Load-independent characterization of trade-off fronts for operational amplifiers

Load-independen cha ac e iza ion o ade-o on s o ope a ional ampli ie s Abs ac —In eme ging design me hodologies o analog in eg a ed ci cui s, he use o pe o mance ade-o on s, also known as Pa e o on s, is a keys one o o e come he limi a ions o he adi ional op-down me hodologies. Howe e , mos echniques epo ed so a o gene a e he on neglec he e ec o he su ounding ci cui y (such as he ou pu load impedance) on he Pa e o- on , he eby making i only alid o he con ex whe e he on was gene a ed. This s ongly limi s i s use in hie a chical analog syn hesis because o he hea y dependence o key pe o mances on he su ounding ci cui y, bu , mo e impo an ly, because his ci cui y emains unknown un il he syn hesis p ocess. We will add ess his p oblem by p oposing a new echnique o gene a e he ade-o on s ha is independen o he load ha he ci cui has o d i e. This idea is exploi ed o a commonly used ci cui , he ope a ional ampli ie , and expe imen al esul s show ha his is a p omising app oach o sol e he issue. Keywo ds - Analog syn hesis; Mul i-objec i e op imiza ion; Pa e o-op imal on s; Hie a chical syn hesis. I. INTRODUCTION Analog in eg a ed ci cui s s ill lag behind in compa ison o hei digi al coun e pa in e ms o Elec onic Design Au oma ion. 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 same pace o e olu ion in sys ema ic design me hodologies. A ho opic in his sense is he sys ema iza ion o hie a chical design o analog ci cui s. This p ocess begins wi h he sys em equi emen s and ends a he de ice le el, wi h he speci ica ion o ansis o sizes, alues o all passi e de ices, and so on. The ac s ha mos building blocks in any analog hie a chy ea u e a mul i-dimensional space o pe o mance cha ac e is ics and ha he mapping be ween design objec i es (o pe o mances) and design a iables (like he W and L o a ansis o ) is an in ol ed p oblem, make analog hie a chical syn hesis a e y complex p oblem. T adi ionally, his p oblem has been add essed by using a op- down design app oach [1], whe e he sys em is hie a chically decomposed in di e en sub-sys em building blocks, down o he de ice le el. A each hie a chical le el, an app op ia e a chi ec u e is selec ed o each block and i s speci ica ions a e ansmi ed in o a sub-se o speci ica ions o each o he sub- blocks. The op-down speci ica ion ansmission p ocess ends up when he de ice le el is eached, i.e., speci ica ion ansmission a ha le el implies ob aining de ice sizes. Howe e , his app oach has wo impo an laws: i s , i does no gua an ee he easibili y o he building blocks (as hei equi emen s a e being de i ed in he speci ica ion ansmission p ocess) since i is unknown i hese equi emen s a e ealizable o no a lowe hie a chical le els; second, he e a e no accu a e es ima es o powe consump ion and a ea occupa ion a he beginning o he speci ica ion ansmission p ocess (a any in e media e hie a chical le el) since hese wo igu es depend also on low-le el de ails, no known a his ea ly s age o he design. In he ecen ly p oposed mul i-objec i e bo om-up (MOBU) app oach [2], he hie a chy is handled in a bo om- up- i s way by means o he concep o Pa e o-op imal on (POF), a p omising esou ce o pallia e he d awbacks o adi ional op-down design me hodologies [3][4][5]. A POF is he se o di e en ins ances o designs (e.g., di e en sizing) o a ci cui block ha bes cha ac e izes he ade-o s be ween compe ing pe o mances, like powe s. speed. Gene a ing he POF is a mul i-objec i e op imiza ion p oblem, ypically sol ed by a popula ion-based op imiza ion algo i hm, coupled o a pe o mance e alua o (such as an elec ical simula o ). POF gene a ion ypically in ol es many housands o simula ions and, hence, qui e long compu a ion imes. The ul ima e po en ial o he POF concep is ha , once gene a ed, i could be used whe e e and whene e necessa y, ha is, in any syn hesis p oblem in ol ing such building blocks. No ice ha he on es ablishes ully de ined, bi-uni ocal ela ions be ween pe o mances and de ice sizes. In his way, i is in p inciple possible o hie a chically compose all he building blocks’s ade-o on s o ob ain he comple e ade-o on o an analog sys em. Then, mapping he sys ems equi emen s becomes a apid, s aigh o wa