GHOSH, RAHUL. Error analysis of Through Reflect Line method for calibrating

Transcription

1 Absrc GHOSH, RAHUL. Error lyss of Through Rflc L mhod for clbrg mcrowv msurms. (Udr h drco of Profssor Mchl B. Sr) Ucry rsmsso l bsd clbro chqus for mcrowv msurms r sudd wh h m of dfyg h opmum clbro codos. Vro h drmd rror prmrs udr cr codos Through Rflc L (TRL) clbro mhod du o vro h ukow rflco coffc r lyzd xprmlly d horclly. I s foud h msurd ucry s mmzd wh h rflco sdrd s hr shor or op. Th TRL clbro mhod s comprd wh h Through-L (TL) clbro procdur. Th mjor cuss of ucrs hr TRL clbro r bs h TL clbro chqu du o h syhszd rflco sdrd.

2 Error Alyss of h Through Rflc L Mhod for Clbrg Mcrowv Msurms By Rhul Ghosh A hss submd o h Grdu Fculy of Norh Crol S Uvrsy prl fulfllm of h rqurms for h Dgr of Msr of Scc Elcrcl Egrg Rlgh 2003 Approvd by:- Prof. Gluc Lzz Prof. Grff Blbro Prof. Mchl Sr, Chr of Advsory Comm

3 Bogrphcl Summry Rhul Ghosh ws bor o 8h Jury 1979 Clcu, Id. H rcvd hs lmry d scodry duco Es Afrc. H rcvd h Bchlor of Scc dgr wh mjor Elcrocs from o Th Pu Uvrsy, Id, From July 2000 ul July 2001 h compld h Pos-Grdu Dplom cours VLSI Dsg Bmppr Igro Tchologs Pv Ld. I Augus 2001 h ws dmd o Norh Crol S Uvrsy o sudy for h Msr of Scc Elcrcl Egrg. Hs rs s h fld of hrdwr dsg.

4 Ackowldgms I wsh o xprss my pprco o my dvsor Dr. Mchl Sr for hs couous suppor, gudc d hours of -dph dscusso h md hs work possbl. I would lso lk o hk Dr. Grff Blbro d Dr. Gluc Lzz for srvg o my comm. I would lk o xprss my pprco o Mr. Jysh Nh d Mr. Mrk Buff for hr hlp d pc durg h lghy hours kg h msurms d clrg my doubs. My spcl hks go o ll of my ohr frds for hr morl suppor d courgm. Ad flly, I would lk o xprss my grud o my ps d prs srucors who ugh m d who provdd m usful formo bou lcrcl grg. Ths work s ddcd o my l fhr Aloy Kumr Ghosh.

5 Tbl of Cos Ls of Fgurs v 1 Iroduco Movo Thss Ovrvw 02 2 Lrur Ovrvw Iroduco o Smh Chrs d VNA Mcrosrp Dvc Msurms Msurm Errors Error Corrco Tchqus Msurms Clbro Sdrds O por Clbro Two Por Clbro SOLT Clbro TSD Clbro TRL Clbro LRM Clbro TL Clbro Cocluso TRL Clbro Sps TRL Clbro TRL D-Embddg Soluo Lmos of TRL Clbro Cocluso Error Alyss of TRL Clbro Proposd rror TRL clbro Alyss of h proposd hr Error TRL Clbro Comprsos bw TRL d TL Clbros.. 35 v

6 4.3 Cocluso Exprml Rsuls Grl Msurm Sps Exprm Sup MATLAB Smulo Rsuls TRL Clbro rsuls obd from usg shor d op sdrds TRL clbro for r flld rsmsso l TRL clbro for 20dB uor TRL clbro for combo of 20 db uor d rsmsso l Comprso of rsuls from TRL clbros for h hr dvcs TRL Clbro rsuls obd from usg 20 db uor log wh shor d op sdrds TRL clbro for r flld rsmsso l TRL clbro for 20dB uor TRL clbro for combo of 20 db uor d rsmsso l Comprso of rsuls from TRL clbros for h hr dvcs wh h 20dB uor cocd o h shor d op sdrd Coclusos 73 Rfrcs 74 Appdx 76 A Dld TRL D-Embddg Soluo B MATLAB Cod for TRL D-mbddg...81 v

7 Ls of Fgurs Fgur 2.1: Rfrc pls o-wfr mcrowv msurm.. 04 Fgur 2.2: O por OSL clbro sdrds Fgur2.3: O por rror modl 07 Fgur 2.4: Two por forwrd rror modl.. 08 Fgur 2.5: Modfd gh rm rror modl.. 11 Fgur 2.6: Two por rror modl for symmrc fxurs 14 Fgur 2.7: Rducd wo por rror modl for symmrc fxurs 15 Fgur 2.8: Sgl flow grph of rducd wo por rror modl for Symmrc fxurs 16 Fgur 2.9 Sgl flow grph of dl shor s rflco coffc. 17 Fgur 3.1 Fcous wo por rror boxs A, B wh h hr Sdrds.. 21 Fgur 5.1 Rflco coffcs of h shor sdrd 47 Fgur 5.2 Rflco coffcs of h op sdrd 48 Fgur 5.3 Rflco coffcs of h mch sdrd.. 49 Fgurs Clculd rflco coffc from TRL lgorhm for r flld rsmsso l. 50 Fgur 5.7 Error fcor for dffr sdrds usd TRL procdur.. 52 Fgurs Clculd rflco coffc from h TRL lgorhm for 20dB uor 54 Fgur 5.11 Error fcor for dffr sdrds usd TRL procdur for h 20dB uor.. 56 Fgurs Clculd rflco coffc from h TRL lgorhm for 20dB uor d r flld rsmsso l.. 57 Fgur 5.15 Error fcor for dffr sdrds usd TRL procdur for h 20dB uor d r flld rsmsso l 59 Fgur 5.16 Rflco coffcs of h 20 db uor d op sdrd. 60 v

8 Fgur 5.17 Fgurs Fgur 5.21 Fgurs Fgur 5.25 Fgurs Fgur 5.29 Rflco coffcs of h 20 db uor d shor sdrd 61 Clculd rflco coffc from h TRL lgorhm for rl.. 63 Error fcor for dffr modfd sdrds usd TRL procdur for h rl 65 Clculd rflco coffc from h TRL lgorhm for 20dB uor wh modfd sdrds 66 Error fcor for dffr modfd sdrds usd TRL procdur for h 20dB uor. 68 Clculd rflco coffc from h TRL lgorhm for 20dB uor d r flld rsmsso l wh modfd sdrds. 69 Error fcor for dffr modfd sdrds usd TRL procdur for h 20dB uor d rl.. 70 v

9 Chpr 1: Iroduco 1.1 Movo Th m movo for hs work s h rc dvlopm of w chqu, h Through L (TL) clbro procdur dvlopd by M. B. Sr, S. B. Goldbrg, G. R, P. D. Frzo, I. Turlk d J. S. Ks [1]. Udr cr codos hs s rplcm for h Through-Rflc-L (TRL) clbro procdur dvlopd by G.F. Eg d C.A. Hor [2]. Ev hough TRL s h mos wdly usd clbro chqu, hs work rs o dlv o h rsos why TL clbro pprs o hv br chrcrscs h h TRL clbro chqu. I prculr, h TRL procdur pprs o hv glchs wh h lgh of h l sdrd s clos o gr mulpl of hlf wvlgh. Ths glchs r o ppr TL clbro. Th TRL clbro chqu suffrs from hr rrors h rsul ucrs h msurm wh h phs dly log h l s mulpl of 180 dgrs. Th TL clbro lso suffrs from hs hr problm, bu h rducd umbr of sdrds rqurd TL clbro s mjor rso for br rsuls s comprd o TRL clbro. Th Rflc sdrd for h TRL clbro d o b kow ccurly. Ths rqurm mks h TRL clbro lss suscpbl o rrors s comprd o ohr clbro chqus lk Through Shor Dly (TSD) h rqurs h Rflco sdrd o b ccurly kow. Ths sll dos o gur h h TRL procdur s rror fr. I hs work rror fcor hr TRL clbro h s bs TL clbro s foud. 1

10 1.2 Thss Ovrvw Chpr 2 prss lrur ovrvw cludg brf roduco o h d for clbro h mcrowv msurm. Th bsc oprg prcpl of vcor work lyzr s dscussd log wh h dffr rmology ssocd wh work lyss. Dffr yps of rrors d chqus of rror corrco ppld o work lyzr msurms r dscussd. Th dffr rror corrcg clbro chqus r xpld log wh hr hr drwbcks d dvgs. Chpr 3 dscusss h TRL clbro procdur dl s hs clbro s h mos populr d wdly usd for o-wfr msurms. Th hr rrors TRL clbro r dscussd dls. Nx h w clbro chqu, TL clbro, s roducd d comprsos r md bw h TRL d TL clbro mhods. Chpr 4 dscusss ohr possbl hr rror, h vro h rflco coffc sdrd s h wo pors of work lyzr h cuss vro h msurd prmrs for h rror boxs h TRL clbro d cosdros o b md slcg h Rflco sdrd o mmz hs vros. Chpr 5 prss h xprml vrfco for h proposd rror prs TRL. Ths cluds MATLAB cod mplmg h TRL lgorhm d h hr rrors roducd for dffr rflco co-ffc sdrds. 2

