Extended S-parameters for imperfect test ports

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1 Home erch Collections Journls About Contct us My IOPscience Extended -prmeters for imperfect test ports This content hs been downloded from IOPscience. Plese scroll down to see the full text. 25 Metrologi 52 2 ( View the tble of contents for this issue, or go to the journl homepge for more Downlod detils: IP Address: This content ws downloded on 3//25 t 8:6 Plese note tht terms nd conditions pply.

2 Metrologi 52 (25) 2 29 Extended -prmeters for imperfect test ports Metrologi doi:.88/26-394/52//2 J Hoffmnn, M Wollensck, J Ruefencht nd M Zeier Federl Institute of Metrology META, Lindenweg 5, 33 Bern-Wbern, witzerlnd E-mil: johnnes.hoffmnn@mets.ch Received 9 October 24, revised December 24 Accepted for publiction 9 December 24 Published 9 Jnury 25 Abstrct Reflection nd trnsmission of microwves in coxil devices re usully described by -prmeters. The current definition of -prmeters requires tht the reference plne is in section of idel wve guide. Due to this, tremendous effort is necessry to fcilitte the dissemintion of stndrds, for the comprison of mesurement vlues nd for cscding devices. These processes cn be simplified by extending the definition of -prmeters to reference plnes in sections of non-idel wve guide, e.g. in connectors. Extended -prmeters cn be pproximted with conventionl simultion progrms. Prcticl experiments show tht extended -prmeters cn be compred nd cscded without effort. Keywords: vector network nlyzer, -prmeter, connector (ome figures my pper in colour only in the online journl). Introduction Trnsmission nd reflection of microwves in coxil lines re often specified s scttering-prmeters (-prmeters). The -prmeters of device under test (DUT) re usully mesured with vector network nlyzer (VNA). Mesurement results should be trceble to I units nd directly comprble between different lbortories. A problem which complictes compring -prmeters is tht their current definition [] requires tht the reference plne is in section of stright wve guide. In figure the reference plne F is between two hlf-connectors nd thus would be prohibited by the current definition of -prmeters. There re two possibilities where -prmeters cn be defined ccording to the current definition: reference plne nd 2. These reference plnes re situted in single mode regions. Clibrting VNA t the reference plne requires tht the connector pirs C C 2 of test port nd stndrds re mechniclly chrcterized nd their respective -prmeters re computed. For reference plne this cn be done with pseudo -prmeters [] becuse the connections cn be treted lwys s connector pirs. The -prmeters mesured with VNA clibrted in this wy contin the connection C C 3 of test port hlf-connector nd DUT hlf-connector. ee figure. The C hlf-connector prohibits direct comprison of such -prmeters to results from VNA with nother test port connector. For the sme reson it is not possible to directly compute from such -prmeters the cscding of devices. The bove mentioned restrictions cn be overcome if ll DUT hlf-connectors re mechniclly chrcterized nd the -prmeters of the individul connector pirs of DUT nd test port re computed. Overll, using reference plne requires stndrds nd test ports with chrcterized connectors nd connector chrcteriztion of the DUTs. Note tht mechnicl chrcteriztion of connectors requires dvnced 3D coordinte mesurement mchines nd computing the corresponding -prmeters requires softwre nd expertise. Clibrting VNA t reference plne 2 requires tht the connectors C 2 of ll stndrds re mechniclly exctly the sme. Computtion of the -prmeters of the connector pirs of test-port nd stndrds is not necessry. Agin, one cn use pseudo -prmeters during clibrtion becuse ll connectors re treted s pirs. After clibrtion the error terms of the VNA contin C C 2. Mesuring DUT with hlf-connector C 3 yields results which contin the difference of the two connections, [ C2 C ][ CC 3 ]. Note tht both connections re still treted s connector pirs, indicted by the squre brckets nd tht this difference of connections vnishes if C 3 = C 2. The difference term in the -prmeters of the DUT prevents direct comprison of results tht were not obtined with the /5/2+9$ BIPM & IOP Publishing Ltd Printed in the UK

