ISSN : On optimization of manufacturing of a multi-level invertor to increase density of elements

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1 SSN : olume ssue Nao Siee a Nao eholog A ia Joural NSNAJ () 6 [55-66] O opimiaio o mauaurig o a muli-level iveror o irease esi o elemes E..araov * E.A.Bulaeva Nih Novgoro Sae Uiversi 3 Gagari aveue Nih Novgoro 6395 (RUSSA) Nih Novgoro Sae Uiversi o Arhieure a ivil Egieerig 65 l is sree Nih Novgoro 6395 (RUSSA) elp4@mail.ru ABSRA his paper we iroue a approah o irease iegraio rae o elemes i a hree level iveror. he approah base o ereasig o imesio o elemes o he iveror (ioes a bipolar rasisors) ue o mauaurig o hese elemes b iusio or io implaaio i a heerosruure wih speii oiguraio a opimiaio o aealig o opa a raiaio ees. 6 rae Siee. - NA NROUON he prese ime oe a i ireasig o esi o elemes o iegrae iruis ue o ireasig o esi o heir imesios a opimiaio o ehologial proesses. Oe a also i ireasig o spee o hese elemes. Oe wa o erease imesios o elemes o iegrae iruis is opig o require areas o maerials b iusio or io implaaios a usig laser or mirowave pes o aealig [-3]. Usig hese pes o aealig leas o geeraio ihomogeous isribuio o emperaure. his siuaio oe a i ihomogeei o iusio oeiie a aoher parameers o proesses ue o Arrheius law. he ihomogeei o hese parameers gives a possibili o erease imesios o elemes o iegrae iruis. Aoher wa o hage properies o ope maerials is raiaio proessig [45]. A aleraive approah o erease imesios o elemes o iegrae iruis is usig ihomogeei o heerosruures [6-9]. his paper we osier a hree-level iverer whih has bee presee i he igure. he iverer has bee mauaure ramewor he heerosruure rom igure. he heerosruure osis o a subsrae a a epiaial laer. he epiaial laer ilues io isel several seios whih were mauaure b usig aoher maerials. hese seios were ope b iusio or io implaaio o mauaure he require pes o ouivi so as i is show o igure. Aer his opig i is require aealig o opa a/or raiaio ees. Mai aim o he prese paper is aalsis o reisribuio o opa a raiaio ees o eermie oiios whih orrespo o ereasig o elemes o he osiere iverer a a he same ime o irease heir esi. Meho o soluio We eermie spaio-emporal isribuio o oeraio o opa b solvig he ollowig bouar problem

2 O opimiaio o mauaurig o a muli-level iveror o irease esi 56 NSNAJ () 6 Nao Siee a Nao eholog A ia Joural () wih bouar a iiial oiios () ()= (). Here () is he spaio-emporal isribuio o oeraio o opa; is he emperaure o aealig; is he opa iusio oeiie. alue o opa iusio oeiie epes o proper- igure a : Sruure o iveror. iew rom op igure b : Heerosruure wih wo laers a seios i he epiaial laer

3 E..araov a E.A.Bulaeva 57 NSNAJ () 6 Nao Siee a Nao eholog A ia Joural ies o maerials spee o heaig a oolig o heerosruure (wih aou Arrheius law). epeees o opa iusio oeiies oul be approimae b he ollowig uio [4] * * (3) where () is he spaial (ue o eisig several laers wi iere properies i heerosruure) a emperaure (ue o Arrheius law) epeees o opa iusio oeiie; () is he limi o solubili o opa; parameer oul be ieger ramewor he ollowig ierval [3][] ; () is he spaio- emporal isribuio o oeraio o raiaio vaaies; * is he equilibrium isribuio o oeraio o vaaies. oeraioal epeee o opa iusio oeiie have bee isusse i eails i []. shoul be oe ha usig iusio pe o opig i o leas o geeraio raiaio ees a = =. We eermie spaio-emporal isribuios o oeraios o poi ees have bee eermie b solvig he ollowig ssem o equaios [4] (4) wih bouar a iiial oiios ()= () (5) Here =; () is he spaio-emporal isribuio o oeraio o raiaio iersiials; () is he iusio oeiies o raiaio iersiials a vaaies; erms () a () orrespo o geeraio o ivaaies a iiersiials; () is he parameer o reombiaio o poi raiaio ees; () are he parameers o geeraio o simples omplees o poi raiaio ees. We eermie spaio-emporal isribuios o oeraios o ivaaies ( ) a

