Op Amp Noise in Dynamic Range Maximization of Integrated Active-RC Filters

Transcription

1 Op Amp Nose Dyamc Rage Maxmzao of Iegaed Acve-R Fles N MARAOS* AND M MLADENO** * Naoal echcal Uvesy of Ahes Dep of Eleccal ad ompue Egeeg 9 Ioo Polyechou S ogafou 577 Ahes eece ** Depame of heoecal Elecoechcs Faculy of Auomacs echcal Uvesy of Sofa Asac: - I hs pape a mehod s poposed fo maxmzg he dyamc age of acve-r ad MOSFE- fles so ha I mplemeao cosas ae sfed ad a gve asfe fuco s ealsed Boh op amp geeaed ose ad esso hemal ose ae ake o accou he calculao of dyamc age By oducg a specal asfomao of vaales a almos-explc soluo of he dyamc age maxmzao polem s oaed he opmal ccu s scaled ad makes exhausve use of he avalale capacace Key Wods: - Iegaed aalog ccus acve fles dyamc age maxmzao op amp geeaed ose Ioduco he dyamc age of acve-r ccus s popooal o he squaed amplude of he maxmum pu sgal ha guaaees leay of he ccu ad vesely popooal o he oal oupu ose powe Lea opeao of he ccu s lmed y op amp oupu volage uao oal oupu ose acve-r ccus s made up of op amp geeaed ose ad esso hemal ose Whe cosdeg dscee-compoe ccus esso geeaed ose ca pcple e lmed o ay desed level y usg suffcely lage capacos ad coespodgly small essos Whe howeve such fles ae egaed o a sgle chp ehe decly acve-r fom o coveed o MOSFE- fom he oal capacace allowed s lmed y chp aea cosas ofe o a few huded pf hs mples ha essace values ad cosequely ose powe ca e odes of magude lage ha hose of dscee-compoe fles hus oh esso geeaed ose ad op amp geeaed ose mus e cosdeed whe desgg hgh dyamc age egaed acve-r ccus he leaue coas seveal classcal esuls eg []-[4] o acve ccu ose aalyss ad desg Alhough useful hose esuls ae mosly lmed o pacula opologes ad do o ake o accou I mplemealy cosas Rece wok [5]-[9] does ake o accou such cosas; howeve he esuls [5] ad [6] ae lmed o cea ypes of opologes ad hose [7] ad [8] hough of geeal applcao have he fom of lowe ouds fo geeaed ose I [9] a geeal mehod s poposed fo maxmzg he dyamc age of a ccu howeve op amp geeaed ose s egleced I hs pape we popose a mehod fo maxmzg he dyamc age of acve-r ad MOSFE- fles so ha : a a pesced asfe fuco s ealsed oal capacace s ouded Boh esso geeaed ose ad op amp geeaed ose ae ake o accou Esseally he poposed mehod s a exeso of he esuls of [9] wheey op amp geeaed ose s o loge egleced I s assumed ha he ccu opology s gve ad ha elogs o a cea class of opologes o e desced lae whle he opmzao execse ams a selecg appopae ccu eleme values he mehod s ased o modfed odal fomulao of he ccu equaos [] calculaes ose y he ado sysem mehod [] ad makes use of cea smple esuls fom sgula lea sysem heoy [] A specal asfomao of vaales s oduced whch eales he devao of almosexplc fomulae fo opmal eleme values Fomulao of he Polem I s assumed ha a ccu opology s gve ad ha a asfe fuco p s pesced fo he ccu o ealse Desgale ccu eleme values coaed a veco x ae o e seleced so ha

