P. Bruschi - Notes on Mixed Signal Design

Transcription

1 Chaptr 1. Concpts and dfinitions about Data Acquisition Systms Elctronic systms An lctronic systms is a complx lctronic ntwor, which intracts with th physical world through snsors (input dvics) and actuators (output dvics). This dfinition is graphically rprsntd in Fig.1.1, whr th arrows indicat information fluxs. Th aim of th lctronic systms is gnrally to gt information about slctd aspcts of th physical world and, optionally, us th information to modify th physical world in a usful way. Th main oprations that an lctronic systm can prform ar listd in Fig.1.1. physical world snsors actuators Elctronic systm: -) Acquisition -) Procssing -) Display -) Storag -) Communication -) Actuation Fig.1.1 Th gnral architctur of an lctronic systm is shown in fig.1.2. Th snsors ar handld by th DAS sub-systm. Th DAS output is a digital signal which is rad by a digital procssor. Th aim of th lattr is complting th acquisition procss by stimating th quantitis of intrst. Th information is furthr procssd in ordr to ma it suitabl for bing transmittd, stord or displayd. Th digital procssor also uss th stimatd quantitis to calculat th corrct action to b applid to th physical world in ordr to obtain th dsird function (.g. to st th ambint tmpratur to a givn st-point). Data transmission is prformd by a communication intrfac, which implmnts th rquird protocol. Othr subsystms (priphral intrfacs) allow communication with th storag and display dvics, providing th corrct protocol and powr. Th analog powr signals which ar typically rquird for driving th actuators ar also gnratd by spcial subsystms (actuator drivrs). Th dsign of a complx lctronic systm gnrally rquirs diffrnt sills. Th DAS involvs advancd analog dsign tchniqus; th digital procssor rquirs dsignrs with both nowldg of informatics and automatic digital synthsis; th actuator drivrs and priphral drivrs involv powr lctronics dsign sills, whil th communication intrfac may rquir th sill of an RF dsignr. 1

2 Elctronic Systm physical world snsors actuators DAS: Data acquisition systm actuator drivrs powr managmnt Digital procssing systm Communication Intrfac(s) prifral intrfac(s) commun. mdium Data acquisition systms (DAS). powr lin battry Fig.1.2 storag display Th aim of a DAS is acquiring information on ral world objcts by masuring th physical and chmical quantitis of intrst. For simplicity, in th following part of this documnt w will simply indicat ths quantitis with physical quantitis or, whn thr is no ambiguity, only with quantitis. Th objct, to which th quantitis ar prtinnt, will b indicatd as physical world or physical domain. Th acquird quantitis ar convrtd into an analog lctrical signal, gnrally a voltag. In most cass it is ncssary to convrt th analog signals into digital signals in ordr to facilitat procssing, storag, visualization and transmission. Fig. 1.3 shows th various transformations that th information (quantitis and Y) undrgos during th acquisition procss. At th nd of th procss, w gt two numbrs m and Y m that rprsnt stimats of th quantitis and Y. W hav also introducd a third quantity, Z, that w suppos can b calculatd from and Y, so that it is not ncssary to build an acquisition channl also for Z but th Z stimat (Z m ), can b simply drivd from m and Y m. With this choic, m and Y m ar indpndnt quantitis, whil Z m is a dpndnt quantity. 2

