly ex apola ed o some o he condi ions. M4 M3 M2 M1 M6 Cc M7 Vss Vdd M5 Vin Mbn Ib Vdd Vip Vo ZL Figu e 4 Mille wo-s age ope a ional ampli ie . Figu e 5. Dc gain s. uni y-gain equency ade-o o he Mille ope a ional ampli ie o a load impedance equal o 1pF (black ci cles) and he pa allel connec ion o a 2-pF capaci ance and a 10-k esis ance. 7 As discussed in Sec ion 1, Pa e o-op imal pe o mance on s o di e en opologies can be used o selec he mos app op ia e a chi ec u e o some gi en speci ica ions. Fo ins ance, Figu e 6 shows he bes dc gain s uni y-gain equency ade-o ha can be ob ained o he Mille - compensa ed ope a ional ampli ie ( ed do s) in Figu e 4 and he olded-cascode ope a ional ampli ie (blue do s) in Figu e 2, espec i ely, when bo h a e loaded wi h a 1-pF capaci ance. Acco ding o hese pe o mance on s, and assuming ha only hese wo pe o mances a e o in e es , he e is a ange o pe o mance alues o which a Mille ope a ional ampli ie would be p e e able. A olded-cascode opamp, on he o he hand, would be he p e e ed op ion o o he pe o mance alues. Assume now ha he ampli ie load impedance is gi en by he pa allel connec ion o a 2-pF capaci ance and a 50-k esis ance. In his case, he Pa e o-op imal pe o mance on o he Mille ampli ie is plo ed wi h g een do s and ha o he olded-cascode ampli ie is plo ed in magen a do s. In his second case, all poin s o he olded-cascode ampli ie on a e domina ed by poin s o he Mille ampli ie on , hence, he Mille ampli ie would be he p e e ed op ion wha e e he equi ed pe o mances a e. The e o e, hese expe imen s exempli y ha knowledge o he pe o mance on s a he igh load condi ions is also essen ial o p ope selec ion o ci cui opology. 4 Me hodology o con ex -dependen ans o ma ion o Pa e o-op imal pe o mance on s o ope a ional ampli ie s Some o he pe o mances usually conside ed in an op imiza ion p ocess o an ope a ional ampli ie depend on he load condi ions, o ins ance dc gain 0 A , uni y-gain equency u , o phase ma gin PM. Howe e , he load condi ions a e no known un il he design p ocess is being pe o med, ha is, un il any o he ci cui y a ound he ampli ie is known. Since he Pa e o on gene a ion p ocess is a leng hy p ocess, he objec i e o ou esea ch is o gene a e ade-o Figu e 6. Dc gain s. uni y-gain equency ade-o o he Mille and olded-cascode ope a ional ampli ie s o a load impedance equal o 1pF and he pa allel connec ion o a 2- p F capaci ance and a 10-k esis ance. 8 in o ma ion a p io i and easily and e icien ly ans o m his in o ma ion in o he Pa e o on s o he pe o mances when he load condi ions a e known. The ope a ional ampli ie can be iewed as a wo-po , like he one shown in Figu e 7. Vol age 1 and cu en i1 ep esen he di e en ial inpu ol age and cu en , espec i ely. Vol age 2 and cu en i2 ep esen he ou pu ol age and cu en , espec i ely. A p io i, any wo-po ma ix cha ac e iza ion o he ampli ie can be conside ed. Howe e , he wo-po ma ix pa ame e s mus be selec ed in elligen ly, so ha a load-independen cha ac e iza ion o he ope a ional ampli ie can be ob ained mo e easily, acco ding o he pe o mances o in e es o he block. Le us conside he hyb id-2 pa ame e s [29] o cha ac e ize he wo-po : 1111122 2211222 ih hi h hi           (2) In his equa ion, pa ame e 11 h ep esen s he inpu impedance, 12 h  is he in e se cu en gain, 21 h  ep esen s he ol age gain o he ampli ie wi hou any load and 22 h  is