Frequency Domain Crosstalk Analysis Utilizing Network Matrices

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1 Frequency Doman Crosstalk Analyss Utlzng Network Matrces Brett Bymaster Abstract Increasngly complex PCB and IC geometres along wth hgh frequency analog and dgtal sgnals demand quck crosstalk analyss for arbtrarly complex structures. hree dmensonal planar electromagnetc smulators combned wth network matrces analyzed n the frequency doman are well suted to accomplsh ths task. Methodologes of evaluatng crosstalk are presented. In partcular, these methods are appled to arbtrarly loaded complex n-port networks. Crosstalk performance s evaluated n and - Parameters through a varety of technques desgned to provde both a computatonally exact soluton, and ntuton buldng approxmatons. An example s presented to llumnate the process of usng frequency doman network parameters to obtan tme doman results. Index erms Crosstalk, catterng Parameters, Parameters, Network Analyss, Frequency Doman Analyss, Fourer ransforms N I. INODUCION EWOK analyss has, unfortunately, been relegated to mcrowave analyss n carefully mpedance matched stuatons. Lower frequency applcatons have focused on developng lumped models for use n transent smulators such as PICE. A vacuum has been left between these two applcatons. here are a number of rch uses of network analyss n non-mcrowave dgtal and analog applcatons. hese applcatons are, however, typcally not mpedance matched, whch requres specal attenton when dealng wth scatterng parameters and scatterng transmsson matrces. Desgn engneers are requred to quckly evaluate potental geometres on both PCBs and ICs. Although many geometres do have analytcal solutons [][], complexty n understandng the bounds of these approxmatons makes accurate analyss lengthy for evaluatng a large number of geometres. A unform method s needed to allow desgners to compare the performance of dfferent geometres. hs method should be both expedent and consstent for any potental geometry. he method should allow the desgner to ntutvely understand and optmze tradeoffs such as parastc loadng and crosstalk. Network parameters n varous forms elegantly ft ths requrement.

2 Freely avalable planar electromagnetc smulators, such as onnet (yracuse, NY) provde the desgner wth surprsngly quck solutons to varyng geometres. Most smple geometres can be solved wthn a matter of mnutes. he output from these electromagnetc (EM) smulators s typcally provded n the form of scatterng parameters. he am of ths paper s to provde a mathematcal framework for evaluatng crosstalk usng dfferent network parameter representatons. eferences regardng the accuracy of EM smulators can be found elsewhere, and hence wll not be covered. he processng begns wth 5Ω normalzed scatterng parameters (-Parameters). hese -Parameters are converted to the approprately chosen network parameter representaton (whether t be or re-normalzed Matrces). he converted parameter of choce s loaded wth the proper source and termnaton mpedances (allowng arbtrary frequency dependent complex loads). he crosstalk rato s then dsplayed. For analog desgns, ths s typcally suffcent for evaluatng the geometry. For dgtal desgns, however, tme doman nformaton s generally more useful. For dgtal desgns, the Inverse Fast Fourer ransform (IFF) s appled wth the proper forcng functon, resultng n a voltage vs. tme output at the desred vctm port. It s key to note that the presented method provdes a computatonally precse soluton wthout approxmatons. It has been well establshed that -Parameters can model essentally any lnear system. he EM smulator s the only source of naccuracy. Gven the proven accuracy of the EM smulators, ths provdes a hgh level of precson over a large varety of geometres and frequences. Freedom from evaluatng the accuracy of analytcal solutons allows the desgner to quckly evaluate and compare dfferent geometres of nterest. here s, however, a prce to pay for the ncreased accuracy. he method presented here s not computatonally effcent. It s not sutable for detectng crosstalk problems on a system wde scale. Instead, t s ntended to provde desgn rules that prevent back-end re-routng. It can also be used to evaluate more computatonally effcent approxmatons. Network Parameters provde a convenent form of post-desgn feedback. Network ector Measurement systems allow the predcted performance to be easly compared wth the actual performance. II. -PAAMEE ANALYI Parameters provde a convenent format for smple crosstalk analyss. For the purpose of ths analyss, the aggressor and vctm traces are arranged as shown n Fg.. Note that - are assumed to be complex, frequency dependent mpedances.