3 Metrologi 52 (25) 2 Figure. The upper prt of the figure shows three loctions of reference plnes between VNA nd one port stndrd. The results of DUT mesurement t these three reference plnes re shown below. The boxes with thick lines depict connectors which re treted s pirs nd the connecting lines re regions of single mode propgtion. C,C 2 nd C 3 denote the hlf-connector of the test port, the stndrd nd the DUT, respectively. sme test port nd the sme clibrtion kit. Cscding of -prmeters is not directly possible. Agin these restrictions cn be overcome if the connectors of test port, stndrds nd DUT re mechniclly chrcterized nd the -prmeters of the connections re computed. Using reference plne 2 requires connector chrcteriztion for comprison nd cscding but not for clibrtion. There re ccounts for the use of reference plnes nd 2 for highly ccurte mesurements up to 8 GHz in coxil systems [2] nd t very high frequencies in wveguides. ee [3 nd 4]. As lredy explined, it is quite difficult to compre -prmeters from different lbortories which re mesured t reference plne. The sme holds for reference plne 2. This is the min reson why it is desirble to use reference plne F. Clibrting VNA t reference plne F requires breking up connector pirs nd chrcterizing the hlf connectors of the stndrds. pecificlly, it requires mechnicl chrcteriztion nd -prmeter computtion of the hlf connectors. Pseudo -prmeters re, however, not suitble to describe hlf connectors due to the higher modes in reference plne F. An lterntive is multi-mode -prmeters s in [5], but they re not suitble either becuse the discontinuities re too close to the reference plne. In other words, the convergence of the multi-mode expnsion is such tht prohibitive number of modes would be needed to chieve cceptble ccurcy. There re industril lbortories which use reference plne F with lower ccurcy nd without worrying bout higher modes [6]. Below, the theory of -prmeters is extended in such wy tht pir of connectors cn be mthemticlly divided in two hlf connectors. Clibrting VNA t reference plne F hs severl dvntges. DUT mesurements from mesurement systems with differing test port hlf-connectors C re directly comprble to ech other nd cscding of devices is lso possible without previous computtions of connector -prmeters. Using reference plne F requires stndrds with definitions tht contin connector effects. These stndrd definitions cn come from ntionl metrology lbs. Comprison nd cscding do not require connector chrcteriztion. Note tht this pproch hides the complexity of connector chrcteriztion from the end user. In this rticle, the reltion between electromgnetic fields nd -prmeters in reference plne F, see figure, is explined. In the following, these -prmeters will be clled extended -prmeters becuse their definition is n extension to the concept of pseudo -prmeters []. Extended nd pseudo -prmeters re identicl for cses tht cn be described with pseudo -prmeters. Another commonlity is tht the field problem in the vicinity of the reference plne lwys hs to be the sme for comptibility of -prmeters. If the field problem in the vicinity of the reference plne is not the sme, n dditionl uncertinty is introduced for extended -prmeters. The tolernce rnge (geometry nd mteril prmeters) of the field problem determines the resulting uncertinty. Thus the uncertinty cn be evluted by computing extended -prmeters for ll possible field problems. Detils re given in section 3. There is n interesting property of extended -prmeters for cses which cn not be treted with pseudo -prmeters. The computtion of power from extended -prmeters is different from wht is known for pseudo -prmeters. Mthemticl detils re given in ppendix A. With extended -prmeters, one cn directly compre mesurement results from mesurement systems with differing test port hlf-connectors C. This is prcticlly demonstrted by clibrting 2.4 mm nd.85 mm test port with the sme.85 mm stndrds nd compring DUT mesurements of both test ports. With extended -prmeters one cn cscde DUTs. This is prcticlly demonstrted by compring computed cscding of.85 mm dpters nd mesured cscding. 2. Theory Extended -prmeters cn be defined in reference plnes with rbitrry surrounding geometry. In first step the electromgnetic fields in the reference plne F in figure re defined. It is ssumed tht ll mterils close to the reference plne hve liner mteril prmeters. Amplitude nd phse of the electromgnetic fields re represented by complex vribles. A complex exponentil time dependence e jωt is used where t is the time nd ω = 2π f with f being the frequency. The electric nd mgnetic fields in reference plne with rbitrry surrounding geometry re described by the complex vribles EF, HF () for forwrd opertion nd ER, HR (2) 22

