UNIVERSITY OF TRENTO MEASUREMENTS OF TRANSIENT PHENOMENA WITH DIGITAL OSCILLOSCOPES. Antonio Moschitta, Fabrizio Stefani, Dario Petri.
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1 UNIVERSIY OF RENO DEPARMEN OF INFORMAION AND COMMUNICAION ECHNOLOGY 385 Povo reno Ialy Via Sommarive 4 hp:// MEASUREMENS OF RANSIEN PHENOMENA WIH DIGIAL OSCILLOSCOPES Anonio Moschia Fabrizio Seani Dario Peri June 4 echnical Repor # DI-4-55
2 .
3 Measuremens o ransien Phenomena Wih Digial Oscilloscopes Anonio Moschia Fabrizio Seani Dario Peri Universià degli Sudi di Perugia Diparimeno di Ingegneria Eleronica e dell Inormazione Phone: Fax: moschia@diei.unipg.i Universià degli Sudi di reno Diparimeno di Inormaica e elecomunicazioni Phone: Fax: peri@di.unin.i Absrac In his paper he eecs o sampling upon rise ime measuremens wih a digial oscilloscope are considered. In paricular he use o linear inerpolaion or esimaing signal rise imes is discussed and is eecs are analyzed or various sep signals. A simple expression is derived which accuraely models he sampling and linear inerpolaion conribuions o he overall rise ime measuremen error. Using hese resuls a correcion ormula is proposed and is applicabiliy is discussed. I. INRODUCION Digial Sorage Oscilloscopes DSOs are nowadays widely used in many measuremen processes oen replacing analog ones. Such devices aer condiioning and A/D convering he inpu signal perorm mos measuring processing ino he digial domain hus oering high and reproducible perormances. In order o evaluae he uncerainy associaed o measuremens o wideband signals or ransien phenomena i is very imporan o characerize he dynamic perormance o boh analog and digial oscilloscopes. o his aim i has been shown ha he knowledge o analog circuiry bandwidh and he use o some empirical rules usually applied o esimae is eec on a ransiion ime measuremen are no ully jusiied []. Moreover when a DSO is employed he hreshold crossing imes o a signal are oen esimaed by means o linear inerpolaion. hus measuremens o ransien phenomena are aeced by a urher source o error which o he bes o he knowledge o he auhors has no been adequaely addressed ye. In his paper he eecs o sampling on he rise ime measuremens are analyzed or various sep signals. In his regard i should be noiced ha he sampled signal is no he DSO inpu signal bu he oupu o he analog circuiry o he insrumen [][][3]. Firs simple sep signal analyical models are considered. hen more realisic waveorms are achieved as he oupu o a second-order sysem ed by simple sep signals []. In boh cases he error inroduced on he rise ime measuremen by sampling and linear inerpolaion is evaluaed by means o simulaions. A ormula is hen derived and is useulness in compensaing measuremen error or a wide class o inpu signals is discussed.
4 Linear Sep Exponenial Sep Gaussian Sep.. ε / Rm s Fig. : Relaive error ε due o sampling and linear inerpolaion on rise ime measuremens II. EFFEC OF SAMPLING AND LINEAR INERPOLAION he rise ime o a sep signal is usually deined as he dierence beween he insans and 9% in which he signal crosses wo hreshold values convenionally assumed as and 9% o he sep ampliude respecively. A DSO measures he rising ime o a sep signal by evaluaing he dierence beween he esimaed values or and 9%. o his aim he insrumen oen perorms a linear inerpolaion beween he las colleced sample lower han he reerence hreshold level and he ollowing sample. his process inroduces an esimaion error which depends also on he delay beween he signal ransiion and he sampling insans. A irs hree signals are kep ino accoun: a linear an exponenial and a Gaussian sep signal respecively. he linear and exponenial seps are oen used o model physical phenomena while he Gaussian one is oen adoped o describe he overall sep response o a se o cascaded linear ilers and ampliiers []. Each sep signal has been normalized hus presening a uniary ideal rise ime and uniary ampliude. he relaive error ε Rm- R / Rm where R is he signal rise ime and Rm is he measured rise ime is evaluaed by means o boh meaningul simulaions and heoreical analysis which is repored in he appendix. Fig. obained or various sep signals repors ε as a uncion o he raio beween Rm and he sampling period S. For each considered waveorm wo curves are ploed which represens he maximum error ε max and he minimum error ε min respecively achieved by varying he delay beween he signal ransiion and he sampling insans. When he Rm / S raio is low he error grows quickly because he measured ime ends o he sampling period. For higher values o Rm / S he error magniude ends o zero due o he improved resoluion in esimaing and 9% and can be upper bounded wih a hyperbolic law see he appendix. In paricular Fig. shows ha in order or ε max o be below Rm / S should exceed 4. Moreover i can be observed ha or he exponenial sep a low Rm / S raio may lead o a negaive error.