11 Chpr 2: Lrur Rvw 2.1 Iroduco o Smh Chrs d VNA Ths sco gvs ovrvw of h fudml prcpls of vcor work lyss d Smh Chrs [7]. Nwork lyzrs ccurly msur ros of h cd, rflcd, d rsmd rgy,.g., h rgy h s luchd oo rsmsso l, rflcd bck dow h rsmsso l owrd h sourc (du o mpdcs msmch), d succssfully rsmd o h rmg dvc. Th mou of rflco h occurs wh chrcrzg dvc dpds o h mpdc h h cd sgl ss. Ths mpdc c b rprsd by rl d mgry prs. Isd of plog mpdc drcly, h complx rflco coffc s dsplyd vcor form. Th mgud of h vcor s h dsc from h cr of h dsply, d phs s dsplyd s h gl of vcor rfrcd o fl l from h cr o h rgh-mos dg. Smh chrs r ormlly o rprs hs. As mod [8], Smh Chr s polr plo of h complx rflco coffc or lso kow s h 1-por scrg prmr S 11, for rflcos from ormlzd complx lod mpdc; h ormlzd mpdc s complx dmsolss quy obd by dvdg h cul lod mpdc Z L ohms by h chrcrsc mpdc Z o (lso ohms, d rl quy for losslss l) of h rsmsso l. O h Smh Chr, loc of cos rssc ppr s crcls, whl loc of cos rcc ppr s rcs. Impdcs o h Smh chr r lwys ormlzd o h chrcrsc mpdc of h compo or sysm of rs, usully 50 ohms for RF d mcrowv sysms d 75 ohms for brodcs 3

12 d cbl-lvso sysms. A prfc rmo pprs h cr of h Smh chr Mcro-srp dvc msurms Mcrosrp dvcs co b cocd drcly o h coxl pors of work lyzr. Th dvc udr s (DUT) mus b physclly cocd o h work lyzr by som kd of rso work or fxur. Th Fgur 2.1 blow shows ypcl block dgrm of h dffr rfrc pls mcrosrp dvc msurm. I ordr o g h ccur msurm w d o rmov ll rrors up o h fxur rfrc pl. Ths ms h h rl Vcor Nwork Alyzr (VNA) s wll s coxl cbl rrors mus b rmovd. Clbro for fxur msurm mcro-srp prss ddol dffculs. O-wfr Rfrc Pl VNA d Cbl Errors FIXTURE ERRORS DUT FIXTURE ERRORS VNA d Cbl Errors Co-xl Rfrc Pl Fgur 2.1: Rfrc pls o-wfr mcrowv msurm. Clbro h coxl pors of h work lyzr rmovs h ffcs of h work lyzr d y cbls or dprs bfor h fxur; howvr, h 4

13 ffcs of h fxurs hmslvs r o ccoud for. Thus clbro sdrds r rqurd h prob ps. 2.2 Msurm Errors Errors work lyzr msurms c b sprd o hr cgors [9]: 1) Sysmc rrors r h mos sgfc sourc of msurm ucry RF d Mcrowv msurms, cusd by mprfcos h s qupm d s sup. Ths rrors c b chrcrzd hrough clbro d mhmclly rmovd durg h msurm procss. Sysmc rrors courd work msurms r rld o sgl lkg, sgl rflcos, d frqucy rspos. Th sx sysmc rrors h forwrd drco r drcvy, sourc mch, rflco rckg, lod mch, rsmsso rckg, d solo. Th rvrs rror modl s mrror mg, gvg ol of 12 rror rms for wo-por msurms. 2) Drf rrors occur wh s sysm s prformc chgs fr clbro hs b prformd. Thy r prmrly cusd by mprur vro d c b rmovd by ddol clbro. 3) Rdom rrors vry s fuco of m. Sc hy r o prdcbl hy co b rmovd by clbro. Th m corbuors o rdom rrors r srum os, swch rpbly, d cocor rpbly Error Corrco Tchqus Vcor rror corrco [9] s h mos wdly usd pproch o rmovg sysmc rrors. Ths yp of rror corrco rqurs work lyzr cpbl of msurg 5

14 phs s wll s mgud, d s of clbro sdrds wh kow, prcs lcrcl chrcrscs. Th vcor-corrco procss chrcrzs sysmc rror rms by msurg kow clbro sdrds, sorg hs msurms wh h lyzr s or corollr s mmory, d usg hs d o clcul rror modl whch s h usd o rmov h ffcs of sysmc rrors from subsqu msurms. Ths clbro procss ccous for ll mjor sourcs of sysmc rrors d prms vry ccur msurms. Howvr, msurm ccurcy s lrgly dpd upo clbro sdrds, d s of clbro sdrds s of suppld s clbro k. 2.3 Msurm Clbro Sdrds Th wo m yps of vcor rror corrco r h o-por d wo-por clbros O Por Clbro Ths procdur cosss of msurg srs of kow clbro sdrds. Bsd o h msurms of hs kow sdrds, rror modl of h prformc chrcrscs of h mcrowv hrdwr rl o h work lyzr c b cosrucd. Afr hs rror modl hs b grd from h msurm of h clbro sdrds, h work lyzr rmovs h sysmc rrors from h ohr dvcs. 6

15 DUT SHORT OPEN LOAD Fgur 2.2: O por OSL clbro sdrds. Th hr sdrds usd for 1-por clbro r h shor, op, d mchd rmo s show Fgur 2.2. Bsd o h msurms md for ch of hs sdrds, h vlus of h compos h sysmc rror modl (00, 01 d 11), s show Fgur 2.3, c b compud. Oc hs vlus r foud, h S prmrs of h DUT c b corrcd for hs o-dls. IDEAL NETWORK ANALYZER Clbro Sdrds Fgur 2.3: O por rror modl. I h rror modl 00 s h Drcvy rror 01, 10 r h Rflco rror 11 s h Sourc Mch rror Ths mhod s grlly usd for rflco msurms. Th mhod ds good rmo for hgh ccurcy wh wo-por dvcs. 7

16 2.3.2 Two Por Clbro I clbrg wo-por dvc howvr, o por clbro ssums good rmo o h uusd por of h DUT. Ths my cus ccurcs f h uusd por of h DUT s o prfcly mchd. To ob hghs ccurcy wo-por rror corrco s usd. Th wo-por rror corrco ylds h mos ccur rsuls bcus ccou for ll of h mjor sourcs of sysmc rror. Th rror modl show Fgur 2.4 for wo-por dvc rvls h four S-prmrs msurd h forwrd d rvrs drcos. Oc h sysm rror rms hv b chrcrzd, h work lyzr ulzs four quos o drv h cul dvc S-prmrs from h msurd S-prmrs. Bcus ch S-prmr s fuco of ll four msurd S-prmrs, work lyzr mus mk forwrd d rvrs s swp bfor updg y o S-prmr. EX ETT 1 ED ES S11 S21 S22 EL b2 b1 ERT S12 2 Por 1 Por 2 Fgur 2.4: Two por forwrd rror modl. I h rror modl: ED s h Forwrd Drcvy rror ES s h Forwrd Sourc Mch rror ERT s h Forwrd Rflco Trckg rror EL s h Forwrd Lod Mch rror ETT s h Forwrd Trsmsso Trckg rror EX s h Forwrd Isolo rror 8

17 Th mos commoly usd wo-por clbro chqus r: Shor Op Lod Through (SOLT) Through Shor Dly (TSD) Through Rflc L (TRL) L Rflc Mch (LRM) Ad Through L (TL) Ths clbro chqus r dscrbd h scos blow Shor Op Lod Through (SOLT) Clbro Clbro h coxl pors of h work lyzr rmovs h ffc of h work lyzr d y ohr cbl bfor h fxur. Mos work lyzrs lrdy co sdrd clbro k dfo fls h dscrb h chrcrscs of vry of clbro sdrds. Ths clbro k dfos usully covr h mjor yps of coxl cocors usd for compo d crcu msurms. Ths mhod s smplr d vry ccur s log s h sdrds r ccurly dfd. Th SOLT clbro chqu s of prfrrd wh S-prmrs r msurd wh rspc o dl chrcrsc mpdc such s 50 ohm. M chrcrscs of Shor Op Lod Though Clbro r h ll sdrds op, shor, lod mus b prfcly kow. Th op sdrd s ypclly rlzd s urmd rsmsso l. Elcrcl dfo of dl op hs uy rflco wh o phs shf. Th cul modl for h op, howvr, dos hv som phs shf du o frgg cpcc. Th lcrcl dfo of dl shor s uy rflco wh 180 dgrs of phs shf. All of h cd rgy s rflcd bck o h sourc, prfcly ou of phs wh h rfrc. A smpl shor 9

18 crcu from sgl coducor o groud mks good shor sdrd. For xmpl, h shor c b vs (pld hrough hols) o groud h d of mcro-srp rsmsso l. If coplr rsmsso ls r usd, h shor should go o boh groud pls. To rduc h ducc of h shor, vod xcssv lgh. A good RF groud should b r h sgl rc. A dl lod rflcs o of h cd sgl, hrby provdg prfc rmo ovr brod frqucy rg. W c oly pproxm dl lod wh rl rmo bcus som rflco lwys occurs som frqucy, spclly wh o-coxl cul sdrds Through Shor Dly (TSD) clbro TSD ws dvlopd by N. R. Frz d R. A. Spcl [11] d s bsd o boh rflco d rsmsso msurms d offrs wo dvgs. Frs, mchd lod sdrd o rqurd d scod uss mchclly smpl sdrds. TSD s o whou dsdvg. Th dly sdrd s ssumd losslss d orflcg.. h propgo cos ws pur dly d h mpdc ws h msurm sysm or 50 ohms. Also, TSD rqurs formo from wo dlys; h hrough coco d h cul dly. If h phs dffrc bw hs wo s r zro or 180 dgrs h lgorhm fls o rur vld rsul. Ths s lmo of y d-mbddg chqu usg dly yp sdrd Through Rflc L (TRL) Clbro For -fxur clbro hgh quly Shor-Op-Lod-Thru (SOLT) sdrds r o rdly vlbl for full 2-por clbro of h sysm h dsrd msurm pl of h dvc. I mcrosrp, prccl shor crcu or v s ducv, prccl op crcu rds rgy d hgh-quly purly rssv 10

