Electronic noise in a constant voltage anemometer

Transcription

1 REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 75 NUMBER 5 MAY 2004 Elctronic nois in a constant voltag anmomtr Julin Wiss a) Institut für Arodynamik und Gasdynamik Univrsität Stuttgart Stuttgart Grmany Gnvièv Comt-Bllot b) Cntr Acoustiqu LMFA Ecol Cntral d Lyon Ecully Franc Rcivd 24 Sptmbr 2003; accptd 28 January 2004; publishd 26 April 2004 Th lctronic nois and th signal-to-nois ratio in a constant voltag anmomtr CVA ar analyzd in trms of th main constitutiv lmnts of th circuit. It is shown that th output voltag du to lctronic nois dcrass with th wir rsistanc prmitting on to know th nois uppr limit by using th rsults of th unhatd wir. Th nois powr spctrum incrass at high frquncis as f 2 lik in othr anmomtrs bcaus of th nd to compnsat for th thrmal lag of th hot wir thus lading to a rducd signal-to-nois ratio at high frquncis. Explicit formulas ar givn in trms of wir CVA and flow quantitis. Masurmnts of lctronic nois in a CVA prototyp confirm th thortical analysis and illustrat som intrsting issus concrning masurmnts of nois and low lvls of flow fluctuations Amrican Institut of Physics. DOI: / I. INTRODUCTION a Prsnt addrss: Dpartmnt of Mchanical Enginring and Matrials Scinc Duk Univrsity Durham NC 27708; lctronic mail: julin.wiss@duk.du b Elctronic mail: gnviv.comt-bllot@c-lyon.fr Th constant voltag anmomtr CVA is a rcnt typ of hot-wir anmomtr dvlopd by Sarma. 12 It prsnts svral valuabl attributs for flow fluctuation masurmnts: Easy mod of opration high bandwidth stability and insignificant cabl capacitanc ffcts. 23 In particular th CVA has bn shown to b spcially advantagous for masurmnts of high-spd transitional and turbulnt boundary layrs. 3 5 Rcnt dvlopmnts includ a procdur for automatic stpping of ovrhat for masurmnt in short duration wind tunnls. 6 For vry masurmnt tchniqu th signal-to-nois ratio is a vry important paramtr sinc it dtrmins th lowst signal that can b rsolvd. In analog systms lik hotwir anmomtrs th major sourcs of nois consist of th lctronic nois gnratd by th oprational amplifir and th thrmal nois associatd with th rsistiv componnts in th circuit. A thortical analysis of th signal-to-nois ratio in th constant tmpratur anmomtr CTA and in th constant currnt anmomtr CCA has bn prformd by Frymuth. 78 In both cass th hot wir was includd in a Whatston bridg. Th lctronic nois was found to lad to a svr dcras of th signal-to-nois ratio at high frquncis bcaus of th nd to compnsat for th thrmal inrtia of th hot wir. In particular Frymuth showd that at high frquncy th powr spctrum of lctronic nois riss lik f 2 f is th frquncy for both a CTA and CCA so that thr is always a frquncy abov which th nois dominats. This frquncy can in fact limit th usabl bandwidth in th masurmnt of low turbulnc lvls as th turbulnc spctra fall off rapidly with th frquncy according to f 1 or vn f 5/3 laws Bstion t al. 12 compard th nois prformanc of a CCA and CTA in high-spd flows and arrivd at th conclusion that th two systms ar basically quivalnt thus corroborating th thortical work of Frymuth. 78 Earlir rfrncs on th CVA considrd th lctronic nois only incidntally but showd that this nw typ of anmomtr prsnts a lowr lvl of nois whn compard to othr systms. 