3 snsor E AFE -ADC D stimator m Y Z=f(,Y) Y snsor E Y Y AFE Y Y-ADC DY Y stimator Y m physical domain analog domain digital domain Z stimator Z m Fig.1.3 Each indpndnt quantity is acquird by a radout channl. A channl is composd by th cascad of svral lmnts: Considring th radout channl w hav: Snsor. Th snsor intracts with th physical quantity of intrst (), causing an lctrical quantity E to dpnd on. In ordr to unambiguously stimat th valu, th rlationship from to E should b monotonic in th whol rang of vals of intrst. Not that E can b any lctrical quantity; typical cass ar: voltag, currnt, rsistanc and capacitanc. AFE (Analog Front End), also namd intrfac. Th aim of this bloc is to convrt th quantity E producd by th snsor into a signal (typically a voltag), suitabl to b asily convrtd into a digital signal. Th AFE may includ svral functions such as convrsion (i.. from rsistanc to voltag), amplification, filtring (to liminat out of band nois and intrfring signals), tmpratur compnsation. For many snsor catgoris, th AFE should also provid th snsors with propr stimuli, ncssary for th convrsion mchanism. In th cas that th snsor rspons is strongly non-linar, it might b convnint to oprat linarization in th analog domain (i.. insid th AFE), although linarization as wll as othr non linar oprations ar prfrably prformd in th digital domain. Analog-to-digital convrtr (ADC). This bloc is ncssary to produc a digital rprsntation of, that w will indicat with D. Clarly, th output of th ADC is a cod (digital numbr), but this numbr rprsnts th input quantity through th ADC transfr charactristics. D is diffrnt from du to th quantization rror and to othr ADC rrors (offst, gain and nonlinarity rrors).. -stimator. This bloc implmnts a numrical algorithm or formula that gts th data from th ADC and calculats th numbr m, consisting in an stimat of. Dpndnt quantitis, such as Z, can b drivd from m and Y m through a propr bloc (Z.stimator) that producs th stimat Z m. implmnting th function f( m,y m ). 3

4 Signals. Information is carrid by signals. Th lattr can b of thr diffrnt typs, as shown in Fig.1.4. Analog signal ar continuous in magnitud, sinc thy can idally assum all th valus within thir rang. Digital signals ar quantizd in magnitud, sinc thy can assum only a finit numbr of valus. As a rsult, also th information carrid by a digital signal is quantizd Analog and digital signals can b furthr dividd into continuous tim and discrt tim signals. A continuous-tim signal is valid for any instant of th intrval of intrst. Discrt tim signals assum valid valus only at instants that can b rprsntd by a progrssion of intgr numbrs. Sinc, in all ral cass, th tim intrval of intrst is finit, thn also th numbr of instants at which th signal is valid is also finit. Digital signals ar always discrt-tim, whil analog signals can b ithr discrt or continuous tim. For xampl, discrt tim analog signal ar prsnt in switchd capacitor circuits. digital signals analog signals discrt tim discrt tim continuous tim Systm Prformanc Fig.1.4 Th prformanc of a DAS is givn by th closnss of th stimats ( m,y m.zm...) with rspct to th corrsponding physical quantitis (,Y,Z...). Mor prcisly, th following rror can b dfind: x (1.1) m This is quivalnt to considr that a complt acquisition channl of a DAS (.g. channl ) can b modld as an idal systm (dfind by th idntity bloc I in Fig. 1.5), with th addition of th disturbanc x at th input. S I x Fig. 1.5 In a systm formd by th cascad of svral blocs, th total rror x is th sum of th contribution of all blocs. Fig. 1.6 shows th bloc diagram of th DAS, with th rrors highlightd. Each bloc is modld as an idal (rrorlss) bloc with an addr that sums up an rror trm at th output. Th numbrs 0, 1, 2, 3, 4 idntify th points whr th signal is considrd. Th numbr 0 idntifis th 4

5 original quantity to b masurd, th numbr 1 th quantity at th output of th snsor and so on. Th trms x1, x2, x3 ar rror trms that ar considrd to b injctd at positions 1, 2, 3, rspctivly. In ordr to calculat th contribution of a singl rror trm to th ovrall rror x (s fig. 1.5) it is ncssary to dfin th snsitivity. If w considr th signal at th i-th position, th snsitivity from th input quantity to that position is givn by: i 0, i (1.2) whr xi is th signal at position i-th. Th snsitivity can b considrd as th small signal gain from th input position to th considrd output position. It is possibl to dfin th snsitivity from whichvr position, not only from position 0. Exampls of snsitivitis ar rprsntd in Fig.1.6 by mans of paths conncting th input to th output position E D S S S x1 snsor AFE ADC stimator m x2 1,2 0,1 0,2 0,3 x3 Fig. 1.6 Each rror contribution can b rfrrd to th input quantity through th corrsponding snsitivity. Considr Fig. 1.7, whr th i-th position is xamind. W suppos that th rror xi is summd up to th signal coming from prvious blocs. 0 position i-th position S i 1 0,3 xi Fig.1.7 5