he ou pu impedance. The ol age gain and ou pu impedance when a ce ain load L Z is added can be ob ained om (2) and he cons i u i e equa ion: 22L Zi   (3) yielding he ollowing exp essions: 21 22 22 () () () 1() () () L o hs As hs Z s Z shs      (4) Equa ion (4) allows o ob ain he hyb id-2 pa ame e s 21 h  and 22 h  om he ol age gain, () A s, and ou pu impedance, () o Z s, o some known load condi ions, and ice e sa, o ob ain he ol age gain () A sand ou pu impedance () o Z s o some load om he hyb id-2 pa ame e s 21 h  and 22 h. Mo eo e , a pa icula case is when L Z. In his case, 21 h  is iden ical o () A s. This equa ion is he key o de eloping he me hodology ha is p oposed in his pape o ans o m a Pa e o on o some known load condi ions o a bi a y new load condi ions. Le us assume ha we wish o gene a e he Pa e o-op imal pe o mance on o he ou pu impedance, dc ol age gain, uni y-gain equency, and phase ma gin o some a bi a y load condi ions. The gene a ion me hodology would p oceed as ollows: 1) Gene a e he Pa e o-op imal on o he pe o mances o in e es o some known load condi ions by using a mul i-objec i e op imiza ion algo i hm wi h a nes ed elec ical simula o as pe o mance e alua o . The pe o mances abo e (ou pu impedance, gain, uni y-gain equency 1 i 2 i 2 1     L Z Figu e 7. Two-po wi h a bi a y load. 9 and phase ma gin) can be easily ob ained by p ocessing he ou pu o he AC ci cui analysis pe o med by he elec ical simula o . 2) Fo each sample o indi idual o his Pa e o on , s o e pole and ze o loca ions o he ou pu impedance () o Z s and he ol age gain () A s, bo h being equency dependen unc ions, as well as hei dc alues. This in o ma ion can be e ie ed om common elec ical simula o s. 3) Use (4) o ex ac he hyb id pa ame e s 21()hs  and 22 ()hs  o each sample; his means ex ac ing dc alues and bo h, poles and ze os. No e ha , om basic ci cui heo y, he poles o 21()hs  and 22 ()hs  a e iden ical. 4) Apply (4) o ob ain he ol age gain () A s o he new a bi a y load condi ions (new () L Z s) om he p e iously calcula ed hyb id pa ame e s. 5) Ob ain he pe o mance pa ame e s (dc gain, uni y-gain equency and phase ma gin) by simple p ocessing o he magni ude and phase o he ne wo k unc ion () A s. No e ha his p ocedu e can be applied o any ini ial known load condi ions. The i s s ep o his me hodology has he hea ies compu a ional e o by a , bu no e ha he esul s o s ep 3 a e independen o he applica ion, i.e., independen o he inal load condi ions. The e o e, he i s h ee s eps can be pe o med be o ehand, as only he ci cui opology and he pe o mances o in e es a e equi ed. The esul s can be s o ed and used whene e and whe e e necessa y. 5 Expe imen al Resul s 5.1 Folded-cascode ope a ional ampli ie The p oposed me hodology will be i s applied o a single-s age ope a ional ampli ie , he olded- cascode ampli ie in Figu e 2. In his case, he op imiza ion p ocess will be ca ied ou o h ee di e en pe o mances: dc gain, 0 A , uni y-gain equency, u , and phase ma gin, PM. Acco ding o s ep 1 o he me hodology in Sec ion 4, a Pa e o-op imal on wi h a capaci i e load o 1pF was i s gene a ed o he h ee objec i es. The design a iables in his case a e he wid h and leng h o he ansis o s, as well as he bias cu en s. Se e al cons ain s a e imposed in o de o ob ain alid ci cui s, while a he same ime explo e all he easibili y ange o he pe o mances ( o