235 Howvr ths rsults somtims xhibitd diffrncs whn nois spctra wr compard. Thrfor th goal of th prsnt articl is to giv a thorough analysis of th lctronic nois and th signal-to-nois ratio in th CVA. 13 Th articl is organizd as follows: In Sc. II th CVA output voltag du to lctronic nois is stablishd with th hlp of circuit quations drivd by Sarma. 2 Th signal-tonois ratio is thn obtaind by comparing th anmomtr rspons to flow fluctuations with th lctronic nois. In Sc. III xprimntal data confirming th thortical rsults ar prsntd. Finally th prsnt rsults as wll as spcific issus concrning th masurmnt of lctronic nois and low turbulnt signals ar discussd in Sc. IV. II. THEORETICAL ANALYSIS A. Constant voltag anmomtr circuit A schmatic diagram of th CVA is availabl in Sarma. 2 Lik in typical op-amp circuits th nois voltag at th circuit output is du to th thrmal nois which occurs across th rsistors composing th circuit as wll as th lctronic nois in th op-amp itslf Th output nois voltag can thus b calculatd by modling all rsistors by a prfct noislss rsistanc in sris with a fluctuating voltag sourc and by modling th op-amp nois with a finit numbr of nois sourcs. Figur 1 shows th circuit obtaind whn this mthod is applid to th CVA circuit of Sarma. 2 Nois voltag sourcs /2004/75(5)/1290/7/$ Amrican Institut of Physics

2 Rv. Sci. Instrum. Vol. 75 No. 5 May 2004 Constant voltag anmomtr nois 1291 FIG. 1. CVA modl including Nyquist Johnson nois sourcs. ar includd in th original circuit to modl th thrmal nois producd by th rsistors. Ractiv lmnts ar assumd noislss. Th lctronic nois in th op-amp is modld by th noninvrting input nois voltag n and th invrting input nois currnt i n. This is a typical way of modling th nois in amplifir circuits vn if a larg numbr of opamps ar usd to build a complx amplifir ntwork In addition svral rmarks must b considrd: 1 Th tim constant of th wir in th CVA is compnsatd by a rsistor capacitor RC T ntwork. To simplify th analysis this T ntwork is transformd to its quivalnt ntwork. 2 2 Th rsistancs R a and R b of th original circuit ar rnamd rspctivly R 2a and R 2b in Fig According to Sarma 2 R 2 Z b and Z c ar rlatd to R 2a and R 2b by R 2 R 2a R 2b R 2a 1 1 T c s Z b R 2 1T c s Z c R 2b 1 1 T c s T c CR 2aR 2b R 2 s is th Laplac variabl and T c is th tim constant compnsation includd in th CVA. 4 Th rsistanc R d which is in paralll with R 2 in th original circuit introducs a damping trm that limits th gain at vry high frquncy. Sinc R d is much largr than R 2 its thrmal nois will b nglctd. Typical valus of th diffrnt rsistors involvd in th circuit ar givn by Sarma: 2 R R and R w 5. Th diffrnt nois sourcs in Fig. 1 ar rlatd to th rsistanc and tmpratur of th corrsponding rsistors by th Nyquist Johnson xprssions: R1 2 4kT 0 R 1 f RF 2 4kT 0 f R2a 2 4kT 0 R 2a f R2b 2 4kT 0 R 2b f R2 2 4kT 0 R 2 f and Rw 2 4kT w R w f. In ths rlations k J/K is Boltzmann s constant T 0 is th room tmpratur T w is th hot-wir tmpratur and f is a frquncy intrval. It is assumd that th nois in ach componnt is indpndnt of th frquncy whit nois. As discussd by Frymuth 7 th usual incras of nois in lctrical componnts at low frquncy pink nois is not of primary importanc for hot-wir anmomtry. It should b notd that th modl in Fig. 1 dos not show th totality of th CVA circuit which is propritary of Tao Systms Inc. 12 Nvrthlss th authors bliv that th prsnt simplifid modl is sufficint