4 Metrologi 52 (25) 2 Figure 2. The electromgnetic fields in reference plne F in forwrd opertion, EF, HF, re defined by n idel source nd bsorber, which both hve zero reflection. The source nd the bsorber re plced in sections of plin wveguide with mono-mode opertion. The electromgnetic fields in reverse mode, E, H R R, re obtined by exchnging the positions of source nd bsorber. for reverse opertion. Forwrd opertion refers to setup with n idel source (no reflection, field pttern equls mode) nd n idel bsorber, s shown in figure 2. Reverse opertion refers to n inverse setup of source nd bsorber. The totl electric nd mgnetic fields in plne F re E = ce + F + ce R (3) H = c+ HF + c H R. (4) c + nd c re rbitrry complex-vlued fctors, which determine the mplitude nd phse of the forwrd field nd reverse field, respectively. The electromgnetic field E, H is liner superposition of forwrd nd reverse field. Note tht in contrst to this definition of electromgnetic fields, it is common to restrict the electromgnetic fields in reference plne to eigenfunctions of the wveguide []. Now, the quntities' power, current nd voltge need to be defined in the wveguide. The power flux through the reference plne is = p W E H n d. (5) is the reference plne nd n is the norml vector, see figure 3. The complex conjugte is denoted by *. The current in the wveguide is the integrl over the mgnetic field long closed curve in the reference plne i = H d l W. (6) l represents closed rbitrry curve in the reference plne, see figure 3. In most cses the current in the wveguide depends on the choice of this curve. This mbiguity rises s well in the definition of pseudo -prmeters in plin lossy coxil line nd is usully solved by conventions, e.g. the integrtion curve is on the surfce of the inner conductor of coxil line, see [7] nd figure 3. The sme convention is used here. The voltge in the reference plne is n integrl over the electric field long n open curve v = W E d s. (7) s represents n rbitrry open curve in the reference plne, see figure 3. As for the current, the choice of the curve influences the voltge in the reference plne. For the definition of pseudo -prmeters in plin lossy coxil lines this Figure 3. A reference plne in plin coxil line is depicted s n exmple. The integrtion curves l nd s re in the reference plne. The norml vector n is orthogonl to the reference plne. mbiguity is usully voided by using only current nd power. v W is used here only for computing the wve mplitudes c + nd c in the wveguide. Thus the only requirement for the integrtion curve is tht it yields non-zero voltge v W for non-zero electric field E. The wveguide voltge v W is superposition of the forwrd nd reverse wveguide voltges v WF nd v WR. These voltges re defined by E F, E R nd (7). imilrly, the wveguide current i W is superposition of the forwrd nd reverse wveguide currents i WF nd i WR. These currents re defined by H F, H R nd (6). As lredy explined, the setup is ssumed to be liner nd thus the wveguide voltge nd current re v = c v + c v W + WF WR (8) iw = c+ iwf+ c i WR. (9) Inversely c + nd c cn be determined from v W nd i W, if v WF, v WR, i WF nd i WR re known. The equtions (5) (7) serve s link between electric nd mgnetic fields in the reference plne nd voltge nd current in the trnsmission line model. A trnsmission line model is two-port network tht consists of complex impednce per unit length Z nd complex dmittnce per unit length Y, see figure 4. In this network forwrd nd reverse wves exist. The complex reference impednce is rbitrrily defined s v Z Zref = = i Y. () with Re(Z ref ) >. The next eqution defines the rel current of unit wve in the trnsmission line model W i = () Re ( Zref) nd the corresponding complex voltge W v = Zref. (2) Re ( Z ) ref 23

5 Metrologi 52 (25) 2 Figure 5. Junction of two coxil lines with 5 Ω chrcteristic impednce but different dimeters. The resulting mechnicl steps led to non-uniqueness for extended -prmeters defined t plne 23. Figure 4. The trnsmission line model is n bstrct representtion of plin wveguide. It describes wves with voltge v nd current i insted of electric nd mgnetic fields. The totl voltge nd current in the trnsmission line model re v = v+ bv (3) i = i bi. (4) nd b re complex fctors of forwrd nd reverse wves in the trnsmission line model. The trnsmission line model is very similrly defined s for the pseudo wves in []. The only difference is tht nd b do not hve physicl units. The chosen voltge nd current do not hve link to the electromgnetic fields yet. The link is estblished through the power flux p WF (5) nd the current i WF (6), which depend on the electromgnetic fields E F nd H F. etting wveguide power nd current equl