Network Analyzer Error Models and Calibration Methods

Transcription

1 Ntwork Anlyzr Error Modls nd Clirtion Mthods y Doug Rytting Pg This ppr is n ovrviw of rror modls nd clirtion mthods for vctor ntwork nlyzrs.

2 Prsnttion Outlin Ntwork Anlyzr Block Digrm nd Error Modl ystm Error Modl for Error-Corrction On-Port Error Modl nd Clirtion Two-Port Error Modls nd Clirtion 2-Trm Mthod 8-Trm Mthod 6-Trm Mthod Pg 2 A systm rror modl will drivd from gnric ntwork nlyzr lock digrm. This rror modl will thn simplifid to th stndrd on-port nd 2-trm two-port modls usd th pst yrs. Nwr 8-trm nd 6-trm modls will thn introducd nd th modrn clirtion pprochs dscrid. 2

3 Ntwork Anlyzr Block Digrm nd Error Modl Pg First th lock digrm for ntwork nlyzr is dscrid nd th hrdwr flow grph is dfind.

4 Ntwork Anlyzr Block Digrm RF ourc IF IF LO ourc IF IF Port - Port - 2 Cl DUT 2 Cl 2 Pg 4 This is gnric lock digrm of 4 chnnl ntwork nlyzr. Th sourc cn switchd to xcit port- or port-2 of th dvic undr tst (DUT). Th switch lso provids Z trmintion for th output port in ch dirction. Dirctionl couplrs r usd to sprt th incidnt, rflctd nd trnsmittd wvs in oth th forwrd nd rvrs dirction. Mixrs r usd to down convrt th RF signls to fixd low frquncy IF. Th LO sourc is tund to th frquncy of th RF + IF. Th s-prmtrs of th DUT cn dfind s follows: /, switch in forwrd dirction 2 2 /, switch in forwrd dirction 2 / 2, switch in rvrs dirction 22 2 / 2, switch in rvrs dirction 4

5 Forwrd Dirction Block Digrm Port - Port - 2 DUT 2 RF ourc LO 2 LO Dt Dt Dt IF IF A/D A/D A/D Pg 5 This lock digrm shows th msurmnt systm switchd to th forwrd dirction. Ech of th IF signls r dtctd nd digitizd nd th rl nd imginry trms r msurd. From this dt th mgnitud nd phs cn clcultd. In most modrn ntwork nlyzrs th A/D digitizs dirctly t th IF nd th dtction is don in th digitl domin. Th rsultnt digitizd vrsions of th DUT wvs (,, nd ) r scld vrsion of th ctul wvs t th DUT (,, nd 2 ). 5

6 Error Modl for Forwrd Dirction L - Port - Cl & Connctor DUT 2 Cl & Connctor Port - 2 L C L C M M M C M C 2 22 M C M C M 2 L - L C 2 L C L - L - L - L - 2 L 2- N L N L N L L - +A +N H IF +A +N H +A +N H IF Brnchs With No Ll Pg 6 From th lock digrm flowgrph cn dvlopd showing ll th possil signl pths. Ths pths not only includ th min dsird signls ut th loss, mtch rrors, nd lkg rrors, of th ntwork nlyzr long with th cls, connctors, or pros tht connct to to DUT. Also includd in this modl r th IF, A/D nd dtctor non linritis nd th systm nois. 6

