Prgress In Electrmagnetics Research Letters, Vl. 39, 81 95, 213 A NEW QUALITY INDICATOR FOR WIDEBAND UN- TERMINATION BY USING REFLECTIVE STANDARDS Jsé Fays-Fernández *, Antni Lzan-Guerrer, and Juan Mnzó-Cabrera Departament de Tecnlgías de la Infrmación y las Cmunicacines, Universidad Plitécnica de Cartagena, Plaza del Hspital 1, Cartagena 322, Spain Abstract In this paper a new quality indicatr fr tw-tier calibratin prcedures that use nly reflectin standards is presented and applied t caxial-t-waveguide transitins. The quality indicatr is based n the algebraic cnditining f the system f equatins slved fr btaining transitin characteristics. The study has been carried ut in a wide bandwidth as a difference with previus wrks. The btained results indicate that a threshld value fr this indicatr arund 1% can be established. Fr values belw this limit the errr grws t unacceptable values. Additinally, it has been shwn that an expnential relatinship between quality indicatr and the errr can be predicted. 1. INTRODUCTION Caxial-t-waveguide transitins are emplyed in waveguide measurements such as permittivity estimatin, characterizatin f waveguide devices, etcetera, since vectr netwrk analyzers (VNA) emply caxial interfaces. Therefre, it is necessary t have a gd characterizatin f these structures in rder t perfrm precise measurements in waveguide technlgy. Hwever, the measurement f transitin scattering parameters is nt straightfrward and these must be characterized frm measurements made at the VNA reference plane when knwn standards are embedded in the waveguide prt f the transitin. This prcess is referred t as unterminating [1]. Received 8 March 213, Accepted 8 April 213, Scheduled 14 April 213 * Crrespnding authr: Jsé Fays-Fernández (jse.fays@upct.es).
82 Fays-Fernández, Lzan-Guerrer, and Mnzó-Cabrera There are several prcedures fr unterminating depending n the standards types and errr minimizatin prcedures. Fr instance penshrt-lad (OSL) r thru-reflect-line (TRL) standards may be used [1 3] fr unterminating. Hwever, several studies based n iterative appraches [1, 4] shw that it is pssible t use redundant standards in rder t increase accuracy versus cnventinal calibratin prcedures. Freeware sftware packages such as MultiCal and StatistiCAL develped at the Natinal Institute f Standards and Technlgy (NIST) implement calibratin algrithms based n different studies such as [1, 2, 4 7] that perfrm bth ne and tw-tier deembedding. The tw-tier calibratin prcedure can be used t electrically characterize prbe heads r ther cmpnents such as caxial-twaveguide transitins and can handle up t 4 different standard types. In [8] is presented a calibratin prcedure specifically applied t caxial-t-waveguide transitins knwn as the three-cavity technique. It makes use f three cavities and their input reflectin cefficients fr btaining three linear equatins with which the tw-prt scattering matrix f the transitin can be calculated, by impressing the incident wave in the caxial line nly. The main drawback f this methd is a restrictin regarding the phases f the reflectin cefficient f the shrtcircuited waveguide sectins, which must nt have 36 differences at a given frequency. The authrs suggest using phase differences f 12 and 24. In [9], a tw-tier inverse technique fr characterizing caxial t waveguide transitins based n the use f genetic algrithms and the gradient descent methd is described and cmpared t different well-knwn calibratin techniques prviding gd results. In [1] this prcedure is extended and bth the transitins and the device under test are simultaneusly characterized. Nne f previus wrks, hwever, prvide estimatins r indicatins f hw gd the calibratin prcedure is and n clue is prvided abut the selectin criteria fr the different standards. In fact, this is a key issue that has nt been studied in depth althugh sme attempts have been made. Bauer and Penfield [2], fr instance, shw in a DC study that the calibratin precisin is better as the number f standards grws. Her [11] studies the best length fr a precisin transmissin line fr calibrating netwrk analyzers and cncludes that lines with phases near 18 r multiples shuld be avided in rder t achieve a well-cnditined equatin system. Williams [12] uses the nrmalized variance f the unterminating prcedure as indicatr fr the calibratin quality as a functin f the number f thrus and shrts and cncludes that the best calibratin prcess is achieved when the maximum number f available shrts is used. Maury et al. [13] recmmend that fr line-reflect-line (LRL) calibratin prcedures lines
