Analog Circuit Sizing using Adaptive Worst-Case Parameter Sets

Transcription

1 Analog Crcut Szng usng Adaptve Worst-Case Parameter Sets R. Schwencker 1,2, F. Schenkel 1, M. Pronath 1, H. Graeb 1 1 Insttute for Electronc Desgn Automaton 2 Infneon Technologes Techncal Unversty of Munch P.O. Box Munch, Germany Munch, Germany Abstract In ths paper, a method for nomnal desgn of analog ntegrated crcuts s presented that ncludes process varatons and operatng ranges by worst-case parameter sets. These sets are calculated adaptvely durng the szng process based on senstvty analyses. The method leads to robust desgns wth hgh parametrc yeld, whle beng much more effcent than desgn centerng methods. 1 Introducton In ntegrated crcut technologes wth ever shrnkng feature szes and growng performance requrements, the nfluence of process varatons on the behavor and yeld of analog crcuts cannot be neglected. In order to be able to desgn robust crcuts, random process fluctuatons and also varatons of the operatng condtons (e.g. temperature and supply voltage must be taken nto account as early as possble n the desgn cycle. Furthermore a hgh degree of automaton s needed for analog crcut desgn n order to cope wth the demand of an ever shorter tme-to-market [9]. Powerful tools for nomnal desgn, e.g. [5, 14, 15], were developed and some are commercally avalable. Nomnal desgn usually does not consder process fluctuatons and varatons of the operatng condtons. Therefore, nomnal desgn can only guarantee that the gven specfcatons are fulflled for the typcal process and nomnal operatng condtons. Due to the growng nfluence of process fluctuaton and changes n the operatng condtons, desgn centerng s necessary n addton to nomnal desgn n order to ensure a hgh producton yeld. Many approaches to desgn centerng, based on statstcal, e.g. [2, 11], and determnstc methods, e.g. [1, 6, 12], were presented. Usually desgn centerng algorthms are computatonally very expensve. Hence the desgn centerng process should be started from a good nomnal desgn n order to keep the computatonal cost small. Ths can be acheved by ntroducng worst-case parameter sets for process and operatng condtons nto nomnal desgn [4, 6]. In the dgtal doman, process fluctuatons are beng consdered by means of slow and fast worst-case parameter sets. These are calculated for a gven process and typcal crcut performance lke delay, but ndependent from a specfc crcut. For dgtal cell lbrares, these parameter sets gve a good estmaton of the nfluence of random fluctuatons on the relevant dgtal crcut performances, delay and power. However t s well known that such dgtal corners cases are not suffcent for analog desgn [6, 13]. Usng dgtal corner cases for analog crcut desgn bears a hgh rsk of leavng yeld problems undetected untl producton. For analog desgn, worst-case parameter sets therefore need to be calculated for each crcut topology and each crcut performance ndvdually. Usng such performance-specfc worst-case parameter sets for crcut szng faces the problem that they depend on the nomnal desgn parameter set and hence vary durng the szng process. In ths paper, a new effcent szng algorthm s presented that smultaneously consders process fluctuaton and operatng condtons. It features: calculaton of ndvdual worst-case parameter sets for each performance, adaptve calculaton of worst-case parameter sets n each teraton step of the szng process based on smple and fast senstvty analyses. effcent trust regon optmzaton algorthm usng szng rules [10]. The paper s structured as follows: The next secton formulates worst-case parameter sets. The new szng algorthm s dscussed n Secton 3, and the results are presented n Secton 4. 2 Worst-case parameter sets and yeld For a gven topology, a crcut can be descrbed by ts parameters and performances. Three types of parameters can be dstngushed: Desgn parameters d R n d (e.g. nomnal transstor wdths and lengths can be adjusted by the crcut desgner. Process fluctuatons for nstance at oxde thckness, threshold voltage, or transstor wdth varaton, are modeled by statstcal parameters s R ns and ther dstrbuton functon. As shown n [7], all practcally mportant parameter dstrbutons can be transformed nto a