2 he dyamc age of he ccu s maxmzed suec o cea cosas asg fom lmaos of I echology ad ccu pefomace he dyamc age of a ccu s defed as [8]: max x max DRx Nx ou x whee max x s he amplude of he maxmum pu sgal whch ca e pocessed y he ccu whou dsoo Nx s he oal ose powe a he ccu oupu ad ou x s he ccu asfe fuco I mplemealy of he ccu ad ccu pefomace ae guaaeed y he followg wo cosas: a he pesced asfe fuco p mus e ealsed y he ccu e ou x p I mplemealy cosa: he oal capacace used he ccu mus o exceed a gve uppe oud e whee he summao o he lef sde of cludes all capacos of he ccu hus he dyamc age maxmzao polem ca e fomulaed as follows: DRx : ou x p max x P A moe wokale expesso fo he dyamc age s oaed y makg use of he eal ccu asfe fucos x fom he ccu pu o he oupu of he h op amp he sgal dsoo s avoded y equg ha x whee s he amplude of a susodal pu sgal of fequecy he uao volage of op amps ad he ume of op amps he ccu heefoe he amplude max of he maxmum udsoed pu sgal s gve y max x max max x 4 Susuo of equao 4 eq yelds he followg expesso fo dyamc age DRx max max x Nx max ou x 5 ece he dyamc age maxmzao polem P s equvale o he followg mmzao polem Nx max x : max p m oux p P x whee s a ew depede vaale whose value a he opmum s max max x max max x p ad he addoal cosas esue lea max p opeao of he ccu op amps ve aay eleme values x all he quaes volved he fomulao of polem P ca e eadly calculaed y pefomg a aalyss of he ccu wh eleme values x I should e oed ha polem P s a olealy cosaed opmzao polem wh complcaed cosas ad oecve fuco Soluo of P y suale eave umecal algohms s possle u compuaoally expesve Isead of solvg polem P decly we popose o eplace he desg vaales x y a ew se of vaales ad efomulae he polem hs efomulao leads o susaal smplfcao ad yelds a almosexplc soluo fo polem P cu opologes osdeed he class of fle opologes we exame cosss of ccus wh he followg popees: a he ccu cosss of essos capacos ad deal op amps ccu oupu s he oupu emal of oe of s op amps

3 c ccu pu s a volage souce coeced ewee he ccu pu ode ad goud d apa fom he ccu pu ode evey ohe ode of he ccu s ehe a pu o a oupu emal of oe of s op amps e oe of he pu emals of each op amp he ccu s gouded Seveal well kow hee-amplfe quads eg ow-homas sfy hese popees ghe ode fles usg such quads also elog o he class of opologes examed I s a mpoa class of opologes especally fo full egao as gves se o fully alaced ccus [] 4 Nose Modellg oal oupu ose Nx s made up of wo compoes: esso-geeaed ose N R x ad op amp-geeaed ose N A x e Nx N x N x 6 R Nose due o a sgle esso R k of he ccu s modelled as a ose cue souce J k paallel o he esso wh specal desy 4k B k whee k B s Bolzma s cosa s he asolue empeaue ad k /R k s he coducace of R k he powe specal desy psd of oupu ose due o all essos he ccu s A S R x 4k B k Ĥ k x 7 k whee Ĥ k x s he asfe fuco fom he ose cue souce J k o he ccu oupu wh he ccu pu shoed he oal oupu ose mea squaed value due o all essos he ccu s N R x S R xd 8 whee [ ] [ s he fequecy age of ees Followg [] we model ose due o he h op amp of he ccu as a comao of a ose volage souce e A wh amplude E A sees o he o-gouded op amp emal ad a ose cue souce A wh amplude I A coeced ewee he o-gouded op amp emal ad goud Fgue depcs oe op amp of he assumed opology ogehe wh s ose souces ad all possle passve ach coecos o s emals As [] we shall assume ha he ose volage ad ose cue specal deses of e A ad A ae fequecy depede ad ae desced y: whee E ea E A I A IA E ad I ae /f coe fequeces ad/sec hus he powe specal desy of op amp geeaed ose a he oupu of he ccu wll e gve y E S A x E A x I I x A 9 whee x ad x ae asfe fucos fom e A ad A especvely o he ccu oupu he he oal oupu ose mea squaed value due o all he op amps of he ccu s: N A x SA xd 5 cu ad Nose Aalyss he class of ccu opologes cosdeed hs pape has he geeal sucue show Fg whee he h op amp of he ccu s depced ogehe wh s ose volage ad cue souces ad ogehe wh all possle admace aches y ad y coecg s emal odes k l o ohe odes of he ccu cu pu s he volage souce coeced ewee he ccu pu ode e ad goud he admace y coeced ewee he pu ode k of he h op amp ad he oupu ode l of he h op amp s assumed o e ehe a coduco y o a capaco y s o a paallel comao of a