4 O opimiaio o mauaurig o a muli-level iveror o irease esi 58 NSNAJ () 6 Nao Siee a Nao eholog A ia Joural iiersiials () b solvig he ollowig ssem o equaios [] (6) wih bouar a iiial oiios (7) ()= () ()= (). Here () are he iusio oeiies o omplees o raiaio ees; () are he parameers o ea o omplees o raiaio ees. We eermie spaio-emporal isribuios o oeraios o opa a raiaio ees b usig meho o averagig o uio orreios [3] wih erease quai o ieraio seps [4]. ramewor he approah we use soluios o Eqs. () (4) a (6) i liear orm a wih average values o iusio oeiies as iiial-orer approimaios o he require oeraios. he soluios oul be wrie as e e e e

5 E..araov a E.A.Bulaeva 59 NSNAJ () 6 Nao Siee a Nao eholog A ia Joural e where ep e u w wv v u v v u () = os ( / ). he seo-orer approimaios a approimaios wih higher orers o oeraios o opa a raiaio ees we eermie ramewor saar ieraive proeure [34]. ramewor his proeure o alulae approimaios wih he -orer oe shall replae he uios () () () () ( ) i he righ sies o he Eqs. () (4) a (6) o he ollowig sums + - ( ). As a eample we prese equaios or he seo-orer approimaios o he osiere oeraios (8) (9)

6 O opimiaio o mauaurig o a muli-level iveror o irease esi 6 NSNAJ () 6 Nao Siee a Nao eholog A ia Joural () egraio o he le a righ sies o Eqs.(8)-() gives us possibili o obai relaios or he seo-orer approimaios o oeraios o opa a raiaio ees i ial orm (8a)

7 E..araov a E.A.Bulaeva 6 NSNAJ () 6 Nao Siee a Nao eholog A ia Joural (9a) (a) We eermie average values o he seo-orers approimaios o he osiere oeraios b usig he ollowig saar relaios [34]. () Subsiuio o relaios (8a)-(a) io relaio () gives us possibili o obai relaios or he require average values Ñ ()

8 6 Nao Siee a Nao eholog A ia Joural O opimiaio o mauaurig o a muli-level iveror o irease esi A A A A A A A A 4 A A A A NSNAJ () 6 B3 A B3 B B3 A 4B4 (3b) B4 4 A 4B4 i j where Aabij a b B 3 A A A A A B 3 A A A A A A A A A A A A A A A A A A 4 4 B A A A 4A A A A A A A A A A A A A A A 4A 4A A A A A A A A A A A A A A A A A A A A A A A A A A A A A B A A A A 8A A A A A A A A A A A 4A A A A B A A A A A A A A A A A A A A A A 4A A A A A A A (3a)