2 Gaussan dstrbuton wth zero mean value and covarance matrx C: s N(0, C. The probablty densty functon pdf (s s then gven by: ( pdf (s = (2π ns 2 (det C 1 2 exp β2 (s (1 2 β 2 (s = s T C 1 s. (2 In ntegrated crcut desgn, most statstcal parameters appear as transstor model parameters (e.g. tox, or vth0 and cannot be adjusted by the crcut desgner f a frozen, fully qualfed producton process s assumed. Operatng parameters θ R n θ (e.g. supply voltage, temperature descrbe the crcut s operatng condtons. The crcut must satsfy ts performance specfcaton for a gven range T θ of the operatng parameters, defned by ther lower bounds θ L and upper bounds θ U : T θ = {θ θ L θ θ U }. (3 The crcut s performance values f (e.g. gan, delay can be calculated for a gven parameter set usng an analog crcut smulator: (d, s, θ f. Dependng on the performance, a crcut smulaton means solvng a system of nonlnear equatons (DC- and AC-analyss or ntegratng a system of nonlnear algebro-dfferental equatons (transent analyss. For the crcut performances, lower and/or upper specfcaton bounds f b,, = 1,..., n b are defned that have to be met for a correctly operatng crcut: f (d, s, θ f b, f(d, s, θ f b. (4 =1,...,n b Here, upper bounds are ncluded by f f b, where f b, < 0. Consderng varatons n the operatng condtons, the specfcaton bounds have to be met for the whole operatng range: f(d, s, θ f b. (5 θ T θ The parametrc yeld Y s the percentage of crcuts that satsfy (5: Y (d = pdf (s ds. (6 {s f(d,s,θ f b} θ T θ Durng szng, process and operatng varatons can be consdered by means of worst-case parameter sets that represent a certan standard devaton β max of process varatons [3]: For each performance specfcaton bound f b,, a worst-case parameter devaton s determned accordng to (s, θ = argmn f (d, s, θ s,θ subject to β 2 (s β 2 max and θ T θ. For nstance, β max = 3 corresponds to a 3σ desgn. Satsfyng (5 at the worst-case parameter sets guarantees a mnmum parametrc yeld (7 Y Y l 1 (n s = F ns (β 2 max, (8 Performance nomnal 3σ slow/fast 3σ worst-case Delay [ns] (+42% 1.92 (+42% Delay [ns] (+33% 1.90 (+33% Hysteress [mv] ( 8.0% 441 ( 28% Table 1: Performance values for a Schmtt trgger compared at slow/fast parameter sets and at worst-case parameter sets accordng to Eq. (7. where F ns s the probablty functon of a χ 2 -dstrbuton wth n s degrees of freedom. If f s monotonous wth regard to s, then another loose lower bound can be gven by Y Y l 2 (n f = 1 Φ( β max n b, (9 where Φ s the probablty functon of the normal dstrbuton. For example, f β max = 3 then Y l 2 (n f 100% 0.135% n b. (10 Snce Y1 l depends on the number n s of statstcal parameters, and Y2 l depends on the number n b of bounds, ether lower bound can be the greater one. For dgtal crcuts, t turned out that the s of delay and power consumpton are practcally ndependent from szng and topology. Therefore t s common practce to calculate a set of slow/fast worst-case parameter sets once for a manufacturng process and then use t for all dgtal cells. The worst-case operatng ponts θ are also practcally ndependent from szng and topology, but T θ s part of the crcut specfcaton, not of the process. Therefore, the θ of delay and power are to be determned once for T θ. For analog crcuts, slow/fast sets are nsuffcent. Frst they do not ncorporate non-dgtal performances. Tab. 1 compares the performance values of the Schmtt trgger buffer of Fg. 2 at the slow/fast corners of a 0.18µm-process wth the performance values at the actual worst-case parameter sets calculated drectly for ths topology and szng accordng to Eq. (7. As can be seen, the slow/fast performance values of the delays conform to the actual worst-case performance values. In contrast to that, the 3σ worst-case value of the thrd performance hysteress s n fact much worse than ndcated by a smulaton at the slow or fast parameter set. Usng these parameter sets to verfy a specfcaton regardng hysteress wll therefore pretend an unrealstcally hgh robustness. Secondly, worst-case parameter sets depend on the analog crcut s topology. Assume for nstance a process consstng of only two random varables, s n that nfluences solely NMOS transstors and s p for the PMOS transstors. Further gven are a current mrror completely bult of NMOS transstors and another one completely bult of PMOS transstors. It s then obvous, that the frst crcut s only affected by s n, whereas the second one s affected only by s p. Hence the 3σ worst-case parameter sets wll be orthogonal. Thrd, worst-case parameter sets depend on szng, see Sec. 4.

