Distributed RC and RLCInterconnectsGSI ClassECE6458What about inductance for globallines?[ ]1( )ddVI ss sL RsC=+ ++-Vdd uo(t)RCLI[t]21 1 1[ ] [ ]1 1( ) ( )dd ddV VV s I ssC sC sCs sL R s L RssC C= = =+ + + +2[ ]1( )ddVV sRLCs s sL LC=+ +1 2[ ]( )( )ddVV sLCs s s! !=" "211 42 2R RL L LC!" #= + $% &' (221 42 2R RL L LC!" #= $ $% &' (01 21 2 1 2[ ]( )( )dd ddV V KK KV sLCs s s LC s s s! ! ! !" #= = + +$ %& & & &' (21[ ] ( )RT s s sL LC= + +[ ] (2 )d RT s sds L= +11 11(2 )KRL! !=+22 21(2 )KRL! !=+0K LC=211 42 2R RL L LC!" #= + $% &' (221 42 2R RL L LC!" #= $ $% &' (01 21 2[ ]ddV KK KV sLC s s s! !" #= + +$ %& &' (2 21 4 1 42 2 2 21 1 2 21 1[ ](2 ) (2 )R R R Rt tL L LC L L LCddVV t L C e eR RLCL L! ! ! !" # " #" # " #$ % $ %& + & & & &$ % $ %$ % $ %' ( ' (' ( ' (" #$ %= + +$ %$ %+ +' (Transient Solution for Lumped Model2 21 4 1 42 221 1 2 21 1[ ] 1(2 ) (2 )R RRt ttL LC L LCLddV t V e e eR RLC LCL L! ! ! !" # " #" # " #$ % $ %& & &" #$ % $ %&$ % $ %$ %' ( ' (' ( ' ( ' (" #" #$ %$ %= + +$ %$ %$ %$ %+ +$ %' (' (240RL LC! "# <$ %& '240RL LC! "# >$ %& '240RL LC! "# >$ %& 'Under-dampedCriticallydampedOver-damped2LRC<2LRC=2LRC>Calculation of inductance!LI!=11SB d S! ="!" !"iFlux Linkage1d S!"1vlc=21lcv=Single Line Global Interconnect Model+-Rtrx=Lx=0ΔxrΔxlΔxcΔxGround PlanesVdd uo(t)Open Circuit Terminations1jwCrLL>>•Discuss lossless solution•Discuss low -loss solutionOn caulk board !2!x2V x,t( )= rc!!tV x,t( )+ lc!2!t2V x,t( )!2!x2V x,s( )= V x,s( )lcs s +rl"#$%&' !2!x2V x,t( )= lc!2!t2V x,t( )+ rc!!tV x,t( ) V (x,s) = Ae!x lc s s +rl"#$%&'+ Bex lc s s +rl"#$%&' J. A. Davis and J. D. Meindl, “Compact Distributed RLC Interconnect Models Part I: Single LineTransient, Time Delay, and Overshoot Expressions,” IEEE Transactions on Electron Devices, Vol.47, No. 11, pp. 2068-2077, November 2000.J.A. Davis and J. D. Meindl, “Compact Distributed RLC Interconnect Models Part II: CoupledLine Transient Expressions and Peak Crosstalk in Multilevel Networks,” IEEE Transactions onElectron Devices, Vol. 47, No. 11, pp. 2078-2087, November 2000.Find Solutions to this:VINFx,t( )= VddZoRtr+ Zoe!Rt2LI0R2Lt2! x LC( )2"#$$%&''+12t ! x LCt + x LC()*+,-12ke!Rt2LIkR2Lt2! x LC( )2"#$$%&''4 ! .k !1. + 1( )2( )k =1/0()******+,------u t ! x LC"#%&Infinite Line Solution Vinf(x,t ')Vdd!ZoRtr+ Zoe"rx2Zot 'uot '"1( )+12e"rx2Zot '4erx4Zot '"1( )"(1+ #)2#erx4Zo# t '"1( )" 4 "1+ #( )2#$%&&'())*+,,-.//uot '"1( )tttf' =whereNear Wave-front Approximation (tf < t < 3tf)Slow Rising -‘RC’ PortionFast Rising - ‘LC’ PortionVgenx,t,m( )= VddZoRtr+ Zot ! x LCt + x LC"#$%&'m2e!Rt2LI0R2Lt2! x LC( )2()**+,--+12t ! x LCt + x LC"#$%&'12k + m( )e!Rt2LIk + mR2Lt2! x LC( )2()**+,--4 ! .k !1. + 1( )2( )k =1/0"#$$$$$$$%&'''''''u t ! x LC()+, VFIN!,t( )= 2Vgen!,t,m = 0( )+2e!R2Ltn n ! 