Modeling and Analysis of Crosstalk for Distributed RLC Interconnects using Difference Model Approach

Transcription

1 Modeling and Analysis of Crossalk for Disribued RLC Inerconnecs using Difference Model Approach J.V.R.Ravindra Cener for VLSI and Embedded Sysem Technologies (CVEST), Inernaional Insiue of Informaion Technology, Gachibowli, Hyderabad , India ABSTRACT On-chip inducive effecs are becoming predominan in deep submicron (DSM) inerconnecs due o increasing clock speeds, circui complexiy and decreasing inerconnec lenghs. Inducance causes noise in he signal waveforms, which can adversely affec he performance of he circui and signal inegriy. The radiional analysis of crossalk in a ransmission line begins wih a lossless LC represenaion, yielding a wave equaion governing he sysem response. This paper proposes a difference model approach o derive crossalk in he ransform domain. A closed form soluion for crossalk is obained by incorporaing iniial condiions using difference model approach for disribued RLC inerconnecs. Simulaion resuls show ha he effec of inducive coupling for long inerconnecs is significan bu is almos negligible for local inerconnecs. I is also shown ha when inducance is negleced, he proposed model reduces o a lumped RC model. Also, he analyical model response agrees very well ha obained wih SPICE. All he experimens have been carried ou for 90nm echnology node using Cadence s Dynamic Circui Simulaor SPECTRE c. Caegories and Subjec Descripors B.7.1 [Types and Design Syles]: Very Large Scale Inegraion(VLSI); B.7.2 [Design Aids]: Simulaion General Terms Design Keywords Circui, Disribued RLC, Inerconnec, RC, RL, SPICE 1. INTRODUCTION Crossalk is of considerable imporance in high-speed digial circuis. Many echniques such as wire sizing, gae sizing, and buffer inserion have been proposed o reduce he Permission o make digial or hard copies of all or par of his work for personal or classroom use is graned wihou fee provided ha copies are no made or disribued for profi or commercial advanage and ha copies bear his noice and he full ciaion on he firs page. To copy oherwise, o republish, o pos on servers or o redisribue o liss, requires prior specific permission and/or a fee. SBCCI'07, Sepember 3-6, 2007, Rio de Janeiro, Brazil Copyrigh 2007 ACM /07/ $5.00. M.B.Srinivas Cener for VLSI and Embedded Sysem Technologies (CVEST), Inernaional Insiue of Informaion Technology, Gachibowli, Hyderabad , India srinivas@iii.ac.in crossalk [1-4] ha are based on Elmore delay model [5]. Techniques for crossalk and delay minimizaion under he lossy ransmission line model have also been sudied exensively in he lieraure [6-10]. Gao and Wong [6] applied coninuous wire sizing o minimize he delay while Ismail and Friedman [7] compued a uniform buffer size and he number of buffers o opimize he delay of a circui pah. Roychowdhury e al. [8] proposed algorihms for ransien response under lossy ransmission lines. Gupa e al. [9] modeled he lossy ransmission line using mehod of characerisics. Yu e al. used momens for modeling ransmission line. Wang and Dai [11] adoped he S-parameer macro delay model o minimize he delay and skew based on finie difference approximaion which requires expensive compuaion. In his paper, a new closed form expression for crossalk is obained by incorporaing boundary condiions for RLC inerconnecs using difference model approach. The res of he paper is organized as follows: Theory of crossalk, gliching, even and odd modes of a ransmission line are discussed in secion 2. The difference model approach of disribued RLC circui is explained and also a closed form equaion for delay is derived in secion 3. Simulaion resuls are given in secion 4, while conclusions are made in secion THEORY 2.1 Crossalk Crossalk, which is he coupling of energy from one line o anoher, will occur whenever he elecromagneic fields from differen srucures inerac. In muli-conducor sysems, crossalk can cause wo derimenal effecs: firs, crossalk will change he performance of he ransmission lines in a bus by modifying he effecive characerisic impedance and propagaion velociy. Second, crossalk will induce noise ono oher lines, which may furher degrade he signal inegriy and reduce noise margins. In VLSI circuis, crossalk affecs muual inducance as well as iner-wire capaciance. When he connecors in high speed digial designs are considered, he muual inducance plays a predominan role compared o he iner-wire capaciance. The effec of muual inducance is significan in DSM echnology since he spacing beween wo adjacen bus lines is very small. The muual inducance induces a curren from an aggressor line ono a vicim line which causes crossalk beween connecor lines. 207

