Realizable Parasitic Reduction For Distributed Interconnects Using Matrix Pencil Technique

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1 Realzable Parastc Reducton For Dstrbuted Interconnects Usng atrx Pencl Technque Janet Wang Prashant Saxena Omar az Xng Wang Unversty o Arzona STL Strategy CAD Lab Intel Corporaton Unversty o Arzona Unversty o Arzona wml@ece.arzona.edu prashant.saxena@ntel.com ohaz@emal.arzona.edu xwg@ece.arzona.edu Abstract - Wth the ncreasng desgn complexty ntegratng realzable reducton technques nto desgn lows has shown more advantages than the tradtonal model order reducton methods. In ths paper we propose a realzable parastc reducton method or RLGC dstrbuted nterconnects. The proposed method obatans a reduced order model based on a moded matrx pencl method. By usng a set o analytc ormulas ths method provdes synthesed RLGC elements. Ths new model s appled to power grd and antena crcuts nvolvng trangular ut waveorms lossy transmsson lnes and dscontnutes o nterconnects. The results show better reducton rato than the standard macromodels and good accuracy compared wth the theoretcal values. I Introducton Wth ncreasng desgn complexty the hgh capacty problem caused by the huge sze o nterconnect networks has pushed tmng/nose analyss and transstor lever smulators to the lmts. odel Order Reducton (OR based technques are the only ways to solve ths hgh capacty ssue. Furthermore the current desgn lows call or realzable reducton technques so that ncremental analyss optmzatons and ECO can be ntegrated nto the desgn lows easly. owever recent work n OR technques such as asymptotc waveorm evaluaton (AWE [] complex requency hoppng (CF [] Pade approxmaton va Lanczos process (PVL [] and Passve Reduced-Order nterconnect macromodelng algorthm (PRIA [4] have been ocused on developng ether passve or stable macromodels nstead o realzable models that can be drectly ed nto general crcut smulators. Cheng and Qn [] proposed a generalzed - transormaton based realzable reducton algorthm. In [] Ismal and hs co-authors developed a RLCK crcut crunchng algorthm. Both o these methods work well or lumped RC/RLCK crcuts but are not sutable or dstrbuted nterconnects because they are generally descrbed as macromodels to start wth. In ths paper we propose a realzable reducton algorthm or dstrbuted nterconnects. Frst a new moded matrx pencl technque s developed to provde the macromodel. Ths new technque ntroduces a phase sht model to reduce the number o terms needed n modelng the dstrbuted nterconnects. Then wth a set o analytc ormulas the developed macromodel s realzed as RLGC lumped crcut elements. At ths pont methods n [] and [] can be appled to urther reduce the order. Ths paper s organzed as ollows: the new phase sht model n dscussed n Secton II. A moded matrx pencl technque s ntroduced n Secton III. A set o analytc equatons s presented n Secton IIII. Secton V provdes the mplementaton detals. Secton VI gves the expermental results. Fnally Secton VII concludes the paper. II. Phase sht model Standard OR macromodels approxmate the mpulse response o hgh-speed nterconnects usng a seres o complex exponentals n the tme doman or complex poles n the requency doman. The tme-doman and requency-doman ttng models can be expressed as: ( t = p ( t + ( t = R exp( st U( t + ( t ( = R F( s = F + F = s s p + = where (t and F(s F ( s ( represent the observed responses n the tme doman and requency doman respectvely resdues s are poles terms and U (t R are and F stand or non-pole denotes a unt step uncton. Such classes o technques or extractng parameters or ( ( rom requency or tme doman data can also be called odel-based Parameter Estmaton (BPE []. One o the dcultes wth standard OR macromodels s that they requre a large number o terms to model nterconnects that are electrcally long. It s necent to model the porton o the waveorm that occurs pror to the arrval o the waveorm.e. a large number o ponts are requred to model the porton o the waveorm that s equal to zero because o causalty. By ncorporatng phase shts n the standard requency-doman macromodel macromodel can be expressed n the requency doman as: R F( s = = exp( t s s s + F ( s ( where t model the phase shts (tme delays. Ths phase sht model s reerred as Delay Reduced-Order odel (DRO n the rest o ths paper. III. Reducton wth the atrx Pencl Technque As stated above n order to approxmate nterconnects response wth ewer terms DRO ncorporates phase shts (tme delays n addton to the poles. The tme-doman expresson or DRO s derved by takng the nverse Laplace transorm o (

