Analytical Modelling of Parameter Fluctuations with Applications to Analogue VLSI Circuits and Systems

Transcription

1 Prceedg f the 4th WSEAS/IASME It. Cf. Sytem Scece ad Smulat Egeerg, eerfe, Spa, December 6-8, 5 (pp63-67) Aalytcal Mdellg f Parameter Fluctuat wth Applcat t Aalgue VLSI Crcut ad Sytem HJ KADIM Schl f Egeerg Lverpl JM Uverty Lverpl L3 3AF UK h..kadm@lvm.ac.uk Abtract - Aalgue crcut are etve t parameter fluctuat. Such etvty may caue a aalgue crcut t have lw peratg perfrmace. I th paper, a etmat methd f parameter fluctuat that tegrate deg ad tet prped. It baed aalytcal mdellg equat t etmate the etvty f aalgue crcut t parameter fluctuat a well a the ucertaty bud fr rbut perat, depedet f mulat. Key-Wrd: - Aalgue, etvty, ucertaty, tet Itrduct he fuct f a aalgue crcut are cmplex ad cat be adequately decrbed mple term. he great varety f pble put ad utput gal, ad the umber f pble crcut cfgurat whch a aalgue crcut may be ued make t extremely dffcult t determe whch f the may crcut parameter are mprtat. Furthermre, accurate aalgue mulat ha t be perfrmed at the devce level ad th very demadg f CPU tme ad memry. Cequetly t ca ly practcally be appled t mall aalgue deg; therwe the aaly effrt ad tme requremet wuld be prhbtve. echque that are ether amed at tetg r at detfyg fault that have effect aalgue crcut behavur [], r amed at ythe-fr-tetablty [][3] geerally have hgh cmputatal ct eve fr relatvely mall crcut becaue f the umber f varable ad parameter vlved. I a attempt t vercme me f thee prblem, the methd preeted th paper t replace the mulat wth a aalytcal prce. I ctrat wth mulat th prce eed t be de ly ce, ad therefre tet-terface ad tme requremet fr tmul geerat ad applcat ca gfcatly be reduced. he fault f aalgue crcut are clafed a parametrc fault [4][5][6] ad hard fault [][7]. Parametrc fault are caued by lght varat parameter value, wherea hard fault are caued by catatrphc varat parameter value. A devce dme ctue t cale dw, crcut perfrmace becme creagly etve t devat the devce fabrcat prce. Hece, parametrc fault have becme mre mprtat. echque dealg wth the tetg f aalgue crcut ca be clafed t tw ma grup. Oe grup cder pecfcat tetg [6], ad the ther make ue f meauremet t geerate a tet fr a crcut [7]. he drawback acated wth meauremet-baed techque the exteve umber f mulat tral requred ad the determat f the put tmul. Wth referece t pecfcat tetg, () t extremely dffcult t emulate the peratal evrmet ad () the deg am t make a pecfed perfrmace rbut t parameter fluctuat. h dcate that certa cae chage parameter cat be detected a the crcut uder tet perfrm atfactry the preece f uch chage. Furthermre, the dyamc behavur f a crcut the prduct f a umber f traet repe that crrepd t ple the cmplex plae []. I referece [] t wa hw that abrmalty e traet repe due t parameter chage may be maked by ther traet, cequetly uch abrmalty may t be detected at the fuctal level. herefre, t accurately ythee the degree f effect f parameter fluctuat crcut behavur, the fluctuat have t be defed wth the dyamc behavur. Parameter

2 Prceedg f the 4th WSEAS/IASME It. Cf. Sytem Scece ad Smulat Egeerg, eerfe, Spa, December 6-8, 5 (pp63-67) fluctuat may caue chage the lcat f ple, whch tur reult chage ther crrepdg traet ad ubequetly the verall behavur. If fr a maxmum bud f fluctuat whch drft a crcut cler t t margal tablty, ptmal cmpet value ca be che. h allw the tegrat f deg ad tet by cuplg dyamc crcut behavur t parameter fluctuat. h paper preet a aalytcal mdel t: () etmate the etvty f a crcut t parameter chage term f the pt f the ple; () etmate the maxmum bud f parameter fluctuat that mata a rmal fuctal behavur. Mathematcal Mdellg Fr a udrve ytem the tate-pace repreetat f a aalgue crcut gve: x ' = Ax(t) () where x(t): tate vectr ( elemet), A: ytemtercect matrx (.). wth the ple plymal p() = det(i A) () where : cmplex varable, I: ut matrx. he dagal matrx f A (.e. A d ) ha t ma dagal repreetg the dtct ple f the crcut, whch ca be real r cmplex (λ ; :,,.., ). A d = ( λ ) (3) where λ = whe. I referece [], t wa hw that fault caued by parametrc fluctuat trduce a chage the elemet λ f A d - thu a chage ther } = crrepdg traet repe e λ t = Fluctuat parameter uch a cmpet value, temperature, -learty f a amplfer ad varat pwer upply may caue a aalgue crcut t devate frm t dered repe. If all parameter that affect crcut behavur are lumped tgether ad repreeted by p the the trafer fuct F f a aalgue crcut - uder parameter fluctuat - ca be defed a []: F = + (4) F p where F : mal trafer fuct. he relathp betwee F ad p hw Fg.. Fg. he relathp betwee F ad p. Cder that the chage parameter relatve magtude: F - F = F (5) r F = F + F (6) wth W F = W (7) Re-arrage - = W W F = (8) Cder V = W P (9) ut ad V = W P () where = V F W () he utput V ut, uder parameter fluctuat, gve by: V = V F + W F () r V ut ut V = V W F + W F (3) I a matrx frm: Vut Vp = (4) M wth the trafer matrx gve by M M M = (5) M M wth M = F, M, ad = W F M = W M = W W F F W V ut