19 lod s dffcul o produc ovr brod frqucy rg. TRL Clbro s dvlopd by Eg d Hor [1], usd oly o dly sdrd wh hrough coco d rbrry rflco o clbr hr work lyzr. Th dly sdrd could b rbrry, wh s propgo cos ukow, howvr, h l ws ssumd orflcg. Th rflco sdrd could b y rpbl rflcg lod d hy suggsd hr op or shor crcu. Th Through Rflc L (TRL) 2-por clbro s lrv o h rdol SOLT full 2-por clbro chqu h ulzs smplr mor cov sdrds, for dvc msurms h mcrosrp vrom. Th TRL mhod s mor subl for obg S-prmrs wh rspc o h mpdc of o wfr rsmsso ls. Wh ccur o-wfr L sdrds r vlbl, h TRL mhod usully offrs br ccurcy h h SOLT chqu. A TRL 2-por clbro k uss ls hr sdrds o df h clbrd rfrc pl. For TRL wo por clbros, h 12 rms rror modl s roducd rlr s smplfd o gh rms rror modl s show Fgur 2.5. A ol of 10 msurms r md o qufy h gh ukows. Fgur 2.5: Modfd gh rm rror modl [12]. I h rror modl ε00 s h Forwrd Drcvy rror Produc of ε10 d ε01 s h Rflco Trckg rror ε11 s h Lod d Sourc Mch rror Produc of ε10 d ε32 s h Trsmsso Trckg rror 11

20 A ssumpo s md h: ε11 = Forwrd Sourc Mch rror = Rvrs Lod Mch rror. ε22 = Rvrs Sourc Mch rror = Forwrd Lod Mch rror. Th m chrcrsc of Though Rflc L Clbro r h h TRL chqu rqurs oly hrough-l (Through) d rsmsso l offs (L) s rfrcs. Th TRL mhod lso mploys hghly rflcg mpdcs (Rflcs) s sdrds, bu hr xc lcrcl chrcrscs r o rqurd. Howvr, Rflc sdrds mus b ccurly rpd ch s por. Th TRL clbro uss h chrcrsc mpdc of h lgh of h rsmsso l sdrd o s h rfrc mpdc. Ths pproch rsrcs h rg of frqucy h c b msurd s h sro phs of h l should l bw 20 d 160 dgrs. Bcus of hs rso, mos of h L sdrds c oly b usd ovr 8:l frqucy rg. For brodbd msurms svrl L sdrds my b rqurd. A low frqucs, L sdrds c bcom covly log. Aohr cocr s h chg of h rfrc mpdc du o h ffcs of loss d dsprso h TRL L sdrds. Th ccurcy of h mhod dpds o h sm of h chrcrsc mpdc of h L sdrd Lod Rflc Mch (LRM) Clbro Th sdrds for h LRM clbro mhod r o-zro lgh Through l, Rflc d Mch. Th LRM clbro mhod s smlr o h TRL clbro mhod, xcp h prfc mch o ch por s subsud plc of h L sdrd. Th mchd sdrd c b s s fly log l. Th LRM 12

21 rflc s h sm s h TRL rflc sdrd, h vlu of h rflco sdrd d o b kow ccurly, bu mus b h sm for ch por. Th LRM L sdrd corrspods o h TRL Through sdrd. Th LRM clbro offrs umbr of dvgs. Frsly, h Mchd sdrd s h oly mpdc h ds o b ccurly kow hrby rducg h rrors cusd by mproprly dfd op or shor sdrds usd SOLT clbro mhod. Th dvg ovr h SOLT clbro ls h fc h LRM clbro uss lssr sdrds h rsuls fwr sdrd dfos d h rrors rsulg from mprfc sdrd dfo r mmzd. Scodly, h us of Mchd sdrd o drm h rfrc mpdc rducs h vro h rfrc mpdc vlu s obsrvd TRL mhod d bls brodbd frqucy covrg Through L Clbro Th Through L (TL) clbro s wo r clbro procdur. Th TL mhod ulzs msurms of h hrough d l followg pproxm Op Shor L clbro. Th Through L clbro s mplmd hr sps. Th frs sp s o pply Op Shor Lod (OSL) clbro o ch of h wo s pors. Th oly rqurm for h OSL clbro s h h sm rfrc mpdc b usd h wo s pors. Th scod sp s o prform h hrough d l msurms. Th hrough msurm ylds h rflco coffc of dl shor plcd h fxur rfrc pl. Ths rmovs h d for rbrry rflco sdrd s TRL procdur hus rmovg h mjor sourc of mbguy. 13

22 Fgur 2.6 blow shows h covol wo por rror work d s corrspodg sgl flow grph. I s composd of wo rror works A d B whr h dvdul sgl phs S j d S jb r uqu forcg h fxur S prmrs S jf o b uqul o ch ohr. Ths rprss symmrc fxur. As xpld [13], h fxur s symmrc f h por 1 d 2 rflco coffcs r qul d h fxur s rcprocl,.. f S 11f = S 22f d S 21f = S 12f. Rcprocy s ru wh h fxur s pssv d o-mgc d hs symmrcl quls r possbl wh wo ordrs of symmry; frs d scod ordr. Fgur 2.6: Two por rror modl for symmrc fxurs [13]. From h sgl flow grph of h symmrc fxurs Fgur 2.6, pplyg Mso s rul ylds S21S S 11f = S S S22S S 22f = S 22b + 1- S S S 21b 11b 11b S S 12b 11b (2.1) (2.2) S 21f = S21S 1- S22S 21b 11b (2.3) 14

23 Ad S 12f = S12S 1- S22S 12b 11b (2.4) From quo 2.1 d 2.2 w g S 11 (1- S 22 S 11b ) + S 11b S 2 21 = S 22b (1- S 22 S 11b ) + S 11b S 2 21 (2.5) As rsul of pproxm OSL clbro, h clbrd fxurs of h s sysm r dcl. Hc hrough coco h mbddd dvc udr s s symmrcl. Symmry prms h syhss of TRL rflco sdrds. A frsordr symmrc fxur hs dcl fxur hlvs h r symmrc s show Fgur 2.7. Th por wo rror work, B, s ow por rvrs of A, or B = A R. Ths s ru f S ll = S 22b, S 22 = S llb. Morovr, hs codos ssfy h symmry codo quo (2.5). Mos mporly, h sgl flow grph Fgur 2.8 shows symmry rducs h umbr of rror rms o hr, h A work oly. I ohr words, symmry rducs h umbr of rror rms for wo por fxur. Fgur 2.7: Rducd wo por rror modl for symmrc fxurs [13]. 15

24 Fgur 2.8: Sgl flow grph of rducd wo por rror modl for symmrc fxurs. Syhsz Rflco Sdrds I h sgl flow grphs of Fgur 2.9, w cosdr h hrough coco S prmrs gv by S llf (= S 22f ) d S 21f (= S 12f ), d h cul prmrs of h rror work A bg δ, α, γ (for S ll, S 21 = S 12 d S 22 rspcvly). I ddo, h pu rflco coffc wh dl shor crcu plcd por 1b of A s ρ sc = δ α 2 / (1- γ) (2.6) From h sgl flow grphs w ob S llf = δ + α 2 γ /(1- γ 2 ) (2.7) S 21f = α 2 /(1- γ 2 ) (2.8) From (2.6), (2.7) d (2.8) w g ρ sc = S llf - S 21f (2.9) Smlr rsuls hold for ρ oc = S llf + S 21f (2.10) 16

25 Fgur 2.9: Sgl flow grph of dl shor s rflco coffc. Comprd o TRL, TL clbro rducs h umbr of sdrds dd d rqurs fwr cocos o h mcrosrp pors mprovg h rpbly of hs coco. Sc h rflco coffcs r drvd mhmclly, s possbl o "sr" dl op d shor crcus wh o-sr bl mdum such s dlcrc lodd wvgud. Th TL procdur lso drms h frqucy dpd chrcrsc mpdc of h l sdrd. Th chrcrsc mpdc of h l sdrd s drmd from h scrg prmr msurms d h msurd fr spc cpcc of h rfrc l. Hc h wo r clbro log wh h frqucy dpd chrcrsc mpdc of h l sdrd vods h prodc glchs hr h TRL procdur. 2.4 Cocluso Ths chpr dscussd h dffr yps of rrors roducd mcrowv msurms d h clbro chqus vlbl o rmov hs rrors. Th o por d wo por rror corrco chqus r roducd log wh hr rspcv rror modls d h lmos of o por rror modl wr sd. A brf ovrvw of h populr wo por rror corrco mhods r dscussd log 17

26 wh hr dvgs d dsdvgs. I h x chpr h Through Rflc L clbro s dscussd dl log wh h drvo of h TRL lgorhm. 18

27 Chpr 3: TRL Clbro Ths chpr dscusss h TRL clbro chqu [3] dph d lyzs h dffr sps volvd hs clbro d why hs mhod s h mos populr of ll. Th hr dvg of TRL clbro s comprd o ll ohr clbro chqus dscussd so fr s h h vlu of h rflco sdrd d o b kow. Th m furs of TRL clbro hv lrdy b dscussd h prvous chpr. I hs chpr h mhmcl lyss of h TRL dmbddg procdur s prsd s hs s cssry o udrsd h proposd hr rror roducd du o h rbrry rflco coffc. 3.1 Sps TRL Clbro For h hru sp, h s pors r cocd oghr drcly (zro lgh hru) or wh shor lgh of rsmsso l (o-zro lgh hru). Th rsmsso frqucy rspos d por mch r msurd boh drcos by msurg ll four S-prmrs. For h rflc sp, dcl hgh rflco coffc sdrds (ypclly op or shor crcus) r cocd o ch s por d msurd ( S 11 d S 22 ). For h l sp, shor lgh of rsmsso l dffr lgh from h THRU) s srd bw por 1 d por 2 d g h frqucy rspos d por mch r msurd boh drcos by msurg ll four S- prmrs. I ol, msurms r md, rsulg dpd quos. Howvr, h TRL rror modl hs oly gh rror rms o solv for. Th rflc sdrd d h propgo cos of h l sdrd r drmd. Bcus hs rms r solvd for, hy do o hv o b spcfd lly. Th 19