to the trnsmission line vlues yields the wve mplitudes F, b F. They hve to fulfill pwf = v + bv i F F (5) WF iwf = F i bf i. (6) Note tht the forwrd field results in forwrd nd reverse wve in the trnsmission line model. Thus the term forwrd field, which refers to electromgnetic fields, should not be confused with the term forwrd wve, which refers to the trnsmission line model. imilrly, one cn define the wve mplitudes R, b R. Wveguide voltge v W is not used directly for estblishing the link becuse the usul method of constructing trnsmission line model from electromgnetic fields in coxil wveguide is to define power flux nd current, see [7 nd 8]. With (8) nd (9), c + nd c re computed for n rbitrry mix of forwrd nd bckwrd electromgnetic fields. This is possible becuse the electric field nd mgnetic field re liner to the wveguide voltge v W nd wveguide current i W. With (5) nd (6) the forwrd nd reverse mplitudes in the trnsmission line model cn be computed = c+ F+ c R (7) b = cb + F+ cb R. (8) Four complex coefficients re needed to link the mplitudes nd b to the mplitudes c + nd c of the electromgnetic field. For pseudo -prmeters, only two such complex prmeters re needed s forwrd nd reverse electromgnetic fields re identicl, F = b R nd R = b F, see s well (45) (48) nd (53) (54) in []. The complex fctors nd b re derived from electromgnetic fields with the voltge, current nd power integrl. The voltge integrl is used only for the seprtion of the forwrd nd reverse electromgnetic wve. Finlly, one clcultes from nd b the reflection prmeter t reference plne m nd trnsmission prmeters between different reference plnes, m nd n. bm mm = (9) bm mn = (2) n The power of the wves nd b is normlized with respect to the trnsmission line model but cnnot be normlized to the electromgnetic field in the port plne. The method of computing the power in the port plne is described in ppendix A. Note tht extended -prmeters re reciprocl for devices which consist of isotropic mterils nd for which the forwrd nd reverse fields cn be expressed by one vector field for electric fields nd one for mgnetic fields per port, see s well ppendix D in []. The properties of cscding extended -prmeters originte from their dependence on the problem-specific forwrd nd reverse fields. Cscding extended -prmeters, which do not depend on the sme forwrd nd reverse fields, is enbled by the non-uniqueness uncertinty. 3. Cscding nd non-uniqueness A junction of two coxil trnsmission lines, see figure 5, is used to illustrte cscding nd non-uniqueness of generlized -prmeters. In both lines the rtio of outer conductor dimeter nd inner conductor dimeter is the sme nd the left line hs n outer conductor with dimeter of 2.4 mm. The right outer conductor hs dimeter between.2 mm nd 2.4 mm. To ensure mono-mode opertion, reference plnes nd 4 re set sufficiently fr wy from the junction. Reference plnes 2 nd 3 re t the sme loction in the junction. Extended -prmeters re defined by computing the electromgnetic forwrd nd reverse field. The wve fctors nd b, which re computed from the forwrd field, re mrked with F nd the fctors from the reverse field re mrked with R. The extended -prmeters of the left prt re m 24

6 Metrologi 52 (25) 2 The right prt will be 44 b = 2 34 F b2 = ( ) F b = 2 F b2 = 2 R b3 = 3 F b4 = 3 R b3 = ( ) 4 R b4 = 4 R. (2) (22) (23) (24) (25) (26) (27) (28) Note tht plnes 2 nd 3 hve opposite directions. Cscding the left nd the right -mtrices mens setting 2 = b 3 nd 3 = b 2. This in turn ensures tht the forwrd field in plne 2 equls the reverse field in plne 3 nd vice vers becuse of (7) nd (8). Thus one obtins the -prmeters of the complete junction from cscding the individul prts l r [ ] = [ ] [ ]. (29) Here l nd r re the -prmeter mtrices of the left nd right prt of the junction. The sign denotes cscding nd stnds for the -prmeters of the complete junction. With the exmple of the coxil junction, one cn s well illustrte tht the extended -prmeters re not unique. They depend on the device tht is connected. In this exmple they depend on the dimeters of the right outer conductor. If the dimeter is.2 mm, the -prmeter is lrge. If the dimeter is 2.4 mm, the -prmeter is smll (ssuming n pproprite choice of Z ref ). The non-uniqueness of cn be pproximted by the difference between both cses. Note tht the -prmeters of the right side re unique becuse they connect lwys to the sme left side. In generl the computtion of the non-uniqueness of -prmeters requires tht ll possible combintions of left nd right side geometries re evluted. The non-uniqueness of generlized -prmeters is inherent to the problem nd cn be treted s n dditionl uncertinty. Using extended -prmeters is big dvntge in coxil systems becuse the non-uniqueness is usully very smll for devices tht re within specifiction. On the other hnd, the differences in extended -prmeters between different coxil connectors cn be quite big. In other words, extended -prmeters cn cpture the differences in device behvior tht stem from different connector geometries, but they cnnot cpture the differing interction mechnisms with the connected device. Thus extended -prmeters re useful for systems where the electromgnetic fields in the reference plne depend mostly on the device itself nd not on the connection. 4. Prcticl implementtion Current versions of commercil simultion progrms for electromgnetic fields cnnot compute extended -prmeters. Nonetheless the concept cn be tested with commercil field solvers tht compute pseudo -prmeters. A simplified model of slotted connector, figure 6, is used to explin the technique. The problem is split t plne F nd ech side is substituted with structures tht convert the rbitrry electromgnetic field t the plne F to mono-mode fields. The subfigures in figure 6 depict the resulting configurtions. In the following, the converting structures re clled mode dpters. Typiclly it is ssumed tht the mode dpter hs zero reflection nd its trnsmission is depending only on the length of the mode dpter. Obviously these ssumptions dd uncertinty to the extended -prmeters of hlf-connector. Other uncertinty contributions re the non-uniqueness of extended -prmeters nd the error of the simultion. Ech configurtion in figure 6 is described by extended -prmeters becuse higher modes in plne F prohibit the use of pseudo -prmeters. () The extended -prmeters in configurtion () describe the complete connector pir between plne nd plne 2. They re denoted s c. (b) In configurtion (b) the femle connector is replced by femle mode dpter. The extended -prmeters m describe the section between the thick lines corresponding to plne nd 2. The left port plne hs to be shifted by l d to pproximte the extended -prmeters of the mle hlf connector between plne F nd plne 2 m 2γ = e fld 2γ l e f d m. (3) The sign denotes cscding nd γ f denotes the propgtion constnt in the femle mode dpter. Note tht (3) is only n pproximtion becuse there re higher modes present in the connector. The left port plne is shifted becuse this enbles the use of commercil simultion progrms for electromgnetic fields, which require mono-mode fields in the port plne. (c) Repeting procedure (b) with the role of femle nd mle prts interchnged yields the extended -prmeters of the femle hlf-connector f, see configurtion (c). teps () (c) yield the extended -prmeters of the complete connector pir nd pproximte the extended -prmeters of the hlf-connectors. The ltter should depend to mximum on the geometry of the hlf-connector nd not on the geometry of the complete connector pir. Although this gol cn never be chieved completely, it cn be lrgely pproched by setting plne F to resonble loction. In 25

7 Metrologi 52 (25) 2 Tble. Dimensions of.85 mm connector. d d 2.82 mm.8 mm p f µm p m 4 µm Figure 6. ub-figure () shows simplified cross section of n.85 mm connector. The colors light gry, middle gry nd drk gry indicte mle prt, femle prt nd mode dpter, respectively. The port plnes nd 2 re indicted by thick solid lines nd their shifted versions re indicted by thick dshed lines. Plne F is pproximted by the mode dpters nd the shifted versions of plnes nd 2 in (b) nd (c). figure 6, plne F is set to the mting plne of the outer conductors. This is the idel loction for plne F becuse reflections from the ends of inner nd outer conductor re ttributed to the correct hlf-connector. The definition of the mode dpters is governed by two prdigms tht my contrdict ech other. First, the electromgnetic fields in plne F should be s similr s possible in ll three configurtions of figure 6. In other words, the problems due to non-uniqueness re minimized. This would be idelly stisfied by using the originl geometry close to plne F nd dding section of trnsmission line fr wy. econd, the mode dpters themselves should induce only smll errors. The cscded mode dpter should hve low reflections nd idel trnsmission. These two prdigms cn be expressed s two residuls Δ = f m c (3) Δ 2 = fm mm t. (32) fm mm denotes the generlized -prmeters of the femle mode dpter connected to the mle mode dpter nd t is perfect thru. The residuls re computed with qudrtic norm nd should be s smll s possible. This my serve s