2 2.2 Gliching A glich may be induced in connecor j in which he signal is saic, due o neighboring connecor lines in which he signal is varying [12]. This is given by he eq.(1) V j glich = X j dj k ±L jk j k (1) d where L jk represens muual inducance beween j h and k h connecor. The sign of he coupled volage is posiive or negaive depending on wheher he k h neighboring connecor undergoes a rising or a falling ransiion. Moreover, he crossalk noise in VLSI connecors is daa-paern dependen. 2.3 Odd Mode When wo coupled ransmission lines are driven wih volages of equal magniude and 180 ou of phase wih each oher, odd mode propagaion occurs. The effecive capaciance of he ransmission line will increase by wice he muual capaciance, and he equivalen inducance will decrease by he muual inducance. In Fig.1, a ypical ransmission line model is considered where he muual inducance beween aggressor and vicim connecor is represened as M 12. L 11 and L 22 represen he self inducances of aggressor and vicim nodes while C C, C S, and C L denoe he coupling capaciance beween aggressor and vicim, self capaciance and load capaciance respecively. Figure 1: A Typical 2 line Transmission line model Referring o Fig.1, assuming ha L 11 = L 22 = L 0, he currens will be of equal magniude bu flow in opposie direcion [12]. Thus, he effecive inducance due o oddmode of propagaion is given by Eq (2). L odd = L 11 L 22 (2) The magneic field paern of he wo conducors in odd- Figure 2: Magneic Field in Odd Mode mode is shown in Fig Even Mode When wo coupled ransmission lines are driven wih volages of equal magniude and in phase wih each oher, evenmode propagaion occurs. In his case, he effecive capaciance of he ransmission line will decrease by he muual capaciance and he equivalen inducance will increase by he muual inducance. Referring o Fig. 1, assume ha L 11 = L 22 = L 0. Thus, in even-mode propagaion, he currens will be of equal magniude and flow in he same direcion [12]. The effecive inducance, due o even mode of propagaion is hen given by eq.(3). L even = L 11 + L 22 (3) Figure 3: Magneic Field in Even Mode 3. MODELING OF CROSSTALK IN RLC IN- TERCONNECT 3.1 Difference Model The frequency-domain difference approximaion [13] procedure is more general, because i can direcly handle lines wih arbirary frequency-dependen parameers or lines characerized by daa measured in frequency-domain. The imedomain difference approximaion procedure should be employed only if ransien characerisics are available [13]. For a single RLGC line, he analyical expressions are obained for he ransien characerisics and limiing values for all he modules of he sysem and device models. The difference approximaion procedure is applied o boh he characerisic admiances and propagaion funcions and he resuling ime-domain device models have he same form as he frequency-domain models. The difference approximaion procedure involves an approximaion of he dynamic par of he sysem ransfer funcion, given by eq. (12), wih he complex raional series or disored par of he ransien characerisic wih he real exponenial series. This crierion resuls in simple and efficien approximaion algorihms, and requires a minimal number of he original-funcion samples o be available, which is imporan if he line is characerized wih delay and crossalk. 3.2 Analysis of Crossalk using Difference Model The volage (V) and curren (I) equaions are defined for any ypical disribued RLC circui as dv = (R + jωl) I (4) di = (G + jωc) V (5) 208