1+ j( )!i ! j ! n ! i( )!!1( )i"n! i + jj =0#$i =0n$n=1q$Vgenx = (2n + 1)!,t,m = i + j( )Finite Line SolutionWhere Vgen is:Infinite Line First ReflectionRemaining ReflectionsHSPICE Verification1 lumped RLC ElementZo = 266.4Ω r = 37.86 Ω/cm εr = 2.0 l=3.6cm Rtr =0.5ZoHSPICE Verification10 lumped RLC ElementsZo = 266.4Ω r = 37.86 Ω/cm εr = 2.0 l=3.6cm Rtr =0.5ZoHSPICE Verification50 lumped RLC ElementsZo = 266.4Ω r = 37.86 Ω/cm εr = 2.0 l=3.6cm Rtr =0.5Zo500 lumped RLC ElementsZo = 266.4Ω r = 37.86 Ω/cm εr = 2.0 l=3.6cm Rtr =0.5Zo!2!x2VQx,t( )= R C + Cm( )!!tVQx,t( )" RCm!!tVAx,t( ) !2!x2VAx,t( )= R C + Cm( )!!tVAx,t( )" RCm!!tVQx,t( ) !2!x2V1x,t( )V2x,t( )V3x,t( )"#$$$$%&''''= r2C + Cm(Cm0(Cm2C + 2Cm(Cm0 (Cm2C + Cm"#$$$%&'''!!tV1x,t( )V2x,t( )V3x,t( )"#$$$$%&''''+L11L12L13L12L22L23L13 L23L33"#$$$%&'''2C + Cm(Cm0(Cm2C + 2Cm(Cm0 (Cm2C + Cm"#$$$%&'''!2!t2V1x,t( )V2x,t( )V3x,t( )"#$$$$%&''''RLC Partial Differential EquationsAQCmCC123CCCCmCmC CCΔxRΔxL11ΔxV1(x,t)V2(x,t)V3(x,t)L13ΔxL22ΔxL33ΔxL12ΔxL23ΔxSakurai’s RC Partial Differential EquationsRtr+-Rtr+-VQ(L,t)RtrDistributed rlc lines VQ!,t( )=23VC _ FIN!,t,L =12C!2,C = 2C"#$%&'(VC _ FIN!,t,L =12C + 3Cm( )!2,C = 2C + 3Cm"#$%&'"#$$%&'' VA!,t( )=43VC _ FIN!,t,L =12C + 3Cm( )!2,C = 2C + 3Cm"#$%&'(13VC _ FIN!,t,L =12C!2,C = 2C"#$%&'Zo+ = 266.4ΩZo- = 88.77Ωr = 37.86 Ω/cmεr = 2.0l=3.6cmRtr =0.0HSPICE Verification with 1RLC ElementHSPICE Verification with 10 RLCElementsZo+ = 266.4ΩZo- = 88.77Ωr = 37.86 Ω/cmεr = 2.0l=3.6cmRtr =0.0HSPICE Verification with 50 RLCElementsZo+ = 266.4ΩZo- = 88.77Ωr = 37.86 Ω/cmεr = 2.0l=3.6cmRtr =0.0HSPICE Verification with 500RLC ElementsZo+ = 266.4ΩZo- = 88.77Ωr = 37.86 Ω/cmεr = 2.0l=3.6cmRtr =0.0Minimum Feature Size 50nmOn-chip local Clock Frequency 10GhzOn-chip global Clock Frequency 3GhzNumber of Levels 9 levelsASIC Chip Size 13cm2ASIC Transistor Densities 100M/cm2Number of Transistors 1.3 Billion!Example: Design for Global InterconnectNTRS 2012 ASIC DESIGNASSUMPTIONS3.6 cm3Ghz Global Clock Wiring Tier3.6 cmCompare RC and RLC model results!3Ghz RC Bus Design2.1µm2.1µm2.1µm2.1µmRC Models PredictVn = 0.195Vddt = 302.3psRLC Models PredictVn = 0.317Vddt = 315psL = 3.6cm εr = 2.0 ρ =1.67e-7 [Ω-cm]AssumptionsRtr ⎟ 0.0 W=Hε=Hρ=S Vn=0.2VddResults3Ghz RC and RLC Crosstalk00.10.20.30.40.50.60.70.80.910.0E+00 2.0E-10 4.0E-10 6.0E-10 8.0E-10 1.0E-09Time [sec]Normalized Quiet Line Voltage, VQ(L,t)/VddNew Distributed RLC ModelsDistributed RC ModelsRLC Models PredictVn = 0.204Vddt = 296ps3Ghz RLC Bus Re-DesignIncrease Wire Spacing because of inductive crosstalk2.1µm2.95µm2.1µm2.1µmL = 3.6cm εr = 2.0 ρ =1.67e-7 [Ω-cm]Repeater Design(10 repeaters)PhysicalParameterRC ModelsPredictCompact RLCModels PredictPercentDifferencePeakCrosstalk50% TimeDelay0.195Vdd302.3ps0.317Vdd315.0ps62.56%4.2%0.2Vdd27.7ps0.323Vdd29.4ps61.5%6.12%RC ModelsPredictCompact RLCModels PredictPercentDifferencePhysicalParameterRC Design Compact RLCRe-DesignPercentDifferenceRC Design Compact RLCRe-DesignPercentDifferenceW=Hρ=HεS2.1µm2.1µm2.1µm2.95µm-40.47%0.73µm0.73µm0.73µm1.05µm-43.8%Single Driver DesignRC Design of 3GHz BusRLC Re-design of 3Ghz BusRepeater Design(10 repeaters)Single Driver DesignDensity increases by a factor of 3 with repeaters!!InterconnectParameterOptimalValuePreviousValueSource Resistance 6.24 Ohms 6.24 OhmsMetal Width 0.5 micron 0.73 micronMetal Thickness 1.5 micron 0.73 micronDielectric Thickness 0.3 micron 0.73 micronMetal Spacing 0.8 micron 1.05 micronMetal Pitch 1.3 micron 1.78 micronTime Delay 29.5ps 29.4psPeak Crosstalk 0.2Vdd 0.2VddOptimal Scaling Results Design(minimize wire pitch for fixed delay and noise)Length