prcticl guide for the design of mode dpters, which is s well influenced by interfce stndrds like [9] or []. In the exmple given in figure 6, the mode dpters re just perfect conductors with nominl dimeters ccording to []. They end t plne F, which ensures tht (32) is optimlly fulfilled. The pproximtion of the electromgnetic field in plne F is quite good becuse the mgnetic field is determined in both, the originl setup nd the mode dpter setups, by the current flowing in the center conductor. The originl setup refers to configurtion () in figure 6 nd the mode dpter setups to configurtions (b) nd (c). The electricl field in the originl setup is wek in the gp becuse the gp is very nrrow. The mode dpter produces in the gp region zero electricl field nd outside of the gp region rdil electricl field s in the originl setup. Note tht there re certin restrictions to the ppliction of generlized -prmeters. They rely on the sequentil configurtion of femle nd mle connectors. This mens tht it must be possible to drw stright plne tht seprtes the femle nd mle prt of the connector to the lrgest extent possible. The design of the type-n connector with its displced center conductor is less well suited for this technique. 5. Experimentl verifiction The procedure outlined in the previous section is pplied to.85 mm connector under the ssumption of perfect conductors nd vcuum s dielectric. These re simplifictions to mke the simultion results more relible. The simultions re mde with n in-house finite difference time domin code, which hs been vlidted previously with HF from Ansoft nd CT Microwve tudio. Note tht the following description of connector geometry is incomplete for resons of simplicity, but still llows for reproduction of results with resonble ccurcy. More detiled descriptions of connector geometry cn be found in []. The dimensions of the connectors re specified in tble. The nottion refers to figure 6. All connectors hve four slots with slot width of µm. Inner conductor, outer conductor nd mle pin dimeter hve dimensions s given in []. Figure 7 shows the bsolute vlues of the reflection coefficients of the.85 mm connector. It is interesting to see tht the vlues for the mle nd femle hlf-connectors re different. 26

8 Metrologi 52 (25) Complete Femle Mle Figure 7. The bsolute vlues of the reflection coefficients seen from the left side of slotted.85 mm connector, see figure 6. olid, dshed nd dshed-nd-dotted lines re ttributed to the connector pir, the femle hlf-connector nd the mle hlfconnector, respectively. 4 x Num Num 2 2 Idel Figure 8. The trces show bsolute vlues of complex differences between -prmeters of the connector pir (, 2 ), -prmeters of the cscded femle nd mle hlf-connectors ( Num, Num 2 ) nd section of coxil line ( Idel 2 ) with the sme physicl length s the connector pir. The smll differences between connector pir nd cscded hlf-connectors indicte tht the concept of extended -prmeters is suitble. This is minly due to the different pin depths (p f nd p m in figure 6). Another noteworthy fct is the bsolute mplitude of the connector reflection of.4 t 65 GHz. The connector investigted in this exmple does not hve optimized dimensions but its reflection coefficients lie well in the rnge of commercilly vilble products. Figure 8 shows the difference between the extended -prmeters of the connector pir nd the extended -prmeters of cscded mle nd femle hlf-connectors. The very smll differences in reflection nd trnsmission show tht the reference plne nd the mode dpters re chosen suitbly nd thus the non-uniqueness error due to the mode dpters is very smll. Another contributor to this difference re computtionl errors of the field simultor. of cscded dpters Computed Mnul Difference x 3 Figure 9. Two dpters with.85 mm connectors re mesured individully nd mted. The individul mesurements re computtionlly cscded (Computed) nd compred to the mted mesurements (Mnul). hown re the mgnitude of the reflection versus frequency nd the mgnitude of the complex difference between the two trces (Difference). Figures 7 nd 8 cn be used for comprison of uncertinties between the extended -prmeters nd the ssumption tht the connector is idel. Holding on to the ssumption tht the connector is idel requires n dditionl uncertinty contribution of up to.5 nd.4 in reflection nd trnsmission, respectively. By using extended -prmeters nd defining the llowble connector geometries correctly, one cn expect uncertinties below. for reflection nd trnsmission. Therefore it is possible to drsticlly increse the ccurcy of clibrtion stndrds by ccounting