to undrstand th ffct of lctronic nois and its consqunc on th signal-to-nois ratio of th anmomtr. As will b shown in a furthr sction xprimntal data support th prsnt approach vry wll. B. Govrning quations Th following analysis is prformd in th frquncy domain. Lowr cas variabls writtn with a prim dnot small fluctuating quantitis and th sam variabls without a prim dnot th spctral dnsity of th corrsponding quantity. For xampl if ( f ) is a small voltag fluctuation thn ( f ) is th corrsponding spctral dnsity in V/Hz 0.5. Th govrning quations of th CVA circuit dscribd in Fig. 1 ar th following: 1 Ohm s law btwn A and B: n R1 R 1 I 1 V Ohm s law btwn B and C: n V w I F RF 3 Kirchoff s law at B: I F i n I 1 4 Ohm s law btwn C and D: V w R w I w Rw 5 Ohm s law btwn C and E: V w I a R2a 5 6 Ohm s law btwn S and C: V s V w Z b I 2 R2 7 Kirchoff s law at C: I 2 I a I w I F. C. Elctronic nois in th constant voltag anmomtr as Using Eqs. 1 3 th wir voltag V w can b writtn V w V R 1 n 1 n R 1 1 n R R1 i n RF. 9 1 Equation 8 shows that th wir voltag V w is hld constant by th circuit xcpt for small voltag fluctuation n du to lctric nois in th diffrnt componnts. 8

3 1292 Rv. Sci. Instrum. Vol. 75 No. 5 May 2004 J. Wiss and G. Comt-Bllot An xprssion of th output voltag V s as a function of V w can b obtaind by combining Eqs. 2 and 4 7. This givs V s 1 Z b Z b Z b R w V w Z b n R2a Rw RF R w R2. 10 W now prform a small prturbation analysis of Eq. 10 baring in mind that th diffrnt nois sourcs can alrady by considrd as small voltag prturbations. This yilds s 1 Z b Z b Z b n R2a R w w Z b Rw RF R w I R w r w Z b w R2. 11 This xprssion of th fluctuating output voltag s can b sn as th noisy countrpart of Eq. 3 in Rf. 2. For frquncis lowr than th cutoff of th anmomtr w can assum that V w is hld constant xcpt for th fluctuating nois voltag n of Eq. 8. Insrting w n in Eq. 11 yilds s s t s n 12 D 1 Z b Z b Z b R w E Z b R w F Z b. Th quantity ( s ) n rprsnts th nois voltag at th CVA output assuming that th op-amp works idally up to an infinit frquncy. In rality at high frquncy th circuit gain will dcras bcaus of th finit bandwidth of th systm. This dcras in gain will affct th output voltag fluctuation ( s ) n du to th nois as wll as th output voltag fluctuation ( s ) t du to fluctuations in th flow. Hnc in a ral systm a damping trm must b addd to Eqs. 13 and 15 to account for th dcras in gain. According to Sarma 2 a CVA bhavs lik a scondordr systm with a natural frquncy f n and a damping ratio. f n and dpnd on th gain bandwidth product of th op-amp as wll as on R 2 R w T c and R d. In addition th CVA output is filtrd by a fourth-ordr Buttrworh filtr to rduc aliasing whn digitizing th output signals. Th cutoff frquncy f Bw of this filtr is highr than th natural frquncy f n of th CVA circuit itslf. As a consqunc th damping trm which nds to b addd to Eqs. 13 and 15 may b writtn as s t Z b R w I w r w 13 is th fluctuating output voltag du to flow fluctuations and s n 1 Z b Z b Z b R w R 1 1 n R R1 i n RF 1 Z b n R2a Rw RF R w R2 14 is th fluctuating output voltag causd by lctronic nois in th componnts of th circuit. Rordring th diffrnt sourcs of nois th xprssion of ( s ) n can b writtn in th form: s n A n B R1 C RF D i n E Rw F R2a R2 A 1 R 11 Z b Z b Z b R w Z b B R 1 1 Z b Z b Z b R w C 1 Z b Z b R w 15 B 4 f Bw s