3 Where R, L, G, C are he resisance, inducance, conducance and capaciance per uni lengh respecively of a parallel ransmission line. Assuming conducance G is zero, and he ransmission line is along z-axis. d 2 V 2 di = (R + jωl) = (R + jωl) jωcv (6) Obaining he soluion for above differenial equaion V = Ae γz + Be γz (7) Assuming V = V R, I = I R and V = V s, I = I s a z=0 respecively (where V R are I R volages and currens a receiving and sending end respecively and z = l ) and solving eq. (7), we obain 4. SIMULATION RESULTS Mos exising crossalk models and reducing echniques consider only capaciive coupling [2-6]. However, a high operaing frequencies, inducive-crossalk effecs can be subsanial and should be included for complee coupling-noise analysis. Figure 4 shows waveforms for capaciive-coupling, inducive-coupling and capaciive+inducive coupling noise separaely. The waveforms show ha inducive noise (volage) is comparable in magniude o he noise (volage) due o capaciive coupling, and hence, neglecing inducance in noise analysis can be highly inaccurae. V s = V R cosh γl + I RZ 0 sinh γl (8) I s = I R cosh γl + VR Z 0 sinh γl (9) where l is he lengh of he ransmission line. The above se of equaions can also be represened in marix form as «««cosh γl Z0sinhγl VR = 1 I s cosh γl Z 0 sinhγl I R Vs (10) Assuming inpu as a sep funcion,v s(s) = s, where is he inpu ampliude, oupu response V R(s) can be obained as, V R(s) = psc (11) s cosh (R + sl) For low frequencies, where, R ωl,he above equaion reduces o V R(s) = (12) s cosh (src) Applying he inverse Laplace ransform, he ime domain response may be obained as v R() = 2 "1 erf v R() = 4 RC Π r!# RC 4 i h1 e RC RC (13) RC (14) For high frequencies, where, R ωl,he equaion reduces o Figure 4: Noise volage waveforms for capaciive, inducive, and capaciive+inducive coupling for a Transmission line Typical values of R, L, and C for differen wire lenghs and for 90nm echnology node have been exraced for long (10mm) and local (1mm) inerconnecs from Arizona s Sae Universiy Predicive Technology Model (PTM) [14] and are summarized in Table 1. The exraced values of inducance and capaciance are hen used o calculae crossalk noise. I is assumed ha he inpu for all experimens is a pulse waveform wih 1.1V V P P and 2ns pulse widh. Table 1: Exraced Values of Long and Local Inerconnecs for 90nm Long (10 mm) R= Ω L=21.668nH Cs= fF Local (1 mm) R=488.9Ω L=1.779nH Cs=24.71fF V R(s) = s cosh s (15) LC Cc=682.49fF Cc=68.32fF Applying he inverse Laplace ransform, ime domain response may be obained as v R() = "1 cos r! # 2 LC (16) 4.1 Crossalk in Local vs Long Inerconnecs Figure 5 illusraes crossalk due o long and local inerconnecs. I can be seen ha while crossalk is negligible in shor inerconnecs, i is significan in long inerconnecs due o inducive naure of he line. 209

4 Table 2: Volage response of a disribued RLC/RC line under a sep inpu exciaion of magniude Mehod Accuracy Time Domain Volage Response Simple Lumped Mehod Approximae V0 (1 e RC ) Brown s 2-por Model [16] Approximae V0 ( e RC e RC 0.023e RC ) Sakurai s Model [15] Approximae Diffusion Model [17] Exac Proposed Model Exac V0 ( e RC e RC ) q V0 (1 erf ( RC )) 4 q i h 2 ) V0 1 cos( LC 4.3 Crossalk Calculaions considering R, L and C To beer undersand he proposed analyical model, he fourh order RLC circui shown in Fig. 1 is considered. The values of R, L and C of his circui are as given in Table 1. I is observed from Fig. 7 ha inducive effecs will be dominan for long inerconnecs. Figure 5: Crossalk due o long and local inerconnecs 4.2 Crossalk Calculaions considering only R and C - Comparison wih Oher Models The crossalk responses of various models [15-17] considering only R and C are shown in Fig 6. I can be seen ha he crossalk prediced by he proposed model is closer o ha of diffusion model han oher models. The resisance and capaciance values exraced are for 90nm echnology node. Noe ha he eq (16) is obained from eq (12) by neglecing inducance(ωl R) for his analysis. The ime domain volage responses for oher models [15-17] is abulaed in Table. 2. Please observe ha he oher models are given only for RC models. Figure 7: Wires Effec of Inducive Coupling for Long Experimen resuls have also been conduced for he ramp inpu. I is observed from he Fig 8. ha he proposed model is also suiable for ramp inpu. The simulaed crossalk is ploed for a sep and a ramp inpu shown in Fig. 8. Figure 6: Uni Sep response for lumped RC model and oher models Figure 8: Comparison wih Ramp inpu 210