6 Th corrsponding input quantity, x i is givn by: x xi i (1.3) 0, i In ordr to calculat th total rror it is ncssary to sum up th input rfrrd valus of all rror componnts that ar injctd into th rad-out channl by th various blocs. Not that, if th rspons of th systm is non-linar (for xampl bcaus of snsor non-linarity), th snsitivity is not constant ovr th input quantity intrval. Thrfor, if w considr an rror sourc that is not dpndnt on th input signal (for xampl nois form an amplifir), this rror will rsult in a largr input rfrrd uncrtainty in intrvals of th input quantity whr th snsitivity is smallr. Error on dpndnt quantitis. W hav sn that a dpndnt quantity is not dirctly masurd but its valu is calculatd starting from th stimats of othr quantitis, which, on th contrary, ar masurd (indpndnt quantitis). It is important to calculat how th rrors on th indpndnt quantitis affct th rror on th dpndnt on. Considring, for xampl, Z=f(,Y) and suppos that th function f is implmntd by th Z- stimator with infinit prcision. Thn th Z stimat will b Z m =f( m,y m ). But: Using th first ordr Taylor approximation: Z m f Z m f, Y) x Th rror on Z, qual to ZZ m, is givn by: f, Y ) f ( x, Y y ) (1.4) ( m m f y Y Z Z x Z y Y ( (1.5) Z x Z y Y z (1.6) This xprssion can b asily xtndd to th cas of mor than two indpndnt variabls. Typs of rrors. Thr ar thr typs of rrors that can b distinguishd: 1. Quasi static rrors. Ths rrors can b considrd to b constant during th whol obsrvation priod. Th obsrvation priod is th tim during which th systm is monitord (from svral sconds to svral hours). 2. Dynamic rrors. Ths ar rrors that xist only during transints and ar du to th slownss of th systm. 3. Nois. This trm indicats all ind of unwantd tim varying systms that ar suprimposd to th signal. It includs random nois, which is du to phnomna involving th charg carris (lctrons, hols and othrs) at th microscopic scal, as wll as disturbancs du to xtrnal sourcs, gnrally rfrrd as intrfrnc. 6

7 Quasi static rrors can b dividd into th following catgoris: -) Offst rror -) Gain rror -) Non linarity rror. Th offst rror is rsponsibl for th fact that, whn an input zro quantity is applid, th output is not qual to th convntional zro. Mor prcisly, for a DAS, th input offst io is th valu of quantity that producs a zro stimat ( m =0). Th Gain rror can b asily undrstood if w considr a linar radout systm. If w tas into account position i-th in th radout chain, than, nominally: i nom nom (linar systm) (1.7) whr x-nom is th gain of th systm from th to th i-th position in nominal conditions. Du to an rror on th gain, w hav that th actual gain is: ral nom rr whr -rr is th gain rror trm. With this valu w hav: i ral nom rr Sinc w do not now th actual gain, w hav to stimat th input quantity by using th nominal gain. Thus: i ral rr m nom nom Comparing this quation with Eqn.(1.1), w obtain that th input rfrrd rror du to th gain rror is x gain rr (1.8) W not that: (1) th rror is proportional to th input quantity. Thrfor, th rror du to gain inaccuracy is not an additiv trm, sinc it dpnds on th input signal. (2) th rror gain is a rlativ rror, sinc th ratio -r r/ -non appars in th input rfrrd rror contribution. Mthods for rducing th gain rror: In ordr to obtain prcis gains (and thus small gain rrors) it is important that th gain dpnds only on two typs of contributions: nom 7