example, dc gain is se o be la ge han 20dB, and phase ma gin 90º >PM >10º). The Pa e o on was gene a ed by using he elec ical simula o HSPICE as pe o mance e alua o wi hin he mul i- objec i e e olu iona y op imiza ion algo i hm NSGAII 3. The popula ion size and he numbe o gene a ions we e 1500 and 100, espec i ely. The gene a ion o his Pa e o on ook 1 hou o CPU ime on a 2.2 GHz p ocesso . The esul is 1500 sample poin s o he 3-dimensional Pa e o on , which can be seen in Figu e 8 (2 dimensional p ojec ions o his on a e included in Figu e 3 Al hough all expe imen al esul s in his pape ha e been ob ained by applying he mul i- objec i e op imiza ion algo i hm NSGAII, i should be highligh ed ha he ans o ma ion p ocedu e in oduced in Sec ion 4 is comple ely independen o he op imiza ion algo i hm used o ob ain he ini ial Pa e o-op imal on . 16 Fu u e wo k will add ess he gene a ion o pe o mance on s o se e al ini ial load condi ions o imp o e he di e si y o he ans o med pe o mance on s. 7 Acknowledgemen s This esea ch wo k has been suppo ed by he TIC-2532 P ojec , unded by Conseje ía de Inno ación, Ciencia y Emp esa, Jun a de Andalucía, and he TEC2010-14825 P ojec , unded by he Spanish Minis y o Science and Inno a ion and ERDF. 8 Re e ences [1] Ru enba , R., Gielen, G. & Roychowdhu y, J. (2007). Hie a chical modeling, op imiza ion, and syn hesis o sys em-le el analog and RF designs. P oc. o he IEEE, 95(3), 640-669. [2] Ma ens, E. & Gielen, G. (2008). Classi ica ion o analog syn hesis ools based on hei a chi ec u e selec ion mechanisms. In eg a ion – he VLSI Jou nal, 41(2), 238-252. [3] Medei o, F., Rod íguez-Macías, R., Fe nández, F.V., Domínguez-Cas o, R., Hue as J.L., & Rod íguez-Vázquez, A. (1994). Global design o analog cells using s a is ical op imiza ion echniques. Analog In eg a ed Ci cui s and Signal P ocessing, 6(3), 179-195. [4] Phelps, R., K asnicki, M., Ru enba , R.A., Ca ley, L.R., & Hellums, J.R. (2000). Anaconda: simula ion-based syn hesis o analog ci cui s ia s ochas ic pa e n sea ch. IEEE T ans. on Compu e -Aided Design, 19(6), 703-717. [5] Alpaydin, G., Balki , S., & Dunda , G. (2003). An e olu iona y app oach o au oma ic syn hesis o high-pe o mance analog in eg a ed ci cui s. IEEE T ans. on E olu iona y Compu a ion, 7(3), 240-252. [6] Ba os, M., Guilhe me, J., & Ho a, N. (2010). Analog ci cui s op imiza ion based on e olu iona y compu a ion echniques. In eg a ion – he VLSI Jou nal, 43(1), 136-155. [7] Sonmez, O., & Dunda , G. (2011). Simula ion-based analog and RF ci cui syn hesis using a modi ied e olu iona y s a egies algo i hm. In eg a ion – he VLSI Jou nal, 44(2), 144-154. [8] Cas o-López, R., Gue a, O., Roca, E., & Fe nández, F.V. (2008). An in eg a ed layou - syn hesis app oach o analog ICs. IEEE T ans. on Compu e -Aided Design, 27(7), 1179- 1189. [9] Liu, B., Fe nández, F.V., Gielen, G., Cas o-López, R., & Roca, E. (2009). A meme ic app oach o he au oma ic design o high-pe o mance analog in eg a ed ci cui s. ACM T ans. on Design Au oma ion o Elec onic Sys ems, 14(3). [10] Ru enba , R.A.(1989). Simula ed annealing algo i hms: an o e iew. IEEE Ci cui s and De ices Magazine, 5(1),19-26. [11] Michalewicz, Z. (1996) Gene ic algo i hms + da a s uc u es = e olu ion p og ams. Sp inge . [12] P ice, K., S o n, R.,& Lampinen, J. (2005) Di e en ial e olu ion. A p ac ical app oach o global op imiza ion. Sp inge . [13] Deb, K., P a ap, A., Aga wal, S., & Meya i an, T. (2002). A as and eli is mul i-objec i e gene ic algo i hm: NSGA-II. IEEE T ans. on E olu iona y Compu a ion, 6(2), 181-197. 