9 NSNAJ () 6 E..araov a E.A.Bulaeva 63 A A A A A A A A A A 3 B B B 4B B3 B q p q q p q q B B3 8B 6 B 7 p 3B B 8B 3 B A 8 B 3 4B A (4) A. he osiere subsiuio gives us possibili o obai equaio or parameer. Soluio o he equaio epes o value o parameer. Aalsis o spaio-emporal isribuios o oeraios o opa a raiaio ees has bee oe b usig heir seo-orer approimaios ramewor he meho o average o uio orreios wih erease quai o ieraive seps. he seo-orer approimaio is usuall eough goo approimaio o mae qualiaive aalsis a obai some quaiaive resuls. Resuls o aalial alulaio have bee hee b ompariso wih resuls o umerial simulaio. SUSSON his seio we aale spaio-emporal isribuio o oeraio o iuse (see igure a) a implae (see igure b) opas i he osiere epiaial laer. Aealig ime is he same or he eah urve ramewor eah igure. reasig o umber o urve orrespos o ireasig o ieree bewee values o opa iusio oeiie i laers o heerosruure. he igures show ha ierae bewee laers o heerosruure gives a possibili o irease absolue value o graie o oeraio o opa i ireio whih is perpeiular o he ierae. his siuaio oe a i ereasig o imesios o ioes a rasisors ramewor he osiere iverer. A he same ime wih ireasig o absolue value o he above graie oe a i ireasig homogeei o isribuio o oeraios o opas i ope areas. o hoose aealig ime i is eessar o ae io aou ereasig o absolue value o graie o oeraio o opa ear ierae bewee epiaial laer a subsrae wih ireasig o value o aealig ime. ereasig o value o aealig ime leas o ireasig o ihomogeei o isribuio Nao Siee a Nao eholog A ia Joural

10 64 O opimiaio o mauaurig o a muli-level iveror o irease esi NSNAJ () 6 igure a : isribuios o oeraio o iuse opa i heerosruure rom igs. a i ireio whih is perpeiular o ierae bewee epiaial laer subsrae. reasig o umber o urve orrespos o ireasig o ieree bewee values o opa iusio oeiie i laers o heerosruure uer oiio whe value o opa iusio oeiie i epiaial laer is larger ha value o opa iusio oeiie i subsrae igure b : isribuios o oeraio o implae opa i heerosruure rom igs. a i ireio whih is perpeiular o ierae bewee epiaial laer subsrae. urves a 3 orrespos o aealig ime =.48( + + )/. urves a 4 orrespos o aealig ime =.57( + + )/. urves a orrespos o homogeous sample. urves 3 a 4 orrespos o heerosruure uer oiio whe value o opa iusio oeiie i epiaial laer is larger ha value o opa iusio oeiie i subsrae o oeraio o opa (see igures 3a or iusio pe o opig a 3b or io pe o opig). We eermie he ompromise value o aealig ime ramewor reel iroue rierio [5-]. ramewor he rierio we approimae real isribuio o oeraio o opa b iealie sep-wise uio (). Aer ha we eermie he require ompromise value o aealig ime b miimiaio o he mea-square error U Nao Siee a Nao eholog A ia Joural. (5)

11 NSNAJ () 6 E..araov a E.A.Bulaeva 65 igure 3a : epeees o imesioless opimal aealig ime or opig b iusio whih have bee obaie b miimiaio o mea-square error o several parameers. urve is he epeee o imesioless opimal aealig ime o he relaio a/ a = = or equal o eah oher values o opa iusio oeiie i all pars o heerosruure. urve is he epeee o imesioless opimal aealig ime o value o parameer or a/=/ a = =. urve 3 is he epeee o imesioless opimal aealig ime o value o parameer or a/ =/ a = =. urve 4 is he epeee o imesioless opimal aealig ime o value o parameer or a/=/ a = = igure 3b : epeees o imesioless opimal aealig ime or opig b io implaaio whih have bee obaie b miimiaio o mea-square error o several parameers. urve is he epeee o imesioless opimal aealig ime o he relaio a/ a = = or equal o eah oher values o opa iusio oeiie i all pars o heerosruure. urve is he epeee o imesioless opimal aealig ime o value o parameer or a/=/ a = =. urve 3 is he epeee o imesioless opimal aealig ime o value o parameer or a/ =/ a = =. urve 4 is he epeee o imesioless opimal aealig ime o value o parameer or a/=/ a = = epeees o opimal aealig ime are presee o igure 3 or iusio a io pes o opig respeivel. shoul be oe ha i is eessar o aeal raiaio ees aer io implaaio. Oe oul i spreaig o oera- io o isribuio o opa urig his aealig. he ieal ase isribuio o opa ahieves appropriae ieraes bewee maerials o heerosruure urig aealig o raiaio ees. opa i o ahieves a ieraes urig a- Nao Siee a Nao eholog A ia Joural