Ds s/2 f n 2 2s/2 f n 1 16 B (4) f (s) is th transfr function of th Buttrworth Bw filtr. D. Simplifid xprssion of s n Equation 15 togthr with th damping trm dfind by Eq. 16 is an xprssion of th fluctuating output voltag of th CVA causd by th diffrnt sourcs of nois in th circuit. Although rlatd quantitis lik th root man squar rms or th powr spctral dnsity PSD of ( s ) n can b asily computd from Eq. 15 on a prsonal computr it is hlpful to simplify this xprssion to gain mor insight into th paramtrs that mostly affct th signal-to-nois ratio. Th first stp consists of noticing that is largr than both R w and R 2a so that it is possibl to nglct th trms in Z b / in Eq. 15. Th xprssion of ( s ) n thn bcoms s n 1 Z b Z b R w n Z b R Rw Z b w Z R2a a R2. 17 In th scond stp th last thr trms on th right-hand sid rhs of Eq. 17 ar found to b ngligibl compard to th first on so that s n 1 Z b Z b R w n. 18 Equation 18 is an approximation of ( s ) n that is much mor tractabl than th original xprssion 15. Th trm n

4 Rv. Sci. Instrum. Vol. 75 No. 5 May 2004 Constant voltag anmomtr nois ms. Th amplifir nois charactristics wr assumd to b typical for rcnt low-nois high-prformanc opamps: n 1.0 nv/hz 0.5 and i n 0.8 pa/hz 0.5. According to Eq. 21 ths valus yild a whit nois spctral dnsity of n 2.7 nv/hz 0.5. It can b sn in Fig. 2 that th two curvs ar vry clos to anothr justifying th simplifications mad abov. On advantag of Eq. 22 is that it shows wll th functional rlation btwn th nois rms and R w. In particular plotting R w ( s ) n rms should giv a straight lin th slop and intrcpt of which only dpnd on CVA paramtrs lik R 2 R 2a T c th bandwidth of intrst and only on nois-rlatd quantity. This provids a convnint way to chck th validity of th prsnt thory against xprimntal data as will b don in a furthr sction. FIG. 2. Thortical rms of CVA nois voltag in th bandwidth khz. dfind by Eq. 9 is a whit nois sourc that dpnds only on th rsistors and R 1 and on th nois modl of th op-amp. Whn f 1/(2T c ) on can assum that R 2a and Z b 2R 2 T c jf so that Eq. 18 can b furthr simplifid to giv s n 2R 2 T c jf 1 R 2a 1 R w n. 19 Thus th PSD of th nois signal riss lik f 2 at high frquncy lik in th CCA and th CTA. 710 In addition it is intrsting to notic that th nois signal dos not dirctly dpnd on V w. E. Estimation of th nois root-man squar Sinc th most nrgtic part of th nois spctrum will b prsnt at high frquncy th nois rms can b obtaind by squaring Eq. 19 and intgrating it btwn two frquncis f 1 and f 2 that ar lowr than th natural frquncy f n. This yilds s n 2 2 f 24 2 R 2 2 T 2 c f f 1 R 2a R w 2 n df 2 R 1 2 R 1 20 Th spctral dnsity n of th input whit nois can b obtaind from Eq. 9 by assuming that th diffrnt nois sourcs ar uncorrlatd: 2 n 1 2 n 2 R1 R 2 F i 2 n 2 RF. 21 Using th notation ( s ) n rms ((s ) n ) 2 on can finally writ s n rms 2R 2 T c 1 R 2a 1 R w f f 1 3 n. 22 A plot of th nois rms against th wir rsistanc R w is prsntd in Fig. 2. Th solid lin in Fig. 2 was obtaind by intgrating th original xprssion 15 as th dottd lin was obtaind using th simplifid xprssion 22. Th PSD was intgratd btwn f 1 1 khz and f khz and typical valus wr usd for th rsistors in th CVA: R R 2a 50 R and T c F. Signal-to-nois ratio Th signal-to-nois ratio in th CVA can b obtaind by comparing th output voltag ( s ) t that is causd by fluctuations