7 Error Modl Dfinitions Incidnt ignl t Port- Rflctd ignl t Port- 2 Incidnt ignl t Port-2 2 Trnsmittd signl t Port-2 ourc Port Msurd Incidnt Port Msurd Rflctd Port Msurd Trnsmittd Port L - Loss from ourc to Port- L - Loss from Port- to ourc L - Loss from ourc to L - Loss from ourc to (Dirctivity) L - Loss from Port- to (Dirctivity) L - Loss from Port- to L 2- Loss from Port-2 to L - Loss from to (Lkg) L C Loss of Cls Rfl Cof of DUT t Port- 2 Forwrd Trns Cof of DUT 2 Rvrs Trns Cof of DUT 22 Rfl Cof of DUT t Port-2 M Mtch t Port- M 2 Mtch t Port-2 M Mtch of ourc M C Mtch of Cls N L Low Lvl Nois t N L Low Lvl Nois t N L Low Lvl Nois t N H High Lvl Nois t N H High Lvl Nois t N H High Lvl Nois t A Dynmic Accurcy t (Linrity) A Dynmic Accurcy t (Linrity) A Dynmic Accurcy t (Linrity) Pg 7 Th ov tl givs th dscription of ch of th rnchs nd th ky nods for th flow grph. This provids vry complt modl for th ntwork nlyzr. Howvr it is possil to rduc th flow grph without ny loss in ccurcy. This rducd flow grph is much sir to nlyz nd will discussd nxt. 7

8 ystm Error Modl for Error-Corrction Pg 8 Th simplifid systm rror modl is dscrid. This systm modl will usd to dvlop th rror corrction procdur. 8

9 ystm Error Modl for Forwrd Dirction Port - DUT 2 Port Brnchs With No Ll Dirctivity L L Lkg L Rflction Trcking L L L L L L 2 ( L ) Trnsmission Trcking 2 L L L 2 Port- Mtch M L L 2 L Port-2 Mtch M 22 2 Thr r lso Errors cusd y th Convrtr, IF, Cls nd Connctors Thr r lso ix Trms in th Rvrs Dirction Pg 9 Th rsultnt systm rror modl is th forwrd portion of th wll known 2-trm rror mod. Ech of th rnchs hv n ccurt rltionship to th originl hrdwr orintd flow grph prsntd rlir. Th 6 forwrd trms dscrid ov show simplifid st of qutions rlting th two flow grphs. Th dirctivity rror is cusd primrily y th couplr lkg or couplr dirctivity. This rror is lso incrsd y cl nd connctor mtch rrors twn th msurmnt couplr nd th DUT. Th rflction nd trnsmission trcking is cusd y rflctomtr nd mixr trcking s wll s cl lngth imlnc twn th msurd ports. Th mtch rror is th mthmticl rtiod port mtch rror tht is not ncssrily th rw port mtch. Th lkg rror is through th LO pth of th mixrs. It is not th lkg of th switch nd this modl ssums th switch lkg is ngligil. 9

10 Improvmnts with Error-Corrction ERROR REMOVED ERROR REMAINING L - Loss from ourc to Port- L - Loss from Port- to ourc L - Loss from ourc to L - Loss from ourc to (Dirctivity) L - Loss from Port- to (Dirctivity) L - Loss from Port- to L 2- Loss from Port-2 to L - Loss from to (Lkg) L C Loss of Cls M Mtch t Port- M 2 Mtch t Port-2 M Mtch of ourc M C Mtch of Cls N L & N H Nois t N L & N H Nois t N L & N H Nois t A Dynmic Accurcy t (Linrity) A Dynmic Accurcy t (Linrity) A Dynmic Accurcy t (Linrity) Drift ftr Error-Corrction tility ftr Error-Corrction Rptility Errors of Clirtion tndrds Pg A linr clirtion procdur is thn pplid to rmov s mny of th rrors s possil. Th loss nd mtch rrors cn grtly rducd dpnding on th ccurcy of th clirtion stndrds usd. Howvr, th nois nd linrity rrors cn not rducd using simpl linr clirtion procdur. If fct th nois nd linrity rrors incrs smll mount. Onc th ntwork nlyzr is clirtd th drift, stility, nd rptility rrors will dgrd th systm prformnc. This usully mns tht th systm will nd to rclirtd t som intrvl dpnding on th systm usg, nvironmnt nd rquird ccurcy.

11 On-Port Error Modl nd Clirtion Pg Th on-port modl will first dvlopd. This will thn usd to furthr dvlop th two-port modl.