Prgress In Electrmagnetics Research Letters, Vl. 39, 213 83 shuld shw phase differences between 3 and 15. All these studies are carried ut at single frequencies r narrw bandwidths. Hwever, the selectin f standard number r phases is nt clear. Fr instance, Adalev et al. [14] shw that using a greater number f standards des nt guarantee that accuracy is increased. On the cntrary, Marks [4] shws that the additinal infrmatin prvided by redundant standards minimizes the effects f randm errrs, such as thse caused by imperfect cnnectr repeatability. The resulting methd exhibits imprvements in bth accuracy and bandwidth ver cnventinal methds. Cmmercial calibratin kits, which cnsist f 3 reflectin standards typically, might be used fr unterminating caxial-twaveguide transitins. Nevertheless, adding sme extra standards fr the unterminatin prcess might increase the calibratin quality [4]. In this paper, we present a simple indicatr fr standard selectin and we relate this indicatr t calibratin errrs and final accuracy. We exclusively use reflectin standards in rder t calibrate caxialt-waveguide transitins and we assess the calibratin quality in terms f the algebraic cnditining f the system f equatins slved fr btaining transitin characteristics. As a difference frm previus wrks, we carry ut this study in a 1 GHz bandwidth in rder t evaluate the utility f the prpsed calibratin quality indicatr as a functin f frequency. In additin, in the result sectin a cmparisn f the system f equatins quality when using a cmmercial calibratin kit standalne and when using it supplemented with extra standards is presented. 2. CALIBRATION PROCEDURE AND QUALITY ESTIMATION The bjective f the calibratin prcedure is t accurately estimate the scattering parameters f the caxial-t-waveguide transitin. This kind f devices can be mdeled as a 2-prt micrwave netwrk. In this study, it is assumed that the prt 1 is linked t its caxial interface, whereas prt 2 is t its waveguide interface. Let us represent the transitin thrugh its scattering parameter matrix: [ ] S11 S [S] = 12 (1) S 21 S 22 This investigatin used a cllectin f waveguide shrt standards with different lengths that were cnnected t the prt tw f the transitin. This means that measurements were nly available at the caxial prt. Let us call Γ i t the ith reflectin cefficient measured at prt ne
84 Fays-Fernández, Lzan-Guerrer, and Mnzó-Cabrera when the ith shrt standard is applied t the waveguide prt f the transitin and Γ Li as the theretical reflectin cefficient f the ith shrt standard with i [1, N]. N is the number f standards used during the calibratin prcess. The reflectin cefficients f the shrted guides can be cmputed as Γ Li = e j2βl i, where β is the waveguide prpagatin cnstant and L i are the lengths f the shrted guides. Figure 1 summarizes this explanatin. VECTORIAL NETWORK ANALYZER Γ i L i [, 41.96 ] cm TRANSITION COAXIAL-TO-WAVEGUIDE SHORTED WAVEGUIDES (STANDARDS) Γ L i Figure 1. Scheme shwing the different magnitudes emplyed during calibratin prcedure. By expanding the methdlgy emplyed in [8], ne can express the equatins relating Γ i, Γ Li and [S] as expressed in Equatin (2) when N shrts are emplyed as standards: 1 Γ 1 Γ L1 Γ L1 1 Γ 2 Γ L2 Γ L2 [ 1 Γ 3 Γ L3 Γ L3... 1 Γ N Γ LN Γ LN S 11 S 22 S 12 S 21 S 11 S 22 ] = Γ 1 Γ 2 Γ 3. Γ N Suppsing that each shrt prvides an independent equatin in (2), the system can be easily slved by algebraic prcedures when its matrix has a full rank and N = 3. Nevertheless, when N > 3 the system f equatins is verdetermined, and it must be slved by applying Least Square Methd (LSM) as implemented in Matlab TM sftware. In rder t assess the quality f the calibratin prcess, it was used the 2-nrm cnditin number f the first matrix in Equatin (2) (2)