2 ( t = R + = exp( s ( t t U( t t (4 where (t represents smulated or measured tme-doman data. owever the matrx pencl method cannot be mplemented drectly or DRO because o the tme delay terms. Thereore (4 needs to be moded beore applyng the matrx pencl method. It should be noted that the tme delays t are not necessarly dstnct rom each other. Assume that a transent waveorm whch ranges rom to the mum tme t exhbts n derent tme delays. We partton (4 nto n tme segments so that we may utlze the matrx pencl method to deal wth each tme segment separately. Each tme segment can be modeled as a sum o exponentals n the same way as the standard macromodels. The expresson or each tme segment and ther summaton can be wrtten as: n ( ( t = (5 = ( = R s t t + Utt Utt+ = ( exp( ( [ ( ( ]. (6 Expresson (6 represents the porton o the waveorm or the tme t <t<t + where s the model order correspondng to the tme segment. The two unt step unctons guarantee ( s zero when t<t or t>t +. Snce the matrx pencl technque requres unormly sampled tme-doman data cubc splne nterpolaton can be appled to make the samplng nterval unorm. In order to represent the tme delay terms n the orm requred or the matrx pencl method we replace t-t wth x and obtan ( ( x = ( R exp( s x + [ U( x U( x ( t + β= t ] where < x < t + - t. In the ollowng paragraphs we ocus on extractng the poles and resdues or the th term. The non-pole terms wll be handled n Secton V. Other terms n (5 can be obtaned n the same way. Frst sample the tme-doman waveorm and let Z = exp( s δ where δ x x s the samplng nterval. Ths leads to ( ( ( k = ( kδ x (8 k = R z ; k = N where N s the total number o samples. Accordng to the matrx pencl method [] and [] two matrces are dened as ( ( (... ( L = ( ( ( N L... ( N ( N L L (7 and (9 ( ( (... ( L = ( ( ( N L... ( N ( N L L ( where L s reerred to as the pencl parameter. Then the matrx pencl s created as λ ( where λ s a scalar parameter. It s easy to prove that when λ = Z where =... the rank o the matrx ( wll be reduced. ence Z may be ound as the generalzed egenvalues o the matrx par { ; } whch s an ll-condtoned (N-L L matrx. Followng [] and [] the sngular value o the decomposton o s taken as ollows: = uv = UDV. ( = Snce the contamnated data compute the truncated pseudo-nverse has ull rank one can + = VD U where U = u u ] V = v v ] and D [ u = [ v. ( The model order can be changed gradually to observe how well the macromodel matches wth the smulaton data. Another condton used to determne the model order s p = where p s the number o sgncant decmal dgts n the data. I p s chosen to be then the sngular values below - are assumed to be sngular values assocated wth nose and are gnored. It s used as a condton n our computer code so that the model order can be determned automatcally. It was proven n [] that the soluton requres the computaton o the matrx egenvalue problem: ( Z z I za = (4 where Z = D U V z are egenvalues and za are egenvectors. The computaton cost s reduced to the calculaton o an egenvalue problem or a matrx. Our code computes the egenvalues usng (4. Once and z are known the resdues R are obtaned by solvng the ollowng least-squares problem

3 ( ( ( ( ( z = N ( N z z z N R z R. (5 N z R relatonshp between the TIR and I TIR s ( s I ( s ( s t ( s = e =. (9 V ( s t t Up untl now we have concentrated on the calculaton o the poles and resdues or (6. The same steps are taken to extract the parameters rom other terms o (5 untl all the poles and resdues are obtaned. IV. Analytc Formulas or Realzable Reducton DRO admttance parameters can be analytcally derved as: = Z Z ( Z Z( Z + Z + Z + Z( Z + Z std [ ]exp ( Z + Z ( Z + Z + Z + Z ( Z + Z = = Z Z Z ( Z + Z ( Z + Z + Z + Z ( Z + Z (6 std ( + TD [ ]exp = Z Z ( Z Z ( Z + Z + Z + Z ( Z + Z ( Z + Z ( Z + Z + Z + Z ( Z + Z std [ ]exp (7 (8 where TD and TD are the tme delays o the two dstrbuted lnes respectvely. atched loads Z that are equal to the characterstc mpedance o transmsson lne are placed on both ends o the crcut to elmnate the multple relectons. The