17 [14] Zi zle , E., Laumanns, M., & Thiele, L. (2001). SPEA2: Imp o ing he s eng h Pa e o e olu iona y algo i hm. TIK-Repo 103, Swiss Fede al Ins i u e o Technology. [15] Zhang Q., & Li, H. (2007). MOEA/D: A mul iobjec i e e olu iona y algo i hm based on decomposi ion. IEEE T ans. on E olu iona y Compu a ion, 11(6), 712-731. [16] De Smed , B., & Gielen, G. (2003). WATSON: Design space bounda y explo a ion and model gene a ion o analog and RF IC design. IEEE T ans. on Compu e -Aided Design, 22(2), 213- 224. [17] Chu, M., Alls o , D.J., Hua d, J.M., & Wong, K.Y. (2004). NSGA-based pa asi ic-awa e op imiza ion o a 5GHz low-noise VCO. In P oc. Asia and Sou h Paci ic Design Au oma ion Con ., (pp. 169-174). [18] Tiwa y, S.K., Velu, S., Ru enba , R.A., & Mukhe jee, T. (2004). Pa e o op imal modeling o e icien PLL op imiza ion. In P oc. NSTI Nano echnology Con . (pp. 195-198). [19] S eh , G., G aeb, H.E., & An eich, K.J. (2007). Analog pe o mance space explo a ion by no mal-bounda y in e sec ion and by Fou ie –Mo zkin elimina ion. IEEE T ans. on Compu e -Aided Design o In eg a ed Ci cui s and Sys ems, 26(10), 1733-1748. [20] Cas o-López, R., Roca, E. , & Fe nández, F.V. (2009). Mul imode pa e o on s o design o econ igu able analogue ci cui s. Elec onics Le e s, 45(2), 95-96. [21] Eeckelae , T., McConaghy, T., & Gielen, G. (2005). E icien mul iobjec i e syn hesis o analog ci cui s using hie a chical pa e o-op imal pe o mance hype su aces. In P oceedings o he Design Au oma ion and Tes in Eu ope Con e ence (DATE’05) – VOLUME 2, (pp. 1070-1075). [22] Eeckelae , T., Schoo s, R., S eyae , M., Gielen, G., & Sansen, W. (2008). A con inuous- ime del a-sigma modula o o 802.11a/b/g WLAN implemen ed wi h a hie a chical bo om- up op imiza ion me hodology. Analog In eg a ed Ci cui s and Signal P ocessing, 55(1), 37- 45. [23] Deb, K. (2001). Mul i-objec i e Op imiza ion Using E olu iona y Algo i hms. Wiley. [24] Coello-Coello, C., Lamon , G.B., & Van Veldhuizen, D.A. (2007). E olu iona y algo i hms o sol ing mul i-objec i e p oblems. Second edi ion. Sp inge . [25] HSPICE ci cui simula o . h p://www.synopsys.com/Tools/Ve i ica ion/AMSVe i ica ion/Ci cui Simula ion/HSPICE/Pa ges/de aul .aspx. Accessed 15 Feb ua y 2011. [26] Ruiz-Amaya, J., de la Rosa, J.M., Fe nández, F.V., Medei o, F., del Río, R., Pé ez-Ve dú, B., & Rod íguez-Vázquez, A. (2005). High-Le el Syn hesis o Swi ched-Capaci o , Swi ched- Cu en and Con inuous-Time  Modula o s Using SIMULINK-based Time-Domain Beha io al Models. IEEE T ans. on Ci cui s and Sys ems, Pa I, 52(9), 1795-1810. [27] Roca, E., Velasco-Jiménez, M., Cas o-López, R., & Fe nández, F.V. (2010). Con ex - independen pe o mance modeling o ope a ional ampli ie s using Pa e o on s. In P oc.o he XI h In . Wo skhop on Symbolic and Nume ical Me hods, Modeling and Applica ions o Ci cui Design, (pp.1-4). [28] Fe nández, F.V., Es eban-Mulle , J., Roca, E., & Cas o-López, R. (2010). S opping c i e ia in e olu iona y algo i hms o mul i-objec i e pe o mance op imiza ion o in eg a ed induc o s. In P oc. IEEE In . Con . on E olu iona y Compu a ion. 18 [29] Chua, L.O., Desoe , C., Kuh, E.S. (1987). Linea and Non-linea Ci cui s. McG aw-Hill.