Prgress In Electrmagnetics Research Letters, Vl. 39, 213 85 as: 1 Γ 1 Γ L1 Γ L1 1 Γ 2 Γ L2 Γ L2 C = cnd 1 Γ 3 Γ L3 Γ L3... 1 Γ N Γ LN Γ LN 1 Γ 1 Γ L1 Γ L1 1 Γ 1 Γ L1 Γ L1 1 Γ 2 Γ L2 Γ L2 1 Γ 2 Γ L2 Γ L2 = 1 Γ 3 Γ L3 Γ L3 inv 1 Γ 3 Γ L3 Γ L3...... 1 Γ N Γ LN Γ LN 1 Γ N Γ LN Γ LN In Equatin (3) the peratr indicates the 2-nrm f the matrix, and inv is the inverse matrix peratr. C yields a value f 1 fr well-cnditined systems r it will tend t infinity as the cnditin system gets wrse tward an undetermined system. Therefre, it seems mre cnvenient t express the quality f the system f equatins as a percentage frm % t 1%. The fllwing relatinship was chsen in rder t estimate the quality f the slved system f equatins at each frequency pint: (3) Q i (f i ) = 1/C(f i ) (%) (4) where f i is the ith frequency pint where the quality indicatr is cmputed. The prcedure f slving the system f Equatin (2) and btaining the quality indicatr was simultaneusly applied t all the frequency pints f the measurement by using vectr peratins and n iterative prcedures were emplyed. The Equatin (5) was used fr averaging values f the quality indicatr within the cnsidered bandwidth [2, 3] GHz with k sampled frequencies. Q i (f i ) i=f 1 Q average = (5) k Fr ill-cnditined systems the quality indicatr will tend t %, and fr well-cnditined systems it will apprach t 1%. This indicatr system must be carried ut fr each frequency pint within the cnsidered bandwidth. It is difficult t estimate a threshld value fr Q i in rder t decide whether the calibratin is precise enugh r nt. Hwever, a limit value f 1% was set due t multiple bservatins carried ut with the experimental data. Cnsequently, the calibratin prcess is cnsidered having a lw quality fr Q < 1%. f k
86 Fays-Fernández, Lzan-Guerrer, and Mnzó-Cabrera Fr validatin purpses, the errr was cmputed at each frequency pint as shwn in Equatin (6): Errr(f i ) = Γ VNA n (f i ) Γ CAL n (f i ) (6) where Errr(f i ) is the errr cmputed fr the ith frequency pint, Γ VNA n (f i ) the reflectin cefficient f the nth standard chsen fr the validatin measured at the caxial prt f the transitin, and Γ CAL n (f i ) the cmputed reflectin cefficient calculated as the scattering parameters f the unterminated transitin cascaded t the chsen standard. The average errr within the cnsidered bandwidth with k sampled frequencies has been cmputed as: f k Errr(f i ) i=f i Errr average = k (7) Scattering parameters (mdulus) 1.9.8.7.5.3.2.1 Unterminated Caxial-t-Waveguide transitin using standards: 21 22 39 S 11 S 12 S 21 Quality 1 9 8 7 5 3 2 1 System equatin quality (percentage) Scattering parameters (phase) 18 135 9 45-45 -9-135 -18 Figure 2. Extracted S-parameters fr the caxial-t-waveguide transitin using a bad cmbinatin f standards fr its unterminatin. S 11 S 12S 21
Prgress In Electrmagnetics Research Letters, Vl. 39, 213 87 3. EXPERIMENTAL SETUP A Rhde&Schwarz ZVA 67 VNA has been used in rder t measure the scattering matrix frequency behavir f the caxial-t-waveguide transitins when using different standards. The reflectin standards were built as a cmbinatin f a shrtplate r any f tw ffset shrt-circuited waveguides (lengths f 1.82 and 5.46 cm) with fur different waveguide sectins available fr building up the transmissin standards. As a result, up t 48 different reflectin standards with physical lengths ranging frm cm up t 41.96 cm were available fr shrt-circuiting the waveguide prt f each transitin as shwn in Figure 1. The cmplete list f the shrt-circuited waveguide sectins used in the calibratin prcess and their physical lengths are prvided in Appendix A. In each measurement, 11 pints were sampled hmgeneusly distributed within the [2, 3] GHz frequency range, which yields a 1 khz frequency reslutin. Therefre, it was ensured that waveguide cmpnents wrked nly with the TE 1 main mde. A Rhde & Schwarz ZV-Z32 PC 3.5 fixed matched calibratin kit 2 1.8 1.6 1.4 1.2 1.8.6.4.2 18 135 9 45-45 -9-135 -18 Verificatin with shrtlad #48 Unterminatin standards used: #21 #22 #39 S VNA 11 Std.#48 S EST 11 Std.#48 Verificatin with shrtlad #48 Unterminatin standards used: #21 #22 #39 S VNA 11 Std.#48 S EST 11 Std.#48 Figure 3. Verificatin f the untermined transitin using a bad cmbinatin f standards using standard #48 as lad.
