Gigascale Integration ECE 6458
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This 0 page Class Notes was uploaded by Cassidy Effertz on Monday November 2, 2015. The Class Notes belongs to ECE 6458 at Georgia Institute of Technology - Main Campus taught by Staff in Fall. Since its upload, it has received 8 views. For similar materials see /class/233931/ece-6458-georgia-institute-of-technology-main-campus in ELECTRICAL AND COMPUTER ENGINEERING at Georgia Institute of Technology - Main Campus.
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Date Created: 11/02/15
Distributed RC Interconnects GSI Class ECE645 8 Two types of devices transistors and wires M4 M3 M2 M1 silicon wafer surface Interconnect Devices x y orthogonal pair via x y orthogonal 7 Side view and zoom of IBM Copper Process Top view IBM Copper Process Serni G aba lntemonnec s S g z Global Interconnects CLASS OUTLINE GOAL Understand Interconnect Device and Circuit Models Common Mathematical Techniques to Deriving Transient Models Example Simple Lumped RC Model Example Distributed RC Model without transistor driver effects oComplete single line model with real drivers Reading oCoupled Line Solutions Interconnect Modeling Lumped RC Model R lt l C Vdd Uot 1 T electrical model driver receiver Assume for the moment that driver transistors are perfect switches Assume for the moment that transistor receivers have negligible load capacitance power I 34 p lJlJLJLJLJLJlJLJLJLJlJLJ Models for Wire Resistance and Capacitance W Rp 4 H3 W i He parale plate approximation Models for Wire Resistance and Capacitance parale plate plus fringing field approximation H Czergo 115128 quot L H8 H8 resistance per unit length and capacitance per unit length defined H 622880 p L WH L H8 H8 p Lumped RC Model for Transient Expression R lint 39I39 Vdd uot C T int C dt lemma KO C chU R dt Laplace Transforms SingeSided Laplace Transforms FS ne Std 0 Example 2 711 sr L oolt sgt o 1 FS 26 dt Se e S Exampe at 4 H 1 ltxgt a S 0 Z 1 FS e dt aSe e Sa Laplace Transforms ntegration in the time domain d j ft e d 0 dt Di ulvl WW 39I u 39tvt WW juzv39zdz j u39tvtdt j utv39tdt utvt j u39tvtdt m e stdr fte ff TfUXs fooe s f0e0 sjfxe dz 0 a W f0 Lumped RC Model Example Laplace Domain VddW Ic t C chU R R 0391 EL Cchltsgt SR ms dd 1 InverseLaplace Transforms PartiaFraction Expansion Technique 1 A B SSCl S SCl 1ABSaA ABO 1aA A B 1 A21 32 a 1 l L 1 SSCl Cl S Sa Transient Expression using Lumped RC Model V 1 V S 2 5 c RC 1 S S RC KltsgtVdd 11 s RC Vct2Vdd1eR uo Interconnect Modeling Lumped RC Model time delay 50 and 90 t t VctVdd1 eRC VVdd Vdd1eRC v fraction of supply voltage VcNdd 1 tv RC1n 05 0693RC 09 23RC using parallel plate model shows the problem with global interconnects 2 105 20693 pL M 20693 M WHp H8 HEHp Interconnect Modeling Distributed RC Line x0 xL Vdd u0t rAx I39A39I39A39L I39I39A39AIL A39A39A39A39 T A T T X Vxt voltage along the line xt current along the line Distributed RC Line rAx Ix0 Ix Ax ch T Ixt IxAxt CAXVXZ VxA 0 VxA 0 KCL I xl I xAxl a a 11mAHO A IxlcEVxt KVL hm VxAxt Vxt 8Vxtlzlx9t AXO rAx 8x r Distributed RC Line m121x31 8x r a a 1 z V t 82 a y Vx t m Vx t Find solutions to this partial differential equation Distributed RC Line Use Laplace Transform 2 8 2 Vx S rcs Vx S 0 8x General Solution Vx s A1 