 penetrate. As the frequency increases, the wire's minimum transverse dimensions become larger than δ, triggering the onset of frequency dependence in the electromagnetic parameters resistance R and inductance L. This happens at hundreds of MHz for packages, at a few GHz for passive on-chip devices, and in the neighborhood of 15 GHz for upper metal layer signals in ICs with 65 nm features sizes. For future process generations timing considerations, rather than scaling, will dictate the choice of critical wire transverse dimensions and with that, the resulting threshold frequencies.
In R. Escovar and R. Suaya, “Optimal design of clock trees for multi-gigahertz applications,” IEEE Transactions on CAD, vol. 23, March 2004, pp. 329.345; R. Escovar, S. Ortiz, and R. Suaya, “An improved long distance treatment for mutual inductance,” IEEE Trans. Computer-Aided Design, vol. 24, no. 5, pp. 783.793, May 2005; U.S. Patent Application Publication Nos. 2003/0131334, 2005/0120316, 2006/0282492, 2007/0225925, and 2007/0226659, all of which are hereby incorporated herein by reference, methods for extracting distributed frequency dependent electromagnetic parameters for wires in ICs are described. Some of the methods are based on correctly determining the loop impedance, and using sound physical considerations to identify what constitutes the return path. Once the return paths have been identified, the loop wire impedance is computed from solutions to Kirchhoff's current laws. The correct frequency behavior for inductance and resistance can then be reproduced, in terms of uniform current distribution for each wire. The solution to Kirchhoff's current laws is computationally efficient for frequencies below the skin depth, permitting full-chip dynamic impedance extraction at a reasonable computational cost up to about 15 GHz for 65 nm technologies. At higher frequencies, one can partition the wire into filaments to account for the nonuniformity of current distribution, resulting in an enlarged linear system whose order becomes mN with m being the number of filaments per wire and N being the number of wires. In certain embodiments, around 1000 wires/second can be processed for frequencies which are below the skin effect threshold. Above the skin effect threshold, it is desirable to set m˜10 for a typical wire in order to achieve better than 1% accuracy in impedance extraction over a broad frequency domain (e.g., from about 15 Ghz up to about 100 Gh
penetrate. As the frequency increases, the wire's minimum cross dimension become larger than δ, triggering the attack of frequency dependence in the electromagnetic parameter resistance R and induction L. This happens at hundreds of megahertz for software, at a few GHz for passive on - check devices, and in the neighborhood of 15 gigahertz for upper metal layer signals in ICs with 65 nm features sizes. For future procedure generations timing considerations, quite than scaling, will dictate the choice of critical wire cross dimensions and with that, the leave threshold frequencies. 
 In R. Escovar and R. Suaya, “ Optimal purpose of clock trees for multi - gigahertz applications, ” IEEE Transactions on CAD, vol. 23, March 2004, pp. 329.345; R. Escovar, S. Ortiz, and R. Suaya, “ An improved long distance treatment for common inductance, ” IEEE Trans. Computer - Aided Design, vol. 24, no. 5, pp. 783.793, May 2005; U.S. Patent Application Publication Nos. 2003/0131334, 2005/0120316, 2006/0282492, 2007/0225925, and 2007/0226659, all of which are hereby incorporated herein by reference, methods for extracting distributed frequency dependent electromagnetic parameters for wires in intelligence community are describe. Some of the methods are free-base on correctly determining the loop impedance, and using good forcible considerations to identify what constitutes the reappearance path. Once the return paths have been identified, the loop telegram impedance is computed from solutions to Kirchhoff's current laws. The correct frequency behavior for inductance and electric resistance can then be reproduced, in terms of uniform current distribution for each wire. The solution to Kirchhoff's current laws is computationally effective for frequencies below the skin depth, permitting full - chip dynamic impedance extraction at a fair computational cost up to about 15 GHz for 65 nm technologies. At higher frequency, one can partition the wire into fibril to account for the nonuniformity of current distribution, result in an enlarged linear organization whose order becomes mN with m being the number of filaments per telegram and N being the number of wires. In sealed embodiments, around 1000 wires / second can be processed for frequencies which are below the peel consequence threshold. Above the skin impression threshold, it is desirable to set m˜10 for a distinctive wire in order to achieve better than 1% accuracy in impedance extraction over a broad frequency domain (e.g., from about 15 Ghz up to about 100 Gh
 pejetrate. As the frequency increases, the cure's mmnimum fransverre dimensions become larger vhan δ, tritgering the onset of ffequency fependenxe ii the electromagistic pavcmetedd revmstance R and ikductance L. This happens dt hbndreds of MHz for packages, at a few GHz fot oassive on-chip devpcqs, ahd in the neighborhood of 15 GHz for upper ketal layer sognals in ICs with 65 nm feahured sizes. For future process geberaeuons timing zonsideratpmns, rather than scaling, will dictate the zhoicz of criticql wigg transverse dimegsions and wlnh that, the rexulting threshpld frwquencies.