elements Z Z and Z can be resstors nductors or capactors. By choosng derent elements t s possble to model varous dscontnutes wthn the nterconnect. The equvalent crcut s depcted n Fgure. Snce n many cases such as power lnes and antennas the ut sgnal s trangular. ere we use trangular waveorm as our uts to explan the realzaton procedure. Assume the trangular mpulse response s denoted as TIR. I the rse tme o the voltage source s very small.e. t then there s an approxmate Fgure. An equvalent crcut model or a dscontnuty wthn an nterconnect In the real applcatons the tme delays o the two dstrbuted lnes are chosen to be the same ( TD = TD. ns. Thus because o the symmetry and = recprocty we get TIR = TIR TIR = TIR and ther overall tme delay terms are equal ( τ = TD =. ns. The mpedances or the lnes and the matched termnatons are chosen to be Z = 5Ω. PSPICE s used to smulate the currents at port and port when mpulses wth varous rse tmes are used to excte port. In order to very the accuracy o the extracted parameters we compare them wth the values assocated wth the theoretcal admttance parameters or the va crossover and bend cases. Iout we use the moded current = accordng to t (9. It s observed that as the wdth o the trangular mpulse decreases the tme sht between the TIR and the admttance becomes small and gradually the smulaton data approaches the theoretcal admttance waveorm as well. As prevously descrbed n the second secton the TIR can be obtaned by shtng the current I out by t.e. TIR( t = I out ( t + t. Due to the artcal smulaton data the tme delay τ =. ns searched n the begnnng s n the mddle o the rse slope. V. Implementaton Consderng the ratonal orm o a transer uncton a constant term called a non-pole term can be taken out when the orders o the numerator and denomnator are both the same. In ( the non-pole terms are dened or DRO. The non-pole terms may cause a trangular waveorm wth a same tme wdth as the trangular mpulse n the TIR. For example the analytcal expresson or o a smple sngle transmsson lne wth matched loads s shown below : = exp( s Z τ ( where Z s the characterstc mpedance o the

4 transmsson lne and τ s the tme delay. The non-pole term n the admttance.e. n ( accounts or the Z waveorm seen n the TIR plotted n Fgure. I the TIR samplng nterval s equal to the rse tme o the source then the matrx pencl technque s unable to extract a pole representaton or the trangular waveorm whch contans only three ponts snce the model order cannot be greater than the total samplng number. I the samplng nterval s chosen to be / o the rse tme then applcaton o the matrx pencl technque to ths porton wth p chosen to be yelds a model order o 8. owever ths obvously leads to a more complcated model because t takes many terms to approxmate the waveorm.e. the non-pole term. It s also lkely that the poles ound usng the matrx pencl method wll not appear n complex conugate pars. Furthermore the samplng number ncreases then the matrx sze becomes large and the computaton speed wll slow down or even worse the sotware may not be able to compute the response or such a large amount o data. Thereore n the ollowng we wll handle the non-pole porton o the waveorm n a derent way rom the matrx pencl technque. One prerequste or TIR smulatons s that the wdth o the trangular mpulse should be small enough so that we can dstngush the non-pole terms apart rom pole terms. TIR Tme(s x - Fgure. TIR o smple matched sngle transmsson lne For most cases the spectrum ampltude o the pole terms decays to small value at a relatvely low requency. Thus t s reasonable to assume that the bandwdth o the pole terms s much smaller than the mum requency = where t s the rse tme o the trangular t mpulse. Snce the spectrum ampltude o the non-pole term s constant wth requency theoretcally the bandwdth o the waveorm aected by the non-pole term s much larger than the porton that contans only pole terms. Thereore whether a non-pole term exsts or not or a certan tme segment can be determned by lookng at the waveorm s bandwdth. I the bandwdth s comparable to the mum requency (e.g. 7% o wecan tell that a non-pole term exsts. On the contrary the bandwdth s small then there are only pole terms. Two methods can be used to acqure the ampltudes o the non-pole terms. The rst method s to read