for their hlf-connectors with the help of extended -prmeters insted of just ignoring them. The very smll non-uniqueness error shown in figure 8 explins s well why the connectors of clibrtion stndrds hve to be well known while the connectors of DUTs nd test ports re not s criticl. A quite shrp seprtion of effects is possible. The effect of the hlf-connector of the DUT cn be ttributed to the DUT nd the hlf-connector of the test port cn be ttributed to the error model of the VNA. In system with big non-uniqueness error, it would not py off to ccurtely chrcterize the hlf-connectors of the stndrds. A prmeter of test ports nd DUTs tht hs to be controlled well is the mle pin dimeter becuse this my mechniclly chnge the complementry hlf-connector. Another importnt prmeter of test port nd DUT connectors is the pin gp (p f + p m in figure 6). If it is too smll, both hlf-connectors influence ech other significntly. ee []. The lower limit is 5 µm for the.85 mm connector. The electromgnetic fields in connectors with too smll pin gp depend strongly on very smll geometricl chnges. Inevitble vritions of the connector geometry cuse big non-uniqueness errors in such cse. Figures 9 nd show the differences between mesurement of two mted dpters nd computed cscding of mesurements of the two individul dpters. The dpters re insertble femle to mle dpters from Agilent, prt number 2 Com Mn 27

9 Metrologi 52 (25) 2 2 of cscded dpters Computed Mnul Difference Com Mn 2 2 of mismtch mm.85 mm Difference Figure. The trnsmission coefficient from the sme experiment s discussed in figure 9 is plotted versus frequency The differences between mnul nd computed cscding re smll becuse the reference plne of the clibrtion stndrds is the mting plne of the outer conductors. This is only possible with extended -prmeters. Using other reference plnes like nd 2 in figure without computtionl correction for connector effects results in complex differences tht re typiclly 5 times lrger thn given in figure 9. Nonetheless the differences between the two methods of cscding re not zero for reference plne F. There re severl resons like cble movement, instrument drift, connector repetbility nd stndrd definitions which prevent the differences from being zero. The experiment, described in the following, is very similr to the one described in [2]. In order to prcticlly demonstrte comprbility, two clibrtions with mle.85 mm stndrds re executed on 2.4 mm nd.85 mm test port. The.85 mm nd 2.4 mm test ports hve pin depths of 22 µm nd 5 µm, respectively. Figure shows the reflection coefficient of mismtch mesured with both test ports nd the mgnitude of the complex difference between these two results. Additionlly flush short is mesured on both testports, see figure 2. Note tht the definition of the stndrds hs only second order influence on the difference between results. The non-uniqueness error explins why the differences re very smll in figure nd rther big in figure 2. In comprison to the forwrd nd reverse fields present during clibrtion, the flush short lters the forwrd nd reverse fields t the reference plne of the 2.4 mm test port. This is becuse the flush short does not exhibit step in the outer conductor. For the mismtch the fields during clibrtion nd mesurement re very similr. Nonetheless smll differences in e.g. the chmfers of the outer conductors of clibrtion stndrds nd mismtch cuse the non zero difference. The smll differences in figure despite the big differences in test port dimeter indicte tht using extended -prmeters for compring devices between different lbs with slightly differing test-ports is possible. Higher modes re present in the beds of the 2.4 mm test port t frequencies bove 5 GHz. All other components,.85 mm test port dpter nd stndrds, work without higher Figure. hown re the mgnitude of reflection coefficients of mle.85 mm mismtch versus frequency. The mesurements were mde on.85 mm test port (.85 mm) nd 2.4 mm test port (2.4 mm) tht hd previously been clibrted with the sme.85 mm clibrtion stndrds. The mgnitude of the complex difference between the two trces is shown s well (Difference). of flush short Figure 2. The DUT is flush short mesured with the sme setup s for figure. The differences re lrger thn for figure becuse the forwrd nd reverse field re different during clibrtion nd DUT mesurement. This difference rises from the step in the outer conductor which is present during clibrtion but not during the mesurement of the flush short. modes up to 67 GHz. If the DUT chnges the forwrd nd reverse field, higher modes will be visible in the result. This chnge of fields in the reference plne ffects s well the electromgnetic fields in the test port dpter. This is the reson why the higher modes re visible in the results of the flush short. In the cse of the mismtch, the higher modes re brely noticeble becuse there is not too much chnge in the electromgnetic fields in the reference plne. The effect of the higher modes is s well proportionl to the reflectivity of the DUT. 6. Conclusion 2.4 mm.85 mm Difference Extended -prmeters re suitble for device chrcteriztion t reference plnes close to or even in regions with nonidel wve guide. They present n extension of the concept of 28