28 chrcrsc mpdc of h l sdrd bcoms h msurm rfrc d, hrfor, hs o b ssumd dl (or kow d dfd prcsly). 3.2 TRL D-Embddg Soluo Th bsc TRL clbro ould [1] s bsd o h gh rm rror modl. Th procdur h cosss of fdg h S prmrs for h wo wo-por rror mrcs. A mor dld drvo of h TRL clbro quos s gv Appdx A. Th s of msurms for h uclbrd S prmr of h DUT c b wr rms of R (T prmrs) mrcs s: R M = R * RDUT * R b whr RM rprss h msurd T mrx, DUT R rprss h cul T mrx of h DUT (Dvc Udr Ts) d R d Rb r h rror box T mrcs. Oc h clbro s compl d h rms of h rror mrcs drmd, h T prmrs of h DUT c b sly drmd from R DUT = v( R) * RM * v( Rb) whr v s h mrx vrs. Ths rsuls xplc xprssos for h d-mbddd S prmrs. Th clbro procdur cosss of msurg h S mrcs for h hr sdrds, mly Thru, Rflc d L, whch gvs hr kow mrcs. Th hr mrcs obd gv h scrg prmrs of h fcous wo pors s show h Fgur 3.1 h x pg d h scrg prmrs of h dvdul rror boxs A d B. 20

29 Fgur 3.1: Fcous wo-por rror boxs A, B wh h hr sdrds. [3] Th mrg wv mpluds b 1, b 2 pors 1 d 2 r rld o h cd wvs 1, 2 by h wll-kow scrg quos: b 1 = S111 + S12b2 (3.1) b 2 = S S 22b2 (3.2) whr h S j r h scrg coffcs. Dvdg h frs of hs by l, h scod by 2, d h lmg h ro l / 2 bw hm ylds: w 2S11 + w2s 22 = w1w2 (3.3) whr = S11S 22 S12S 21 (3.4) Equo (3.5) shows h covrso bw h T d h S prmrs. R R R R = S S S S 22 S 21 S 12 S 11 S11 S 21 1 S 21 (3.5) Frs covr h msurd S prmrs o hr quvl T prmrs s Equo (3.5). L h cscdg mrx b rprsd by h mrx R. L h 21

30 cscdg mrcs of h rror wo-pors A d B b dod by R d R b rspcvly, whl R rprss hr cscd hru coco d R d rprss h cscd L msurms d R l1 rprss h cscd mrx of h srd l. Th T mrx for h Though coco c b rprsd by R = RRb (3.6) whl h T mrx for h l coco c b rprsd by Rd = RRl1Rb (3.7) whr r R mrx b rprsd by: R = r r r or R = r 22 c b 1 r11 / r 22 =, b = r12 / r 22 d c = r 21 / r 22. r R b mrx b rprsd by: R b = r 11b 21b r r 12b 22b α β or R b = r 22 b γ 1 α = r11 b / r 22b, β = r12 b / r 22b dγ = r 21 b / r 22b. Solvg quo (3.6) for R b d subsug o quo (3.7) gvs TR = RRl 2 whr Rd T = R -1 (3.9) Th quo (3.9) cosss of four quos wh fv ukows. Howvr hr r oly hr dpd soluos o Equo (3.9). Ths soluos gv / c = ( r 11 / r 21), b = ( r 12 / r 22) 22

31 d xp ( 2 γl ) = ( r ( r / r / r ) + ) (3.10) Up o hs sp w hv solvd for wo rms of h rror mrx A d h propgo cos of h rsmsso l. Usg hs formo w c solv for R b usg quo (3.6) o ob wo prmrs of h rror mrx B mly β / α dγ. Nx w mk us of h rflco sdrd. Th rflco coffc msurd por 1, w 1, c b wr s fuco of h rflco coffc of h rmo Γ l d h S prmrs of h rror box A s w 1 = Γl + b cγl +1 (3.24) or = w1 b 1 w1 c / ) Γl( * (3.25) Smlrly, w c wr for h msurd rflco coffc h rror box B s w 2 = αγl γ βγl +1 (3.26) or α = w2 γ. (3.27) 1 w2 β / α) Γl ( + * Th ukow rflco coffc Γ l s lmd from Equo (3.25) d (3.27) o yld = ± [( w1 b)(1 + w2β / α)( d bf )]/[( w2 + γc / )(1 c / )] (3.28) Thus h d for h vlu of rflco coffc o b xcly kow s lmd. Oc h vlu for for wo por rror mrx A s foud, h rmg prmrs of h rror mrx A d B c b sly foud s 23

32 γ = f dc / 1 c / (3.30) β = α α = b d bf d bf 1 c / (3.31) (3.32) A hs po h TRL clbro dvlopm s compl. Th S prmrs of h rror mrx A r gv rms of h msurd qus by: S11A r12 = b = (3.33) r 22 S 22 A = r r = r r r r c = * (3.34) S12 A * S 21A = r r r r r r = 1 b * c / (3.35) As s vd from h bov quos, h TRL clbro solvs oly for h rror mrx produc S A * S 21A 12. Th rsos bhd h populry of h TRL lgorhm s comprd o h ohr wo por clbro chqus, mly SOLT (Shor Op Lod Through) clbro, OSL (Op Shor Lod) clbro d TSD (Through Shor Dly) clbro r: Frsly h TRL clbro dos o d Shor d Op sdrds. Ths sdrds lm h ccurcy of h msurm, s h vlus of hs sdrds d o b ccurly kow o drm h rror prmrs. Scodly h mos mpor dvg of h TRL lgorhm s h vlu of h Rflc sdrds usd for clbro d o b kow o drm h prmrs of h rror mrx. Thus h clbro procdur s o dpd o h ccur drmo of h sdrds usd. Hc h TRL clbro procdur s sll h mos populr d wdly usd clbro procdur. Howvr h TRL clbro s 24

33 o whou s drwbcks. I h followg sco som of h drwbcks of h TRL clbro procdur r dscussd. 3.3 Lmos of Thru-rflc l (TRL) Clbro Th mjor lmo of h TRL chqu s h lmd bdwdh of L sdrds. Mos L sdrds c oly b usd ovr lmd frqucy rg. For brodbd msurms, svrl L sdrds my b rqurd. A low frqucs, L sdrds c bcom covly log. Th lgorhm rqurs h chrcrsc mpdc of h l sdrd b qul o h msurm sysm mpdc. If o, s vlu hs o b drmd. Also, h lgorhm ssums rl chrcrsc mpdc wh o frqucy dpdc. If h fxur o b clbrd s composd of dsprsv rsmsso ls,.g. mcrosrp, bcoms cssry o drm h complx chrcrsc mpdc s fuco of frqucy. Th sro phs of h l mus o b sm s h hru (for zro or ozro lgh). Ths pus mpor rqurm h h dffrc bw h lgh of h L sdrd (sy l 2 d h lgh of h hru sdrd (sy l 1 mus b bw (20º d 160º) ± 180º. Idlly h dffrc bw h lghs s k o b ¼ wvlgh or 90º of h sro phs rlv o h hru h mddl of h dsrd frqucy sp. Hc h rg of frqucs ovr whch h msurms r o b k dcds h cr frqucy, whch ur dcds h dffrc bw h wo lghs. 25

34 3.4 Cocluso Ths chpr dscussd dl h TRL clbro procdur. Th dscusso cludd h rqurms of h sdrds usd h TRL clbro, h mhmcl lyss of h TRL d-mbddg lgorhm, h dvgs of hs clbro procdur s comprd o h ohr wo por clbro mhods. I h d h hr drwbcks d lmos of hs populr procdur r dscussd. 26

35 Chpr 4: Error Alyss of TRL Clbro Ths chpr prss h rdr wh h w work do o vsg why h TL (Through L) clbro, udr cr codos provds br lrv o h TRL (Through Rflc L) clbro. I h followg scos h dscrpcs roducd h clculd prmrs of h rror mrcs du o h rbrry rflco sdrd s drvd. Th hr ssumpo bou h rflco sdrd vlu h TRL d-mbddg lgorhm s s o cus vros h obd prmrs of h rror mrcs. From h drvo of hs rror fcor, h ucrs r foud o b dpd o h vlu of h slcd rbrry rflco sdrd. Aohr phomo, mly h dscrpcs msurm wh h l dffrc pprochs gr mulpl of hlf wvlghs, s lso obsrvd TL clbro chqu. Mhmcl lyss s prsd o dc h cus of hs dscrpcs h TL d TRL clbro d h cocluso s drw h h dscrpcs wll b obsrvd ll clbro procdur h uss h hrough d dly sdrds s pr of h clbro procdur. 4.1 Proposd Error TRL clbro Thr hv b umbr of pprs vsgg h vrous cuss of rror hr TRL clbro. Mrks [16] uss ls m squr sm d rdud rsmsso l sdrds o mmz h ffcs of rdom rrors roducd by mprfc cocor rpbly. Th msurd modl hs ppr s rprsd by cscd of h fxur rror mrcs d h rsmsso l mrx. Th fxurs 27