8 -) Ratios btwn quantitis that hav th sam dimnsions, rlating to objcts fabricatd with th sam tchnology. Ths ind of dimnsionlss ratios ar affctd only by matching rrors, which can b rducd to lss than 0.1%. In addition, tmpratur variations affct in th sam way th two trms of th ratio, which, as a rsult, rmains unchangd. -) Prcis quantitis (not dimnsionlss). In natur, thr ar quantitis that can b mad vry prcis, such as th output frquncy of a quartz oscillator. Clarly, if th input and output quantity ( and i, rspctivly, in Eq.1.7), hav not th sam dimnsions, th gain cannot involv only dimnsionlss ratios, but should includ at last on nondimnsionlss quantity, which, should b mad as prcis as possibl. Il all cass whr th abov ruls cannot b rspctd, th gain can b affctd by larg rrors. For xampl, in intgratd circuit, if a gain is proportional to th rsistanc of an on-chip rsistor, gain rrors mor than 10 % can b xpctd. In ths cass, it is ncssary to individually trim ach fabricatd dvic. Ratiomtric systms. In ths systms th gain is purposly mad to b proportional to th powr supply voltag dd. At first glanc this appars to b an inaccurat approach, sinc th dd is gnrally providd by powr voltag rgulators that ar not as prcis as voltag rfrncs. This drawbac is compltly ovrcom if w combin th ratiomtric radout channl with an ADC that uss dd as its rfrnc voltag (s Fig 1.8) dd ratiomtric radout channl a DD rf ADC D n D 2 rf Fig. 1.8 Th gain of th ratiomtric systm is givn by a dd, whr a is a constant. Many systms, such as Whatston bridgs usd to rad rsistiv snsors, ar intrinsically ratiomtric. Th output cod of th ADC is indicatd with D, whil n is th ADC rsolution. Substituting th xprssion of into th output cod, w obtain a rsult that is indpndnt of th dd valu. In this way it is possibl to us dd as a voltag rfrnc with no pnalty in trms of accuracy. Non linarity rrors. Th rrors driv from th us of an approximat law to modl th snsor rspons. In th stch of Fig. 1.9 w show th output voltag of a radout channl ( ) as a function of th input quantity. Th ral snsor rspons is rprsntd by th curv (actual curv). In ordr to obtain a prcis stimat, th stimator (s Fig. 1.3) should hav a transfr charactristic qual to th invrs of th actual rspons. If a computational unit is not availabl, a possibl solution is to us a linar approximation, 8

9 which can b asily implmntd with vry simpl logical blocs. Clarly, this introducs an rror which is shown in th figur by th diffrnc btwn th actual ral valu of th input quantity () and its stimat ( m ) obtaind starting from th output voltag valu *. A non linarity rror may still b prsnt also whn mor prcis approximations, such polynomial or xponntial functions ar usd (rsidual non linarity rror). It should also b obsrvd that th nonlinarity of th rspons varis among th diffrnt sampls of a particular systm. Ths rrors will rmain vn if th nominal curv usd to intrprt th snsor rspons is virtually xact. linar approximation * actual curv m Fig. 1.9 Dynamic rrors. Fig (a) shows th rspons of a DAS to a stp variation of th input quantity. Th figur includs also (quasi)-static rrors, which caus th rspons to start from and sttl to a valu which is diffrnt from th actual valu. Not that th stimat dos not rach th final valu immdiatly, but thr is a transint intrval during which th stimat significantly diffrs from th final valu. During this transint tim th diffrnc btwn th stimat and th actual valu can b much largr than th static rror, so that th output of th DAS will b invalid during this tim. An important paramtr is th sttling tim, t st, dfind as th tim ncssary for th stimat m to gt closr than a crtain margin to th final valu. In practic, aftr t st, th diffrnc from th final valu stays within a givn rror band placd across th final valu. A typical rror magnitud usd to dfin th sttl tim is ±1%. Th rason of th systm slownss is du to inrtial lmnts, which, in th lctronic domain, ar mainly capacitatancs and, lss frquntly, inductancs. An important contribution may also driv from th snsor, whr non-lctrical lmnts (mchanical and thrmal masss, diffusion and adsorption phnomna) ar lily to play an important rol. Th rspons spd of a DAS can b xprssd by two paramtrs: -) Frquncy bandwidth, B S -) Slw rat, s r. Th bandwidth rfrs to th linar bhavior of th systm, which gnrally occurs whn th input variations ar small (.g. small magnitud stps) or slow. Th slw rat is th imum valu of th tim drivativ of m. For xampl, in th cas of larg stps, th drivativs is no mor proportional to th stp magnitud, but saturats to a imum valu s r. 9