in th flow to th output voltag ( s ) n du to lctronic nois. According to Comt-Bllot 16 th voltag fluctuation ( s ) t causd by a vlocity fluctuation u at frquncy f can b writtn as s t 12 j ft c a w BU 12 j fm CVA 212a w ABU u U V s 23 U is th man vlocity and a w is th corrsponding wir ovrhat ratio A and B ar th constants dpnding on th wir in th classic King s form of th hat xchang law of th wir R w I 2 w /(R w R a )ABU. In practic lctronic nois is an important issu only for frquncis highr than th wir natural frquncy 1/(2M CVA ) which is usually clos to th frquncy 1/(2T c ) of th compnsation ntwork and lowr than th natural frquncy f n. Thus th first factor in th rhs of Eq. 23 may b simplifid by T c /M CVA. In addition th man output voltag V s (1R 2 / R 2 /R w )V w may b approachd by V s V w R 2 /R w. Using ths simplifications Eq. 23 can b writtn as s t T c a w V w R 2 BU M CVA 212a w R w ABU u U. 24 At th sam frquncy th amplitud ( s ) n causd by th whit nois voltag fluctuation n in th circuit can b obtaind from Eq. 19: s n 2 fr 2 T c 1 R 2a 1 R w n. 25 Dividing Eqs. 24 and 25 and using th spctral dnsitis u and n as xplaind at th bginning of Sc. II B yild an xprssion of th signal-to-nois ratio:

5 1294 Rv. Sci. Instrum. Vol. 75 No. 5 May 2004 J. Wiss and G. Comt-Bllot (SNR CVA s t s n 1 R 2a a w BU 2 fm CVA R 2a R w 212a w ABU V w n u U. 26 For high-spd flow BU/(ABU)1 and xprssion 26 may b approachd by 1 R 2a a w V w u (SNR CVA 2 fm CVA R 2a R w 212a w n U. 27 Equation 27 xprsss th signal-to-nois ratio in th CVA at a frquncy f in trms of xplicit quantitis that ar usually known to th xprimntalist. In particular it shows that th signal-to-nois ratio dcrass if th frquncy incrass if th nois lvl in th circuit incrass or if th lvl of turbulnc dcrass. For low turbulnt signals th maximum usabl frquncy may for xampl b calculatd from Eq. 27 by putting (SNR CVA 1 and solving for f. For th M CVA valu a usful rlation has bn stablishd by Comt-Bllot and Sarma: 3 M CVA 1a w d 2 w c w a w 4k a ABR d d is th wir diamtr w and c w th dnsity and spcific hat of th wir matrial k a is th thrmal conductivity of th ambiant fluid at tmpratur T a R d is th wir Rynolds numbr and A and B th constants in th gnral nondimnsional hat transfr law of th wir: Nu d A BR d A0 at high vlocity and B0.56. Taking Eq. 28 into account in Eq. 27 it can b shown that th signal-to-nois ratio can b improvd by oprating th wir at a high ovrhat ratio and using a thin wir to minimiz thrmal inrtia. FIG. 3. Exprimntal rms of CVA nois in th bandwidth khz. 670 khz. Th lctronic nois in th circuit was obtaind by dirctly masuring th fluctuating output voltag of th CVA. Whn rsistors wr usd this mthod of opration is obvious. Whn ral hot wirs wr usd car was takn to put th wir in vry still air in ordr to limit th contribution of th output signal du to flow fluctuations. Nvrthlss natural convction ovr th hot wir was unavoidabl but contributd to th signal at a frquncy much lowr than th frquncy of intrst in th prsnt masurmnts. B. Nois root-man squar Figur 3 prsnts a plot of th masurd valu of th nois rms ( s ) rms n. In ordr to compar ths data with th simpl xprssion 22 th rms was computd within 1 khz and 400 khz to nsur that only frquncis lowr than th natural frquncy f n wr takn into account. For th data obtaind with th rsistors it was chckd that th