36 α β usd r ssumd o b symmrc d rprss h rror cscd mrx d γ δ δ γ s rvrs cscd mrx s. I ordr o k o cosdro h rrors β α roducd h msurm du o mprfco h rsmsso ls, h rsmsso l mrx s furhr rprsd s cscd of hr mrcs T = [ I + δ 1] L [ I δ 2] + whr L rprss h dl rror fr rsmsso l mrx d δ rprss h mprfcos rsmsso l d h rror s ssumd o b smll. Th wo s of msurms k r rprsd by h wo mrcs s M = X T Y b d M j = X T j Y b whr X rprss h frs rror mrx of h fxur d Y b rprss h rvrs cscd rror mrx. Solvg h wo s of quos o lm Y b rsuls M j X = X T j whr M j = M j * M -1 d T j = T j * T 1. Sc h mrcs M j d T j hv dcl g vlus, h ffc of prurbos o mrx M j c b lyzd v hr ffcs o T j. Th mhmcl lyss h follows h ppr rsuls h rsmsso l mrx rprsd by -1 E1 T j L j + ε j whr ε j s h lr dffrc rm, L j = L j L = 0 0 E2 E1= xp [ γ ( l1 l2) ] d E2 = xp [ + γ ( l1 l2) ]. Th off dgol lms of h rror rm ε j rprss h rrors msurm of h rflco coffc of h l d h dgol lms rprs rrors h msurm of h rsmsso coffcs of h l. Th rror rm ε j vshs wh h rsmsso l s losslss d rror fr. Th rror lyss do hs work sms h dscrpcs h TRL msurms cusd du o losss h rsmsso l sdrds d ppls h ls m squr lgorhm o sm h vro h propgo cos γ bsd o h ε j rror mrcs. 28

37 Hor [15] dscusss h sysmc rror msurg h rflco coffc sgl por. To drm h sysmc rrors roducd, h rflco coffc s msurd for dffr combos of uor cocd wh dffr rflco sdrd of vlus ( Γ l ) qul o 0.01, 0.05, 0.1 d 1.0. Ths msurd rflco coffc Γ m r h comprd wh h rflco coffc clculd from h dvdully msurd S prmrs of h dffr uors usd d h rflco sdrds ( Γ l ) usd for h msurms from h sgl flow grph s S12S 21Γl Γc = S11 +. Th dffrc bw h msurd vlu Γ m 1 S 22Γl d h clculd vlu Γ c s dco of h sysmc rrors roducd. Wllms d Mrks [18] lyz h bouds o h vros h msurd S prmrs roducd du o h dffrcs h rfrc mpdc or h rfrc pl. Th msurd cscd mrx s modfd o k o ccou h djusd rfrc mpdc or h rfrc pl. Hc h cscdd mrx of T M = X o T o Y ob s modfd, o k o ccou h vro du o h rfrc mpdc, o T M = [X o R(Z, l)] T o [Y ob R(Z, l) b ] whr h rror mrcs r rprsd by X o d Y ob d R(Z, l) 1 xp[ γ ( l lo)] 1 Γ xp 2 Γ( xp[ γ ( l lo)]) Γ( xp[ γ ( l l xp[ γ ( l lo)] )]) o rprss h vros wh h rfrc pl chgs o l or h rfrc mpdc chgs o Z d Γ = Z - Zo. Th wors-cs dvos of h msurd S prmrs d h Z + Zo mpdc d rfrc pl rsformd S prmrs r drvd o b B δ x δ δ δ + δ 11 x x21 + x12 y21 29

38 whr δ x = [Xo R (Z, l)]- I d δ y = [Yo R (Z, l)] - I. Ths rsul shows h vro bw h msurd d h cul S prmrs of h DUT for chg h rfrc mpdc or h rfrc pl. Ksr d Wllms [17] lys dffr sourcs of sysmc rrors h coplr wvgud TRL clbro. Amog h sourcs of sysmc rrors lyzd, h symmry h symmrc shor sdrd s lso cosdrd. Th ppr mks us of h rsul B = T 11 T T 21 + T 12 drvd by Wllms d Mrks [18] o sblsh mxmum vro bw h msurd d h cul S prmrs wh h shor sdrd s o por hs offs of l. Th uppr boud of h rror s s o b drcly proporol o h mou of offs l. Th lyss do [17] d [18] sblsh h dscrpcs cusd h TRL clbro du o chg h rfrc pl d offs h shor sdrd. Summrzg h work do o sm h sourcs of rror TRL clbro du o h vro h rflco coffc sdrd cluds work by Mrks [16] h sms h rrors du o h losss h rsmsso l sdrd, work by Hor [15] h sms h sysmc rrors roducd sgl por by h rflco coffc sdrd d h work by Wllms d Mrks [18] d Ksr d Wllms [17] h sms h rrors du o h chg h rfrc pl d offs h shor sdrd. I h followg sco ohr w sourc of rror du o h rbrry rflco sdrd s roducd, mhmclly drvd d s mplcos dscussd. 30

39 4.1.1 Alyss of h proposd hr Error TRL Clbro Ths sco roducs h proposd rrors roducd h TRL msurms du o h rbrry rflco sdrd. Wh rfrc o [1], h TRL clbro dos o ssum symmrc d dcl fxurs d hc hs dffr rror mrcs for boh h fxurs h crss h umbr of sdrds dd o fd h prmrs of h rror mrcs. Th TRL clbro d-mbddg procdur by Eg d Hor [1] mks us of hr sdrds, h hrough, rbrry rflco sdrd whos vlu d o b kow, d lgh of rsmsso l o d-mbd h prmrs of h rror boxs. Th hrough d h l sdrds yld wo s of quos h r solvd o fd h propgo cos d wo prmrs of o of h rror mrx. I h x sp, h wo s of msurms obd by cocg h ukow rflco sdrd h wo pors r solvd smulously o lm h ukow rflco sdrd d hc mkg h TRL d-mbddg procdur dpd of h ukow rflco sdrd vlu. I h TRL d-mbddg procdur mpor ssumpo hs b hrly md, mly h h rflco sdrd s h wo pors s xcly sm. Ths ssumpo co b ovrlookd s bsd o hs ssumpo h wo ss of msurms, obd by cocg h ukow rflco sdrd h wo pors, r smulously solvd o lm h ukow rflco sdrd. Hc for h TRL d-mbddg procdur o b ccur, h ukow rflco sdrd s h wo pors hv o b xcly sm. I h followg scos lyss s do o s h mplcos o h obd prmrs of h rror mrcs f h rflco sdrd s h wo pors s o xcly sm. L us ssum h h rbrry rflco sdrd usd h TRL clbro procdur suffrs from rpbly rrors d h ukow 31

40 rflco sdrd s h wo pors r o xcly h sm vlu. L h rflco sdrd s o por b Γ l d h rflco sdrd vlu s h ohr por b ( Γl + ε ). From h TRL d-mbddg procdur mod h drvo chpr 3 d Appdx A, h modfd TRL d-mbddg procdur s prsd blow. Wh h modfd rflco sdrd ( Γl + ε ) s por 2, Equo (A.27), dcl o Equo (3.27), s modfd from α = w2 ( 1 w2 * Γl + γ β / α) o w2 γ α =. No h h oly modfco s h h rflco ( Γl + ε ) * (1 + w2 * β / α) sdrd Γ l s rplcd wh ( Γl + ε ) whr h rm ε shows h vro h rflco sdrd vlu, s s por 2 of h work lyzr, rlv o h rflco sdrd prsd por 1. Equo (A.25), dcl o Equo (3.27) rms uchgd s = w1 b 1 w1 c / ) Γl( * s hs oly dpd o h rflco sdrd cocd por 1, d hs rms uchgd. Th x sp h TRL d-mbddg lgorhm lms h rflco sdrd by kg ro of α. Th modfd vlu of α chgs h ro of ( w1 b)(1 + w2β / α) = α (1 w1c / )( w2 γ ) o α m ( Γl + ε )( w1 b)(1 + w2β / α) =. Thus h ro of ( Γl)(1 w1c / )( w2 γ ) h ws dpd of h rflco sdrd s modfd o α m h s ow α dpd o h rflco sdrd vlu s wll. Th x sp h TRL dmbddg mks us of h ro of α d h produc of d α obd from Equo (A.32) o lm h α rm d ob h quo for. Th quo 32

41 o drm h vlu of obd from h orgl TRL d-mbddg lgorhm s modfd du o h chgd quo of m o clud h rflco sdrd α vlu. Hc h quo for chgs from = ± [( w1 b)(1 + w2β / α)( d bf )]/[( w2 + γc / )(1 c / )] obd from h orgl TRL d-mbddg lgorhm o = ± [( 1+ ε / Γl )( w1 b)(1 + w2β / α)( d bf )]/[( w2 + γc / )(1 c / )] If h rflco sdrd vlu s por 1 d 2 s o xcly h sm, h h proof bov prdcs h h rm for wll b dpd o h vlu of h rflco sdrd s wll s o h vlu of ε whch dos h vro h rflco sdrd vlu s h wo pors. From h bov drvo s s h h vlu of vrs by h rror fcor 1+ ε / Γl. I s clrly s h h mou of rror roducd h orgl TRL d-mbddg lgorhm s drcly proporol o h vlu of ε whch rprss h mou of vro of h rflco sdrd vlu s por 2 from h rflco sdrd vlu s por 1. Th rror fcor s lso vrsly proporol o h vlu of h slcd rflco sdrd. Th smllr h vlu of h slcd rflco sdrd, h grr wll b h rror roducd prdcg h corrc vlu of. I ordr o mmz h rror fcor, s sly obsrvd h h rflco sdrd wh mgud of 1 wll rsul ls vro of h prdcd vlu of. Hc o mprov ccurcy h rflco sdrd should b chos o b hr shor or op. Th ccurcy h msurm of s vry mpor bcus h TRL dmbddg lgorhm mks us of h vlu of o drm h ohr prmrs 33