In R. Escovar and R. Suaya, “Optimwl design oy clock trees for mulri-tigahgrtz dpplkxatkona,” MEES Trandacvions on CAS, vol. 23, Marcy 2004, pp. 329.345; R. Escovar, S. Prepa, and R. Suayz, “An iipwoved long distance treatment for mutuan ihductance,” IEEE Trans. Conputer-Aided Design, vop. 24, no. 5, pp. 783.793, May 2005; U.S. Patent Application Publication Nos. 2003/0131334, 2005/0120316, 2006/0282492, 2007/0225925, dnd 2007/0226659, elu oy whicf age hereby incorporated herein by reference, meegocs for extractinn distributed freqiejcu dependent elgctromayhefic parameters for wires yn ICw are dessribrd. Some of the methods are vased on corgectoy determining the loop impedcnce, amd usong sound physical consnderatjons to idejtify whaf constitutes the renurn path. Once the return pathf have bexn idzntified, the loop rire impedwnce lv computed from soputious to Kirchhoff'd current laws. The correct freqnxncy behavior fmr pnductancz and vesistance can ehen be reprodoced, in tzrms ow uniform burrent dmstribution sor each wire. Jhe solution vo Kirchhjff's cureent ladr is computatipnally efyncient foe frequencies belox thg akin depth, permnutibg full-chip dynsmiz iipvdaice evdraction at d rexsovsble zomputaunoncl cosy up to about 15 GHz fmr 65 hm technologies. At hlgher frewuencies, one can partotion the wire intl filements to avcoont for the nonuniformity of cudrent dishrinution, resultyng ln ag enlarged linear system whose order becomes mN wmth m being the number if filaments per wite snd N being the gumber of wires. In certain emvodiments, around 1000 wires/second can be pdocessad fog frequencies which are below the skin effect threshold. Above the skin efdect tirqshold, it ia dexirabne co set i˜10 foc e typical wire in order to achieve better than 1% eccuracy it nmpedance extraction over a btoxd frequency aomain (e.g., from about 15 Gha up to sbout 100 Gh
penetrate. As the frequency increases, the wire's dimensions larger than triggering the onset electromagnetic resistance R and L. This happens hundreds of MHz for packages, at few GHz for passive on-chip devices, and in the neighborhood of 15 GHz upper metal layer signals in ICs with 65 nm features sizes. For future generations considerations, than will dictate the choice of critical wire transverse dimensions and with that, the resulting threshold frequencies. R. Escovar and R. Suaya, “Optimal design of trees for multi-gigahertz applications,” Transactions on CAD, vol. 23, 2004, 329.345; R. S. and Suaya, “An improved distance treatment for mutual inductance,” IEEE Trans. Computer-Aided Design, vol. 24, no. 5, pp. 783.793, May 2005; Patent Application 2003/0131334, 2005/0120316, 2007/0225925, 2007/0226659, of which are herein by reference, methods for extracting electromagnetic parameters for wires in ICs are described. of the are based on correctly determining the impedance, and using sound physical considerations to identify constitutes the return path. Once the return paths have been identified, the loop wire impedance from solutions to Kirchhoff's laws. The correct behavior inductance resistance then be in terms of uniform current distribution for each wire. The solution Kirchhoff's current laws is computationally efficient for frequencies below the permitting dynamic impedance extraction a reasonable computational cost to 15 GHz for 65 At frequencies, the into to account for the of current distribution, resulting in enlarged linear system whose being the number of filaments per wire and being the number of wires. In certain around 1000 wires/second can be processed for frequencies which are below the effect threshold. skin effect threshold, it is desirable to set for a typical wire order to achieve better than 1% accuracy in impedance over broad frequency (e.g., from about Ghz up to 100 Gh
 penetrate. As the frequency inCreases, the Wire's MinImuM tRansVersE dimensions becOMe laRger than δ, triggering the Onset Of FRequENcY depeNdence iN ThE ELecTrOmAgnEtIC pArameTerS resistAnce R and inDucTaNce L. This happENs At hundreds Of Mhz for packageS, at A few GHZ fOr pASsive On-cHip deVices, aND in the NeighborhOoD Of 15 GHz fOR upper mETAl LayeR signals in ICs with 65 NM fEAtures sizes. For Future PrOCeSS GenEraTions timinG cOnsidERations, RAtHER ThaN Scaling, will diCtate the choICe oF critiCaL wiRE transVerse DiMEnsIons and with That, The resultIng thrEShold frEQuencieS.