sinhx src B1 coshx src 1x s Ml 76 A1 coshxM B1 sinhxM 8x r Distributed RC Line Boundary Conditions Ideal Driver and Load x0 FL Vdd u0t rAx 39vl39A39J A39I39A39A39 CAX J39A39A39A39 T T T Ax time domain Lapace domain Vltx0rgtVddultrgt mzo gkh S 1xLt0 IxLSO Distributed RC Line Use Boundary Conditions to solve for B1 V06 0 s A1 sinh0 B1 cosh0 Q S BZQ 1 S Use Boundary Conditions to solve for A1 S x L s m A1 COShLM B1 SinhLM 0 739 A1 coshLsrc B1 sinhL src 0 Vdd SinhL src S coshL src Laplace Transform for Distributed RC Line Transfer function for any position x Vx s Qw sinhx src Q coshx src S coshL src S Transfer function at end of line Vdd sinhLM2 Vdd T COSmLM TcoshLM V coshLM2 sinhLM2 V x L S S coshLsrc coshLsrc coshLM2 sinhLM2 1 Q 1 VX LS S VxLs Time Domain Expansion Technique Assuming that Ts can be approximated by a power series expansion 1 K0 K1 K2 K3 Kk K quot sTs S s 61 S 52 5 53 s5k 3 5quot 28 This can be transformed into a time domain expansion 1 51t 521 5t STS K0 Kle K3c Kle 36 Where 6k is a complex root of Ts s 5k is a solution to Ts O To find the K coefficients multiply both sides of 28 by sTs K K K 1 T K0 1 V V n b 5TltA5 51 3T0 5 50 5TltSgtltS52J lt29 Setting s0 gives Z 1 1 TOK0 K0 TO Time Domain Expansion Technique To solve for the kth coefficient lets approach 8k STSKk 1 1111 55 S Using L hopital s rule gives dT V T lszhlbn de SiJr 5 S gtk Solving for Kk gives 2 1 6k T5k Is Time Domain Expansion for Distributed RC Line Q 1 VXL S S COShLM Solving for 8k gives 5 5k is a solution to Ts O Ts coshLM 0 LiSkrc j2m1m 0123 2 Sf i 12m1 km1m0123 rc 2L 1 2 7r 5 2k 1 39k123 k rc2L 1 7 7 Time Domain Expansion for Distributed RC Line Q 1 VX LS S Solving for K s gives 2 K 4 K 70 k 5k39r5k 1 1 K0 2 2 1 TO cosh0 iTs icoshLsrc 2 LR i sinhLsrc ds ds 2 J 1 1 Kk 6k Lfdlaiksinh d rc 1 K k 7r2k1 Wsmhggek 4 Time Domain Expansion for Distributed RC Line Using the identities for sinhx gives 1 75 75 smh 39 2k 1 51n 2k 1 1k1 J M gtjgt lt2lt gtgt lt gt The following are the Kk and SK values 1k 1 7r 2 K021 Kkz k123 5k 2k 1 k123 Z2k 1 rc 2L 1 K0 K1 K2 K3 Kk K quot SS S s 61 s 52 3 53 s5k s 5n 28 11e 2e i 4quot e agek wi V aV dd scoshLsrc dd 7 37 m 2k1 4 Time Domain Expansion for two models Distributed Model Vx t Vdd 1 ie gj ig enjz mie 72k 1T 7t 37 2k 1 Lumped Model t mm Vdd 1 wsz Distributed RC Model with R5 and CL Vdd u0t I C L rAX T 39I39J quotquot T Ax Differential E uations Bounda conditions 2 V a Voc zgtrc3m z 1ltXO SgtRsvltxosgt 8x2 7 a Ix Ls CLsVx Ls Voltage in Laglace Domain Vin5 1 M sithsRC cosh SRCRACL5 1 Distributed RC Model with RS and CL Kk 1 5k coshJSRC CTRTRCRSCLEZRCCTRTRSCLS1 R C 5 1 tathakRC S L k J5kRCCTRT VOW Z Vdd1 K1651 Kke5k Kne5quot Sakurai makes the substitution Where 5k is strictly postive Using this new de nition the solutions for Kk and ok come from the following equations K 1k 2 kRlgtquotkC gt k J0k UkR1Ukci CTRT1quot ltRTCT 1 RTCTO39k t 2 8mm may RT Sakurai salient contribution are approximations to rst coef cient and pole RTlCTll K1 104 01 2 RTCTRTCT His Approximate expression is given by Z z K16IVdd Normalized Voltage V l E1 05 Comparison to Simulation I V r b Formula ColC1 F 39 viiw39 l Vailt 1 2 nutRC Time Delay Expression for Distributed RC Model with R8 and CL zv RCRTCTRTCT1 V 01RC 09 23026RSCL RSC RCL 10332RC 105 RCL look familar