3 3 Szng and adapton of worst-case parameter sets Worst-case szng s the task of fndng a desgn parameter vector d that guarantees a mnmum yeld accordng to Eqs. (8, 9. Snce the worst-case parameter sets depend on the szng n a non-lnear manner, we propose an teratve numercal optmzaton. Snce the exact calculaton of the worst-case parameter sets (s, θ s computatonally expensve, we ntroduce an approach to relaxed calculaton of worst-case parameter sets based on lnear approxmatons. An update of the approxmated worst-case parameter sets s performed at each teraton step. Begnnng wth µ = 0, the followng actons are performed n each teraton step of the algorthm, The steps 1 through 6 are performed for each specfcaton f b, (see Fg the correspondng performance f s lnearzed wth respect to the operatng parameters θ at ts respectve worst-case parameter set ( s (µ 1 (µ 1, θ and desgn parameter set d (µ 1 of the prevous step : = θ f. (11 (µ 1, s (µ 1, θ (µ 1 2. Then, the components j of the worst-case operatng parameter set g θ ( θ = j are calculated: { ( (θl j f (θ U j else g j 0. (12 3. Performance f s lnearzed wth respect to the statstcal parameters s at the worst-case parameter set from the prevous teraton step ( s (µ 1 and at the worst-case operatng parameter set of the current teraton step ( θ h 4. Thereafter, s s defned by = s : = s f. (13 (µ 1, s (µ 1, θ h β 2 max T C h C h. (14 5. Performance f s then lnearzed wth respect to desgn parameters d at the worst-case parameter set of the current teraton step ( s, θ 1 : k = d f (µ 1, s, θ (15 1 Please note that each teraton step s dvded nto 3 update steps accordng to the 3 parameter types. Frst, the worst-case operatng parameter (µ 1 set s updated: θ wc θ wc. Thereafter ths updated worst-case operatng parameter set θ wc s already used for the senstvty calculaton that leads to an updated worst-case parameter set s (µ 1 wc s wc. The updated worst-case parameter set s then used for the senstvty calculaton leadng to an updated desgn parameter set d (µ 1 d. Usng the latest avalable parameter sets for each update step contrbutes to the effcency of the algorthm. µ := 0, d := d (0 µ := µ + 1 For each specfcaton f b, Determne g, θ Determne h, s Determne k Mnmze ϕ (d subject to d F d Untl f (d, s, θ > f b, Fgure 1: Structure of optmzaton algorthm. f (d = f (d (µ 1, s, θ +k T (d d. (16 6. The parameter dstance functon α (d [15] s then defned as α (d = f (d f b,. (17 k The objectve functon ϕ (d s calculated based on the parameter dstances α for the specfcatons f b,, = 1,..., n b : n b ( ϕ (d = exp a α (d, a > 0. (18 =1 The postve constant factor a s a weghtng factor. Hgh values of a make the optmzer focus stronger on volated specfcatons. 7. The objectve functon ϕ (d s then mnmzed by means of a trust regon method presented n [15]: d = argmn ϕ (d subject to d F, (19 d where F s the feasblty regon that guarantees the basc functonalty and robustness of a crcut [10]. The constrant d F ensures that functonal constrants lke all transstors must be n saturaton are fulflled durng the szng. The feasblty regon F s the subset of the desgn space where all functonal constrants are fulflled. Consderng F s crucal for automated szng of crcuts: The result of the szng has to be feasble n order to represent a techncally correct crcut. Only parameter vectors d F are techncally vald solutons. Most performances are only weakly nonlnear n the feasblty regon. Therefore, the reducton of the desgn space to the feasblty regon sgnfcantly mproves the precson of the used lnearzed performance models.