88 Fays-Fernández, Lzan-Guerrer, and Mnzó-Cabrera was emplyed in rder t calibrate the VNA at prt 1 f the transitin. The caxial-t-waveguide transitin that was used t carry ut the study belngs t a Cntinental Micrwave WCK34-HP waveguide calibratin kit [15]. All the waveguide sectins used t build the ffset shrt standards accmplished the WR-34 standard. 4. RESULTS Figure 2 shws a calibratin prcedure using shrts {#21, #22, #39}. The btained quality indicatr f the system f equatins Q i is near zer fr all frequencies, and bth the phase and magnitude values shw a nisy and errneus behavir. In this case, the phase values and phase slpes fr shrts #21 and #22 were very similar and therefre the equatin system had nt enugh infrmatin t prperly carry ut the characterizatin f the transitin. Figure 3 shws the cmparisn f the measured reflectin cefficient f a validatin standard that was nt used during the calibratin prcedure and the reflectin cefficient f that standard cmputed by using the btained values f the scattering parameters f the transitin. The standard #48 was chsen fr the prcedure Scattering parameters (mdulus) 1.9.8.7.5.3.2.1 Unterminated Caxial-t-Waveguide transitin using standards: 1 2 5 S 11 S 12 S 21 Quality 1 9 8 7 5 3 2 1 System equatin quality (percentage) Scattering parameters (phase) 18 135 9 45-45 -9-135 -18 Figure 4. Extracted S-parameters fr the caxial-t-waveguide transitin using a gd cmbinatin f standards fr its unterminatin. S 11 S 12S 21
Prgress In Electrmagnetics Research Letters, Vl. 39, 213 89 validatin. It can be bserved that the transitin characterizatin is cmpletely errneus: magnitude and phase values were nt crrectly cmputed due t an imprecise calibratin prcedure. Figure 4 shws a calibratin prcedure using shrts {#1, #2, #5}. In this case, the calibratin quality indicatr stays abve the 1% threshld alng the bandwidth, s the calibratin prcess can be cnsidered precise. In fact, nisy values fr the [S] matrix f the transitin are n lnger bserved. This is cnfirmed in Figure 5, where the magnitude f the measured reflectin cefficient fr validatin standard #48 is cmpared t the cncatenatin f the scattering matrix f the transitin btained with the calibratin prcess and the theretical value f the reflectin cefficient f the validatin standard Γ L48. The agreement is cmplete bth in magnitude and phase fr the cnsidered bandwidth. Therefre, it can be bserved that there is a strng crrelatin between the quality indicatr prpsed in this paper and the precisin f the calibratin prcedure. Fr values f Q greater than 1% the calibratin prcess culd be cnsidered f acceptable, whereas errneus calibratins are expected t ccur fr values lwer than this threshld. 1.9.8.7.6.5.4.3.2.1 18 135 9 45-45 -9-135 VNA S 11 Std.#48 EST S 11 Std.#48 Verificatin with shrtlad #48 Unterminatin standards used: #1 #2 #5 Verificatin with shrtlad #48 Unterminatin standards used: #1 #2 #5 VNA S 11 Std.#48 EST S 11 Std.#48-18 Figure 5. Verificatin f the untermined transitin using a gd cmbinatin f standards using standard #48 as lad.
9 Fays-Fernández, Lzan-Guerrer, and Mnzó-Cabrera.12.1.8 Verificatin errr with shrtlad 48. Unterminatin stds.: 1 4 5 6 16 2 24 27 46 47 Verificatin errr with shrtlad 48. Unterminatin stds.: 2 3 6 7 14 15 17 18 21 22 25 26 29 3 31 32 38 39 43 44.12.1.8 S 11.6.4 11 S.6.4.2.2 Scattering parameters (mdulus) 2 2.2 2.4 2.6 2.8 3 Unterminated Caxial-t-Waveguide transitin using standards: 1 4 5 6 16 2 24 27 46 47 1 1.9 9.8 8.7 7.5.3.2.1 (a) S 11 S 12 S 21 Quality 3 2 1 2 2.2 2.4 2.6 2.8 3 (c) 5 System equatin quality (percentage) Scattering parameters (mdulus) 2 2.2 2.4 2.6 2.8 3 Unterminated Caxial-t-Waveguide transitin using standards: 2 3 6 7 14 15 17 18 21 22 25 26 29 3 31 32 38 39 43 44 1 9 1.9.8.7.5.3.2.1 (b) S 11 S 12 S 21 Quality 2 2.2 2.4 2.6 2.8 3 Figure 6. Cmparisn f (a), (b) validatins and (c), (d) system qualities f tw different calibratin prcesses by using (a), (c) 1 and (b), (d) 2 standards. (d) 8 7 5 3 2 1 System equatin quality (percentage) Figure 6 shws a cmparisn fr tw calibratin prcedures where 1 and 2 shrts are emplyed. In the first case, shwn in the left side f the figure, 1 shrts with minimum phase cincidences within the cnsidered bandwidth were chsen, which are {#1 #4 #5 #6 #16 #2 #24 #27 #46 #47}. That case is cmpared t a calibratin prcedure where shrts {#2 #3 #6 #7 #14 #15 #17 #18 #21 #22 #25 #26 #29 #3 #31 #32 #38 #39 #43 #44} were emplyed, shwn at the right side f the Figure 6. It can be bserved that bth calibratin prcedures have very similar values fr the quality indicatr. In fact, the errr fr bth prcedures when cmparing t the validatin standard number #48 is very similar fr bth cases. Mrever, it can be stated that using a greater number f standards des nt necessarily guarantee that accuracy is increased. Figure 7 shws the frequency-averaged quality indicatr Q average and the average errr Errr average as a functin f the number f standards by fllwing different shrt selectin strategies. In the first selectin strategy, the standards were sequentially cllected in
