Traceability and uncertainty for phase measurements

Transcription

1 Traceablty and ncertanty for phase measrements Karel Dražl Czech Metrology Insttte Abstract In recent tme the problems connected wth evalatng and expressng ncertanty n complex S-parameter measrements have been often dscssed and many artcles on the theme have been pblshed. The man am of ths paper s to show that the reflecton coeffcent ncertanty regons n the complex plane are of the dfferent shape for TL and OSL calbraton method. The problem of traceablty s brefly mentoned becase of contnty of explanaton. Usng of the law of propagaton of ncertanty n matrx form for one-port calbraton s descrbed. Uncertanty of reflecton coeffcent for both TL and OSL calbraton method are calclated and ts geometrcal representaton s shown. Traceablty for reflecton measrements For measrements of reflecton coeffcent the error model can be represented as follows where M = D TA + M A M reflecton coeffcent vale ndcated by the calbrated VNA A actal vale of the reflecton coeffcent D T M resdal error terms of the calbrated VNA. The resdal errors of the calbrated VNA are sally evalated by the rpple method. The traceablty to SI nts s establshed by the arlne wth electrcal characterstcs calclable from ts geometrcal dmensons. In ths case the lne s sally sed only as verfcaton standard. The rpple method yelds only the nformaton abot the magntde of resdal error terms D and M. These resdal error terms case the errors n both magntde and phase measrement wth sgn + or n dependence on the vale of measrand whle the resdal error term T represents the systematc offset n both magntde and phase. In order to evalate the resdal reflecton trackng T.e. to make the measrements completely traceable to SI nts the addtonal calclable hgh reflecton standard s needed. However the vale of resdal reflecton trackng specfed by the manfactrer for calbraton kts s sally lower than several hndredths of db. Hence the calclable reflecton standard s not necessary for magntde measrement. A specal case s the TL (LL calbraton. The TL algorthm allows to set the reference plane by the defnton of the reflecton standard (SET EF EFLECT opton n AGILENT (HP VNA s or by the defnton of the THU standard (SET EF THU opton. When the SET EF EFLECT opton s chosen for the TL calbraton the phase of the resdal reflecton trackng (.e. phase error s the same as the phase devaton of the reflecton standard from the nomnal vale. The TL method s based on the assmpton that both reflecton standards are dentcal. Hence when the SET EF THU opton s sed the phase of resdal reflecton trackng s a half of the dfference between the phase devatons of the both reflecton standards. When the same reflecton standard s sed at both measrement ports the phase of resdal reflecton trackng T s (n deal case.

2 The conclsons from the above are as follows: The traceablty for the transmsson medm wth sexless connectors can be establshed wth TL lne. The traceablty for the connectors wth the same sex can be establshed wth two (nonnsertable lnes ( THU and LINE. The traceablty for the connectors wth dfferent sex can be establshed wth one TL lne and (at least one calclable reflecton standard. Converson between mag/phase and e/ form of ncertanty expresson When the ncertanty analyss s performed the converson between both forms of ncertanty expresson can be sefl. It can be done by the law of propagaton of ncertanty n matrx form as descrbed n[]. Ths law states: ( Y JV ( X J T V = where V(X s the covarance matrx of the npt vector X V(Y s the covarance matrx of the otpt vector Y and J s the Jacoban matrx. When the ncertanty n the form e/ s to be obtaned the vector of npt qanttes s X where = arg( ( x x = ( = φ φ and vector of otpt qanttes s Y ( y y = ( e( ( = The Jacoban matrx s: J cosφ = sn φ sn φ cosφ When the ncertanty n the form mag/phase s to be obtaned the vector of npt qanttes s X ( x x = ( e( ( = and vector of otpt qanttes s Y ( y y = ( = φ When the ncertanty of reflecton coeffcent s very small n comparson wth ts magntde the Jacoban matrx s: J cosφ sn φ sn φ cosφ =