42 of h fxur rror mrcs. Ay dscrpcs wll ld o ccur prdco of h fxur rror mrcs h wll subsquly ld o ccur clculo of h S prmrs of h dvc udr s. I h x prgrph lyss s do o show how h rror fcor of 1+ ε / Γl roducd h vlu of ffcs h vlus of h ohr fxur rror mrx prmrs. Th TRL lgorhm mks us of h vlu of o clcul ohr prmrs of h rror mrcs. Wh rfrc o Appdx A, fr obg h vlu of, h quo of ( w1 b)(1 + w2β / α) = α (1 w1c / )( w2 γ ) s usd o clcul h vlu of α. Tkg h rror fcor o cosdro, h w corrcd vlu of dod by m s gv by m = ( 1+ ε / Γl ). Ths vro h msurd vlu of cuss h vlu of α o b modfd o αm = α 1+ ε / Γl. Afr fdg h vlu ofα, h quo for α β, Equo (A.31), obd durg h TRL d-mbddg procdur s usd o clcul h vlu of β. Tkg o cosdro h rror fcor h modfd h vlu of α o s gv by αm = α 1+ ε / Γl, h corrcd vlu of β dod by β m β m = β 1+ ε / Γl. Flly Equo (A.23) h s h ro of c s usd o clcul h vlu of c. Tkg o cosdro h rror fcor h modfs h vlu of, h corrcd vlu of c dod by c m s gv by c m = c ( 1+ ε / Γl ). Th rmg prmrs of h rror mrcs mly b d γ do o rqur h vlu of d hc r o 34

43 ffcd by h rror fcor h modfs h vlu of. Th lyss prsd hr shows how corrc prdco of h rror prmr c ld o furhr ccurcs drmg ohr prmrs of h rror mrcs d ulmly ldg o ccurcs drmg h S prmrs of h dvc udr s. I h bov scos w sourc of possbl rror h TRL clbro procdur s roducd d sudd dl. Th followg scos prs xplo h dlv o h rso for h bsc of h rrors roducd du o rbrry rflco coffc w clbro chqu, h Through L (TL) clbro h udr cr codos s rplcm o h TRL clbro procdur. Aohr phomo, h dscrpcs h msurms wh h dffrc bw h hrough d h l sdrd pprochs gr mulpl of qurr wvlgh, obsrvd h TRL clbro s wll s h TL clbro chqu s vsgd. 4.2 Comprsos bw TRL d TL Clbros Th Through L (TL) clbro s wo r clbro procdur. Th frs sp TL clbro s o do OSL clbro followd by wo s of msurms usg h hrough d h l sdrds. Th TL clbro, s comprd o h TRL clbro procdur, ds rducd umbr of sdrds for msurms d rqurs fwr cocos o h mcrosrp pors mprovg h rpbly of hs coco. Th mos mpor dffrc bw h TRL d h TL clbro s h TL clbro h rflco coffcs r drvd mhmclly (s show chpr 2), whch mks possbl o "sr" dl op d shor crcus wh o-srbl mdum such s dlcrc lodd wvgud. Sc h vlu of h 35

44 rflco sdrd s mhmclly drvd, s vlu s drmd ccurly s comprd o h TRL clbro procdur whr s s h slgh vro h rflco sdrd s h wo pors cuss ccurcs h msurms. Aohr hr ssumpo h TL clbro s h h fxurs r ssumd o b symmrc, s comprd o h TRL clbro whr h fxurs r ssumd o b o symmrc. As ws s Chpr 2, symmrc fxurg smplfs h sgl flow grph d mks d-mbddg possbl wh oly wo s of msurms. I cors TRL clbro rqurs hr s of sdrds. Th us of oly wo sdrds TL clbro s comprd o h hr sdrds usd TRL clbro lso rducs dscrpcs cusd msurms du o rpbly d cocvy rrors. Th TL procdur drms h frqucy dpd chrcrsc mpdc of h l sdrd. Th chrcrsc mpdc of h l sdrd s drmd from h scrg prmr msurms d h msurd fr spc cpcc of h rfrc l. O hr drwbck prs boh h TRL d TL clbro procdur s h dscrpcs h msurms wh h l dffrc bw h hrough d h l sdrd pprochs qurr of wvlgh. Th followg sco prss lyss h ccous for hs cus of dscrpcs prs h TRL d TL clbro chqu. Modl d Ch [2] hv prsd mhmcl lyss h dscuss h cus of dscrpcs h TRL clbro procdur. I h followg sco lrv mhmcl lyss s prsd o xpl why hs dscrpcs r lkly o b prs y clbro procdur h mploys dly d hrough s h wo sdrds for clbro. From h TRL clbro procdur h propgo cos s drmd o b 36

45 ± R xp[2γ ( l2 l1)] = whr R = [( 11 22) xp ] m R I h bov quo h lms 11, 12, 21 d 22 rprs h T prmrs of h msurm k TRL clbro. Ths mrx s h produc of h l sdrd msurm of lgh l 2 d h vrs mrx of h hrough sdrd msurm of lgh l 1 (ypclly zro). Ths mrx c b rfrrd o Equo (A.9) Appdx A. Assumg h h msurd S prmrs, S 11 d S 22 (for boh l d hrough sdrd msurms) r 0, h T prmrs 12 d 21 (for boh l d hrough sdrd msurms) bcom 0 s wll. Hc h propgo cos bcoms fuco of h 11 prmrs of ls l1 d l 2 (dod by 11l1 d 11l2 ) d 22 prmr of l l1 d l 2 (rprsd by 22l1d 22l2 ). Ths c b prsd h mrx form s l1 = l1 0 11l2 0 22l2-1 whr 11 = 11l1 / 11l2 d 22 = 22l1/ 22l2. Th bov ssumpo lso modfs h propgo cos quo o ± ( 11 22) xp[2γ ( l2 l1)] = or xp[2γ ( l2 l1)] = or m ( 11 11) Subsug 11 = 11l1 / 11l2 d 22 = 22l1 / 22l2 h bov quos for propgo cos ylds 11l1 22l2 22l1 11l2 xp[2γ ( l2 l1)] = =. 22l1 11l2 22l211l1 Covrg h T prmrs o S prmrs ylds S12l1S 21l1 S12l2S 21l2 xp[2γ ( l2 l1)] = =. S12l2S 21l2 S12l1S 21l1 37

46 Th followg ssumpo s md h S 12 = S 21. Wh hs ssumpo h bov S12l1 S12l2 quos furhr smplfy o xp[ γ ( l2 l1)] = =. S12l2 S12l1 As s from h bov smplfd quos, h propgo cos s drcly proporol o h S12 msurms of h l d hrough sdrds. Afr drvg h quo for h propgo cos, furhr lyss s do h followg prgrphs o show h cus of dscrpcs h msurms wh h dffrc bw h dly sdrds pproch gr mulpl of hlf wvlgh. Th propgo cos drvd for h l sdrd c b rprsd S12l1 mgud d phs oo or s p = * xp[ j( S12l1 S12l2)]. Th S12l2 mgud phs oo c lso b rprsd complx umbr oo s S12l1 p = R + ji whr R = cos( S12l1 S12l2) d S12l2 I S12l1 = s( S12l1 S12l2). S12l2 From h rl d mgry rms of h propgo cos quo, s s h h drmo of h propgo cos s dpd o h sro phs of h dly sdrd. Th sro phs of l s drcly proporol o h frqucy d h l lgh d vrsly proporol o h wvlgh. Hc h slco of h l lgh, whch s h dffrc bw h l lghs of h wo dly sdrds, h hrough d h l sdrd TRL d TL clbro, dcds h sro phs. Wh h l dffrc pprochs qurr of wvlgh, h sro phs pprochs 90º. 38

47 W ow lyz h ffc of vro h sro phs (whch s dcdd by h dffrc of h l sdrds) o h rl d mgry prs of h propgo cos quo. Wh h sdrd lghs l1 d l2 r chos h λ rg of 0 < ( l1 l2 ) <, boh h rl d mgry prs of p r posv. I 4 ohr word h sro phs s bw 0º d 90º, hc boh h cos rm h rl pr d h s rm h mgry pr rm posv. For h l λ λ dffrc h rg of < ( l 1 l 2 ) <, h rl pr s gv d h 4 2 mgry pr s posv. Hc h rgums of p r usd o kp rck of h corrc gulr roo for h xprsso of h propgo cos. Aohr obsrvo s h wh h l dffrc pprochs 0º or 180º, h rl pr of p h rprss h uo cos bcoms vry hgh (h cos of 0º d 180º bcoms ± 1) d h mgry pr h rprss h propgo cos bcoms vry smll d pprochs zro. Th lrg vros h rl d mgry pr of p cusd wh h sro phs pprochs 0º or 180º s h m cus of h obsrvd dscrpcs h TRL d TL clbro procdur. Also h slco of h l dffrc h rg of 4 λ s jusfd bcus hs lgh h sro phs pprochs 90º whch cuss h rl pr o b vry smll hrby rducg rrors h uo cos drmo. 4.3 Cocluso Ths chpr dscussd dls w sourc of rror hr h TRL clbro. Th proposd sourc of rror s bsd o h ssumpo h h 39

48 rflco sdrd vlu s h wo pors of h work lyzr r o xcly sm. Th TRL clbro mhod s comprd wh h w clbro mhod, h TL clbro o xpl why hs clbro, udr cr codos s br lrv o TRL clbro. Th proposd sourc of rror prs TRL clbro s vodd TL clbro bcus mks us of symmry fxurs d syhszd rflco co-ffc h smplfs h rror modl d rducs h umbr of sdrds dd hrby rducg h rpbly rror. Aohr drwbck of TRL mly h ccur drmo of h chrcrsc mpdc s rmovd TL clbro by drmg h mpdc of h l from h scrg prmrs msurms d h msurd fr spc cpcc of h l. Flly h cus of dscrpcs h TRL d TL clbro msurm wh h dffrc h lgh of h hrough d l sdrd pprochs 0 or λ s dscussd. 2 40