4 Intal Fnal Performance Hysteress Delay Delay [V] [ns] [ns] Spec. f b, > 0.50 < 1.75 < 1.75 f (d (Intal, s (Intal Y 2.0% 55.0% 73.5% Y tot 2.0% f (d (Fnal, s (Fnal f (d (Fnal, s (Intal Y 100.0% 99.5% 100.0% Y tot 99.5% Table 2: Results for Schmtt trgger, wth partal yelds Y and total yeld Y tot. The constrants reduce the exploraton space for the optmzaton algorthm and therefore mprove the convergence of the algorthm. Durng the optmzaton, a lnear approxmaton F s used and s updated n each teraton step. 4 Results The proposed method was appled to two example crcuts usng statstcal data of an ndustral fabrcaton process. The frst crcut, a Schmtt trgger (Fg. 2, s a typcal dgtal crcut from a cell lbrary. enabq pad VDDP VDD Fgure 2: Schematc of a Schmtt trgger. out The yeld values lsted n ths secton were all obtaned from a 200 sample Monte-Carlo analyss, wth consderaton of operatonal parameters accordng to Eq. (6. Please note, that ths Monte-Carlo analyss s not part of the algorthm tself, but only to llustrate the yeld mprovement obtaned wth ths approach. The results of the optmzaton are compled n Table 2. As can be seen, the ntal total yeld Y tot was nsuffcent 2.0% for ths crcut, manly due to the low partal yeld Y of the hysteress. The row f (d (Intal, s (Intal unvels that all specfcatons were volated at ther approprate 3σ worst-case parameter set before the optmzaton. After only 5 teratons of the worst-case szng algorthm all specfcatons could be met at ther fnal worst-case parameter sets (d (Fnal, s (Fnal. Ths mprovement led to a total yeld of 99.5%. In Table 2, the performance values for the fnal desgn parameter set d (Fnal have been smulated for both, the ntal worst-case parameter sets s (Intal and the fnal worstcase parameter sets s (Fnal. Obvously, both worst-case parameter sets lead to dentcal performance values. Performance Hysteress Delay Delay (s (Intal, s (Fnal Table 3: Angles between ntal and fnal worst-case parameter sets of the same performance of the Schmtt trgger. bas nn np Cc Cload Fgure 3: Schematc of a Mller operatonal amplfer. In Table 3, the angles between the ntal and fnal worstcase parameter sets are gven for each performance. Apparently, the ntal and fnal worst-case parameter sets are very smlar for the hysteress and nearly dentcal for the two delays. In Table 4, angles of ntal and fnal worst-case parameter sets are compared between the dfferent performances. The dfferences between ntal and fnal angles are around 10 comparng the hysteress wth the delays and almost 0 between rsng and fallng delay. Hysteress Delay Delay Hysteress Delay Delay Table 4: Angles between the worst-case parameter sets of dfferent performances of the Schmtt trgger before the optmzaton (unshaded upper rght trangle and after the optmzaton (lght gray shaded lower left trangle. These experments llustrate, that for dgtal crcuts, worst-case parameter sets are rather ndependent from the szng. The second example, a Mller operatonal amplfer (Fg. 3, s a typcal analog crcut. For the ntal szng all specfcatons but the one for the power consumpton were volated at ther 3σ worst-case parameter sets (row f (d (Intal, s (Intal of Table 5, leadng to a total yeld of 0.0%. Agan after 5 teratons of the szng algorthm, all specfcatons could be fulflled at ther worst-case ponts (d (Fnal, s (Fnal. Unlke the Schmtt trgger, the worst-case parameter sets of the Mller operatonal amplfer change sgnfcantly wth the szng. Conductng the same experment as wth the Schmtt trgger of transferng the ntal worst-case parameter sets s (Intal to the fnal szngs d (Fnal agan, ths tme leads to consderably dfferent performance values compared to the values at the actual fnal worst-case parameter sets (d (Fnal, s (Fnal. The same holds for the postons of the ntal and fnal worst-case parameter sets (Table 6 and 7. Gven these varabltes and the spread of the worst-case parameter sets over the space of the statstcal parameters (Table 7, t becomes apparent that the worst-case condtons for analog crcuts cannot be represented only by pre-