5 I is also worh noicing ha when Rm is an ineger muliple o S he error can be null. Such a behavior can be explained by considering ha in such a siuaion wo samples may exis which equal he and 9% sep reerence levels. Consequenly he corresponding sampling imes are exacly and 9% and no inerpolaion error is inroduced. Furhermore when he measured signal is a linear sep sampling does no inroduce an esimaion error when Rm / S equals or exceeds 8. In ac under such consrain i can be easily shown ha he inerpolaing sraigh lines exacly reproduce he linear sep rising ron. III. ERROR ESIMAION AND CORRECION he measured rise ime Ro o an analog oscilloscope is usually expressed as ollows + Ro R O where O is he oscilloscope rise ime []-[5]. Fig. suggess ha a similar expression can be adoped or he mean value o ε hus expressing he measured rise ime as Rm + α R S where α is a suiable coeicien used o model he eecs o sampling and linear inerpolaion. By applying and he deiniion o ε he ollowing expression can be derived S Rm ε α 3 in where ε in models he mean value o ε as a uncion o he Rm / S raio. hus by properly iing 3 o simulaion resuls i is possible o evaluae he coeicien α which allows o esimae he correcion erm in. Fig. repors he curves ε in obained by iing 3 o he mean value o ε wih he leas squares mehod or he hree considered inpu signals. he relaed values o α provided by he algorihm or Rm / S are repored in ab..
6 .9.8 Linear Sep Exponenial Sep Gaussian Sep.5 Linear Sep Exponenial Sep Gaussian Sep.7.6 /ε in.5 ε in.5.4 ε max / Rm S Fig. : Inerpolaing uncions or he average error ε avg obained or he considered inpu sep signals / Rm S Fig. 3: Envelope o he absolue value o he dierence beween he relaive error ε and he inerpolaed error ε in normalized o ε in. Linear Sep Exponenial Sep Gaussian Sep α. α. α.9 ab. : Fiing parameer α evaluaed or various inpu sep uncions and Rm / S I can be observed ha he inerpolaing curves o Fig. are very close o each oher which suggess ha sampling eecs are weakly aeced by he shape o he sep signal a leas when waveorms wih negligible ripple are considered. Such a resul is validaed by heoreical analysis which provides values o α quie close o he ones obained rom simulaions. hus he bias o he rise ime measuremen error inroduced by sampling and linear inerpolaion may be correced by using he ollowing relaionship: ' Rm Rm Rm α S 4 S ' where is he correced rise ime and he coeicien α is assumed independen o he shape o he sep signal. In paricular Rm he resuls described in he ollowing have been obained by choosing α.. In order o gain beer insighs on he correcion eeciveness he error properies have been urher analyzed. Fig. 3 shows he maximum magniude o he dierence εε-ε in normalized o ε in as a uncion o Rm / S or he considered sep signals. In all o he considered cases ε in is comparable o he maximum magniude o he error. Hence i is expeced ha removing he bias may be an eecive way o improve he measuremen accuracy. o gain a urher insigh he residual error ε Rm - R / Rm obained by applying 4 has been evaluaed and repored in Fig. 4 as a uncion o Rm / S. For each considered waveorm boh he maximum and minimum error curves are repored. As expeced Fig. 4 shows ha he error mean is negligible. However ouside he inerval Rm / S he use o 4 is no advanageous. In ac or values o Rm / S lower
7 Linear Sep Exponenial Sep Gaussian Sep.. ε' / Rm S Fig. 4: Residual error ε obained aer applying he bias removal correcion han he measuremen resuls are dominaed by he sampling period S. Conversely or values o Rm / S greaer han he esimaion error inroduced by sampling and linear inerpolaion is negligible. Finally i should be noiced ha 4 is useul only when α S is no negligible wih respec o he oscilloscope rise ime because he conribuion o he analog circuiry o measuremen errors canno be easily correced []. IV. ERROR ESIMAION AND CORRECION FOR FILERED SEP SIGNALS In order o analyze a more realisic siuaion hree new simuli have been considered obained by eeding wih a linear sep an exponenial sep and a Gaussian sep respecively he ollowing second-order sysem: H ω 5 ω jω + jd + ω ω wih damping acor D / ωπ where is he requency expressed in Hz ω π and /π R. Such a sysem inroduces a moderae disorion on he considered waveorms and i is characerized by a rise ime o abou. R []. he analysis has shown ha he second-order sysem has a regularizing eec generaing quie similar oupu signals even i he inpu sep signals are appreciably dieren. By ollowing he approach described in he previous secion he relaive error has been esimaed. Firs ε has been evaluaed and repored in Fig. 5 showing ha he eec o sampling is quie similar or all o he considered ilered waveorms. hen he mean value o ε has been ied o 3 using he leas squares mehod. For all o he considered waveorms a value o α close o. has been obained. hus 4 becomes: ' Rm Rm Rm S 6 S
