Behavioral modeling of PWL analog circuits using symbolic analysis

Beha io al Modeling o PVVL Analog Ci cui s using Symbolic Analysis F ancisco V Fe ndndez, Bel& Pe' ez-Ve du' and Angel Rod iguez-Vdzquez Ins i u o de Mic oelec dnica de Se illa Cen o Nacional de Mic oelec 6nica-C.S.I.C. Edi icio CICA-CNM, A da. Reina Me cedes sln, 41 0 12-Se illa, SPAIN FAX:: 34 5 4231832 Phone: 34 5 4239923 email: [email p o ec ed] ABSTRACT^ Beha io al models a e used bo h o op-down design and o bo om-up e i ica ion. Du ing op-down design, models a e c ea ed ha e lec he nominal beha io o he di e en analog unc ions, as well as he cons ain s imposed by he pa asi ics. In his sce la io, he a ailabili y o symbolic mod- eling exp essions enable designe s o ge insigh on he ci - cui s, and educes he compu a ional cos o design space ex- plo a ion. Du ing bo om-up e i ica ion, models a e c ea ed ha cap u e he opological and cons i u i e equa ions o he unde lying de ices id o beha io al desc ip ions. In his sce- na io symbolic analykis is use ul because i enables o au o- ma ically ob ain hesi desc ip ions in he o m o equa ions. This pape includes an example o illus a e he use o sym- bolic analysis o op-down design. 1. INTRODUCTION Ci cui analysis is he co ne s one o elec onic ci cui en- ginee ing. On he on1 hand, i p o ides he keys o unde - s anding he in ica e mechanisms unde nea h he ci cui op- e a ion. On he o he ^ hand, designe s use analysis o ob ain models o he ci cui lbeha io - he basis on which his be- ha io can be p edicded. Howe e , manual analysis o e en he simples ci cui s kncoun e ed in p ac ical applica ions is a complica ed, ime-consuming, and e o -p one ask. Sym- bolic analyze s a e in ended o elie e designe s o sys ema - ic manual analysis hsks, hus le ing hem concen a e on c ea i e issues. I The las gene a io? symbolic analyze s a e able o handle up o a ound 400 di b en symbols [ 11. This pe mi s, o ex- ample, o analyze di cui s as complex as he ail- o- ail CMOS opamp o Fig!l(a) 121 wi h he small-signal ansis o model o Fig. l(b) [3]1-[5]. The analyze s capabili ies include he calcula ion o s-dbmain exp essions o all ypes o d i - ing-poin and ans e cha ac e is ics, he simpli ica ion o hese exp essions o e ain only he dominan e ms, he ex- ac ion o hei poles^ and ze oes, e c. Recen ly, di e en au ho s ha e ocused also on he sym- hose weak nonlinea - I bolic analysis o no ci cui s [6][7]. The mos impo - a ew e ms ( ypically 1. This wo k has been pa ially suppo ed by he Spanish C.I.C.Y.T. unde con ac TIC97-0580 ind he EEC in he p ojec ESPRIT AMADEUS. I I I I B Figu e 1. (a) Rail- o- ail opamp; (b) small-signal MOS ansis o model. 3) o a powe se ies expansion. In [6] sys ema ic echniques a e p oposed o he au oma ic calcula ion o he Vol e a ke nels [8] ha cha ac e ize he beha io o hese weakly- nonlinea ci cui s in he equency domain. Howe e , he symbolic analysis o ha dly-nonlinea ci cui s is ye in ex- plo a o y phase 171. To da e, no sys ema ic echnique is a ailable o au oma e he calcula ion o a se o symbolic equa ions go e ning he la ge signal nonlinea beha io , ei- he in he s a ic o in he dynamic case. This lack o nonlinea analysis limi s he modeling capa- bili ies o s a e-o - he-a symbolic analyze s. Howe e , in many p ac ical applica ions hese limi a ions can be o e - come by eso ing o piecewise-linea (PWL) ep esen a- ions. Fo ins ance, hese ep esen a ions su ice o s udy he quali a i e beha io o many p ac ical dynamical ci cui s such as oscilla o s and compa a o s, as well as o app oxi- ma e he ansien esponse o opamps and compa a o s. VI- 17 0~7803-4455-3/98/%10.~0 0 1998 lEEE 2. BEHAVIORAL MODELING A beha io al model is a se o equa ions ha cap u e he ope a ion o a ci cui om i s e minals. Beha io al models can also be gi en as ci cui al ep esen a ions o hese equa- ions. Depending on he na u e on he exci a ions and e- sponses, he se o equa ions and i s associa ed ci cui al ep- esen a ion can be s a ic o dynamic, linea o nonlinea . Fo example, Fig.2 shows di e en beha io al models o an opamp. The i s one is s a ic and he o he s dynamic; he i s wo a e linea and he o he s nonlinea . Ci cui al ep e- sen a ions a e included o each model. The cons uc ion o a beha io al model does no s ic ly equi e o know he de ailed ci cui schema ic. Nei he he model ci cui al ep esen a ion mus e lec he ci cui opolo- gy. Models can be buil by, i s , applying a sui able se o ex- ci a ion- esponse pai s and, hen, inding and uning a ma h s uc u e ha ep oduce