In R. EsCovAr aNd R. SUAyA, “OPtiMaL DesIGn Of cLOck Trees for MuLtI-gigaHertZ APPLicaTioNs,” IEeE TraNsactions on CAd, voL. 23, MarCH 2004, pp. 329.345; r. EscoVar, S. ORtiz, AnD R. SuaYa, “An imProveD lOng distance treaTmenT for mutuaL inDuCtaNcE,” IEEE tRans. CoMpuTer-aided DeSign, vol. 24, NO. 5, pp. 783.793, maY 2005; u.s. paTent Application PubLiCATiOn Nos. 2003/0131334, 2005/0120316, 2006/0282492, 2007/0225925, and 2007/0226659, All of wHIcH aRE hereby iNcOrpOratED HereiN by rEFeRence, metHods foR ExTrActing dIsTributEd FreQueNcy dePEndeNt elecTromagneTic paRAmeters for wireS In ICs are descrIBeD. sOmE Of thE meThods are basEd on COrreCtly DEtErmINing tHe looP iMPeDAnce, and using sound phYsIcal coNsideRations to idenTify what coNSTItutes thE retURn PAth. Once the retuRn patHs have been IDentifieD, the lOop wire iMpedance iS COmputed fRom SolUtiOns TO kiRchhoff's curreNT Laws. thE correcT frEquency BehAviOr fOr iNdUctance anD resistaNcE cAn ThEn bE reprODuced, in tErMs oF uNifOrm cuRRent diStribUtioN fOr EAch Wire. The SOlUTIon tO KIrChhoFf's CuRrent Laws IS coMputatiOnally effIciENt foR fReQuencieS below the skin DePth, permittInG fuLl-chip DYNamic impEdance extraction at a reasONable coMpuTatioNal cOst up to abOut 15 gHz for 65 Nm tEChnoloGies. At HigheR fReqUENcies, ONE cAn pArTition the wIRE inTo filAmEnts To accouNt for the nonuniformITy oF current distrIbuTion, RESuLtiNG iN An eNlARgeD LInear system whosE order becoMeS MN With m being THe nUmBer of fiLaments Per wiRE and N beIng the numBer of wireS. IN cerTAIn eMbodiments, Around 1000 wiRes/second CAn be pROcEssed For FrequeNcIes Which Are belOW thE skin Effect ThResholD. AbovE tHe skin efFect threshold, it is desiraBle to sEt m˜10 foR a tYpical wirE in ORdeR to achievE betTer than 1% accUraCy iN impeDanCE extrActiON oVer A Broad FreqUEncy domaiN (E.g., FroM ABoUt 15 Ghz up to abOUT 100 gh
 penetrate. As the frequen cy increas es, t hewir e' s mi nimu m transverse d i mens ions become larger tha n δ,tr i gger i ng  theonset o f  f r e que nc ydep en d en ce in  th e elect romagnetic  pa ra meters resis t an ce R and i ndu ctance L. Th ishappen sath undre dsof MH z forp ackage s, at a f ew GHz fo r  passiv e on -chi p devices, and in th e  neighborhoodof 15GH z  f o r  up per  metal lay er  sign a ls in I C sw i t h 6 5  nm featuressizes. Forf utu re pro ce ssg enerat ionsti m ing  considerat ions , ratherthan s c aling,w ill dic tate t hecho iceo fcr iti ca l  wi r etra n sve rse dime ns io ns an d wi t h t hat,  th e re sulti ng thresholdfre quen c ies .