3 Aggressor ctm - Fg.. Crosstalk Port Labelng and ermnaton Fg.. -Matrx epresentaton of Fg.. It s necessary to solve for,, and as a functon of the source voltage,. he -port matrx can be wrtten as, I I I I. () We note that, s I, I, I, I. () Expandng, can be wrtten as,

4 . () epeatng for,, and and then collectng terms,. () hs matrx can be represented as: ] ][ [ ] [ B A, hence, ]. [ ] [ ] [ A B (5) For a gven forcng functon, we can then solve for the voltage at any port. An mportant fgure of mert s the trace s characterstc mpedance,. hs can be easly estmated usng a combnaton of the matrx and the Y Matrx. For a lossy transmsson lne, n can be wrtten,. tanh tanh l l γ γ L L n (6) he characterstc mpedance can be obtaned by lookng at the short crcut nput mpedance and open nput crcut mpedance. l l l γ γ γ tanh tanh tanh C (7) l l l γ γ γ tanh tanh tanh OC (8) Multplyng C and OC, C. OC (9) Note that OC and C are equvalent to and Y (where Y [] - ) for port one. Generalzng,. ] [ ] [ Y ()

5 5 hs assumes that couplng mpedance to adacent channels n Fg. s small. For systems where crosstalk mnmzaton s mportant ths assumpton holds vald by desgn. Although the dervaton s frequency ndependent n nature, C and OC can become msbehaved when the lne length, l approaches a quarter wave length. ypcally, due to recprocty,,,,. () hese are the parameters that most drectly effect crosstalk. Fxng all other parameters and load resstors, these parameters are lnear to the vctm output voltage,.e. doublng these parameters wll double the crosstalk voltage on the vctm lne. he dagonal components,, have a useful nterpretaton. /ϖ s the shunt capactance to GND on the gven trace. mlarly, /ϖy s the seres nductance of the trace. he crosstalk components lsted above have a less ntutve nterpretaton. As an example, we shall look at. hs s nterpreted as, v wth,,. () hs mples that s the rato of the open crcut output voltage on port to the current nected nto port. chematcally, can be nterpreted as n Fg.. Fg.. chematcal meanng of. Note that Fg. s equvalent to a p network, where would be gven as, where Crosstalk,. () nce we expect Crosstalk to be much larger than or, s often small n magntude. nce and are known, t s possble to estmate Crosstalk usng (). Knowng the nature of Crosstalk provdes the desgner wth mportant nsghts. Of course,

6 6 as Crosstalk becomes larger, the system s crosstalk performance mproves. In the schematc of Fg., assumptons were made that may not always hold vald. he traces were modeled as lumped capactve elements, whch s only true for certan frequency and load condtons. Note that a smlar nductve analyss can be performed on Y parameters, although the results are not as ntutve to nterpret. III. -PAAMEE ANALYI catterng parameters provde a powerful analyss tool for crosstalk. When termnated n homogeneous real loads, the nterpretaton of the -Parameters s smple. However, when ports are termnated n varyng, complex loads, the analyss can become troubled. In ths paper, the power wave defnton of Kurokawa [] wll be used. Although ths may not be the most effcent defnton when dealng wth re-normalzed complex -Parameters, t s the most commonly used and studed defnton. For revew, the power wave s defned as follows: a I * b () e I e It s key to note s conugated n b. In many sources, ths s omtted for smplcty. Unfortunately, the omsson can lead to sgnfcant msnterpretaton of -Parameter data. Fundamentally, the complex conugate exsts to permt smplfed power analyss when the parameters are normalzed to complex mpedances. It should be noted that dfferent defntons of - Parameters, such as Pseudo Waves, are equvalent for real loads, but dffer when re-normalzed to complex loads. Cauton s warranted n nterpretaton of re-normalzed -Parameters due to confuson on whch defnton was used. For the purposes of crosstalk analyss, we would lke to express the voltage present on the vctm ports as a functon of the voltage drvng the near end of the aggressor lne. hs s done by re-normalzng the -Parameters to the proper port load mpedances (- n Fg., but referred to as - here). hen the vctm output voltage s calculated usng the re-normalzed -Parameters. hs process s summarzed n Fg.. Due to the unque nature of -Parameters, the notaton s dfferent than that used for -Parameters. We wll derve an expresson for the vctm output voltage of an -Matrx wth an arbtrary number of ports. he aggressor port s called port, whle the vctm port s be called port. he voltage at the vctm port s. he source mpedance on the aggressor port s, and the termnaton mpedance on the vctm lne s.