hese pai s wi hin some p esc ibed e o ma gin. Conside o example he OTA shown in Fig.3(a) and assume ha we a e in e es ed only in i s linea AC beha io . This can be modeled by h -pa ame e s, which can be expe imen ally calcula ed in he lab by using a ne - wo k analyze in he con igu a ions o Fig.3(c). Al e na i ely, beha io al models can be cons uc ed h ough he elec ical analysis o he in e nal ci cui schema - ic. Fo ins ance, assuming ha he OTA schema ic o A dd . Figu e 3. (a) Simple OTA; (b) simpli ied small-signal MOS an- sis o model; (c) black box modeling p ocedu e o a wo-po . Fig.3(b) and he ansis o model o Fig.3(c), a las gene a- ion symbolic analyze such as SYMBA [9] would e u n he ollowing h -pa ame e app oxima ed exp essions, hi(s) = "b - + - gm3(gml+ gm2) 2 - - scg.~2gm1gm3 Cgs2[gml(Cgs3 'gs4) + gm3Cgsl] s[gm3(cgsl + 'gs2) + (gml + gm2)(Cgs3 + Cg. dI sCgs2gmlgm3 +S Cgs2[gml(cgs3 + Cg34) + gm3CgslI h, W = + 2 + n-; + FVO Vi-+ u VIP '"b - gdslgm3 +sgd.~l(Cg.sl + cgs3 + Cg.J gmlgml+ S[gml(Cg. 3 + Cgs4) + gm3Cgsll gds2 gm1 +sCgsl gds1gm3 + Sgdsl(Cgs1 + cgs3 + Cgs4) gmlgm3 +s[gin1(cgs3 + cg.~4) +~m3Cg. 1] - - - h,(s) = Vi-+ ,+ i; h,(s) = (1) In p ac ice, designe s use hei expe ience o build modu- la models, whe e unc ional subs uc u es a e used o ep e- sen he nominal ci cui beha io as well as he pa asi ics. Thus, o example, Fig.2(a) cap u es he nominal opamp be- - - Figu e 2. Opamp beha io al models. VI- 18 ha io ; ha o Fig.2 o he dynamics ass1 and Fig.2(c) models y. These h ee mod< o he hand, Fig.2(d) ing-poin beha io s The di e en un ha io al model can su emen s o h oug he case o PWL mc nonlinea subs uc u ed h ough hei syi p esen ed in he sec in e es o hese PW 3. A PWL MOD OPA Conside he SC opamp ep esen ed I ansien obse ed ii he in eg a ing capac pa . The linea pa gml( V,(S) = - C,(C VJS) = - whe e, yc = cic2. yielding, whe e he coe iciei pa ame e s [lo]. Th lowing condi ion is 1) includes a i s -o de app oxima ion :ia ed o any ol age gain mechanism; ie i s s age ansconduc o nonlinea i- ; ocus on he ans e beha io , On he ip u es also o a i s app oach he d i - he ci cui e minals. onal subs uc u es pe aining o a be- e uned ei he h ough blackbox mea- analysis o he unde lying ci cui . In els, measu emen s a e used o une he s while hose linea ones a e ep esen - bolic ans e unc ions. The example )n below illus a es abou he p ac ical models. L FOR TOP-DOWN TRANSIENT [P OOPTIMIZATION i eg a o o Fig.4(a) and assume he he PWL model o Fig.4(b) [lo]. The he ans e ing o he inpu ol age o o con ains a linea pa an a nonlinea in be calcula ed symbolically om, ic, + C2C, + c2c, + c,c, , = ci + c, + c, (3) cp(-a )cos@ + C,, exp(-a )sinp up(-a )cosp + Cllexp(-a )sinp (4) cp(-a )cosp + CUI exp(-a )sinp s a e gi en as unc ions o he model e equa ions emain alid while he ol- Hilled, 8mzI * (q I, (5) VI- 19 Figu e 4. (a) SC in eg a o and, (b) opamp model. A e wa ds, he second model s age en e s in o sa u a ion and he ansien is desc ibed by he ollowing equa ions, whe e. and he emaining coe icien s a e gi en as unc ions o he model pa ame e s [ 101. This second ansien pe sis s while he condi ion (5) holds. A e wa ds, he ansien is gi en by, yielding, ,( ) = A,, +B,,exp(-a )cosp + C,,exp(-a )sinp (10) whe e he coe icien s a e, as in he p e ious exp essions, gi en as unc ions o he model pa ame e s. The model has been alida ed by compa ing i s ou pu wa e o m o ha o ob ained h ough de ailed elec ical sim- ula ion. The ampli ie consis ed o a ully-di e en ial old- ed-cascode OTA whose co e schema ic and summa ized pe - o mance a e gi en in Fig.5. Said ampli ie was designed o ha e small phase ma gin (a ound 4Sdeg) o educe powe dissipa ion. Fig.6(a) shows he in eg a o ou pu ol age du - ing he in eg a ion phase ob ained h ough elec ical simula- ion (HSPICE) and ha ob ained using he model. Model pa- ame e s a e also enclosed in Fig5 A good conco dance be- ween bo h app oxima ions is obse ed. A mo e gene al esul is gi en in Fig.6(b) whe e he di e ence be ween he inal alue o he in eg a o ou pu ol age and i s ideal alue is shown as a unc ion o he inpu le el. V- --I GBW(3.2pF) = 106.7MHz PM (3.2pF) = 42.2O I, = 500pA Figu e 5. Fully di e en ial OTA. 0.4 . . 1 0.3 2 0.2 2 0.1 2 0.0 c c 0 0) +- -0.1 -0.2 - -0.40 5.0e-09 1 .Oe-08 13-08 2.0e-08 2.5e-08 Time (s) (4 - HSPICE 1 --Model 0.5 1 .o 1.5 2.0 -0.1 ‘0 Vol age s o ed in Ci (V) Figu e 6. Simula ed and calcula ed esponses. (b) 4. REFERENCES 11 F.V. 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