3 Prceedg f the 4th WSEAS/IASME It. Cf. Sytem Scece ad Smulat Egeerg, eerfe, Spa, December 6-8, 5 (pp63-67) Oe way f apprachg tetg a aalgue crcut va aeg t tablty uder parameter fluctuat. Cderg that the trafer fuct F repreet a table crcut behavur, the the crcut ad t be faulty-free r rbut t parametrc fault ff (f ad ly f) F rema table fr all allwable hat ~ p < γ ~ p (6) where γ: maxmum bud t eure tablty ad dered perfrmace. Hece, a aalgue crcut may exhbt utable behavur ly f ˆ > γ wth λ ( M ˆ ) > (7) where λ:a et f egvalue f F, ˆ : :chage. he determat a tate-pace frm gve by: det (I A) = (8) Fr a mall varat parameter: a < γ where a a ceffcet f matrx A. Fr mall chage, the etvty S f F wth repect t a gve by: S a = (a / F ) ( F / a ) (9) wth referece t (9), the chage parameter repreet a rage f value, γ m < a < γ max he ceffcet a w repreet a ucerta rage f value. It mprtat t kw hw cle a crcut t tablty. h ca be acheved by mdfyg (7) a fllw: λ M ˆ ) κ ; κ: teger ( that λ ˆ ) (M = λ(m ) κ ˆ ad λ = λ κ () he value f κ, effect, repreet a maxmum allwable fluctuat parameter that reta a atfactry fuctal behavur. 3 Example ad Smulat Depedg ther realat charactertc, aalgue crcut may repd dfferetly uder certa parameter fluctuat. Dfferet parameter ca reult dfferet effect crcut behavur. Hwever, t t alway the cae that parameter Ampltude x - fluctuat yeld faulty behavur. Fr tace, me parameter may cuteract the effect f each ther wth tceable verall chage crcut behavur thu the crcut ad t be rbut t parameter chage. Example I: cder the trafer fuct f a varable tch flter, a hw (). F = ( + ) ( + + ) () wth a =, a = ( / ) + + = () Aume that the cmpet f the flter have bee che t prduce a tch at apprxmately Hz. Hece, = 6, ad = 5. wth the tw ple f the flter are lcated at: = ( ), = ( ) Applyg () = = = = = κ=.7 κ= 3 = = = κ= κ= 5 he effect f chagg κ hw Fg., [] k = k = k = 5 k = me (ec) x - Fg. he effect tep-repe f creag k. Fg. hw that fr k =.3 the cllatry behavur f the crcut very cle t the mal e - thu may t be detected by the cvetal tet techque. Hwever, uch a vahg chage the behavur ha abut 9% creae the phae cmpared t the mal value. Frm Fg. ad fr a rbut perat k <.3. If k che t be, ad ce λ mag. = 6, the fr a rbut

4 Prceedg f the 4th WSEAS/IASME It. Cf. Sytem Scece ad Smulat Egeerg, eerfe, Spa, December 6-8, 5 (pp63-67) Ga (db) x Ga (db) x perat λ/λ ca be calculated ug (): λ/λ = k/λ <.6; Fr ay che value f k, t pble t predct a crcut cmpet value ug equat (8) ad (). It wa fud that a chage f % ether r ha effect the phae ad magtdue f λ. Hece, fr a rbut perat γ <. he frequecy repe f the flter t parameter fluctuat hw Fg.3 (a) ad (b). he value f κ btaed abve ctet wth γ ad al wth the mulat reult hw Fg = 5% = % = 5% = % Frequecy (Hz) = 5% = % = % = 5% = % = % 4 (a) hp = (3) + ( / ) + bp = (4) + ( / ) + lp = (5) + ( / ) + he etvty equat are a fllw: Rtch Rtch S ; S (6) ad R tate var, lp tate var, lp S able ; able S (7) R tate var, bp tate var, bp S able ; able S + (8) tate var, hp tate var, hp S able ; able S (9) Equat (6) hw that the tch flter le etve t parameter chage at lw frequece. Equat (7), (8) ad (9) hw that the tatevarable flter, whe t recfgured a a lw-pa r hgh-pa flter, le etve t parameter varat at lw frequece. Hwever, f th flter recfgured a a bad-pa flter, t etvty creae a chage ad decreae a chage. I th partcular cae, fluctuat parameter may cuteract each ther ad, hece, the crcut cdered t be rbut t parameter fluctuat. Fr tace, uder the ame parameter fluctuat the tate-varable flter exhbt tlerace t parameter fluctuat whe t cfgured a a badpa flter, but the flter exhbt a tceable devat (dted le Fg.4) frm the rmal repe (traght le) f t recfgured a a hgh-pa flter. -8 Frequecy (Hz) Fg.3. Frequecy repe fr the tch flter. 3 (b) 4 Ga (db) - -4 Example II: here a pblty that multaeu r verlapped chage parameter may cuteract each ther ver a perd f tme. Cder the fllwg trafer fuct f a ecd rder tate-varable: Frequecy (Hz) Fg.4 Frequecy repe fr the tate-varable flter.