8 Linear Sep Exponenial Sep Gaussian Sep..5. Linear Sep Exponenial Sep Gaussian Sep Fiing curve..5 ε ε' Rm / s Fig. 5: Relaive error inroduced by sampling and inerpolaion on rise ime measuremens obained or he linear exponenial and Gaussian seps ilered by a second-order sysem Rm / S Fig. 6: Residual error ε obained aer applying he bias removal correcion o he measured rise ime obained or he linear exponenial and Gaussian seps ilered by a second-order sysem and limiing curve. whose applicaion leads o he residual error curves repored in Fig. 6. As expeced he resuls are quie similar or all o he considered simuli and sugges ha he correcion acor α. may be applied o mos inpu signals. Finally by applying iing echniques o he maximum absolue local values o he residual error curves he ollowing hyperbolic law '.5 max Rm R ' Rm Rm / S.3 Rm S. 7 has been derived which provides an upper bound or he residual error magniude and is also repored in Fig. 6. V. CONCLUSIONS he error inroduced by sampling and linear inerpolaion on he rise ime measuremens o a sep signal has been analyzed. In paricular a simple ormula has been proposed which removes he error bias and is useulness and validiy have been discussed. I has been shown ha dierenly rom he analog case a correcion may be applied wih advanage or a large class o inpu signals. APPENDIX: EFFEC OF SAMPLING ON RISE IME MEASUREMENS Le us call he esimaes o and 9% made by he DSO as ˆ and ˆ 9 % respecively. he eec o sampling and inerpolaion on he resul o a rise ime measuremen can be heoreically evaluaed by expressing ˆ and ˆ 9 % as a uncion
9 o S o he delay beween he signal and he sampling insans o he signal shape and o he inerpolaion algorihm. In paricular in he case o linear inerpolaion is given by ˆ < < % % ˆ % + n n n s s A. where is he mahemaical law which describes he sep signal o be sampled n- and n are he wo consecuive sampling imes such ha n- < < n and n - / s may be modeled as a random variable uniormly disribued in [] which keeps ino accoun he lack o synchronizaion beween he signal and he sampling imes. Similarly is given by 9% ˆ ˆ 9% 9% 9% % 9 + m m m s s A. where m- and m are such ha m- < 9% < m and 9% m - 9% / s. I should be noiced ha and 9% are no uncorrelaed. In ac by expressing he signal rise ime R as ollows S R k + A.3 where k is an ineger and [[ i can be shown ha < < + 9% A.4 hus by using A. A. and A.4 he DSO measured rise ime Rm - may be expressed as 9% ˆ % ˆ + 9% n n n m m m s S Rm A A k A k A.5 where A expresses he eec o he signal shape on he rise ime measuremen when linear inerpolaion is used. he measuremen error normalized o Rm can hen be expressed as
10 Rm ε Rm R k + k + A k + k A < < A.6 In order o validae he model accuracy he mean value o ε has been also evaluaed by numerically inegraing A.6. In paricular he relaive deviaion beween heoreical resuls and simulaions is lower han 9-5 or R / S. Finally as he error on he esimaion o boh and 9% is upper bounded by S i resuls ha Rm - R < S. Consequenly he magniude o he relaive error ε expressed as a uncion o Rm / S is upper bounded by a hyperbolic curve according o he ollowing ε / A.7 Rm S REFERENCES [] C. Miermayer and A. Seininger On he deerminaion o Dynamic Errors or Rise ime Measuremens wih an Oscilloscope. IEEE rans. on Insrum. Meas. Vol. 48 No. 6 pp. 3-7 Dec [] ekronix Applicaion Noe 55W_47_ Eecs o Bandwidh on ransien Inormaion available on he Inerne a [3] Agilen Applicaion Noe 596_5E Precision ime Domain Measuremens using he Agilen E43A available on he Inerne a [4] ENV 35:999 Guide o he Expression o Uncerainy in Measuremen. [5] H. W. Johnson and M. Graham High Speed Digial Design-A Handbook o black magic Englewood Clis NJ Prenice Hall 993.
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