Prgress In Electrmagnetics Research Letters, Vl. 39, 213 91 1% 9% 8% 7% 6% 5% 4% 3% 2% 1% % Unterminatin sequence strategy N.1 System Eq. Quality Verificatin errr 3 6 9 12 15 18 21 24 27 3 33 36 39 42 45 Amunt f standards used 1% 9% 8% 7% 6% 5% 4% 3% 2% 1% % Unterminatin sequence strategy N.2 System Eq. Quality Verificatin errr 3 6 9 12 15 18 21 24 27 3 33 36 39 42 45 Amunt f standards used 1% 9% 8% 7% 6% 5% 4% 3% 2% 1% Unterminatin sequence strategy N.3 System Eq. Quality Verificatin errr % 3 6 9 12 15 18 21 24 27 3 33 36 39 42 45 Amunt f standards used 1% 9% 8% 7% 6% 5% 4% 3% 2% 1% Unterminatin sequence strategy N.4 System Eq. Quality Verificatin errr % 3 6 9 12 15 18 21 24 27 3 33 36 39 42 45 Amunt f standards used Figure 7. Frequency-averaged quality indicatr Q average and average errr Errr average as a functin f the number f standards fr different standard selectin strategies. Sequence strategy number 1: {1 2 3 4... 47}, sequence strategy number 2: {9 18 47 27 37 17 23 34 25 46 39 2 4 13 1 16 4 5 31 11 35 42 36 19 45 15 8 32 41 33 28 1 38 3 22 14 24 3 26 12 29 6 21 2 7 43 44}, sequence strategy number 3: {19 24 1 4 47 27 5 6 16 37 45 46 41 42 9 36 1 34 35 4 11 12 13 17 18 28 29 26 25 2 3 32 33 31 3 7 8 44 43 14 15 39 38 2 23 21 22}, sequence strategy number 4: {22 21 23 2 38 39 15 14 43 44 8 7 3 31 33 32 3 2 25 26 29 28 18 17 13 12 11 4 35 34 1 36 9 42 41 46 45 37 16 6 5 27 47 4 1 24 19}. grups f 3, 4,... up t the 47 available standards in ascending length rder. In sequence strategy number 2, the standards were cllected frm a randmly rdered sequence. Fr the unterminatin sequence strategy number 3, thse shrts with less phase cincidences within the cnsidered bandwidth were cllected first and thse with mre phase cincidences last. Fr sequence strategy number 4, the selectin strategy f shrts was the reverse versin f strategy number 3. The calibratin prcedure was cmpared t standard #48 in all cases. Several cmmn behavirs are bserved in all cases. Fr instance,
92 Fays-Fernández, Lzan-Guerrer, and Mnzó-Cabrera Q average increases as the number f standards used fr the calibratin rises as a general trend. Hwever, it cannt be stated that this is a mntnic behavir frm the results bserved when using the unterminatin strategy number 1. In this figure it is nticeable that using 12 standards gives a better cnditining fr the system f equatins rather than using greater amunts starting frm 13 t 3 shrts. Additinally, it can als be bserved that the mre standards used during the calibratin prcess the lwer the average errr Errr average is. Hwever, it is bserved that the reductin f errr with the number f standards is nt linear but fllws an expnential behavir. Furthermre, fr quality indicatr values near r belw 1% the errr grws substantially. Fr higher values f Q average the errr reduces at a much lwer rate when the number f standards used grws. An interesting cnclusin that can be derived frm Figure 7 is that the selectin criterin f the shrts, when the number f standards is lw, is mre critical than in the case f using mre standards. In fact, selectin strategy number 3, where the phase cincidences are minimized, prvides higher Q average values and lwer Errr average values than the ther selectin strategies fr lw numbers f standards. Therefre, it seems that a gd selectin criterin wuld be t minimize phase cincidences amng shrts within the cnsidered bandwidth. Fr higher number f standards phase cincidences grw and the errr and quality indicatr imprve very slwly. 1 9 Errr in verificatin using cntrl standard #48 Vs. Quality f unterminating system equatins Verificatin Errr Vs. Quality Fitting curve 8 7 Errr in predictin (%) 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 1 System Equatin Quality (%) Figure 8. Expnential relatinship between Errr average and Q average.