10 stp P. Bruschi - Nots on Mixd Signal Dsign Thrfor th bandwidth rprsnts th rspons to small signals, th slw rat, to larg signals. In most cass both paramtrs contribut to th sttling tim, sinc th systm is in th slw-rat condition in th initial part of th transint and gts into linar opration in th last part. Th analysis in ths conditions is difficult and strongly dpnds on th systm architctur. It is intrsting to considr th valu of th sttling tim whn only on paramtr dominats. static rror final valu D m m m t st tim (a) (b) Fig.1.10 t st tim For linar rsponss, whn B S dominats, w hav th approximat condition: 1 tst (1.9) BS This xprssion is prcis for th 1 % sttling tim of a systm charactrizd by a scond ordr lowpass Buttrworth rspons. In th cas of first ordr low pass rspons, th t st givn by th prvious quation should b multiplid by If th slw rat dominats, as shown in fig. 1.1(b), w simply hav: t st 0. m D m D (1.10) 99D s r Th important diffrnc hr is that th sttling tim dpnds on th amplitud of th input stp (D) Nois. In lctronics, th trm nois indicats unwantd signals that contaminat th dsird signal. This dfinition includs also capacitiv or magntic intrfrnc from quipmnts placd in th vicinity of th circuit that w ar xamining or disturbancs inducd by incoming lctromagntic wavs. From now on, unlss xplicitly statd, w will us th trm nois only to indicat random signals gnratd by microscopic phnomna occurring insid th sam blocs and dvics that form th DAS. If w considr a constant valu for th input quantity, nois producs random oscillations across th thortical output constant valu. Nois affcts th rsolution of th systm, which is th minimum diffrnc btwn two valus of th input quantity that can b distinguishd. Fig (a) shows a stch of th stimats m producd by th DAS as a function of tim for two diffrnt valus of th input quantity. Th output stimats varis around th man valu (indicatd by th rd lin) for th ffct of nois. Th intrval of possibl valus that can b assumd by th signal around th man valu is calld nois band. Th amplitud of th nois band corrsponds to th pa-to-pa magnitud of 10 s r s r