5 Prceedg f the 4th WSEAS/IASME It. Cf. Sytem Scece ad Smulat Egeerg, eerfe, Spa, December 6-8, 5 (pp63-67) 3. Etmat f cmpet value It t the am f th paper t hw a ytematc way f evaluat f crcut cmpet, but rather t llutrate hw the methd prped ca ad etmat f cmpet value. Cder the charactertc equat f the tate-varable flter. + + = wth = (R/R q ), = (/R C), ad Aume that the flter deged uch that t tw ple are lcated at, = ±, the = 5 ad = Wth κ <.3 ad γ <., the ew value fr ad gve a fllw: = 5.7 ad =.3. he value che fr R, R q, R ad C huld atfy ad, rder t atfy a table behavural repe. If the value che fr the crcut cmpet prvde a mal value fr ad, the tlerace, fr tace, a cmpet value huld mata the value f r wth that btaed relat t κ. Otherwe, a uwated cllatry behavur may ccur. 4 CONCLUSION A aalytcal methd fr etmat f etvty f aalgue crcut t parameter fluctuat preeted. he methd al allw fr etmat f ucertaty bud fr rbut perat, depedet f mulat. Whe tegrated a part f a deg prce, cmpet value fr rbut perat ca be etmated. Referece: [] H.J. Kadm, B.R. Bater, "A Dmat Ple Baed Apprach fr Fault Idetfcat Aalgue VLSI", CEEDA'96 Cferece, UK, 996, pp [] L.Mler, V. Vvaatha, Detect f Catatrphc fault Aalgue Itegrated Crcut, IEEE ra. Cmputer Aded Deg, Vl.8, February 989, pp.4-3. [3] L. Mlr, A. Sagva, Mmg Prduct et tme t Detect Fault Aalgue Crcut, IEEE ra. Cmputer Aded Deg, Vl. 3, 994, pp [4] A. Balvada, J. Che, J. Abraham, Effcet etg f Lear Aalgue Crcut, Iteratal Mxed Sgal etg Wrkhp, Frace, 995, pp [5] H.J. Kadm, "Fault Detect Aalgue VLSI", IASED Iteratal Cferece Mdellg ad Smulat, Phladelpha, NJ, USA, May , paper [6] N. Be Hamda, B. Kamka, Aalgue Crcut etg baed Setvty Cmputat ad ew Mdellg, Prc. IEEE Iteratal et Cf., 993, pp [7] L.Mler, V. Vvaatha, Detect f Catatrphc fault Aalgue Itegrated Crcut, IEEE ra. Cmputer-Aded Deg, Vl.8, February 989, pp.4-3. [8] L. Mlr, A. Sagva, Mmg Prduct et tme t Detect Fault Aalgue Crcut, IEEE ra. Cmputer- Aded Deg, Vl. 3, 994, pp [9] G. Gele, Z, Wag, W. Sae, Fault Detect ad put Stmulu Determat fr the etg f Aalgue Itegrated Crcut baed Pwer-Supply Curret Mtrg, Prc. IEEE It. Cf. Cmputer-Aded Deg, 994, pp [] R. Drf, R. Bhp, Mder Ctrl Sytem, Add-Weley, INC., UK, 995, pp. Pp.8-5, 73-74, pp.86-9, pp [] H.J. Kadm, Dyamc Predct fr Aalgue Crcut Behavur ug State-Space baed raet Fault Aaly, 7 th IEEE Deg ad Dagtc f Electrc Crcut ad Sytem Wrkhp, Stara Lea, Aprl 4, pp [] H.J. Kadm, Etmat f a Maxmum Bud f Ucerta Parameter Fluctuat wth Applcat t Aalgue IP-Cre, Prc. f Iteratal Sympum Sytem--a- Chp, Flad, 4, pp6-64.