7 7 Determne ermnaton Impedances (5) Convert 5Ω -Parameters, [], to -Parameters, [] (6) Convert -Parameters,[], to renormalzed -Parameters, [ ] (6) Calculate Output voltage, Fg.. Procedure for determnng crosstalk wth -Parameters he -Parameters are typcally provded as an output of an EM smulator. It s assumed that these parameters are normalzed to 5Ω. In re-normalzaton, t s convenent to frst fnd the -Parameters, whch are ndependent of load resstance, and then convert back to re-normalzed -Parameters, []. [ ] 5([ I] [ ])( I [ ]) (5) where [I] s the dentty matrx [ ' ] [ F]([ ] [ G] )([ ] [ G]) [ F] (6) where the denotes complex conugaton transpose and [F] and [G] are gven by [ F] eal( ) eal( ) eal( ) eal( ) (7) [ G ] (8)

8 8 Now we are prepared to fnd the vctm output voltage for a gven aggressor nput voltage. By the defnton of -Parameters, b wth ak for k. (9) a Pluggng n the defnton provded from () we arrve at * I e e I. () Aggressor Port Near End ctm Port Far End ctm Port Fg. 5 -Parameter Port and Impedance Numberng. By way of example, t can be seen n Fg. 5 that s the voltage at port plus the voltage drop across, I () where s the source drvng port. hs leads to * I e e. () and I can be broken up nto ther travelng wave components, - and I - I -, gvng, ( I I ) * e e. () By defnton, the transmsson lne s termnated n ts characterstc mpedance, and there s no forward reflected wave, mplyng and I, resultng n

9 9 * e ( I ) e. () nce there s no reflected wave, - and I I -. I can be rewrtten as /. We are now able to fnd an expresson for a vctm port s output voltage for a gven aggressor nput voltage. * e e (5) olvng for, * e e. (6) For homogeneous real termnatons, ths reduces to the commonly sted equaton,. (7) Note that (6) requres that and are dfferent ports. he equatons change slghtly when lookng at the voltage at port due to the voltage source drvng that port (essentally the voltage drop across resstor ). In the above dervaton, port (the vctm port s output voltage) was equated to /. Now, however, the voltage present at port due to the voltage source drvng that same port s gven by I. (8) olvng for usng a smlar process as above, * *. (9) * e I. ANMIION CAEING MAICE A convenent aspect of -Parameters s that smaller blocks can be cascaded to create a larger system. hs s typcally done usng ransmsson Matrces, or -parameters (not to be confused wth ABCD parameters whch are sometmes called -

10 parameters as well). -parameters redefne nput and output travelng waves so that multply blocks can be cascaded. A -port ransmsson catterng Matrx s defned below n (). It should be noted that there are multply defntons commonly used for the transmsson matrx. Here, the output wave (b ) s solved n terms of the nput wave (a ). Hence, when cascaded, the transmsson matrces are multpled from the frst system to the last n a forward order. a b a b b a b a () hs defnton can be used to derve transforms to convert from the scatterng matrx to the transmsson matrx, and back to the scatterng matrx. he results of ths dervaton are presented n the appendx. Fg. 6. Four Port Cascade for use wth ransmsson Parameters a b a b A A A A A B C [ ][ ][ ] b a b a C C C C () A three step process s used to model a system such as s shown n Fg. 6. Frst, [ A ], [ B ], [ N ] are converted to [ A ], [ B ], [ N ] usng the relatons n the appendx. (It s possble at ths pont to re-normalze the matrces before the converson to matrces, but specal care must be taken. he output port s mpedance nsde the cascade must be equal to the complex conugate of t s attached nput port. Hence, t s more expedent to frst convert to the matrx wth the parameters normalzed to 5Ω real, and re-normalze later on n the process.) Next, the matrces are multpled together, A B C. () [ ] [ ][ ][ ]