11 th nois (x np-p ). At this point it should b obsrvd that most cass of random nois ar charactrizd by distributions of valus sprad ovr an infinit intrval (s for xampl Gaussian nois). In practic, it is possibl to considr a finit intrval whr most valus fall, or, mor prcisly, a givn prcntag of valus fall. For a Gaussian distribution, which fits most practical cass with rasonabl prcision, th probabilitis ar shown in Tabl 1.1, whr indicats th standard dviation. Intrval Total intrval width (x np-p ) Probability 1 probability ± (68.3 %) ± (95.4 %) ± (99.7 %) ± ( %) Tabl 1.1 In th following part of this documnt, unlss diffrntly spcifid, w will assum a nois intrval amplitud of 4. Considring Tabl 1.1, this mans that th total signal (idal signal + nois) will spnd 95 % of tim within th nois band. Equivalntly, this mans that, if w sampl th output nois, mor than 95 % of sampls will fall insid th rror band. If w considr two diffrnt valus of th input quantity, as in Fig.1.11(a), sparatd by th diffrnc D, and w suppos that thr ar not gain and non-linarity rrors, th avrag valu of th output stimats ar sparatd just by D. Th nois bands ar shiftd also by D. Th figur shows a situation whr th diffrnc D is so small that th nois bands corrsponding to th two valus of th input quantity ar partially ovrlappd. Th intrsction of th two bands includs valus of th output which ar compatibl with both valus of th input quantity, thrfor, at th instants whn th signal is insid th intrsction rgion, it is not possibl to dcid which on of th two input valus is actually prsnt at th input. In ordr to b abl to distinguish btwn two valus of th input quantity, thr should b no ovrlap btwn th corrsponding nois bands. Th smallr diffrnc that can b distinguishd, i. th rsolution, occurs whn th two nois bands ar adjacnt, as shown in fig.1.11(b). Sinc, in this cas, th two valus of th stimat ar sparatd by two half nois bands, th rsolution is simply givn by th amplitud of th nois band, i.. by x np-p. Considring again how w hav dfind th rror band, if w try to us th systm to distinguish btwn two quantitis which diffrs by just th rsolution, than th answr will b corrct for 95.4 % of all cass. For systm rquiring a lowr rror probability, diffrnt dfinition of th rror band should b adoptd (.g. 6 instad of 4). Clarly, with this diffrnt dfinition, th rsolution of th systm will turn out to b wors (largr minimum D). mas mas x np-p D x np-p D=x np-p tim (a) (b) Fig tim 11

12 Total accuracy of th DAS. In mtrology, accuracy rprsnts th closnss of th rsult of masurmnt with rspct to th actual valu. Strictly spaing, th accuracy is not a numrical valu, but just a quality, which includs svral quantitativ paramtrs usd to dfin th proprty of th rror (prsnc of systmatic componnts, rpatability and rproducibility, varianc of random rrors, tc.). In th practical us, th accuracy is th diffrnc btwn th masurd and actual valus of th input quantity. Gnrally, accuracy dos not includ nois contributions; sinc thy ar zro-man random signal that can b arbitrarily rducd by avraging larg sts of masurmnts. In th cas of a DAS, avraging is quivalnt to apply a low-pass filtr to th signal stram, slowing down th systm. Whn w considr th rrors on th masurmnts, w should assum that th bandwidth has alrady bn rducd to th minimum valu ndd to guarant th rquird systm rspons spd (.g. sttling tim). Thrfor, ach sampl coming out from th masurmnt systm has to b rgardd as a significant sampl and no mor oprations ar allowd. As a consqunc, w will includ also th nois contribution into th dfinition of total accuracy, sinc it has to b intndd as th closnss of ach singl sampl of th output signal (stimats stram, m ), with rspct to th ral valu of th input quantity. Thrfor, th accuracy will b givn by th sum of th imum static rror (absolut valu) and th imum nois rror (absolut valu). As far as nois is considrd, th imum absolut nois valu is th pa valu, i. half th pa-to pa valu. Additiv rrors and dtction limit. In many practical cass, nois and offst do not dpnd on th input signal, thus thy can b considrd as additiv rrors. This proprty is particularly important for dfining th dtction limit of th systm, i.. th capability of dtcting vry small valus of th input quantity. Exampls that show how this paramtr can b important ar rprsntd by th dtctors of harmful gass, which, for crtain substancs such as nitrogn oxids, should b abl to rais a rliabl alarm for concntrations as low as a fw part pr billion. Othr xampls ar givn by flow snsors dsignd to dtct vry small flow rats in pips, in ordr to dtct fluid las. In ordr to find out which paramtrs rally affct th dtction limit, w hav to considr what happns whn a zro input quantity is applid to th systm. From Eq.(1.8) w not that for =0 thr cannot b a gain rror. Also non linarity rrors, occurring at larg valus of th input signal can b nglctd. Thrfor, th only rror sourcs that affct th masurmnt ar offst and nois, i.. th additiv rrors. Fig.1.12 (a) shows th output signal band, whn a zro input signal is applid. Considring Eq. (1.1), th input offst rror ( io ) shifts th masurmnt rsult by io. Nois adds up random oscillations around this valu. All masurmnts valus fall within th nois band, as shown in th figur. 12