5 Intal Fnal Performance A 0 f t Φ m SR p Power [db] [MHz] [ ] [V/µs] [mw] Spec. f b, > 80 > 1.3 > 60 > 3 < 1.3 f (d (Intal, s (Intal Y 95.5% 0.0% 0.0% 0.0% 100.0% Y tot 0.0% f (d (Fnal, s (Fnal f (d (Fnal, s (Intal Y 100.0% 100.0% 100.0% 100.0% 100.0% Y tot 100.0% Table 5: Results for Mller operatonal amplfer, wth partal yelds Y and total yeld Y tot. Performance A 0 f t Φ m SR p Power (s (Intal, s (Fnal Table 6: Angles between ntal and fnal worst-case parameter sets of the same performance of the Mller operatonal amplfer. defned dgtal slow/fast worst-case parameter sets. In consequence one worst-case parameter set has to be determned for each specfcaton at every szng ndvdually. Table 8 comples the computatonal costs for the szng of both crcuts. The results were obtaned on a network of 5 computers (Sun Ultra I for the Schmtt trgger and 500 MHz Pentum III for the Mller operatonal amplfer, usng the Infneon n-house smulator TITAN [8]. One can see that the whole synthess process takes 385 and 685 smulatons respectvely (1 smulaton ncludes DC, AC and transent smulaton, equvalent to 20 mnutes and 5 mnutes elapsed tme (exclusve computer usage. Hence a complete szng can be done at the cost of a Monte Carlo analyss. Concluson A method for analog crcut szng has been presented that performs nomnal desgn at worst-case parameter sets for operatng condtons and for manufacturng varatons. It has been llustrated that these worst-case parameter sets are dependent on the crcut topology, on the crcut performances and of the desgn parameter values. The presented method features a relaxed calculaton of worst-case parameter sets based on performance lnearzatons. In ths way, szng of robust analog crcuts can be acheved at lower smulaton costs than by desgn centerng approaches. A 0 f t Φ m SR p Power A f t Φ m SR p Power Table 7: Angles between the worst-case parameter sets of dfferent performances of the Mller operatonal amplfer before the optmzaton (unshaded upper rght trangle and after the optmzaton (lght gray shaded lower left trangle. Crcut # Smulatons Wall clock tme Schmtt trgger mn (5 Sun Ultra I Mller mn (5 Pentum III 500 MHz Acknowledgments Table 8: Computatonal costs Ths work was supported by the German Federal Mnstry of Educaton, Scence, Research and Technology under grant 01 M 3050 A wthn the MEDEA+ project Analog Enhancements for a System to Slcon Automated Desgn (ANASTASIA+. References [1] H. Abdel-Malek and A. Hassan. The ellpsodal technque for desgn centerng and regon approxmaton. IEEE IS- CAS, [2] S. A. Aftab and M. A. Styblnsk. IC varablty mnmzaton usng a new Cp and Cpk based varablty/performance measure. IEEE ISCAS, [3] K. Antrech, H. Graeb, and C. Weser. Crcut analyss and optmzaton drven by worst-case dstances. IEEE TCAD, 13(1, [4] G. Debyser and G. Gelen. Effcent analog crcut synthess wth smultaneous yeld and robustness optmzaton. In IEEE/ACM ICCAD, [5] M. del Mar Hershenson, S. P. Boyd, and T. H. Lee. Optmal desgn of a CMOS Op-Amp va geometrc programmng. IEEE TCAD, 20(1, [6] A. Dharchoudhury and S. M. Kang. Worst-case analyss and optmzaton of VLSI crcut performances. IEEE TCAD, 14(4, [7] K. S. Eshbaugh. Generaton of correlated parameters for statstcal crcut smulaton. IEEE TCAD, 11(10, [8] U. Feldmann, U. Wever, Q. Zheng, R. Schultz, and H. Wredt. Algorthms for modern crcut smulaton. Archv für Elektronk und Übertragungstechnk, 46, [9] G. G. E. Gelen and R. A. Rutenbar. Computer-aded desgn of analog and mxed-sgnal ntegrated crcuts. Proceedngs of the IEEE, 88(12, [10] H. Graeb, S. Zzala, J. Eckmueller, and K. Antrech. The szng rules method for analog ntegrated crcut desgn. In IEEE/ACM ICCAD, [11] M. Keramat and R. Kelbasa. OPTOMEGA: an envronment for analog crcut optmzaton. In IEEE ISCAS, [12] K. Krshna and S. W. Drector. The lnearzed performance penalty (LPP method for optmzaton of parametrc yeld and ts relablty. IEEE TCAD, 14(12, [13] G. Müller-Lebler. Lmt-parameters: the general soluton of the worst-case problem for the lnearzed case. IEEE ISCAS, [14] R. Phelps, M. Krasnck, R. A. Rutenbar, L. R. Carley, and J. R. Hellums. Anaconda: Smulaton-based synthess of analog crcuts va stochastc pattern search. IEEE TCAD, 19(6, [15] R. Schwencker, F. Schenkel, H. Gräb, and K. Antrech. The generalzed boundary curve A common method for automatc nomnal desgn and desgn centerng of analog crcuts. DATE, 2000.