Prgress In Electrmagnetics Research Letters, Vl. 39, 213 93 Finally, Figure 8 shws Errr average as a functin f Q average fr all the data btained in the previus selectin sequences. An expnential functin f the type f(x) = a e bx + c e dx has been fitted t experimental data in rder t mdel the relatinship between Errr average and Q average. The cefficients btained fr the fitting are a = 91.7, b =.4252, c = 9.584 and d =.7967. Cmmercial waveguide calibratin kits are designed fr a specific peratin bandwidth. Nevertheless, such a wideband culd be expanded cnveniently by adding mre calibratin standards. Figure 9 shws a practical use f this research fr demnstrating that supplementing the Cntinental Micrwave WCK-34 Waveguide Calibratin Kit (identified in this paper as standards #1, #2 and #6) with tw extra standards (standards #11 and #13) imprves the unterminatin quality, while it bradens its peratinal bandwidth cvering the full mnmde waveguide frequency range [1.74, 3.47] GHz. 1 9 Unterminatin quality imprvement fr the cmmercial WCK-34 Calibratin Kit WCK-34 WCK-34 supplemented 8 7 6 Quality (%) 5 4 3 2 1 1 1.5 2 2.5 3 3.5 4 Figure 9. Example f a desterminatin quality imprvement f a cmmercial waveguide calibratin kit by supplementing it with tw additinal standards. 5. CONCLUSIONS The relatinship between the cnditining f the systems f equatins and the calibratin errr when using shrt standards in rder t calculate the scattering parameters f caxial-t-waveguide transitins
94 Fays-Fernández, Lzan-Guerrer, and Mnzó-Cabrera has been studied. Frm the btained results, an expnential relatinship between the calibratin quality indicatr and the calibratin errr has been established. A threshld value fr this quality indicatr has been set t 1%, belw this limit the calibratin errr grws expnentially. The usage f redundant standards has increased the quality indicatr and reduced the errr when cmparing with a validating standard as a general trend. Fr lwer number f standards, nevertheless, the selectin criterin is f utmst imprtance. In this last case, it has als been shwn that chsing thse shrts with less number f phase cincidences within the cnsidered bandwidth prvides better quality indicatrs and lwer errr values. APPENDIX A. 48 Chart f standards fr reflexin 44 4 36 32 Standard N. 28 24 2 16 12 8 4 1 Std.# Std.# Std.# Length (cm) Length (cm) Length (cm) 1 17 33. 17.15 24.81 2 18 34 1.82 17.35 25.67 3 19 35 2. 18.11 27.31 4 2 36 3.82 18.97 29.31 5 21 37 4.5 19.15 3. 6 22 38 5.46 19.17 31.82 7 23 39 6.32 19.35 32. 8 24 4 6.5 2.11 33.82 9 25 41 7.46 2.97 34.5 1 26 42 8.32 21.17 35.46 11 27 43 9.96 21.85 36.32 12 28 44 11.96 22.61 36.5 13 29 45 12.65 22.81 37.46 14 3 46 14.47 23.67 38.32 15 31 47 14.65 23.85 39.96 16 32 48 16.47 24.61 41.96 5 1 15 2 25 3 35 4 45 Length (cm) Figure A1. Number and physical length f the shrt standards used in the calibratin prcess. REFERENCES 1. Williams, D., De-embedding and unterminating micrwave fixtures with nnlinear least squares, IEEE Trans. Micrw. Thery Tech., Vl. 38, 787 791, 199. 2. Bauer, R. F. and P. Penfield, De-embedding and unterminating, IEEE Trans. Micrw. Thery Tech., Vl. 22, 282 288, 1974.
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