13 +( io)+0.5x np-p mas +( ) io io nois band -( ) io total uncrtanty band tim -( io)-0.5x np-p (a) (b) Fig.1.12 Th situation dpictd in Fig (a) rfrs to a singl dvic, whr w can considr th offst as a nown quantity. In many cass th offst of a givn systm cannot b masurd, or, at last, not with sufficint prcision. It should also b considrd that th offst varis for th ffct of tmpratur and dvic aging. As a rsult, in many cass, w hav to considr also th offst as an unnown random quantity to b rprsntd by its statistical proprtis. Th offst is gnrally givn in trms of imum offst, ( io ), which is actually th imum of th absolut valu of th offst. Th possibl offst valus ar gnrally also symmtrical with rspct to zro, so that th rang of possibl offst valu will b [( io ), ( io )]. Sinc th nois band is placd across th offst valus, th total rang of valus that th masurmnt systm may produc whn =0 is shown in Fig. 1.12(b). Thus, th minimum valu of th m masurmnt that can b rliably rfrrd to a non-zro input quantity is givn by (absolut valu): 1 min( m 0) ( io) xnp p (1.11) 2 Not that Eq.(1.1) dos not giv th minimum dtctabl quantity, which, instad should b calculatd considring Fig.1.13, whr th intrval of possibl masurd valus m is givn as a function of th input quantity, in th cas that gain and non-linarity rrors hav bn alrady corrctd. Th lin m = rprsnts th idal cas, whr no nois and no offst ar prsnt. In th ral cas th masurmnts fall in th band indicatd in th figur. 13

14 mas = radings that may b producd with =0 mas minimum rading that rliably mans = 0 dtction limit -( io)-0.5x np-p ( io)+0.5x np-p -( io)-0.5xnp-p Fig.1.13 For an input quantity to b rliably rcognizd as non zro, th intrval of output valus should hav no ovrlap with th intrval of possibl valus producd whn =0. Considring Fig.1.13, it can b asily shown that th absolut valu of th dtction limit should b at last as larg as th complt rror band, i.: dtction limit 2( ) (1.12) io x np p Signal limits and dynamic rang. Th input quantity valus that can b applid to a DAS should stay within a lowr limit ( min ) and an uppr limit ( ) in ordr for th systm prformanc to b maintaind. Th limits ar gnrally du to xcssiv non-linarity occurring whn th signal magnitud bcoms too larg. In particular cass, th limits can b du to th onst of phnomna that can b dstructiv for th systm. Th full scal rang of a systm, indicatd with D FS is givn by th diffrnc: D FS min (1.13) Not that min is not ncssarily a small quantity, sinc it may b a ngativ valu with a larg absolut valu. A typical xampl is givn by symmtrical input rangs, whr min =. Th full scal rang is a masur of th ral xtnsion of th intrval across which th input quantity can vary. An important paramtr that charactrizs th systm prformanc is th dynamic rang (DR). This is a dimnsionlss ratio givn by: D FS DR (1.14) whr is gnrically th smallst quantity that can b dtctd. Not that th dfinition of th DR varis dpnding on th way w dfin. In systms whr th signal bandwidth dos not xtnd 14