3 Uncertanty for OSL calbraton Usng the law of ncertanty propagaton n matrx form the ncertantes for one-port calbraton can be analysed. The vector of npt qanttes s X = ( e( ( e( ( e( ( where are the reflecton coeffcents of the calbraton standards. The twodmensonal vector of otpt qanttes s Y ( e( ( = where s the measred vale of reflecton coeffcent. The covarance matrx of the npt vector s V ( e( ( e( ( ( ( ( e ( ( ( X = ( e( ( e( ( ( ( e( ( ( ( e( ( e( ( ( ( ( ( ( e where the off-dagonal elements represent covarance terms. Covarance terms for ndvdal standards are ( e ( ( = ( ( e( = ( e( ( ( r( e( ( where r(e( ( are the correlaton coeffcents. It s assmed that and are ncorrelated. In [] [4] the expressons relatng the measrement errors to the errors of calbraton standards have been shown. Utlzng these expressons the Jacoban matrx can be wrtten: J = e ( ( ( ( ( ( ( ( e ( ( ( ( ( ( ( ( e ( ( ( ( ( ( ( (.. e ( ( ( ( ( ( ( ( e ( ( ( ( ( ( ( ( e ( ( ( ( ( ( ( (

4 For several locatons of the measred reflecton coeffcent n the complex plane the covarance matrces were calclated. As an example type-n open-short-sldng load calbraton at 8 GHz was consdered. eslts are presented n Tab. and Fg.. For the calclaton npt data were sed as follows: OPEN phase of reflecton coeffcent -. reflecton coeffcent ncertanty: phase:.5 (manfactrer s specfcaton magntde:. (verfed sng TL calbraton no correlaton between magntde and phase assmed SHOT phase of reflecton coeffcent 8. reflecton coeffcent ncertanty: phase:. (manfactrer s specfcaton magntde:. (verfed sng TL calbraton no correlaton between magntde and phase assmed LOAD reflecton coeffcent ncertanty: real part:.8 (manfactrer s specfcaton magnary part:.8 (manfactrer s specfcaton no correlaton between real and magnary part assmed reflecton coeffcent ncertanty magntde phase ( (e( (( r(e(( ( (φ ( Tab. Uncertantes for OSL calbraton

5 eflecton coeffcent ncertanty for TL calbraton For the TL calbraton a resdal error term approach was sed. As the correlaton between T M and D has not been analyzed yet the worst case ncertanty was calclated. The ncertantes for magntde and phase of the reflecton coeffcent can be expressed: ( = D + M + ( T D ( arg( = arcsn + arg( T + M As an example certantes for.5 mm connectors at freqency of abot GHz were calclated. Usng of TL opton SET EF THU and connectors wth opposte sex at test ports are assmed. For the calclaton npt data were sed as follows: -magntde of resdal drectvty D. (manfactrer s specfcaton -magntde of resdal test port match M. (manfactrer s specfcaton -magntde of resdal reflecton trackng T db (manfactrer s specfcaton -phase ncertanty of shorts.7 (manfactrer s specfcaton For ths opton of TL calbraton the worst case phase ncertanty of resdal reflecton trackng arg(t was assmed of the same vale as the maxmm phase ncertanty specfed for both male and female shorts. eslts are presented n Tab. and Fg.. ( (arg( ( Tab. Uncertantes for TL calbraton Conclsons The law of propagaton of ncertanty n matrx form has been appled to calclate the ncertanty for one-port calbraton. The geometrcal representaton of ncertanty for the OSL and TL calbraton method has been compared. Althogh only the systematc errors cased by mperfect standards were taken nto accont and no consderatons abot level of confdence were performed the reslts can be sefl for the next dscssons abot treatment of phase ncertanty.

6 Fg. Geometrcal representaton of reflecton coeffcent ncertanty for OSL calbraton Fg. Geometrcal representaton of reflecton coeffcent ncertanty for TL calbraton

7 eferences: [] N.M. dler M.J. Salter: Propagatng S-parameter ncertantes to other measrement qanttes [] P.. Yong: Propagaton of Uncertanty n One-port ANA Measrements ANAMET eport 7 Agst 998 [] N.M. dler M.J. Salter: Evalatng and expressng ncertanty n complex S-parameter measrements [4] K. Dražl: One-port Calbraton: Non-deal Standards and esdal Error Terms ANAMET eport Janary 999