d p ocess, whe e, bo h easibili y and accu a e es ima es a e gua an eed. Howe e , he e s ill lies a undamen al issue: ha 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 on i s su ounding ci cui y. Tha is, he . gene a ed Pa e o- on depends on he con ex whe e he analog building block is being used. Conside , o ins ance, he load impedance ha a ypical analog block such as he ope a ional ampli ie has o d i e. Fo ins ance, i he ope a ional ampli ie is equi ed o ha e a dc gain o 50dB when a 5kΩ-load is a i s ou pu and he Pa e o on was ob ained wi h a 100kΩ-load, hen he selec ed designs may u n useless, possibly because hei ou pu impedances a e much la ge han 5kΩ. This same si ua ion a ises in hie a chical syn hesis, because, as said abo e, he su ounding ci cui y o he building block is unknown (and so is he load impedance). I is essen ial o s ess he impo ance o his ac , because he use o POFs o sol e he issues o op-down design and imp o e he sys ema ic design o analog ci cui s is hea ily comp omised by his limi a ion. In he applica ion o he MOBU me hodology o a eal-li e desi on s o ana imal on s and he com II. GENERATION OF PARETO-OPTIMAL FRONTS o a ci c gn p oblem, a ΔΣ A/D con e e [6], he p oblems wi h he su ounding ci cui y ha e been ci cum en ed by selec ing only hose designs om he Pa e o on s o he di e en building blocks ha mee ce ain cons ain s: o ins ance, ha he ou pu impedance o he DAC block is highe han he ou pu esis ance o he co esponding in eg a o , o ha he i s non-dominan pole o he ou pu impedance o he DAC is signi ican ly highe han he in eg a o gain-bandwid h p oduc . The p oblem is ha his is an ad-hoc solu ion, ha o e ly cons ains he design p oblem and limi s he e ec i eness o POF-based syn hesis as only a small ac ion o designs o he POF a e e en ually used (imposing he abo e desc ibed cons ain s leads o an impo an educ ion in he numbe o alid solu ions o he con e e , o which he e ini ially we e o e a hund ed alid designs o each building block). Ou p oposed app oach is o gene a e Pa e o log ci cui s ha ci cum en he dependence wi h he su ounding ci cui y. In his pape , we de elop his app oach o ope a ional ampli ie s, o which a me hodology o ans o m he POF o small-signal cha ac e is ics among a bi a y load condi ions is in oduced Sec ion II desc ibes Pa e o-op pu a ional echniques o ob aining hem. Sec ion III in oduces he gene a ion o POFs independen ly o he su ounding ci cui y. Sec ion IV is de o ed o discuss di e en s a egies o he applica ion o hese POFs, mainly o hie a chical syn hesis p oblems. Finally, conclusions a e p esen ed in Sec ion V. Gene a ion o he POF o he selec ed pe o mances ui block can be posed as a mul i-objec i e op imiza ion p oblem. This p oblem is o mula ed by maximizing o minimizing, simul aneously, a se o b design objec i es, { } () (), (), , () 12 b =L y = xxx x , whe e x is a ec o o e o mance cha ac e is ics o sub-blocks a in e media e hie a chical le els), and each i(x)is a pe o mance cha ac e is ic o he block (such as dc subjec o some cons ain s (e.g., slew a e la ge han a ce ain alue). The p oblem can be ma hema ically posed by min ( ) x x design a iables (e.g. de ice sizes, p gain), ollowing he nex o mula ion: (1) () 0 subjec o << ≥ ⎧ ⎨ ⎩LH XxX gx whe e L X and H X a e he lowe and uppe bounds ec o , es co esponds o he use - de i ons s, delimi ble egion. A design o he x p ned c ec i ely. Vec o ain g(x)≥0 ing he easi poin , ∈ aX, is d o domina e ano he design poin , sai ∈ bX, (no ed as pab) i () ()≤ F aand ()< () ii ab o a leas one unc ion i Fb 1. The design poin ais said o be non- domin he e is no o he design ha domina es e non-domina known as he Pa e o-op imal on . All hese concep s a e illus a ed in Figu e 1 o a wo pe o mance on . The compu a ion o he Pa e o on is ypically e icien ly and accu a ely done by using mul i-objec i e e olu iona y s [7], coupled o an ele a ed i i . Th ed se o he en i e easible sea ch space is c ical simula o (e.g., HS algo i hm PICE). These algo i hms s a wi h a andom popula ion o indi iduals ha , a e being e alua ed by he elec ical simula 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 i ness e alua ions (ci cui simula ions) is he main d awback o hese algo i hms [8]. E iciency, con e gence o he ue POF, and di e si y o solu ions a e a eas o in ense and cu en esea ch. An example is he de ini ion o new quali y e alua ion me ics sui able o analog design p oblems p oposed in [9]. Figu e 1. Illus a ing he Pa e o-op imal on concep o a wo- dimensional on . 