12 On-Port, -Trm Error Modl Prfct Rflctomtr Error Adptr Error Trms DUT Port - DUT Dirctivity Γ Port Mtch ( ) Trcking Pg 2 Th on-port clirtion procdur will now dscrid. Th 2- trm modl dscrid rlir simplifis considrly whn onport dvic is ing msurd. Th modl simplifis to just th trms dscriing th dirctivity, port mtch, nd trcking rrors t ch port. Th rrors cn lumpd into fictitious rror dptr tht modifis th ctul DUT rflction cofficint which is thn msurd y prfct rflctomtr. 2

13 On-Port Clirtion Mthod Msurd Actul Γ M - Γ - Γ Γ Γ M - Γ M - - ( ) For rtio msurmnts thr r rror trms Th qution cn writtn in th linr form + ΓΓ M - Γ Γ M With diffrnt known Γ, msur th rsultnt Γ M This yilds qutions to solv for,, nd + Γ Γ M - Γ Γ M + Γ 2 Γ M2 - Γ 2 Γ M2 + Γ Γ M - Γ Γ M Any indpndnt msurmnts cn usd Pg olving th on-port flow grph yilds ilinr rltionship twn th ctul nd msurd rflction cofficint. Th ctul rflction cofficint is mppd or modifid y th thr rror trms to th msurd rsult. This qution cn invrtd to solv for th ctul rflction cofficint knowing th msurd rsult nd th thr rror trms. Th thr rror trms cn dtrmind y msuring thr known stndrds (such s n opn, short nd lod) tht yild thr simultnous qutions. Ths thr qutions cn thn solvd for th thr rror trms.

14 Two-Port Error Modls nd Clirtion 2-Trm Mthod 8-Trm Mthod 6-Trm Mthod Pg 4 Th clssic 2-trm modl will dvlopd first. Thn th mor rcnt 8-trm nd 6-trm modls will dscrid. 4

15 2-Trm Error Modl Forwrd Prfct Rflctomtr Error Adptr DUT, Rvrs 2 2 Pg 5 Mny of th oldr nd lowr cost ntwork nlyzrs us thr couplrs instd of four. This puts th switch twn th couplrs nd if th switch chrctristics chng s th switch is chngd th rror dptr s rror trms will chng. Evn som four couplr ntwork nlyzrs us this mthod s wll, choosing not to us th fourth couplr. This is th originl tchniqu uss in th 96 s for th first utomtic ntwork nlyzrs. It hs supportd th TOL (through, opn, short, lod) 2 trm modl for mny yrs nd is still widly usd tody. 5

16 2-Trm Error Modl Forwrd Modl Prfct Rflctomtr Forwrd Error Adptr DUT [] 2 6 Error Trms 2 Rvrs Modl Prfct Rflctomtr Rvrs Error Adptr DUT [] Error Trms Pg 6 Th two-port cs cn modld in th sm mnnr s th onport. A fictitious rror dptr is plcd twn th two-port DUT nd th prfct rflctomtr msurmnt ports. This rror dptr contins th 6 rror trms for th forwrd dirction. A similr 6 trm modl is usd in th rvrs dirction. 6

17 2-Trm Error Modl FORWARD MODEL Port - DUT 2 Port ( ) ( 2 ) 22 Dirctivity Port- Mtch Rflction Trcking Trnsmission Trcking Port-2 Mtch Lkg M + ( ) 2M + ( 2 ) Pg 7 olving th forwrd flow grph yilds msurmnts M nd 2M. Ths two qutions contin ll four ctul s-prmtrs of th DUT nd th six forwrd rror trms. 7