15 down to DC, th offst is not rlvant and th minimum dtctabl quantity is affctd only by nois, so that coincids with th systm rsolution. This is also th cas of systms whr offst compnsation is fasibl and rliabl. In th cas that th signal bandwidth includs DC and th offst cannot b ffctivly canclld should b considrd to b qual to th dtction limit calculatd in Eq.(1.12). In common practic th DR is calculatd considring only th nois contributions, so that is th rsolution. An intrsting intrprtation of th DR can b found whn w considr that coincids with th rsolution. If w considr min, thn th closst quantity that can b distinguishd from min is min +. Thn, w hav to add anothr incrmnt to rach th nxt valu that can b distinguishd from min +. Procding in this way, w rach aftr ( - min) / stps, i.. aftr a numbr of stps qual to DR. Thn th DR can b considrd as th imum numbr of diffrnt lvls of th input quantity that can b distinguishd by th DAS. This situation is similar (but not idntical) to what happns in an ADC, which can distinguish only 2 n lvls, whr n is th convrtr rsolution (dfind as numbr of bits). For this rason th DR is somtims xprssd in trms of numbr of bits and, on th othr hand, it should b obsrvd that a possibl way to xprss th rsolution of an ADC is to giv th DR. Normally th DR is xprssd in dcibls. It is important to obsrv that th DR of th whol systm dpnds on th DR of th blocs that compos it. Th nowldg of th DRs of th singl blocs is not sufficint to calculat th ovrall DR. In fact, th ovrall DR dpnds also on th way th output rang of ach bloc matchs th input rang of th following bloc. In ordr to undrstand this, lt us considr Fig. 1.14, whr connction btwn bloc A and nxt bloc B is shown. Th signal at th output of A, which coincids with th signal at th input of bloc B, is indicatd with. W focus on th input rang of bloc B, limitd btwn min and. W considr that ths limits ar du only to bloc B, thus strictr rang boundaris can aris from th blocs that follow B. Thn w hav indicatd with min and th limits on th input quantity that driv from all blocs of th DAS (including also bloc B). Whn varis from min to, voltag swings from ( min ) and ( ). Figur 1.14 rprsnts also th lvls, into which th input rang of B is idally dividd. As w hav shown arlir, th lvls driv from th rsolution (i.. nois ) of bloc B. In th cas dpictd in th figur, whn swings across its input rang, voltag xplors only a fraction of th input rang of bloc B. Thn, th actual numbr of lvls that ar involvd is smallr than all thos that can b providd by bloc B. This is a cas that can potntially rsult in dynamic rang dgradation, spcially if th rsolution of B is rlativly low. In ordr to xploit all th lvls of bloc B, an amplifir could b placd btwn A and B, providd that this additional bloc dos not introduc furthr rang limitation or significant nois. A B output swing of bloc A, whn varis across its full scal rang ( ) ( min ) min input rang and lvls of bloc B Fig

16 If th systm is linar, it is possibl to dmonstrat that th DR of th whol systm cannot b largr than th DR of ach individual bloc. Lt us considr bloc B, and dfin its dynamic rang as: min DR B (1.15) whr is th imum rror introducd by B (nois, or nois + offst) rfrrd to th input of bloc B. Clarly, sinc th systm is linar, w can writ: =, whr is a constant corrsponding to th snsitivity of vs.. For simplicity, w will assum that is positiv; this is not a rstriction sinc th procdur can b asily rpatd for < 0. Th rror, rfrrd to th input quantity will b givn by /. Sinc th rror introducd by bloc B is only on componnt of th total rror on, w can writ: (1.16) Lt us now considr th limits of th rang. Clarly, for all valus blonging to th ovrall rang (intrval [ min and ]) bloc B should oprat corrctly, thn should b within th input rang of bloc B. Thrfor: min min min min min min (1.17) Thn w can oprat th following substitution in 1.15: Considring Eq.(1.16) and (1.17), this incrass th dnominator and dcrass th numrator of th DR B xprssion, dfind in Eq.(1.15). Thn: DR B min min min min min DR (1.18) whr DR indicats th ovrall dynamic rang of th systm. This mans that that DR of all blocs is constraind to b largr than or, at last qual to th targt DR of th whol systm. 16