49 Chpr 5: Exprml Rsuls Th proposd dscrpcs roducd h prmrs of h rror boxs A d B du o h rbrry rflco coffc r vrfd xprmlly. Th proposd rror modl s bsd o h ssumpo h h rflco sdrd vlu s h wo pors r o xcly sm. I h xprml sup, hs ssumpo s vrfd xprmlly. Exprml lyss s do usg symmrc fxur o h dvc udr s d h rdgs r procssd wh h TRL lgorhm mplmd MATLAB o ob h prmrs of h rror box A d B. Nx h rror fcor rm roducd du o msmch h rflco coffc s roducd o h MATLAB mplmo of h TRL lgorhm d h w s of prmrs for h rror box A d B r obd. Th bov procss s rpd for dffr rflco coffc sdrds o ob opml vlu of rflco sdrd h rsul ls rror. Th rsuls r vrfd d prsd hs chpr. 5.1 Grl Msurm Sps Th work lyzr msurm sysm cosss of h sourc, h s s, h vcor sgl procssor d h dsply. Ths oghr comprs h smulus d rspos s sysm h msurs h sgl rsmd hrough h dvc d h rflcd sgl from h pu of h dvc. Th d s h procssd o dsply h mgud d phs rsposs. For our msurms Hwl Pckrd 8510 work lyzr sysm s usd. Ths work lyzr cos swpr o provd h RF smulus. Th s s provds h pos whr h dvc udr s s o b cocd. Th sysm uss coxl s ss up o frqucy rg of 50 GHz. Th fro pl of work lyzr cluds h CRT h dsplys h msurm 41

50 rsuls. Th work lyzr hs wo chls for dsplyg h msurms h wo s pors. Chl 1 or Chl 2 slcs h chl dsply. Th kys h Smulus, Prmr, Form d Rspos blocks r dd for h msurm sup, clbro, d rprso d d oupu fucos. Followg r h grl msurm squc volvd whl kg msurms. Th frs sp s o sup h s por cbls d slc subl cocors for h dvc udr s. Th x sp s o sup h work lyzr, whch volvs usg h kys h Smulus block o s h dsrd sr d sop frqucs, umbr of msurd pos d chrcrsc of frqucy swp. Th kys h Prmr block dcd h prmrs o b msurd d dsplyd. Th kys h Form d Rspos blocks r usd for slcg yp of dsply d poso of h rc o vw h msurd d rspcvly. Nx sp s o slc h ppropr clbro sdrd o corrc h sysmc rrors usg h kys h Cl block. Oc h msurms hv b k, h sv ky sors h msurd d rl/xrl sorg. Th Mrkr ky d kob s usd o poso h msurm mrkrs o h pos of rs o h rc d sor hs rsuls whou furhr rclbro. Th bov-mod sps gv brf oul o k bsc msurms work lyzr d roduc h rdr o h work lyzr corol kys. 5.2 Exprm Sup Afr ovrvw of h bsc corol kys d msurm squc, w dscuss dl h fxurs usd for our msurm sup, h compos usd d h procdur mployd gg h corrcd msurd S prmrs of h dvc udr s. 42

51 Bfor srg wh h msurms h work lyzr d o b cofgurd by h usr o dsply h msurms ccordgly. Th fro pl kys mod h bov sco c sly cofgur h work lyzr. Th followg sps mod hr provd ovrvw of h cofguro sgs usd by us kg h msurms. Th kys h Smulus block r usd o s h frqucy swp, h swp m d h umbr of frqucy pos o b msurd o 801 pos. Th sr d sop kys d h cr d sp kys o h Smulus block r usd o s h lms of h frqucy swp. Prssg h Smulus mu ky gvs opos o s h swp m, umbr of pos d swp mods. To slc h umbr of pos, prss h Numbr of pos sofky o slc h 801 pos. Th Rmp, Sp d Frqucy Ls sofkys slc h swp mod. Th kys h Prmr block llow h usr o slc h S prmrs o b msurd. I our xprm ll h S prmrs wr msurd d sord. Th x sp fr cofgurg h work lyzr for msurms s o prform h msurm clbro. Th kys o h Form block provd comprhsv slco of kys o dsply h msurms dffr dsply forms. Th clbro k usd s Hwl Pckrd mllmrs Clbro K. Th clbro sdrd dfos for h bov mod Clbro K r sord h rl mmory of h work lyzr pror o srg h msurms. Th sdrd dfos provd h coss dd o mhmclly modl h lcrcl chrcrscs of h sdrds usd h Clbro K. Th po whr h dvc udr s s cocd s clld h Rfrc Pl, whch s h d of h coxl cbl. I our cs h s s cosss of h coxl cbls d fxurs h coc h work lyzr d h coxl cbl. 43

52 Clbro s cssry bfor kg msurms s h rcocg of h s s coxl cbls flucs h ccurcy of h msurm. I ordr o rmov hs sysmc rrors h Op-Shor-Lod (OSL) clbro mhod s usd o sup h rfrc pl o h coxl prob ps por 1 d 2 of h work lyzr. For br ccurcy low bd lod s usd s h Lod sdrd OSL clbro for frqucs blow 2 GHz d h sldg lod s usd for msurm bov 2 GHz. Sc h frqucy rg for our msurm s up o 18 GHz, sldg lod sdrd s usd h OSL clbro. Th m m of h xprm s o ob h prmrs of h rror box A d B d flly ob h corrc S prmrs of h dvc udr s by mplmg h TRL lgorhms d h modfd TRL lgorhm h ks o cosdro h rror fcor roducd du o rbrry rflco sdrd d compr hr dffrcs. I ordr o roduc hs rrors, symmrcl fxurs r roducd bw h rfrc pl d h dvc udr s. Th symmrcl fxurs r cosrucd from cocg 7mm - SMA yp dpr o 90º bd SAM yp o SMA 7mm yp dpr boh ds of h s s coxl cbls. Th x sp fr h OSL clbro s o slc h Clbro k. I our xprml sup, for mxmum ccurcy, h FULL 2-PORT clbro choc offrd h work lyzr HP 8510 s slcd. Th FULL 2-PORT clbro uss Op, Shor d Lod sdrds ch por 1 d 2 of h work lyzr. Th FULL 2-PORT clbro s slcd h work lyzr from h Cl yp slco mu. I h followg prgrph, h dffr sps volvd h FULL 2-PORT clbro mhod mplmd our xprml sup r lsd. Th frs hr sps volv cocg h shor, op d lod sdrds h wo ds of h cocd fxurs d og h rflco coffc vlu, 44

53 yldg sx ss of msurms. Th x four sps volv cocg 20 db uor d op/shor sdrds h wo rfrc pls, whch r h ds of h cocd fxurs. Smlr msurms r k by rplcg h 20 db uor wh 50 db uor. I h x sp, h 20 db uor s cocd bw h wo fxur ds. For h hru msurm, h wo fxur ds r cocd oghr. Th x sp volvs cocg h 20 db uor d 10 cm r l bw h rfrc pls, whch r h wo ds of h cocd fxurs. Smlr msurms r k by rplcg h 20 db uor wh 50 db uor. Th clbro k usd for our s of msurms dd o co h Rflc sdrd. I ohr words, h TRL clbro k s o uvlbl. Hc h TRL dmbddg lgorhm s mplmd MATLAB d h s of msurms obd from h FULL 2-PORT CALBRATION r fd o h MATLAB cod h procsss h s of msurms, prforms h TRL dmbddg lgorhm d dsplys h rsuls. Th followg sco prss h obsrvd rsuls for h bov s of msurms k d cofrms h horcl ssumpo of dscrpcs h obsrvd rflco coffc s h wo ds of h rfrc pls, whch r h wo ds of h cocd symmrcl fxurs. 45

54 5.3 MATLAB Smulo Rsuls I h followg prgrphs, h msurms obd from h xprml procdur, s mod h prvous sco, r prsd. Th msurd d obd from h bov mod sps r fd o h MATLAB cod h mplms h TRL dmbddg lgorhm. Afr procssg h d, h rflco coffcs s h pors of h work lyzr r s o b dffr. Th proposd sourc of dscrpcs TRL clbro ws bsd o h followg ssumpo h h rflco coffcs s h d of h wo pors of h work lyzr r dffr. Ths ssumpo s vrfd by h MATLAB cod. Th MATLAB cod furhr mplms h modfd TRL dmbddg procdur h sms h mou of rror roducd h rror boxs A d B du o h obsrvd dscrpcs h rflco coffc s h d of h pors of h work lyzr TRL clbro rsuls obd from usg shor d op sdrds Th xprml lyss s do s by mplmg TRL lgorhm o hr dffr dvcs udr s d h comprg hr rsuls. Th frs dvc udr s s r flld rsmsso l cocd bw h ds of h wo fxurs. Th scod dvc udr s usd s 20dB uor. Th hrd dvc usd s combo of 20dB uor d r flld rsmsso l. Fgur 5.1 show h x pg shows h d obd from h work lyzr mgud d phs rspos for h rflco coffc of h shor sdrd cocd h wo symmrcl fxur ds. 46

55 M g u d P h s Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos () Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (b) M g u d P h s Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (c) Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (d) Fgur 5.1: Rflco coffcs of h shor sdrd: () mgud Por 1; (b) phs Por1; (c) mgud Por 2; d (d) phs Por 2. 47

56 Smlrly Fgur 5.2 d Fgur 5.3 blow show h d obd from h work lyzr mgud d phs rspos for h rflco coffc of h op d mch sdrd rspcvly. M g u d P h s Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos () Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (b) M g u d P h s Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (c) (d ) Fgur 5.2: Rflco coffcs of h op sdrd: () mgud Por 1; (b) phs Por1; (c) mgud Por 2; d (d) phs Por 2. 48

57 M g u d P h s Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos () Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (b) M g u d P h s Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (c) Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (d) Fgur 5.3: Rflco coffcs of h mch sdrd: () mgud Por 1; (b) phs Por1; (c) mgud Por 2; d (d) phs Por 2. 49

58 Th followg scos h x pg provd h rsuls for h TRL clbro o h hr dvcs udr ss TRL Clbro for r flld rsmsso l Th r flld rsmsso l s plcd bw h wo fxur ds d ll h four S prmrs r msurd. Th h hru d rl d r fd o h MATLAB cod h mplms h TRL dmbddg procdur. Th TRL dmbddg procdur s mplmd for h hr dffr sdrds, mly shor, op d mch. For ch of hs sdrds, h rflco coffc clculd from h TRL lgorhm s h wo pors s msurd. As pr our ssumpo, s lso xprmlly obsrvd h h rflco coffc s h wo pors r o xcly sm. Ths rflco coffc s h wo pors for h dffr sdrds r show Fgurs blow. M g u d M g u d Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos () Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (b) Fgur 5.4: Clculd rflco coffc from TRL lgorhm for r flld rsmsso l: () mch sdrd Por 1; (b) mch sdrd Por 2. 50