18 2-Trm Error Modl REVERE MODEL DUT Port - Port ( 2 2 ) ( 2 ) 22 Dirctivity Port- Mtch Rflction Trcking Trnsmission Trcking Port-2 Mtch Lkg 22M 2M ( 2 2 ) ( 2 ) Pg 8 olving th rvrs flow grph yilds msurmnts 22M nd 2M. Ths two qutions contin ll four ctul s-prmtrs of th DUT nd th six rvrs rror trms. Th forwrd nd rvrs qutions comin to giv four qutions contining th four ctul s-prmtrs of th DUT nd 2 rror trms. If th 2 rror trms r known ths four qutions cn solvd for th ctul s-prmtrs of th DUT. 8

19 2-Trm Clirtion Mthod TEP : Clirt Port- using On-Port procdur olv for,, & ( ), Clcult ( ) from TEP 2: Connct Z trmintions to Ports & 2 Msur 2M givs dirctly TEP : Connct Ports & 2 togthr 22 M - M - 2 ( 2M - )( - 22 ) Us th sm procss for th rvrs modl Pg 9 Th 2 rror trms will now dtrmind. First solv for th 6 trms in th forwrd dirction. Thn th sm procdur cn usd to solv for th 6 rvrs trms. tp on clirts port- of th ntwork nlyzr using th sm procdur usd in th on-port cs. This dtrmins th dirctivity, mtch, nd rflction trcking t port- (,, nd ). tp two msurs th lkg or crosstlk rror ( ) from port- to port-2 dirctly y plcing lods on ch of th ports. tp thr consists of conncting port- nd port-2 togthr. Thn msur th port-2 mtch ( 22 ) dirctly with th clirtd port- rflctomtr. Thn with th ports connctd, msur th trnsmittd signl nd clcult th trnsmission trcking ( 2 ). 9

20 2-Trm Clirtion Mthod M 2 + M D M + D ( ) M 22M 2M 2M M 2 2 2M 2 22 M + D D D ( ) M M 2 M 22M 2M M 2 22 Pg 2 This is th rsult of solving th four simultnous msurd s- prmtr qutions. Not tht ch ctul s-prmtr clcultd rquirs msuring ll four s-prmtrs s wll s knowing th 2 rror trms. Rfr to rfrncs [] nd [2] for mor dtils. 2

21 8-Trm Error Modl X Error Adptr Prfct Rflctomtr DUT Prfct witch Y Error Adptr Error Trms To rmov th ffcts of n imprfct switch, us th procdur dscrid ltr. Pg 2 Th 8-trm modl cn drivd from th 2-trm modl. First ssum tht th crosstlk lkg trm is zro. Or tht it cn dtrmind in sprt clirtion stp. Thn ssum tht th switch is prfct nd dos not chng th port mtch of th ntwork nlyzr s it is switchd from forwrd to rvrs. This ssumption is vlid if thr r 4 msurmnt chnnls tht r ll on th DUT sid of th switch. Thn it is possil to mthmticlly rtio out th switch. This mthmticl pproch will xplind t th nd of this ppr. Th sm rror dptr pproch cn now usd to dscri th 8 rror trms. 2

22 8-Trm Error Modl DUT On of th 8 rror trms cn normlizd to yild 7 rror trms Pg 22 Th flow grph consists of n rror dptr t th input nd output of th DUT. For rtio msurmnts of s-prmtrs, th numr of rror trms is rducd to 7 sinc th rror trms cn normlizd. 22

23 8-Trm Error Modl Using th cscd prmtrs in mtrix form yilds MEAURED T M T X T T Y ACTUAL T T X - T M T Y - T 2 22 T M 2M M 22M M M M 22M 2M 2M T X X T Y 2 Y 22 X Y T M X ( ) T ( ) ATB 2 Y 22 2 Pg 2 Not tht th flow grph is cscd of th input rror ox (X), th DUT, nd th output rror ox (Y). Th msurd rsult of this cscd is most sily clcultd y using th cscd mtrix dfinition (t-prmtrs). This formultion ws usd y Engn nd Hor in thir clssic TRL dvlopmnt for th six-port ntwork nlyzr []. And is th sm pproch usd in th HP 85 ntwork nlyzr. From th lst qution in th slid ov, th 7 rror trms r sily idntifid. Thr r t port- ( X,, nd ) nd t port-2 ( Y, 22, nd ) nd on trnsmission trm ( 2 ). Th clirtion pproch rquir nough clirtion stndrds to llow t lst 7 indpndnt osrvtions of th msurmnt systm. Rfr to rfrnc [4] for mor dtils. 2