the ampltude o the lat area n the requency doman n whch there s only the eect o non-pole term snce the spectrum ampltude o the pole terms becomes neglgble ater decay. The second method s to read the peak value o the trangular waveorm drectly n the tme doman and then multply t by t. In our tests the non-pole term values obtaned by usng these two methods are close to the theoretcal admttance values except or the multplcaton actor t as long as a very narrow trangular mpulse s used. Once the non-pole terms have been extracted rom the desred tme segment then the remanng waveorm s approxmated by usng the matrx pencl method. In Fgure ater excludng the non-pole term the remanng part s zero whch agrees wth the analytcal admttance expresson (. At the pont correspondng to the arrval o the sgnal the waveorm may rapdly ump to a certan hgh level rom zero. Note that such sharp changes may not result rom the non-pole terms. Snce the tme delays are taken as the startng ponts or each tme segment approxmaton t s possble that the tme delays obtaned are located n the mddle o steep slopes. For such stuatons the tme delays should be adusted so that we can skp the slope porton o the waveorm to avod undesred poles. The ull realzable reducton procedure s shown below:. Smulate the current at port ( I wth a unt trangular mpulse exctaton at port k ( V k.. Search the current I or tme delays τ =... N where τ N + = t.. Use data nterpolaton to create a unormly-spaced tme record or I and let TIR ( t = I ( t + t. k 4. For = to the number o tme delays (N: (a. Perorm a dscrete Fourer transorm on the tme segment rom τ to τ + and gure out ts bandwdth. (b. I the bandwdth.7 then the non-pole term has an ampltude that s s equal to the value o the waveorm at τ multpled by t. Also set the actual startng approxmaton pont t p = τ + t. (c. I the bandwdth <.7 let t p = τ. owever τ s n the mddle o a sharp slope then try t p = τ + t or use an even larger value or t p. (d. Approxmate the waveorm between matrx pencl method. t p and + τ usng the VI. Expermental Results We now nvestgate a va A va n the transmsson lne. A va s modeled by settng Z = Z = sl and Z =. Thus substtutng them nto (6 to (8 yelds the analytcal admttance expressons as shown below: sl = = exp( sτ ( Z Z ( sl + Z = = exp( sτ ( ( sl + Z

5 a TIR and TIR When L = n the output currents are smulated usng a group o trangular mpulses wth varous rse tmes ( t = ps ps5 ps ps. These data together wth the theoretcal admttance are plotted n Fgure. I out '=I out /delta-t x Theoretcal (TI:ps (TI:ps (TI:5ps (TI:ps Tme(s x - Fgure. The smulated output currents and the theoretcal or a va dscontnuty When the trangular mpulse rse tme s ps we take the FFT o the tme segment rom t p to t (=ns. The spectrum o the TIR data or the tme segment rom.ns to ns s plotted n Fgure 4. Snce ts cuto requency s relatvely low compared wth the mum requency = 5Gz we conclude that there are only pole terms n TIR whch agrees wth the theoretcal result n (. DRO or a va extracted rom smulated TIR data can be expressed as below: m R st TIR s TIR s e p ( = ( =. ( = s s Ampltude x -4 Fgure 4. The spectrum ampltude o the TIR (.ns to ns or a va dscontnuty The poles and resdues or DRO can be acqured by applyng the matrx pencl method to ths tme segment. Table lsts the theoretcal DRO admttance poles and DRO TIR poles or two trangular mpulses wth rse 4 5 Frequency(z x tmes o t = ps and t = ps. As expected (see (6 the pole locatons assocated wth the short ( t = ps trangle mpulse agree well wth the theoretcal value. ps (.ns-ns ps (.ns-ns Ideal Impulse (.ns-ns Real t : Imagnary: Table : The theoretcal DRO pole locatons or the admttance parameters and DRO pole locatons extracted rom the smulated TIR or a va dscontnuty The transent response o DRO or the short TIR also agrees well wth the orgnal smulaton TIR as shown n Fgure 5. b TIR and TIR In theory t s mpossble to draw the tme doman waveorm or.e. the mpulse response snce the non-pole terms would blow up. In the smulaton or the ut current I n the ampltude o the trangular waveorm that results rom the non-pole term does not vary wth the wdth o the trangular mpulse. But the portons o the waveorm aected by the pole terms are qute derent as the trangular mpulse wdth changes. It s not easy to compare these two eects n