nois rms dos not dpnd on V w in accordanc with th thory. For th hot wirs a chang of V w rsultd in a chang of R w and a corrsponding chang of th nois rms. It can b sn in III. EXPERIMENTAL RESULTS A. Exprimntal procdur Masurmnts of lctronic nois wr prformd on th CVA prototyp CV01 manufacturd by Tao Systms Inc. Williamsburg VA To covr a wid rang of R w standard rsistors as wll as two hot wirs wr usd. Th hot wirs wr standard tungstn wirs rspctivly 2.5 m and 5 m in diamtr that wr spot wldd to th prongs of a commrcial prob Dantc. Most of th data wr acquird with a Txtronics ight-bit digital oscilloscop at 2.5 MHz and 5 MHz sampling frquncy. For spcial masurmnts that will b dtaild blow a HP35665A digital oscilloscop with 16- bit rsolution and a sampling frquncy of 262 khz was usd. Th tim constant T c of th RC ntwork was st to T c ms a valu usd in most of th CVA xprimnts in suprsonic flows. 35 With thos sttings th natural frquncy of th CVA circuit was f n 470 khz and th cutoff Fig. 3 that th output voltag du to lctronic nois dcrass with th wir rsistanc as prdictd by th thory. Whn th snsor is rmovd from th circuit (R w ) ( s ) rms n gos down to th lowr limit 1.2 mv. Th trnd of th xprimntal data is vry similar to th thortical curvs rportd in Fig. 2 and shows that on can advantagously stimat an uppr limit of th lctronic nois by using th rsults of th unhatd wir. In ordr to prform a mor quantitativ comparison with th thory Fig. 4 prsnts a plot of ( s ) rms n tims R w. It can b sn in Fig. 4 that th masurd data can wll b fittd to a straight lin indpndnt of what typ of snsor is usd. This is consistnt with Eq. 22. Idntifying th paramtrs of th linar fit with Eq. 22 and baring in mind that th quantitis R and T c ms ar known on obtains a valu of R 2a 20 and a whit nois lvl of n 2.6 nv/hz 0.5. This valu of n is similar to th stimat prsntd in Sc. II E and th prsnt masurmnts thrfor frquncy of th Buttrworth filtr was st to f Bw agr vry wll with th thory drivd abov.

6 Rv. Sci. Instrum. Vol. 75 No. 5 May 2004 Constant voltag anmomtr nois 1295 FIG. 4. Product of nois rms bandwidth: khz with wir rsistanc. C. Nois spctra Figur 5 shows a comparison btwn th PSD of th output nois calculatd using th proposd thory and th masurd data for two valus of R w. Th data corrsponding to R w 7.7 wr obtaind with th 5 m wir and a sampling rat of f s 5 MHz. For R w 39 a standard rsistor was usd and th sampling rat was f s 2.5 MHz. Th thortical PSDs wr calculatd using xprssion 18 taking into account th damping trm and th Buttrworth rspons Eq. 16. Th valus R 2a 20 and n 2.6 nv/hz 0.5 obtaind in last sction wr usd. It can b sn in Fig. 5 that th thortical curvs ar flat for f 1/(2T c ) and ris lik f 2 whn f 1/(2T c ). On th othr hand th xprimntal spctra show this f 2 ris only at high frquncis. For frquncis lowr than 40 khz th xprimntal curv corrsponding to R w 39 is much highr than th thortical curv and actually dcrass with th frquncy for f 10 khz. Th sam phnomnon appars for R w 7.7 but th f 2 trnd is prsnt in a largr bandwidth. Nvrthlss th calculatd PSDs ar vry clos to FIG. 6. Comparison btwn thortical and masurd PSD using Txtronics and HP35665A digital oscilloscops; solid lin: HP35665A dashd lin: Txtronics smooth solid lin: Thory. th xprimntal curvs in th largst and most nrgtic part of th spctrum. Th mismatch btwn thory and xprimnts