1 This o mula ion is alid o minimiza ion p oblems. A simple change o sign applies o maximiza ion. In o de o illus a e he po en iali y o POFs, le us conside he Mille ope a ional ampli ie depic ed in , o which he pe o mances o in e es a e: dc gain, Figu e 2 0 A , uni y-gain equency, u , phase ma gin, PM, and ou pu impedance, o Z . The op imiza ion p ocess aims a maximizing he i s h ee pe o mances ( 0 A ,u and PM) and minimizing o Z . Some o hese pe o mances depend on he load condi ions, so when he POF o his block is gene a ed, a load will be included in he e alua ion o he indi iduals o he POF. The design a iables in his case a e he wid h and leng h o he ansis o s, plus he bias cu en and he compensa ion capaci o . Di e en cons ain s a e imposed o he p oblem 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, and phase ma gin 90 . The POF wi h a capaci i e load o 1pF 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 º 10º>PM> [10]. The popula ion size and numbe o gene a ions we e 1500 and 150, espec i ely. Gene a ion o his POF ook abou 1 hou 36 minu es o CPU ime on a 2.2 GHz p ocesso . The esul is ob iously 1500 sample poin s o he 4-dimensional POF. Fo illus a ion pu poses, shows he p ojec ions o he 1500 poin s o his 4-dimensional hype su ace on he dc gain s. uni y-gain equency plane, on he phase ma gin s. uni y- gain- equency plane, and on he dc gain s. ou pu impedance plane. Each o hese poin s ep esen s a sized ci cui showing he bes ade-o s among he ou pe o mances conside ed. Figu e 3 III. LOAD-INDEPENDENT PARETO-OPTIMAL FRONTS A. POF Gene a ion Me hodogy Le us conside he same ampli ie and he same pe o mances used in Sec ion II. As al eady men ioned, some o hese pe o mances depend on he load condi ions. Howe e , he load condi ions a e no known un il he syn hesis 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 POF gene a ion p ocess is a compu a ionally expensi e p ocess, he objec i e o ou esea ch is o gene a e ade-o 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 POFs o pe o mances ( 0 A ,u , PM, and o Z ) when he load condi ions a e known. The ope a ional ampli ie can be conside ed a wo-po , like ha shown in Figu e 4. 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 (in case o single ou pu ) and he di e en ial ou pu ol age and cu en (in case o ully di e en ial ampli ie ). As we need a load-independen cha ac e iza ion o he ampli ie , we may conside , a p io i, any ma ix cha ac e iza ion o wo-po s [11]. Howe e , he wo-po ma ix pa ame e s mus be selec ed in elligen ly, 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 [11] o cha ac e ize he wo-po : Figu e 3. P ojec ions o he POF gene a ed by he mul iobjec i e op imiza ion algo i hm o a capaci i e load o 1pF. 1 i 2 i 2 1 + + − − L Z Figu e 4. Two-po wi h a bi a y load. Figu e 2. Mille ope a ional ampli ie used in his wo k. M4 M3 M2 M1 M6 Cc M7 Vss Vdd M5 Vin Mbn Vdd Ib Vo Vip CL (2) 1111122 221122 ih hi h hi ′′ =⋅+⋅ ′′ =⋅+⋅ 2 In his equa ion, pa ame e ep esen s he inpu impedance, is he in e se cu en gain, ep esen s he ol age gain o he ampli ie wi hou any load and 11 h′ 12 h′21 h′ 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: (3) 2L Z=− ⋅2 i yielding: 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 om he ol age gain, 22 h′() A s, and ou pu impedance, () o Z s, o some known load condi ions, and ice e sa, ob ain he ol age gain () A sand ou pu impedance () o Z s 22 ′ o some load om he hyb id-2 pa ame e s and . Mo eo e , a pa icula case is ha in which . In his case, and a e iden ical o 21 h′ L Z h →∞ 21 h′22 h′() A sand ( o) Z s, espec i ely. This is he key o de eloping he POF gene a ion and ans o ma ion me hodology ha is p oposed in his pape o ans o m a POF o some known loading condi ions o a bi a y new loading condi ions. Le us assume ha we wish o gene a e he POF o he ou pu impedance, dc ol age gain, uni y-gain equency, and phase ma gin o some a bi a y loading condi ions. The gene a ion me hodology p oceeds as ollows: 1) Gene a e he POF o he pe o mances o in e es o some known loading 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 . 