11 where [ ] represents the conglomerate response of A, B, and C. hrd, [ ] s converted to [ ]. [ ] s re-normalzed to the source and load mpedances, and the dfferent parameters of nterest can be found usng the technques of ecton III. he -matrx can be used to save smulaton and analyss tme n crosstalk problems. In planar smulators, such as onnet, the smulaton tme s a functon of the metalzaton area. Hence, long metal runs on a PCB can result n long smulaton tmes. ransmsson matrces can be used to break up a long trace nto multple smaller peces, assumng that the trace s homogenous along ts entre route. In the example above, f [ A ][ B ][ C ], then [ ][ A ]. It can be seen that traces of arbtrary length can be analyzed usng a sngle smulaton. hs method can be extended to fractonal powers for smulaton of non-unt lengths. hs property s useful when a desgner s attemptng to defne desgn rules. A short unt length secton can be smulated once n an EM smulator, and then cascaded untl the crosstalk becomes unacceptable. Hence, the maxmum trace length s defned n an effcent manner. he transmsson matrx converson for systems larger than -ports becomes qute cumbersome. A generalzed n-port cascade technque s presented n [].. EXAMPLE As an example, the crosstalk and pulse dstorton on a hgh speed dgtal flex crcut s presented. Flex crcuts are partcularly dffcult to analyze because of ther poor aspect rato between delectrc thckness, trace thckness, and trace separaton. Plablty requres delectrcs on the order of 5µm (ml). ypcally polymde delectrc s used wth copper traces. In ths example, a data bus wth a fxed data rate s smulated to determne the maxmum crcut length before crosstalk becomes unacceptable. he copper thckness s 7.8µm (½ oz). A two layer flex s smulated, one layer for sgnals and one layer for grounds (a sngle ended communcatons system s mplemented). he delectrc layer and outer coverlay layers are 5µm (ml), ε o. and. respectvely. A sold ground plane would result n a very low characterstc mpedance, so offset GND traces as shown n Fg. 8 are used to brng the mpedance up to around 5Ω. ypcally 5 traces are used for modelng crosstalk. A par of two outsde traces act as aggressors whle the mddle trace acts as the vctm. Note that all four aggressor traces are labeled as port. hese four aggressors are drven n parallel to reduce the system to a -port matrx. Hence, all source and termnaton mpedances for port and port wll be dvded by four to acheve equvalent currents on the trace. he crcut s perfectly symmetrc whch s used to cut smulaton tme n half. he lne of symmetry cuts the vctm trace n half. hs geometry s smulated n onnet over a frequency range from MHz to GHz. A curve ft based on transmsson lne equatons s used to extend ths data down to DC (Hz).

12 he output drver s modeled from an Altera (an Jose, CA) FPGA, tratxgx.v CMO output. All modelng parameters were extracted from an I/O Buffer Informaton pecfcaton (IBI) fle provded by Altera. If PICE models were avalable, a more precse lnearzed model could be generated usng the method presented n [5]. he system as modeled s shown n Fg. 7. he output resstance of the drver s somewhat non-lnear, but can be approxmated as Ω. he packagng s modeled as a pnetwork composed of a seres resstor, shunt capactor and seres nductor. o brng the total source mpedance up to 5Ω, a Ω source termnaton resstor s added. he system s untermnated at the recever to reduce DC power consumpton. he termnaton mpedance s smply a pf capactor, modelng a CMO nput. he system s source voltage ( ) s taken drectly from the IBI fle. he %-9% rse tme of ths output buffer s about ps. he flex crcut was smulated n onnet EM, and the results exported to a Matlab scrpt whch appled the technques outlned above to create a frequency doman voltage transfer functon. An nput forcng functon ( ) was obtaned from the IBI fle, and converted to the frequency doman usng the FF. In the frequency doman, the voltage transfer functon was multpled by and then converted back to the tme doman usng the IFF. It s mportant to nspect ths result, as the fnte Fourer seres can ntroduce errors. Non-causal rngng before the rsng edge (or fallng edge) of the output sgnal s symptomatc of poor samplng or nsuffcent nyquest cutoff frequency. kn resstance and nductance modelng can also cause non-causalty as dscussed by vensson and Dermer [7]. Dense frequency samplng s one dsadvantage of usng data generated n the frequency doman. In ths example, over data ponts are requred for accurate modelng over a range of -Ghz. A full electromagnetc smulaton at each of those ponts would be prohbtvely tme consumng. Fortunately, the smulaton can be run for a few frequency ponts, and then ntellgently curve ft to ncrease the frequency samplng rate. ONNE ncludes a feature called AB (Adaptve Band ynthess) that, n ths example, s able to generate 7 data ponts from only smulated frequences [8]. In Fg. 9d t can be seen that ncreasng trace length degradates rse tme because of skn effect. It s nterestng to note that the overshoot rngng on the recever end s actually caused by packagng parastcs (as can be demonstrated by removng packagng). he voltage at the output of the source n Fg. 9b clearly shows the reflectons n the system (much lke tme doman reflectometry). hese dstortons are largely masked on the recever by the skn effect. In Fg. 9a classc near end crosstalk (NEX) behavor s clearly observed. As the sgnal propagaton tme exceeds the sgnal rse tme, the maxmum voltage of the NEX flattens. nce the recever end s untermnated, the end of the NEX flattop s supermposed wth the reflectons from the recever. In Fg. 9c the recever end shows both capactve and nductvely coupled Far End crosstalk (FEX). he ntal ramp rate of the transmtter s hgh enough n frequency content to generate nductve