24 8-Trm Error Modl Forming th Equtions Diffrntly Yilds: 2 T T2 T T4 2 T T T k X 2 k Y T k 4 k22 k 2 Pg 24 Thr is nothr mthmticl formultion for th 8 trm rror modl. Considr th rror dptr s just on dptr twn th prfct msurmnt systm nd th DUT. Thn modl this rror dptr using th cscd t-prmtrs. This t-prmtr mtrix (T) cn prtitiond into th four su mtrixs T, T 2, T, nd T 4. Th 7 rror trms r now dfind s X, k Y,, k,, k 22, nd k. 24

25 8-Trm Error Modl Msurd -Prmtrs M (T + T 2 )(T + T 4 ) - Actul -Prmtrs (T - M T ) - ( M T 4 - T 2 ) Linr-in-T Form T + T 2 - M T - M T 4 Expnding Yilds: + M - X M (k 22 ) + + M + 2 M - 2 X M (k 22 ) + - 2M k + 2M M (k 22 ) - 2 (k Y ) + 2M + 2 2M + + (k ) M (k 22 ) - 22 (k Y ) - 22M k Pg 25 Using this pproch th msurd s-prmtrs formultion is ilinr mtrix qution. It looks much th sm s th on-port ilinr trnsformtion dscrid rlir. Th qution cn sily invrtd to solv for th ctul s-prmtrs. And most importnt th rltionship cn put in linr form. Expnding this mtrix qution for th two-port cs yilds 4 qutions with 4 msurd s-prmtrs, 4 ctul s-prmtrs, nd 7 rror trms. Not tht ths 4 qutions r linr with rgrds to th 7 rror trms. This pproch is prticulrly ttrctiv for multi-port msurmnt systms. Th mtrix formultion dos not chng t ll s dditionl ports r ddd. Rfr to rfrncs [5] [6] [7] for mor dtils. 25

26 8-Trm Error Modl vn or mor indpndnt known conditions must msurd A known impdnc (Z ) nd port- to port-2 connction r rquird TRL & LRL Thru (T) or Lin (L) with known -prmtrs [4 conditions] Unknown qul Rflct (R) on port- nd port-2 [ condition] Lin (L) with known nd 22 [2 conditions] TRM & LRM Thru (T) or Lin (L) with known -prmtrs [4 conditions] Unknown qul Rflct (R) on port- nd port-2 [ condition] Known Mtch (M) on port- nd port-2 [2 conditions] TXYZ & LXYZ Thru (T) or Lin (L) with known -prmtrs [4 conditions] known Rflcts (XYZ) on port- or port-2 [ conditions] TXYX & LXYX Thru (T) or Lin (L) with known -prmtrs [4 conditions] 2 known Rflcts (XY) on port- [2 conditions] On known Rflct (X) on port-2 [ condition] LRRM Lin (L) with known -prmtrs [4 conditions] 2 unknown qul Rflcts (RR) on port- nd port-2 [2 conditions] Known mtch (M) on port- [ condition] UXYZ Unknown Lin (U) with 2 2 [ condition] known Rflcts (XYZ) on port- [ conditions] known Rflcts (XYZ) on port-2 [ conditions] Pg 26 Using ithr of th two formultions dscrid, thr is numr of clirtion tchniqus tht hv n dvlopd. vn or mor indpndnt conditions must msurd. Thr must known impdnc stndrd trmintion or known trnsmission lin. And port- nd port-2 must connctd for on of th msurmnts. Th list of clirtion pprochs cn much longr thn th ons shown ov. And thr continus to nw nd novl wys to solv for th svn rror trms nd clirt th systm. Th 8 trm rror modl pproch hs yildd mor ccurt clirtion mthods s wll s simplifid th clirtion procss. TRL nd LRL provid th st ccurcy. Th othr mthods simplify th clirtion stps comprd to th oldr 2 trm modl. In on cs (UXYZ ov) th thru stndrd dos not nd to known s long s it is pssiv. 26