4 e ^ y y y l l y k e A A k l y y k k Fgue eeal sucue of he ccu opologes cosdeed showg he h op amp ogehe wh ose souces ad admace aches coeced o coduco ad a capaco y s he same assumpo s made fo he admace y coeced ewee he ccu pu ode e ad pu ode k of he h op amp e y g o y sc o y g sc I s ow que saghfowad o we ewok equaos fo he geeal sucue of Fg he modfed odal appoach [] s used ad he followg equaos ae oaed: [ s ] ĝ sĉ IA [ s dagg sc] EA whee I A ad E A ae -vecos coag ose cue ad ose volage souces of he op amps ad ae x maces wh elemes ad as defed aove ad g ad c ae -vecos wh elemes g ad c as defed aove Equao s a fcous equao sce supeposo of ad he ose souces s o possle oweve y seg o zeo all ohe souces u oe we oa he followg useful asfe fucos: I A E A yelds he veco sx of asfe fucos sx fom he ccu pu o he oupu ode l of he h op amp: [ s] sx ĝ sĉ Equao coespods o sadad ccu aalyss I A e E A whee e deoes he h colum of he dey max yelds a -veco sx wh elemes sx of asfe fucos fom he pu of he h op amp o he oupu odes l of each op amp he ccu: [ s] sx e I A E A e yelds a -veco sx wh elemes sx of asfe fucos fom he ose volage souce e A a he pu of he h op amp o he oupu odes l of each op amp he ccu: [ s] s x [ s]e g sc e 4 v Fo esso geeaed ose aalyss use s made - see eq 7 of he asfe fucos Ĥ k s x fom a ose cue souce J k paallel o he esso R k o he ccu oupu oweve whe cosucg he I-gaph fo he opology of fg he ccu odes e l l l all collapse o he I-gaph goud ode heefoe all he ecessay asfe fucos Ĥ k s x ca e oaed fom he soluos s x of equaos Whou loss of geealy we shall ow assume ha he ccu oupu ode s he oupu emal l of he op amp umeed las e ou ou d e 5 he he ccu asfe fuco sx used 5 ad P s oaed fom ad 5 as: ou s x d s x d [ s] ĝ sĉ 6 Le zsx e he soluo of he ado sysem of equaos []: he [ s] zsx d e 7 ou s x zs x ĝ sĉ 8 Also he asfe fucos s x used 9 ae oaed fom 5 7 as: s x d s x d [ s] e e zs x z s x 9 he asfe fucos sx also used 9 ae oaed fom as:

5 sx d sx d d e ĝ sĉ [ s] e e ĝ sĉ d e e ĝ sx ĝ sĉ sx ĝ sĉ e [ s] zsx e e z sx fo - sĉ z sx Fally he asfe fucos Ĥ sx used 7 ae oaed fom 5 7 as: Ĥ sx d sx d [ s] e e zs x z sx Now y usg 9 ad 9 we oa fo esso geeaed ose: S R x 4kB ĝ z x Also y usg 9 ad 9 we oa fo op amp geeaed ose: E S A x EA ĝ ĉ z x E E A ĝ ĉ z x I I z x A 6 cu asfomao I hs seco a specal ccu asfomao s oduced he effecs of hs asfomao o ccu opology ccu eleme values ad o he eal ccu asfe fucos ae examed Le a ccu e gve ha elogs o he class of ccu opologes examed hs pape Assume ha al ccu eleme values x have ee seleced so ha he pesced asfe fuco p s ealsed y he al ccu e assume ha ou x p 4 Fo he al ccu ewok aalyss ad ado equaos 5 ad 7 ake he followg fom s s x ĝ sĉ 5 ou [ ] x d x e x 6 [ s ] zs x d e 7 whee he maces ad he vecos ĝ ĉ coa he al ccu eleme values x [ L L L ĝ Lĉ L] appopae posos Le M ad N e dagoal maces wh posve dagoal ees m ad k k especvely We shall oduce he followg asfomao of he ccu maces ad vecos ĝ ĉ d: M N 8 M N 8 ĝ Mĝ 8 ĉ Mĉ 84 d N d 85 I s oed ha he ccu asfomao oduced 8 85 s a equvalece asfomao he sese of [] Whe he ccu asfomao 8 wh aay posve defe dagoal maces M ad N s appled o he al ccu he followg popees ae sfed: a he eal ccu asfe fucos s ad zs ae lealy asfomed Ideed follows fom 8 ad 5 ha sx [ s] N heefoe [ s x ĝ sĉ s ] M Mĝ sĉ N s x 9 Also follows fom 7 ad 8 ha zs x [ s] d M heefoe y 7 zs x [ s ] N Nd M zs x he ccu asfe fuco emas vaa I follows fom ad 6 ha ou s x d s x d c cu eleme values NN s x d s x s x ou ĝ ĉ ae asfomed as follows: m m ĝ ĉ ĝ m ĉ m 4 d he ccu opology emas vaa I follows fom equaos ha he asfomed ccu maces ad have he

6 same sucue posos of zeo ad o-zeo elemes as he al maces ad he same holds ue fo he vecos ĝ ad ĉ Fuhemoe sce m > ad > he asfomed ccu elemes wll e posve heefoe he asc ccu opology s vaa ude he asfomao 8 84 I ode fo he ccu oupu o ema vaa as well equao 85 yelds he codo: hus povded ha holds he ccu asfomao 8: leaves ccu opology ad he ccu asfe fuco vaa ad asfoms ccu eleme values accodg o ad eal asfe fucos accodg o 9 ad 7 Refomulao of Polem P We shall use he ccu asfomao 8 ode o efomulae he dyamc age maxmzao polem P ems of he vaales m ad ahe ha ems of x I hs coex le us exame he cosas ad he cos fuco of polem P a asfe fuco ealo cosa: eq hs complcaed olea cosa s omed alogehe sce he ccu asfe fuco emas vaa ude asfomao 8 ad he gve al ccu eleme values x ealze p I mplemealy cosa equaly s: ĉ Usg equaos we we hs cosa as m ĉ 4 c Op amp lea opeao cosas hese cosas ae expessed y he equales max x 5 max p I follows fom 9 ha max x max x heefoe hese cosas ae we as 6 whee he quaes max x 7 max p ae easly calculaed fom aalyss of he al ccu Sce fom ad y defo he h of he cosas 6 ecomes: 8 d os fuco Le z x I d z x d E ĝ ĉ z x d - E ĝ ĉ z x d I he follows fom ad ha: z x d E I m z x d m ĝ ĉ z x d - E ĝ ĉ z x d hus vew of ad he aove he cos fuco ecomes: cos 4kBm ĝ m 4kB IA E A m ĝ I A E A f 9

7 whee m 4 e Addoal cosas ae: 4 ad I s oed ha 6 ad 8 ede he cosas - eduda hus he dyamc age maxmzao polem P s efomulaed as follows: Pˆ c m f m : 8 Soluo of Polem P Kuh-ucke opmaly codos a a soluo * * * of polem Pˆ ae: A B I ĝ 4k 4 ν A E 4 A B I ĝ 4k ĉ ρ 4 8k B - 4 ĉ 44 ĉ 45 ν ν ρ ρ 48 whee ν ρ ae Lagage mulples A soluo * * * of polem Pˆ s oaed ex fom he Kuh-ucke codos 4-48 Fom 4 follows ha > - hece 47 yeld - Mulplcao of 4 y yelds a hd ode polyomal equao fo whch vew of 48 does o deped o ρ hs equao has oly oe eal oo hs oo s he calculaed ad eplaced 44 o oa a olea equao fo ad 4 o oa > ν Upo execuo of he algeac mapulaos skeched aove he followg fomulae ae oaed fo he soluo * * * of Pˆ : * 4 * 44 q p p q p p * * m 45 whee p q α 46 B ĝ 8k α 47 A I 48 ĉ 49 ad s a soluo of he followg equao: α α 5 he opmal value of dyamc age s