10 Metrologi 52 (25) 2 pseudo -prmeters nd cn be pproximted with ordinry simultion progrms for electromgnetic wves by using the mode dpter technique. The key dvntge of extended -prmeters is tht reference plnes for chrcteriztion cn be set to loctions tht mechniclly seprte DUT nd mesurement system, i.e. connectors. As consequence, cscding of -prmeters nd comprisons between mesurements done t different lbortories become possible without further effort. It hs to be noted tht extended -prmeters in connector pirs suffer from n uncertinty. Exchnging one hlf-connector of the pir chnges not only the rtio of forwrd nd reverse field but it slightly chnges the shpe of these fields. It hs been demonstrted with prcticl exmples tht one cn quntify the error of chnging field shpe. More importntly, it hs been shown tht this error is smll compred to just ignoring connector effects. From metrologicl point of view this is highly desirble. The trcebility chin down to the end user cn be estblished with less difficulties nd more ccurtely. The role of the I system is strengthened for one of the fundmentl quntities in rdio frequency nd microwve mesurements. nd iwf = i WR. (A.7) Using (A.6), (A.7) nd (5), (6) one finds tht F = br (A.8) R = b F. (A.9) Using (7), (8), (A.6), (A.8) nd (A.9) on (A.) leds to pw = p ( + b)( b). (A.) The definition of p is p = v i = W Z Re ( Z ). ref Tking the rel prt of (A.) yields 2 2 Z Re ( p ) = + b 2Im( b ) Im ( ref ) W W. Re ( Z ) ref ref (A.) (A.2) This eqution is s well known from []. For pseudo -prmeters only, b nd Z ref re needed to compute the power. Acknowledgments The uthors would like to thnk K Wong from Agilent Technologies, Dyln Willims from Ntionl Institute of tndrds nd Technology nd P Leuchtmnn from Eidgenösische Technische Hochschule Zurich (ETH Zurich) for helpful discussions. Appendix A. Power The power is defined by the power integrl (5). Evluting this integrl yields 2 WF WFR WRF 2 WR (A.) pw = c+ p + c+ cp + ccp + + c p. The powers in (A.) re defined s = p WF EF HF n d (A.2) = p E H n d WFR F R (A.3) = p E H n d WRF R F (A.4) = p WR ER HR n d. (A.5) The forwrd nd reverse field fctors c + nd c re linked to the extended wves nd b by equtions () (8). Eqution (A.) cn be drsticlly simplified for extended -prmeters in reference plnes tht re s well suitble for the definition of pseudo -prmeters. In such cses it is pwf = pwfr = pwrf = pwr (A.6) References [] Mrks R B nd Willims D F 992 A generl wveguide circuit theory J. Res. Ntl Inst. tnd. Technol [2] Hoer Cletus A, Judish R M, Juroshek J R nd Engen G F 25 Theory, uncertinty nlysis, nd sttisticl control for the nist 2 8 GHz dul 6-port utomtic network nlyzer NIT pecil Publiction 25 [3] Willims D F 2 Rectngulr-wveguide vector-networknlyzer clibrtions with imperfect test ports Microwve Mesurement ymp. (ARFTG), 2 76th ARFTG (Clerwter Bech, FL, 3 November 3 December 2) pp 8 [4] Willims D F 22 Comprison of sub-millimeter-wve scttering-prmeter clibrtions with imperfect electricl ports IEEE Trns. Terhertz ci. Technol [5] Whinnery J R, Jmieson H W nd Robbins T E 944 Coxil line discontinuities Proc. IRE [6] Agilent Technologies 2 pecifying clibrtion stndrds nd kits for gilent vector network nlyzers, ppliction note 287- ( pdf/ en.pdf) [7] trtton J A 27 Coxil lines Electromgnetic Theory (Hoboken: Wiley) pp [8] Dywitt W C 99 First-order symmetric modes for slightly lossy coxil trnsmission line IEEE Trns. Microwve Theory Tech [9] IEC Interntionl Electrotechnicl Comission 27 Rdiofrequency connectors IEC 66 9-XX [] IEEE Instrumenttion nd Mesurement ociety 27 tndrd for Precision Coxil Connectors DC to GHz (New York: IEEE) P 287 [] Hoffmnn J P, Leuchtmnn P nd Vhldieck R 27 Pin gp investigtions for the.85 mm coxil connector Europen Microwve Conf. (Munich, Germny, 8 October 27) pp [2] Juroshek J R, Hoer C A nd Kiser R F 989 Clibrting network nlyzers with imperfect test ports IEEE Trns. Instrum. Mes