12 66 O opimiaio o mauaurig o a muli-level iveror o irease esi ealig o raiaio ees i is praiabl o aiioall aeal he opa. his siuaio opimal value o aiioal aealig ime o implae opa is smaller ha aealig ime o iuse opa. A he same ime io pe o opig gives us possibili o erease mismah-iue sress i heerosruure []. ONUSON his paper we moel reisribuio o iuse a implae opas urig ma-uaurig a iverer wih hree levels base o heeroioes a bipolar heerorasisors. Several reommeaios o opimie mauaure he heerorasisors have bee ormulae. Aalial approah o moel iusio a io pes o opig wih aou ourre hagig o parameers i spae a ime has bee iroue. A he same ime he approah gives us possibili o ae io aou olieari o opig proesses. AKNOWEGMENS his wor is suppore b he ora.g o he Russia eeraio Goverme gra o Sieii Shool o Russia he agreeme o Augus 7 3 ¹. No bewee he Miisr o euaio a siee o he Russia eeraio a obahevs Sae Uiversi o Nihi Novgoro a euaioal ellowship or sieii researh o Nih Novgoro Sae Uiversi o Arhieure a ivil Egieerig. REERENE [] K.K.Og K..e.S.ee A..S.Wee X..Wag Y..hog; Appl.hs.e. 89(7) 7-74 (6). [] H..Wag.S.a E..hor; J.Appl.hs. 98(9) (6). [3] Yu..Bov A.G.Yeremeev N.A.Zharova..loiov K..Rbaov M.N.roov Yu.N.roov..Supov; Raiophsis a Quaum Elerois 43(3) (3). NSNAJ () 6 [4]..Kolivs; Moiiaio o semiouors b proo beams (Naua Sa-eerburg 3 i Russia). [5]..iesi G.A.Kholoar ; Raiaive phsis o semiouors.( Nauova uma Kiev 979 i Russia). [6] G.olovih; Moer Elerois -7 (6). [7] A.Keresev.ai; ower Elerois 34 (8). [8] A.O.Ageev A.E.Belaev N.S.Boloves.N.vaov R..Koaova Ya.Ya.Kuri.M.ivi..Milei A..Saheo; Semiouors 43(7) (9). [9] N..oloobisaa.N.Komarov..Mauhia..Resheiov A.A.Rush..alia A.S.Yasrebov; Semiouors 35(8) 3-7 (). [].Sehilpari K.iwaar A.K.Sigh; Ree aes o Naoeholog 3() 6-7 (9). [] Z.Yu.Gora; eholog o miroeleroi evies (Raio a ommuiaio Mosow) (99). [].M.ahe.B.Grii J..lummer; Rev.Mo.hs. 6() (989). [3] Yu..Soolov; Applie Mehais () 3-35 (955). [4] E..araov; he Europea hsial Joural B. 57(3) 5-56 (7). [5] E..araov; Russia Miroelerois 36() (7). [6] E..araov;.J.Naosiee 7(4-5) (8). [7] E..araov; J.omp.heor.Naosiee 7() (). [8] E..araov E.A.Bulaeva; J.omp.heor.Naosiee (4) (3). [9] E..araov E.A.Bulaeva;.J.Miro-Nao Sale rasp 4() 7-3 (4). [] E..araov E.A.Bulaeva;.J.Naosiee (5) (). [] E..araov E.A.Bulaeva; J.omp.heor.Naosiee () 9- (4). Nao Siee a Nao eholog A ia Joural