5 4.4 Comparison wih SPICE The crossalk model has also been esed for accuracy by resoring o exensive SPICE simulaions of RLC ree circuis. Nearly RLC ree neworks have been generaed randomly and analyzed by applying sep and ramp inpus. Simulaions resuls are shown in Fig. 9 ha he delay prediced by he analyical model is very close o ha obained numerically using SPICE wih an error of abou 2 %. Figure 9: Effecs of Inducance on Inerconnec Delay 5. CONCLUSIONS Signal crossalk in a ransmission line is radiionally obained for a lossless LC equivalen and a ime domain represenaion of he sysem response. The soluion is obained in he ransform domain using 2-por parameers, ypically ABCD parameers. In his paper, he auhors proposed a disribued RLC line model of inerconnecs using difference model approach. A new analyic soluion for crossalk for low and high frequencies is given. A low frequencies, he proposed model exhibis a RC behavior close o ha of diffusion model bu a high frequencies has a subsanially differen behavior due o he effecs of inducance. The validiy of he model a high frequencies is demonsraed by he close agreemen of he crossalk prediced by he model wih ha obained numerically using SPICE. 6. REFERENCES [1] Chung-Ping Chen, Yao-Ping Chen, and D. F. Wong, Opimal Wire-Sizing Formula Under he Elmore Delay Model, Proceedings of 33rd Design auomaion conference, pp, June [2] Chris Cho, and D.F.Wong, Closed form Soluion o Simulaneous Buffer Inserion/Sizing and Wire Sizing ACM Transacions on Design Auomaion of Elecronic Sysems, Vol. 6, No. 3, pp July [3] D. Deschach e al., Theoreical Limis for Signal Reflecions Due o Inducance for On-Chip Inerconnecions In Proceedings of ACM Sysem Level Inerconnec Predicion (SLIP) 2000, pp 55-60, Feb [4] Tai-Chen Chen, Song-Ra Pan, and Yao-Wen Chang, Timing Modeling and Opimizaion under he Transmission Line Model IEEE Transacions on Very Large Scale Inegraion (VLSI) Sysems, VOL. 12, NO. 1, pp 28-41, January [5] W. C. Elmore, The ransien response of damped linear neworks wih paricular regard o wide band amplifiers, J. Applied Physics, vol. 19, no. 1, [6] Y. Gao and D. F. Wong, Shaping a VLSI wire o minimize delay using ransmission line model, in Proc. In. Conf. Compuer-Aided Design (ICCAD), 1998, pp [7] Y. I. Ismail and E. G. Friedman, Effecs of inducance on he propagaion delay and repeaer inserion in VLSI circuis, IEEE Trans. VLSI Sys., vol. 8, pp , Apr [8] J. S. Roychowdhury, A. R. Newon, and D. O. Pederson, Algorihms for he ransien simulaion of lossy inerconnec, IEEE Trans. Compuer- Aided Design, vol. 13, pp , Jan [9] R. Gupa and L. Pileggi, Modeling lossy ransmission lines using he mehod of characerisics, IEEE Transacions on Circuis and Sysems I, vol. 43, pp , July [10] Q. Yu and E. S. Kuh, Exac momen maching model of ransmission lines and applicaion o inerconnec delay esimaion, IEEE Transacions on VLSI Sysems, vol. 3, pp , June [11] J. S. H. Wang and W. W. M. Dai, Opimal design of self-damped lossy ransmission lines for mulichip modules, in Proceedings of Inernaional Conference Compuer- Aided Design (ICCAD), 1994, pp [12] Clayon R.Paul, Keih W.Whies, Syed A. Nasar Reading Inroducion o Elecromagneic Fields McGraw Hill [13] D.B. Kuznesov and J. E. Schu-Aine, Opimum ransien simulaion of ransmission lines, IEEE Transacions on Circuis and Sysems I vol. 43, pp , Feb [14] Arizona Sae Universiy Predicive Technology Model hp:// www. eas. asu.edu/ pm/ [15] T. Sakurai, Approximaion of Wiring Delay in MOSFET LSI, IEEE Journal on Solid-Sae Circuis, Aug. 1983, pp [16] R. J. Aninone and G. W. Brown, The Modeling of Resisive Inerconnecs for Inegraed Circuis, IEEE J. Solid Sae Circuis 18, April. 1983, pp [17] Andrew B. Kahng and Sudhakar Muddu Delay Analysis of VLSI Inerconnecions Using he Diffusion Equaion Model, 31 s ACM/IEEE Design Auomaion Conference, June 1994, pp