one plot. Thus we ust plot the smulated ut current I n or varous trangular mpulses n Fgure TI - R x -4 PSPICE DRO Tme(s x - Fgure 5. Comparson o DRO or a va dscontnuty extracted rom TIR smulaton data (.ns-.5ns ( t = ps Next we demonstrate the order derence between the standard macromodel and DRO extracted rom the smulated TIR data or a dscontnuty wthn an nterconnect. Prevously we have acqured the model orders o DRO or the ollowng cases: va (= crossover (= and bend (=. Snce the phase terms account or the propagaton along the transmsson lnes the order o DRO s ndependent o the lengths o the transmsson lnes. owever snce the standard macromodel does not

6 drectly account or these tme delays ts model order wll depend on the lengths o these lnes. To llustrate ths pont we carred out a seres o TIR smulatons wth derent I n TI:ps TI:ps TI:5ps TI:ps Tme(s x - Fgure 6. The smulated ut currents or a va dscontnuty ps (.ns-ns ps (.ns-ns Ideal Impulse (.ns-ns st :.E-.E-. nd : -.788E E- -. Table : The theoretcal DRO non-pole ampltudes or the admttance parameters and the DRO non-pole ampltudes extracted rom the smulated TIR or a va dscontnuty tme delays between two ports. The matrx pencl method was then used to approxmate these smulatons over the entre tme range under the same order determnaton condton =. In Fgure 7 we plot the model order varaton o the standard macromodel n terms o the tme delay. It s observed that the larger the tme delay the more terms are requred to model the waveorm usng the standard macromodel. These results demonstrate that DRO greatly reduces the number o terms requred to model nterconnects especally or electrcally long lnes. 5 ac ro m od el Or de 5 T Standard acromodel DRO Interconnect Delays(ns Fgure 7. Comparson o model order requred by a standard macromodel and DRO or a va dscontnuty VII. Concluson We have descrbed the algorthm or the matrx pencl technque based realzable parastc reducton or dstrbuted lnes. A number o comparsons have shown that the new model greatly reduces the number o terms requred by the standard macromodel. Reerences [] L. T. Pllage R. A. Rohrer Asymptotc waveorm evaluaton or tmng analyss IEEE Transactons on Computer-Aded Desgn vol. 4 pp ay. 99 [] E. Chprout. Nakhla Analyss o nterconnect networks usng complex requency hoppng IEEE Transactons on Computer-Aded Desgn VOL. 4 PP Feb. 995 [] P. Feldmann R. W. Freund Ecent lnear crcut analyss by Pade approxmaton va the Lanczos process IEEE Transactons on Computer-Aded Desgn vol. CAD-4 pp [4] A. Odabasoglu. Celk and L. T. Plegg PRIA: Passve Delay Reduced-Ordernterconnect macromodelng algorthm IEEE Transactons on Computer-Aded Desgn vol. 7 no. 8 pp Aug. 998 [5] R. Sanae E. Chprout. S. Nakhla A ast method or requency and tme doman smulaton o VLSI nterconnects IEEE Transactons on crowave Theory and Technques vol. 4 no. pp Dec. 994 [6] R. Achar. S. Nakhla Ecent transent smulaton o embedded subnetwords characterzed by s-parameters n the presence o nonlnear elements IEEE Transacton on crowave Theory and Technques vol. 46 no. p.56-6 Dec. 998 [7] W. T. Beyene J. E. Schutt-Ane Ecent transent smulaton o hgh-speed nterconnects characterzed by sampled data IEEE Transactons on Components Packagng and anuacturng Technology - Part B Vol. no. pp. 5-4 Feb. 998 [8] R. Prony Essa Expermental et Analytque sur les Los de la Dlatablte de Fludes Elastques et sur Celles del la Force Expansve de la Vapeur de L alkool a Derenctes Temperatures J.l Ecole Polytech. (Pars 795 pp.4-76 [9] T. K. Sarkar O. Perera Usng the atrx Pencl ethod to Estmate the Parameters o a Sum o Complex Exponentals IEEE Transactons on Antennas and Propagaton agazne Vol. 7 No. pp Feb. 995 [] ngbo ua T. K. Sarkar Generalzed Pencl-o-Functon ethod or Extractng Poles o an E System rom Its Transent Response IEEE Transactons on Antennas and Propagaton agazne Vol. 7 No. pp. 9-4 Feb. 989 [] E. K. ller odel-based Parameter Estmaton n Electromagnetcs: Part I. Background and Theoretcal Development IEEE Transactons on Antennas and Propagaton agazne Vol. 4 No. pp. 4-5 Feb. 998 [] Zhanha Qn and Chung-Kuan Cheng Realzable Parastc Reducton Usng Generalzed - Transormaton Proc. o Desgn Automaton Conerence pp. -5. [] Chrayu Amn asud Chowdhury ehea Ismal Realzable RLCK Crcut Crunchng Proc. o Desgn Automaton Conerence pp. 6-.