59 M g u d M g u d Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos () Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (b) Fgur 5.5: Clculd rflco coffc from TRL lgorhm for r flld rsmsso l: () op sdrd Por 1; (b) op sdrd Por 2. M g u d M g u d Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos () Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (f) Fgur 5.6: Clculd rflco coffc from TRL lgorhm for r flld rsmsso l: () shor sdrd Por 1; d (b) shor sdrd Por 2. As ws proposd chpr 4, h vro h rflco coffc s h wo pors of h work lyzr roducs rror fcor h dds ccurcy 51

60 o h msurd vlu of. Tkg h rror fcor o cosdro, h w corrcd vlu of dod by m s gv by m = ( + / Γ 1 ε l ). As c b obsrvd from h bov rror fcor quo ( + / Γ 1 ε l ), h smll vro h mch rflco sdrd s h wo pors wll cus lrg vro h rror fcor s comprd o h shor or op rflco sdrd. Ths obsrvo s vrfd xprmlly for h hr dffr sdrds d s prsd Fgur 5.7 blow. D v o F r o m 1 Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos () D v o F r o m 1 Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (b) Fgur 5.7: Error fcor for dffr lods usd TRL procdur for r flld rsmsso l wh: () mch sdrd; (b) op sdrd. 52

61 D v o F r o m 1 Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (c) Fgur 5.7: Error fcor for dffr lods usd TRL procdur for r flld rsmsso l wh: (c) shor sdrd. Thus s s Fgur 5.7, mmum rror s roducd wh hr op or shor rflco sdrds r usd s comprd o mch rflco sdrd TRL Clbro for 20 db uor I h x sp, h r procdur mod bov s rpd wh h r flld rsmsso l rplcd by 20dB uor bw h wo fxurs. As pr our ssumpo, s g xprmlly obsrvd h h rflco coffc s h wo pors r o xcly sm. Th rflco coffcs msurd from h TRL dmbddg lgorhm, s h wo pors for h dffr sdrds r show h x pgs Fgurs

62 M g u d M g u d Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos () Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (b) Fgur 5.8: Clculd rflco coffc from TRL lgorhm for 20 db uor: () mch sdrd Por 1; (b) mch sdrd Por 2. M g u d M g u d Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos () Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (b) Fgur 5.9: Clculd rflco coffc from TRL lgorhm for 20 db uor: () op sdrd Por 1; (b) op sdrd Por 2. 54

63 M g u d M g u d Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos () Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (b) Fgur 5.10: Clculd rflco coffc from TRL lgorhm for 20 db uor: () shor sdrd Por 1; d (f) shor sdrd Por 2. Tkg h rror fcor o cosdro, s s bfor, h smll vro h mch rflco sdrd s h wo pors wll cus lrg vro h rror fcor s comprd o h shor or op rflco sdrd. Ths obsrvo s vrfd xprmlly for h hr dffr sdrds d s prsd Fgur 5.11 h x pg. 55

64 D v o F r o m 1 D v o F r o m 1 Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos () Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (b) D v o F r o m 1 Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (c) Fgur 5.11: Error fcor for dffr lods usd TRL procdur for 20 db uor wh: () mch sdrd; (b) op sdrd; (c) shor sdrd. 56

65 TRL Clbro for combo of 20 db uor d rsmsso l Th r procdur mod bov s g rpd wh h 20dB uor d r flld rsmsso l cocd bw h wo fxurs. As pr our ssumpo, s lso xprmlly obsrvd h h rflco coffc s h wo pors r o xcly sm. Th rflco coffcs msurd from h TRL dmbddg lgorhm, s h wo pors for h dffr sdrds r show Fgurs blow. M g u d M g u d Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos () Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (b) Fgur 5.12: Clculd rflco coffc from TRL lgorhm for 20dB uor d r flld rsmsso l for: () mch sdrd Por 1; (b) mch sdrd Por 2. 57

66 M g u d M g u d Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos () Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (b) Fgur 5.13: Clculd rflco coffc from TRL lgorhm for 20dB uor d r flld rsmsso l for: () op sdrd Por 1; (b) op sdrd Por 2. M g u d M g u d Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos () Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (b) Fgur 5.14: Clculd rflco coffc from TRL lgorhm for 20dB uor d r flld rsmsso l for: () shor sdrd Por 1; (b) shor sdrd Por 2. 58

67 Tkg h rror fcor o cosdro, h smll vro h mch rflco sdrd s h wo pors wll cus lrg vro h rror fcor s comprd o h shor or op rflco sdrd. Ths obsrvo s vrfd xprmlly for h hr dffr sdrds d s prsd Fgur 5.15 blow. D v o F r o m 1 D v o F r o m 1 Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos () Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (b) D v o F r o m 1 Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (c) Fgur 5.15: Error fcor for dffr sdrds usd TRL procdur for h 20dB uor d r flld rsmsso l wh: () mch sdrd; (b) op sdrd; (c) shor sdrd. 59

68 Comprso of rsuls from TRL Clbro for h hr dvcs As s from h xprml rsuls, h rror fcor roducd h TRL clbro for r flld rsmsso l (s s from Fgur 5.7), for 20dB uor (s s from Fgur 5.11) d for h combo of 20db uor d rl (s s from Fgur 5.15) show smlr rsuls, h s h rror fcor ll h hr css show mmum vro for shor d op rflco sdrd d mxmum vro for mch sdrd TRL clbro rsuls obd from usg 20dB uor log wh shor d op sdrds For h x s of msurms, h op lod cocd h wo fxur ds r rmovd d sd rplcd by 20dB uor d op lod oghr. Th obsrvd rflco coffc msurms for hs w combos r show Fgur 5.16 blow. M g u d P h s Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos () Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (b) Fgur 5.16: Rflco coffcs of h 20 db uor d op sdrd: () mgud Por 1; (b) phs Por1. 60

69 M g u d P h s Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (c) Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (d) Fgur 5.16: Rflco coffcs of h 20 db uor d op sdrd: (c) mgud Por 2; (d) phs Por2. Smlrly, 20dB uor d op sdrd rplc h shor sdrd. Th obsrvd rflco coffc msurms for hs w combos r show Fgur 5.17 blow. M g u d P h s Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos () Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (b) Fgur 5.17: Rflco coffcs of h 20 db uor d shor sdrd: () mgud Por 1; (b) phs Por1. 61

70 M g u d P h s Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (c) Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (d) Fgur 5.17: Rflco coffcs of h 20 db uor d op sdrd: (c) mgud Por 2; d (d) phs Por 2. Th followg sco prss h rsuls from h TRL clbro usg h hr dffr dvcs udr s TRL Clbro for r flld rsmsso l Th r procdur s g rpd wh h r flld rsmsso l cocd bw h wo fxurs log wh h d for h modfd shor d op sdrd log wh h 20 db uor. As obsrvd Fgurs 5.16 d 5.17, bcus of h 20 db uor, h rflco coffc s for h shor d op sdrds r o logr clos o o. As pr our ssumpo, wh hs combo of uor d sdrd, slcg mch sdrd wll yld h sm ccurcy s shor or op sdrd wh h 20 db uor. Ths s vrfd by obsrvg h clculd rflco coffcs from h TRL dmbddg lgorhm s show Fgurs h x pgs. 62

71 M g u d M g u d Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos () Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (b) Fgur 5.18: Clculd rflco coffc from TRL lgorhm for r flld rsmsso l: () mch sdrd Por 1; (b) mch sdrd Por 2. M g u d M g u d Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos () Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (b) Fgur 5.19: Clculd rflco coffc from TRL lgorhm for r flld rsmsso l: () op d 20 db uor sdrd Por 1; (b) op d 20 db uor sdrd Por 2. 63

72 M g u d M g u d Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos () Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (b) Fgur 5.20: Clculd rflco coffc from TRL lgorhm for r flld rsmsso l: () shor d 20 db uor sdrd Por 1; (b) shor d 20 db uor sdrd Por 2. Tkg h rror fcor o cosdro, s xpcd for ll h hr sdrds, mly h mch, 20 db uor wh shor/op sdrd should yld lmos smlr vro h rror fcor. Ths obsrvo s s Fgur 5.21 blow d h x pg. D v o F r o m 1 Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos () 64

73 D v o F r o m 1 Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (b) D v o F r o m 1 Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (c) Fgur 5.21: Error fcor for dffr lods usd TRL procdur for h r flld rsmsso l wh: () mch sdrd; (b) 20 db d op sdrd; (c) 20 db d shor sdrd. 65

74 TRL Clbro for 20 db uor Th r procdur s g rpd wh h r flld rsmsso l cocd bw h wo fxurs rplcd by 20 db uor log wh h d obd for h modfd shor d op sdrd log wh h 20 db uor. Th clculd rflco coffcs from h TRL dmbddg lgorhm r show Fgurs blow. M g u d M g u d Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos () (b) Fgur 5.22: Clculd rflco coffc from TRL lgorhm for 20 db uor: () mch sdrd Por 1; (b) mch sdrd Por 2. 66

75 M g u d M g u d Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos () Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (b) Fgur 5.23: Clculd rflco coffc from TRL lgorhm for 20 db uor: () op d 20 db uor sdrd Por 1; (b) op d 20 db uor sdrd Por 2. M g u d M g u d Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos () Frqucy Rg of 45MHz 10GHz Dvdd o 801 qul pos (b) Fgur 5.24: Clculd rflco coffc from TRL lgorhm for 20 db uor: () shor d 20 db uor sdrd Por 1; (b) shor d 20 db uor sdrd Por 2. 67