may b rlatd to data acquisition issus. It can b sn in Fig. 5 that th PSD lvl at vry high and low frquncis is actually highr than th thortical curvs. At high frquncis th xprimntal spctra sm to rach a platau for f 1 MHz and thir shap is vry irrgular. This phnomnon is du to th inability of th digital oscilloscop to rsolv th low lvl of fluctuations prsnt at ths frquncis. In fact an xamination of Fig. 5 rvals that a PSD lvl of V 2 /Hz is th lowst signal that can b rsolvd by th systm. Similarly th low PSD lvl at low frquncy cannot b rsolvd. In addition it is intrsting to notic th prsnc of paks in th xprimntal spctra at high frquncy ( f 2 5 MHz). Ths paks can influnc th masurd spctra at low frquncy through th phnomnon of aliasing and thus artificially incras th PSD lvl at low frquncy. To illustrat this phnomnon masurmnts wr prformd with th HP35665A digital oscilloscop in addition to th standard data obtaind with th Txtronics. Th sampling frquncy of th HP35665A is limitd to f s 262 khz but a vry sharp antialiasing filtr is includd in th circuit 72 db/oct. In addition th HP35665A is a 16-bit systm in comparison to th ight bits of th Txtronics. A comparison of th spctra obtaind with th two acquisition systms is shown in Fig. 6. Th masurmnts wr prformd with a rsistor of R w 15. Th solid lin in Fig. 6 shows th data obtaind with th HP35665A as th dottd lin shows th data masurd with th Txtronics. In addition th thortical PSD is also shown smooth solid lin. It can b sn in Fig. 6 that th PSD masurd with th HP35665A fits th thortical curv for almost a dcad longr than with th Txtronics. IV. DISCUSSION FIG. 5. Comparison btwn thortical and masurd PSD; solid lin: A thory was proposd to comput th signal-to-nois Thory dashd lin: Exprimnts. ratio in th CVA. Th thory is basd on th comparison of

7 1296 Rv. Sci. Instrum. Vol. 75 No. 5 May 2004 J. Wiss and G. Comt-Bllot output voltag causd by fluctuations in th flow and output voltag du to th lctronic nois coming from th op-amp and th rsistancs in th circuit. Masurmnts of lctronic nois in a CVA prototyp confirmd th proposd thory in a broad rang of frquncis. An xamination of Eq. 27 rvals that th signal-tonois ratio can b improvd by oprating th wir at a high ovrhat and using a thin wir to minimiz thrmal inrtia. In addition th signal-to-nois ratio is larg at low frquncis and whn th lvl of fluctuations in th flow is high. In fact ths rquirmnts ar similar to thos proposd by Fingrson 17 for othr typs of anmomtrs. Anothr vry important paramtr that drivs th signal-to-nois ratio is th ovrall nois lvl n prsnt in th lctronic circuitry givn by Eq. 9. This lvl should b minimizd as much as possibl to improv th signal-to-nois ratio. Som CVA charactristics also hav to b takn into account as can b sn in Eq. 22. Finally it should b notd that xprimntal masurmnts of lctronic nois in hot-wir anmomtrs ar not asy in trms of a data acquisition tchniqu. Although masurmnts of nois may not b considrd to b vry xciting for th fluid dynamicist th sam typ of issus may appar for masurmnts of vry low flow fluctuations for xampl in th fr stram of high-spd wind tunnls. 