27 Exmpl: TRL Pg 27 Th st know clirtion mthod using th 8-trm modl is TRL. W will now rviw this clirtion mthod. Th mth nomncltur is slightly diffrnt in this rviw. Th first stp involvs sprting th systm into prfct rflctomtr followd y 4-port rror dptr. This rror dptr rprsnts ll th rrors in th systm tht cn corrctd. It cn split into two 2-port rror dptrs, X (t port-) nd Y (t port- 2), ftr rmoving th lkg (crosstlk) trms s first stp in th clirtion. inc X nd Y r 2-ports it would ppr thr r 8 unknowns to find, howvr sinc ll msurmnts r md s rtios of th s nd s, thr r ctully only 7 rror trms to clcult. This mns tht only 7 chrctristics of th clirtion stndrds r rquird to known. If thru (4 known chrctristics) is usd s on of th stndrds, only dditionl chrctristics of th stndrds r ndd. 27

28 Exmpl: TRL Pg 28 It is convnint to us t-prmtrs cus it llows on to rprsnt th ovrll msurmnt, M, of th DUT, A, s corruptd y th rror dptrs s simpl product of th mtrixs, M XAY. In similr mnnr, ch msurmnt of thr 2-port stndrds, C, C 2, nd C cn rprsntd s M, M 2, nd M. M XC Y M 2 XC 2 Y M XC Y 28

29 Exmpl: TRL Pg 29 Whil thr r 7 unknowns, msuring thr 2-port stndrds yilds st of 2 qutions. Du to this rdundncy, it is not ncssry to know ll th prmtrs of ll th stndrds. X nd Y cn solvd for dirctly plus 5 chrctristics of th clirtion stndrds. 29

30 Exmpl: TRL Pg All 4 prmtrs of C must known ut only 2 prmtrs for C 2 nd non for C if 22. Th simplst of ll stndrds is through lin, so lt C thru nd C 2 Z mtchd dvic. If ndd, impdnc rnormliztion cn usd to shift to diffrnt impdnc s. Th othr prmtrs of C 2 nd C cn solvd from th dt. For this clirtion mthod thr r svrl comintions of stndrds tht fit th rquirmnts. Howvr, thr r lso choics tht gnrt ill-conditiond solutions or singulritis. In choosing pproprit stndrds, on stndrd nds to Z sd, on nds to prsnt high mismtch rflction, nd on nds to connct port- to port-2. In ddition, ll thr stndrds nd to sufficintly diffrnt to crt thr indpndnt msurmnts.

31 Exmpl: TRL Pg Thr r svrl possil strtgis in choosing stndrds. For th first stndrd (C ), th us of zro lngth thru is n ovious slction. But non-zro lngth thru is lso ccptl if its chrctristics r known or th dsird rfrnc pln is in th cntr of th non-zro lngth thru. This stndrd will dtrmin 4 of th rror trms. Th scond stndrd (C 2 ) nds to provid Z rfrnc. In this solution, only th mtch of this stndrd nds to of concrn. Its 2 nd 2 cn ny vlu nd do not nd to known. In fct, thy will found during th clirtion procss. This opns up th choics to wid rng of 2-port componnts, such s trnsmission lin, pir of mtchd lods, or n ttnutor. This stndrd will dtrmin 2 of th rror trms. For th finl stndrd (C ) only on pic of informtion is ndd. This could n unknown rflction vlu for th sm rflction connctd to ch port ( 22 ). inc th othr stndrds hv n wll mtchd, this stndrd should hv highr rflction. This stndrd dtrmins th lst rror trm. Th tl shows prtil list of possil clirtion configurtions with pproprit thr lttr cronyms.