8 DRx* f * ** E A I A 5 Remaks: he soluo oaed s almos explc Equao 5 mus e solved umecally ode o oa avg he opmal soluo * ** s calculaed decly y usg he fomulae 4 49 Opmal ccu eleme values ae oaed fom * ** y usg equao A he opmum he I mplemealy cosa 4 s sfed as a equaly e exhausve use s made of he avalale capacace desgg he ccu 4 I follows fom 4 ad 44 ha a he opmum he op amp lea opeao cosas 5 hold as equales wh * hs meas ha he opmal ccu s scaled e ha he ampludes of op amp oupu volages have equal maxma ove fequecy 9 oclusos Maxmzao of he dyamc age of egaed acve-r fles wh gve opology assumed o sfy cea o uusual popees s effcely pefomed y makg use of a ccu asfomao whch leaves he ccu asfe fuco ad ccu opology vaa Boh esso geeaed ose ad op amp geeaed ose ae ake o accou he calculao of dyamc age Esseally he poposed mehod s a exeso of he esuls of [9] wheey op amp geeaed ose s o loge egleced I mplemealy s esued y a uppe oud placed o oal ccu capacace ad he ccu s equed o ealze a pesced asfe fuco he poposed mehod s vald fo ccus of aay ode ad fo ay ype of pesced asfe fuco Almos-explc fomulae ae oaed fo he opmal soluo of he asfomed dyamc age maxmzao polem he opmal ccu s scaled sfes he oal capacace cosa as a equaly e makes exhausve use of he avalale capacace ad ealses he pesced asfe fuco Refeeces: [] J Bachle ad W uggeuhl "Nose Aalyss ad ompaso of Secod-ode Newoks oag a Sgle Amplfe" IEEE as cus Sys ol AS-7 pp [] L Buo FN ofmekoff ad D eleave "Nose Pefomace of Low- Sesvey Acve Fles" IEEE J Sold-Sae cus ol S-8 pp [] FN ofmekoff D eleave ad L Buo "Nose Pefomace of R-Acve Quadac Fle Secos" IEEE as cu heoy ol - pp [4] JW asle "Nose Pefomace Lmaos of Sgle Amplfe R-Acve Fles" IEEE as cus Sys ol AS- pp [5] oeewold "he Desg of gh Dyamc Rage ouous-me Fles" IEEE as cus Sys ol AS-8 pp [6] P Bowo ad KA Mezhe "Nose ad Sesvy Opmzao he Desg of Secod-Ode Sgle-Amplfe Fles" I J cu h Appl ol 9 pp [7] L oh opaha N Maaos ad YP svds "Bouds o Nose Iegaed Acve- R ad MOSFE- Fles" Poc ISAS'9 99 [8] L oh Efhvoulds opaha ad YP svds eeal esuls fo essve ose acve-r ad MOSFE- fles IEEE as cus Sys-II vol 4 pp [9] N Maaos ad Ioads A mehod fo dyamc age maxmzao fo a class of egaed acve-r fles Poc IEEE/ISAS 994 vol 5 pp []J lach ad K Sghal ompue Mehods fo cu Aalyss ad Desg a Nosad Rehold New Yok NY 98 [] eghese B Levy ad Kalah "A eealzed Sae-Space fo Sgula Sysems IEEE as Auomac ool ol A-6 pp 8-8 []D Lueege Lea ad Nolea Pogammg Addso-Wesley Readg MA 984 []JM Khouy ad YP svds "Aalyss ad ompeo of gh-fequecy Effecs Iegaed MOSFE- ouous-me Fles" IEEE as cus Sys ol AS-4 pp Ackowledgeme: hs wok was suppoed y NAO ollaoave Lkage a PSL 97579