59 Thrfor th particular point of aliasing will now b discussd. On of th advantags of CVA is its larg bandwidth. As a consqunc a high sampling rat is ndd to masur th fluctuations of its output voltag without aliasing. Of cours th sam typ of rquirmnt is ncssary for standard flow fluctuations but in th cas of nois th nrgtic part of th spctrum is prsnt at high frquncy and its spctrum incrass lik f 2. On th contrary th spctrum of turbulnt flows dcrass with th frquncy. Thrfor aliasing is a mor critical issu for nois masurmnts sinc highfrquncy fluctuations abov th Nyquist frquncy may b foldd back and altr significantly th low-frquncy part of th spctrum. In addition th rsolution of digital data acquisition systms usually dcrass at a vry high frquncy so that th masurd PSD lvl nar th Nyquist frquncy may b highr than th physical nois spctrum. Ths two intrrlatd phnomna ar blivd to b th caus of th diffrnc btwn th nois spctra masurd with two diffrnt data acquisition systms in Fig. 6. To illustrat th issu of aliasing spctra of lctronic nois wr computd using th thory prsntd abov. In addition th ffct of aliasing was simulatd by adding th contribution of th PSD prsnt btwn th sampling frquncy f s and th Nyquist frquncy f s /2. Th rsults ar givn in Fig. 7. Th spctrum corrsponding to a sampling frquncy of f s 2.5 MHz follows th thortical curv almost up to th Nyquist frquncy so that no practical ffct of aliasing is xpctd at this sampling rat. For f s 1 MHz howvr th simulatd spctrum is havily distortd by aliasing: In particular th low-frquncy part of th spctrum cannot b rsolvd and th slop of th spctrum is lowr than th thortical f 2 slop for frquncis lowr that 100 khz. Finally for f s 0.5 MHz th simulatd spctrum is compltly in rror. FIG. 7. Effct of aliasing on lctronic nois masurmnt. Th authors bliv that th ffct of aliasing combind with th usually poor rsolution of digital oscilloscops at a high frquncy ar rsponsibl for th diffrncs in nois spctra prsntd in arlir rfrncs. 35 In addition ths rsults show th nd to us a vry sharp filtr whn flow fluctuations ar of intrst in a limitd bandwidth only. ACKNOWLEDGMENTS Th authors xprss thir many thanks to Dr. Siva Mangalam and Dr. Garimlla R. Sarma for prmitting thm to tst a CVA prototyp. Discussions with Profssor Gorgs Asch and Dr. Jan-Christoph Béra from th Univrsity of Lyon hav bn vry much apprciatd. On of th authors J. W. acknowldgs th support of Profssor Sigfrid Wagnr and Dr. Hlmut Knauss from IAG and th financial support of th Grman Rsarch Foundation DFG SFB259 Stuttgart Univrsity. 1 G. R. Sarma U.S. Patnt No G. R. Sarma Rv. Sci. Instrum G. Comt-Bllot and G. R. Sarma AIAA J J. T. Lachowicz N. Chokani and S. P. Wilkinson AIAA J J. Wiss H. Knauss S. Wagnr N. Chokani G. Comt-Bllot and A. D. Kosinov Procdings of th 41st AIAA Arospac Scincs Mting and Exhibit AIAA 2003 papr J. Norris and N. Chokani AIAA J P. Frymuth Rv. Sci. Instrum P. Frymuth J. Phys. E S. G. Saddoughi and S. V. Vravalli Mas. Sci. Tchnol P. Frymuth and L. M. Fingrson Mas. Sci. Tchnol J. Wiss H. Knauss and S. Wagnr Rv. Sci. Instrum D. Bstion J. Gaviglio and J. P. Bonnt Rv. Sci. Instrum Th CVA has bn shown to b far lss suscptibl to lctromagntic nois than othr typs of anmomtrs J. T. Lachowicz N. Chokani and S. P. Wilkinson AIAA J This typ of nois will not b considrd in th prsnt articl. 14 G. Asch t al. Acquisition d donnés. Du captur à l ordinatur Dunod Paris M. Stffs Burr-Brown Application Bulltin AB G. Comt-Bllot in Handbook of Fluid Dynamics ditd by R. W. Johnson CRC Prss Boca Raton 1998 Chap L. M. Fingrson Rv. Sci. Instrum