32 Exmpl: TRL Pg 2 Th unknown dvic chrctristics cn sily clcultd y knowing th prmtrs of th X rror dptr, th known stndrd C, nd th msurd dt of th tst dvic nd msurd dt for C. 2

33 Exmpl: Unknown T, Known A & B X Y 22 TM T ATB ( ) 2 ( 2) ( ) 2 T M ATB dt [( 2 ) TM ] dt[ ATB] 2 ( 2) M [ ] dtt dt AB, sinc dt T, cus 2 2 Thrfor ( ) 2 ± dtadtb dtt M Pg This xmpl is for th unknown thru clirtion mthod (UXYZ). This is most sily dvlopd using th cscdd t-prmtr formultion. With known stndrds t port- nd port-2, 6 conditions r providd. Th thru stndrd with 2 2 provids th 7th rquird condition. Th ky to solving this pproch is tht th dtrminnt of T is unity for th pssiv thru clirtion connction ( 2 2 ). Rfr to rfrnc [8] for mor dtil.

34 Exmpl: Unknown B, Known A & T X Y 22 TM T ATB ( ) 2 ( 2) ( ) 2 B T A T D, nd D is compltly known ( ) ( ) 2 Y 22 2 ( ) ( ) 2 2 M D D2 D 2 D22 Thrfor ( ) 2 D2 D2 D 22 Y D D D D Pg 4 This xmpl clirtion pproch (TXYZ) rquirs known stndrds on port- ut non on port-2 s long s th thru connction is known. Th known thru connction provids 4 osrvtions. Th known rflction stndrds connctd to port- complt th rquird 7 known conditions. Rfr to rfrncs [9] [] for mor dtils. 4

35 6-Trm Error Modl Prfct witch Prfct Rflctomtr Error Adptr 2 DUT [] 2 6 Error Trms To rmov th ffcts of n imprfct switch, us th procdur dscrid ltr. Pg 5 With th 8-trm modl w ssumd tht thr ws no lkg twn th vrious msurmnt ports. Th 6-trm modl mks no ssumptions out lkg. 5

36 6-Trm Error Modl 2 DUT On of th 6 rror trms cn normlizd to yild 5 rror trms Pg 6 Ths lkg trms dd 8 dditionl rror trms to th modl. Not only is th trditionl crosstlk trm includd, ut switch lkg, signls rflcting from th DUT nd lking to th trnsmission port, common mod inductnc, nd othr lkg pths r includd. In coxil or wvguid systm, ssuming th switch hs high isoltion, ths rrors r smll. But in fixturd or wfr pro nvironmnt ths rrors cn much lrgr. In wfr pro nvironmnt it is importnt tht th rror trms do not chng s th pros r movd round th circuit. If th rror trms chng, th 6-trm modl thn chngs nd th ccurcy will rduc. Agin th rror trms cn normlizd so tht for rtio msurmnts thr r only 5 rror trms. 6

37 6-Trm Error Modl Error Modl 2 T T2 T T4 2 Msurd -Prmtrs M (T + T 2 )(T + T 4 ) - Actul -Prmtrs (T - M T ) - ( M T 4 - T 2 ) Linr-in-T Form T + T 2 - M T - M T 4 With 5 or mor indpndnt osrvtions th linr mtrix qution cn solvd. TRL s wll s OLT clirtion mthods r possil. Pg 7 olving th 6-trm modl cn don th sm wy tht th 8- trm modl ws solvd. In th 8-trm modl thr wr 8 zro trms in th T mtrix with th ssumption of no lkg. For th 6-trm modl ll trms r prsnt. Of cours th st of rsultnt linr qutions contin mor rror trms, ut th pproch is th sm. Rfr to rfrncs [] [2] [] for mor dtils. 7