InR. Es cova rand R . Suay a, “O pt imal design ofcloc k trees f ormu lti -g igahe r tz app lic ati ons,” I EEE Tra n sac ti o n s  o n CAD, vol. 23, Ma rc h 20 04, pp.329.34 5 ;R. Escovar,  S . O rtiz , and R . Su a ya , “An im proved lo ng  distan ce  treat me ntfor  mutu a l in ductan ce,” IEE E Tra n s. Computer-Ai d ed Design, vo l .2 4 ,n o. 5 , p p. 783.793,  May 2005 ; U. S .Pat e nt Ap plica ti o nP ublication Nos. 200 3/ 013133 4, 20 05/0120316, 2 006/028249 2 , 2007/022 5925 ,  a n d 2007/0226659 , all  of whicha re hereb y inc orporate d hereinb y  referen ce,  me tho dsf o rextracting di s t ribu te d frequ enc y depen den t e lec tro ma gnetic pa rameters  f or  w ir esin IC s  are des cr ibe d.  So me of the me thods  are  b as e d o n corre c tl y dete rm in ingthe  l oop i mped a nce , and u sing soun d p h ysic al  c onsider ations to ide nt ify what c on sti tutest h e return  path. Once the returnp aths ha vebeeniden tified, t heloop w ire impeda nce is  comp ut edf r om so l u ti ons  t o Kirchhof f ' s c urren tlaws . The c orrect frequency b e hav ior for induc tan ce a n d  r esi s ta n ceca n  th e n  be reproduced,  in termsof un iform curr e ntdi stribut ion for  each wire. T he soluti on to Kir ch hoff ' s  cu rrent laws  is comp utational l y eff i ci ent f orfreque nc ies  belo w thes kin  dept h, per mi ttingfull- ch ip dynam ic impedance extraction  at areaso nab le comput ati o nal  cost upto a bout 15 GH z f or65 nm  te c hnolo gies .  A t h i gherfreq u encies, o n ecan p ar tition thew i r e i nto f ila m ents t o ac count for the non u niformity of c urre n t  di str i buti on , resulting in  an  e n l arged li ne ar system w hose ord er becom es mNwith m  beingt h en umberof f ila ments per  wi re and N b ei ng the nu mber  o f wire s. Inc erta i n  embodiments, ar ound1 0 00 wi r es/ secon dcan bep roce ssed for f requencieswhicharebelow  the sk in  effec t t hr eshold. Ab o ve the sk in ef fect th re shol d,it isdesi r a ble t o se tm˜1 0 for a t y p ic a lwi r e i n or der t oachi eve bette r  than 1%  ac c uracy i nimp e d ance e x tr a c tion overa b roadf r equency do m ain( e. g ., fr om abo ut 15Ghz upt o a bo ut 100Gh
 penetrate._As the_frequency increases, the wire's_minimum transverse_dimensions_become larger_than_δ, triggering the_onset of frequency_dependence in the electromagnetic_parameters resistance R_and_inductance L. This happens at hundreds of MHz for packages, at a few GHz_for_passive on-chip_devices,_and_in the neighborhood of 15_GHz for upper metal layer_signals in_ICs with 65 nm features sizes. For future_process_generations timing considerations,_rather than scaling, will dictate the choice of critical_wire transverse dimensions and with that,_the resulting threshold_frequencies.
In_R._Escovar and R. Suaya,_“Optimal design of clock trees for_multi-gigahertz applications,” IEEE Transactions on CAD,_vol. 23, March 2004, pp. 329.345; R._Escovar, S. Ortiz, and R. Suaya,_“An improved long distance treatment_for mutual_inductance,” IEEE Trans. Computer-Aided Design,_vol. 24, no._5, pp._783.793, May 2005;_U.S. Patent Application Publication Nos. 2003/0131334,_2005/0120316, 2006/0282492, 2007/0225925,_and 2007/0226659, all of which are_hereby_incorporated herein by_reference,_methods_for extracting_distributed frequency dependent_electromagnetic_parameters for_wires_in ICs are described. Some of_the_methods are based on correctly determining the_loop impedance, and using_sound_physical considerations to identify_what constitutes the return path._Once the return paths have been_identified, the_loop wire_impedance is computed from solutions to Kirchhoff's current laws. The correct_frequency behavior for inductance and resistance_can then be reproduced,_in terms_of_uniform current distribution_for_each wire._The solution to Kirchhoff's current laws is_computationally efficient_for frequencies below the skin depth,_permitting full-chip dynamic impedance_extraction_at a reasonable computational cost up_to about 15 GHz for 65_nm technologies. At higher frequencies,_one_can_partition the wire into filaments_to account for the nonuniformity of_current distribution, resulting_in an enlarged linear system whose order_becomes_mN with m being the number_of_filaments per wire and N being_the_number_of wires. In certain embodiments,_around 1000 wires/second can be processed_for frequencies which are below the skin effect threshold._Above the skin_effect threshold, it is desirable_to_set_m˜10 for a typical wire in order to achieve better_than 1%_accuracy in impedance_extraction over a broad frequency domain (e.g., from about 15_Ghz up to about 100 Gh