13 couplng, seen as a negatve gong spke n the tme doman. As the rse tme s slowed by skn effect, capactve couplng domnates, and a larger postve gong waveform perssts untl the receved sgnal settles. I. CONCLUION A convenent and powerful method for frequency doman crosstalk analyss s outlned n ths paper. wo network analyss methods are covered: -Parameters and -Parameters. he -Parameter method results n straght forward results wth physcal sgnfcance. he -Parameter method, although more complex, allows for n-port analyss. A method for arbtrarly extendng lne length on a -port system s presented wth emphass on desgn rule creaton. A dgtal system analyss example s presented usng a crcut where closed form smplfed solutons are dffcult to achevable due to non-deal geometry. he use of the IFF algorthm s dscussed along wth a dscusson on the results of the analyss.

14 Fg. 7 ystem model used n example. GND GND GND GND GND GND GND GND P P P P P AI Polymde Coverlay Polymde Delectrc Lne of ymmetry AI Polymde Coverlay Fg. 8 Flex cross-secton, showng GND traces, sgnal traces (and assocated port numberng), and delectrc layers. (a) (b) (c) (d) Fg 9 me Doman esults. Near end vctm crosstalk s shown n (a), transmtter waveforms are shown n (b), far end vctm crosstalk s shown n (c), and receved waveforms are shown n (d).

15 5 EFEENCE [] K.Kurokawa, Power Waves and the catterng Matrx, IEEE rans. On M, vol., 965. [] J.C. Coetzee, J. Joubert, Full-wave characterzaton of the crosstalk reducton effect of an addtonal grounded track ntroduced between two prnted crcut tracks, IEEE rans. Crcuts ys. I, vol., 996. [] G.L. Matthae,.I. Long, C.H. hu, mplfed calculaton of wave-couplng between lnes n hgh-speed ntegrated crcuts, IEEE rans. Crcuts ys. I, vol. 7, 99. [] G.. mpson, A Generalzed n-port Cascade Connecton, M- Internatonal Mcrowave ymposum Dgest, vol. 8, 98. [5] G.L. Matthae,.I. Long, C.H. hu, mplfed Lnear epresentaton of Logc Gate ermnal Impedances for Use n Interconnect Crosstalk Calculatons, IEEE Journal of old tate Crcuts, vol., 989. [6]. De Leo, G. Cerr,.Maran Prman, catterng matrx approach for frequency and tme doman crosstalk analyss n multlayer PCBs,IEE Proceedngs cence, Measurement and echnology, vol., 995. [7] C. vensson, G.H. Dermer, me doman modelng of lossy nterconnects., IEEE rans. on Advanced Packagng, vol.,. [8] onnet oftware User Manual elease 8., onnet oftware, yracuse, NY,.

16 6 APPENDIX Equatons for a four port ransmsson Matrces to convert from [] to [] and [] to [] - - [ ] s ss- ss [ ] ss ss s ss- ss - ss ss - sss sss sss - sss - sss sss - ss ss sss- sss sss sss sss - sss s - ss ss - s ss- ss ss - ss sss- sss - sss sss sss - sss - ss ss - sss sss sss - sss - sss sss

17 7 FOONOE Manuscrpt receved January,. hs paper was supported n part by ensant Corporaton. he author attends the Unversty of Iowa chool of Engneerng and s an ndependent contractor for ensant Corporaton, an Leandro, CA, where he has worked as crcut desgn engneer for the past two and a half years.

18 8 LI OF FIGUE Fg.. Crosstalk Port Labelng and ermnaton Fg.. -Matrx epresentaton of Fg.. Fg.. chematcal meanng of. Fg.. Procedure for determnng crosstalk wth -Parameters Fg. 5 -Parameter Port and Impedance Numberng. Fg. 6. Four Port Cascade for use wth ransmsson Parameters Fg. 7 ystem model used n example. Fg. 8 Flex cross-secton, showng GND traces, sgnal traces (and assocated port numberng), and delectrc layers. Fg 9 me Doman esults. Near end vctm crosstalk s shown n (a), transmtter waveforms are shown n (b), far end vctm crosstalk s shown n (c), and receved waveforms are shown n (d).

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