38 Rmoving Effcts of Imprfct witch Forwrd Imprfct witch Prfct Rflctomtr Error Adptr DUT [] Rvrs 2 2 Forwrd Rvrs M + 2M 2M + 22M M + 2M 2M + 22M Pg 8 As sttd rlir thr is tchniqu to rtio out or rmov th ffcts of n imprfct switch. Not tht th msurmnt chnnls r ll on th DUT sid of th switch. This llows th msurmnts of incidnt nd rflctd signls from th switch. Thn w cn us th rflctomtrs to msur th rflction nd trnsmission of th switch nd ffctivly rmov th switch from th systm. Mthmticlly th s-prmtrs of th systm gnrt 4 qutions. Two in th forwrd dirction nd two in th rvrs dirction. Ths 4 qutions cn thn solvd for th 4 msurd s-prmtrs. This gnrl wy of msuring s-prmtrs dos not rquir th DUT to trmintd in Z nvironmnt. 8

39 9 Pg 9 d d d d d M M M M olving th 4 qutions yilds Rmoving Effcts of Imprfct witch For -prmtrs olving th 4 prvious qutions yild th ov rsults. Not tht th qutions r writtn to llow rtio msurmnts y th ntwork nlyzr. Typiclly th ntwork nlyzr is mor ccurt mking msurmnts this wy. Nois nd othr common mod rrors r rducd. Using this mthod for msuring th 4 s-prmtrs rquirs 6 rtio msurmnts. Th dditionl two msurmnts r rquird to rmov th ffcts of th switch.

40 Rmoving Effcts of Imprfct witch For -prmtrs Or y dfining Γ nd Γ 2 M d Γ 2 2M d Γ 2M d Γ 2 22M d Γ d Γ Γ 2 Pg 4 Th qutions cn writtn in slightly diffrnt form. Two of th rtios tht r msuring th switch mtch do not chng whn th DUT is chngd. o ths two rtios cn msurd onc nd stord s Γ nd Γ 2. Thn whn mking DUT msurmnts ths just com fixd constnts in th clcultions nd rduc th ndd rtio msurmnts from six to four. Of cours th switch nds to stl for this pproch to work. If th switch chrctristics r chnging thn ths switch ffcts cn rmovd y msuring ll 6 rtios. 4

41 4 Pg 4 T d T d T d T d d M M M M Rmoving Effcts of Imprfct witch For T-prmtrs Th sm pproch cn usd for msuring th t-prmtrs. This will lso rmov th ffcts of th switch nd rquirs 6 rtio msurmnts.

42 8-Trm to -Trm Convrsion - Forwrd Port - DUT 2 Port Port - 2 DUT 2 2 Port Γ Γ Port - DUT Port Γ 2 Γ Γ Pg 42 Th 2-trm nd 8-trm modls dscri th sm systm. o thr must rltionship twn thm. First lts rduc th 2 trm modl to trms y rmoving th crosstlk trms which cn sily msurd in sprt stp. Thn th 8-trm modl cn modifid s shown ov. First th 4th msurmnt chnnl ( ) is rmovd y dfining th switch mtch s Γ. Thn th forwrd modl rror trms for port mtch nd trnsmission trcking ( 22 nd 2 ) cn clcultd. This givs th stndrd forwrd modl if w form th two products nd 2 nd normliz to. 42

43 8-Trm to -Trm Convrsion - Rvrs Port - DUT 2 Port Γ Γ Port - DUT Port Γ Γ Γ Port - 2 DUT Port Pg 4 Th sm procdur cn usd for th rvrs modl. Thr is constrining rltionship for th 8-trm nd -trm modls. This is th sm s sying tht th 8-trm modl cn rduc to 7 trms whn mking rtio msurmnts. And th -trm modl cn rduc to 9 trms. 8-trm: ( )( 2 2 ) ( 2 )( 2 ) -trm: [ + ( - )][ ( )] ( 2 )( 2 ) Rfr to rfrnc [4] for mor dtils. 4

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