2) Fo each sample o indi idual o his POF, 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. This in o ma ion can be e ie ed om common elec ical simula o s. I his in o ma ion is no a ailable, a educed wo- pole model can be easily ex ac ed om he dc gain, uni y-gain equency, and phase ma gin alues. 3) Use equa ion (4) o ex ac he hyb id pa ame e s 21()hs ′ and o each sample. No ice ha , om basic ci cui heo y, he poles o and a e iden ical. 22 ()hs ′ 21()hs ′22 ()hs ′ 4) Apply equa ion (4) o ob ain he ol age gain o he new a bi a y loading 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, phase ma gin, and ou pu impedance by simple p ocessing o he ne wo k unc ions. No ice ha his p ocedu e can be applied o any ini ial known loading condi ions. This includes he case in which he POFs a e gene a ed o he ci cui wi hou any load. In his case, s eps 2 and 3 a e uni ied in a single s ep. The i s s ep o his me hodology has he hea ies compu a ional e o by a , bu no ice ha he esul s o s ep 3 a e independen o he applica ion, i.e., independen o he inal loading condi ions. The e o e, he i s h ee s eps can be pe o med be o ehand, and he esul s s o ed and used whene e and whe e e necessa y. B. Resul s In his sec ion, he p oposed me hodology will be applied o he gene a ion o he POF o he Mille ope a ional ampli ie , when a esis i e-capaci i e load is applied. Following s ep 1, a POF wi h a capaci i e load o 1pF was i s gene a ed o he ou objec i es (see Sec ion II): dc ol age gain, uni y-gain equency, phase ma gin, and ou pu impedance. F om he elec ical simula ion o he 1500 poin s o he POF, he ne wo k unc ions () A sand () o Z s h o each design poin can be easily ob ained. And by applying equa ion (4) o each o hese poin s, he hyb id-2 pa ame e s 21 ′ and 22 h ′ o he 1500 poin s a e calcula ed. These wo s eps a e pe o med in less han 5 minu es. No ice ha all he s eps pe o med so a a e independen o he inal load condi ions. The e o e, al hough compu a ionally cos ly, hey a e pe o med long be o e he load is known and he o he s eps ha e o be pe o med. Assume ha we need now o gene a e he POF o a load o 2pF and 50kΩ. Using he poin s p e iously s o ed, he new POF is gene a ed by applying s eps 4 and 5 in Sec ion III. Figu e 5 shows he h ee p ojec ions o he 4-dimensional on . The applica ion o hese wo s eps akes only 20 seconds. To assess he p ocedu e, he POF was also gene a ed by coupling he op imize wi h he elec ical simula o o his new 2pF-50kΩ load ( ha is, no ollowing he ans o ma ion p ocedu e p oposed he e). A se o samples o he POF wi h simila quali y cha ac e is ics is ob ained (see Figu e 6). Howe e , as in he gene a ion o he ini ial POF, 1 hou and 36 minu es o CPU a e employed ins ead o he 20 seconds ha we e equi ed by he ans o ma ion p ocedu e desc ibed he e. This is an accep able ime in o de o inco po a e his echnique in o a hie a chical syn hesis low whe e i e a i e e alua ions o ci cui pe o mances a e necessa y. No ice ha he POFs in Figu es 5 and 6 a e simila bu he samples a e di e en due o he ini e popula ion size and he s ochas ic na u e o he mul i- objec i e e olu iona y algo i hm. When compa ing Figu e 3 and Figu e 5, i can be obse ed ha he ans o ma ion p ocedu e implies an impo an mo emen o poin s in he design objec i e space. Densi y o poin s in Figu e 5 migh Figu e 5. P ojec ions o he POF ob ained by he ans o ma ion p ocedu e o a load o 2pF-50kΩ. Figu e 6. P ojec ions o he POF gene a ed by he mul iobjec i e op imiza ion algo i hm o a load o 2pF-50KΩ. become smalle because in he ans o ma ion some poin s may mo e o egions ha a e no o in e es ( o example, e y low phase ma gin o dc gain), o because he ans o ma ion p ocedu e can mo e a poin o a posi ion in he objec i e/pe o mance space whe e i becomes domina ed by o he poin s. Al hough some poin s may become domina ed a e he ans o ma ion, he p ocedu e gua an ees ha a poin o he ans o med POF may no o igina e om a domina ed poin o he ini ial pe o mance space. IV. CONCLUSIONS This pape in oduces a POF ans o ma ion p ocedu e o ope a ional ampli ie s based on a hyb id-2 pa ame e cha ac e iza ion. 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