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inductance has been taken into consideration. No longer can

interconnects be treated as mere delays or lumped RC networks.

In that frequency range, the most accurate simulation model for

on-chip VLSI interconnects is the distributed RLC model.

Unfortunately, this model has many limitations at much higher

of operating frequency used in today’s VLSI design. The reduction

in cross-sectional dimension leads to more tightly couple

interconnects and therefore, a higher probability of unwanted

crosstalk interference. This can lead to inaccurate simulations

if not modelled properly. At even higher frequency, the aggressor

net carries a signal that couples to the victim net through the

parasitic capacitances. To determine the effects that this crosstalk

will have on circuit operation, the resulting delays and logic

levels for the victim nets must be computed. This paper proposes

a difference model approach to derive crosstalk and delay in the

transform domain. A closed form solution for crosstalk and delay

is obtained by incorporating initial conditions using difference

model approach for distributed RLCG interconnects. The

simulation is performed in 0.18μm technology node and an error

of less than 1% has been achieved with the proposed model when

compared with SPICE.

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- 1. ACEEE Int. J. on Information Technology, Vol. 02, No. 02, April 2012 Modelling of Crosstalk and Delay for Distributed RLCG On-Chip Interconnects For Ramp Input 1 V. Maheshwari, 2S. Gupta, 3R. Kar, 4D. Mandal, 5A. K. Bhattacharjee 1 Hindustan College of Science and Technology, Mathura, U.P., India 2 Hindustan Institute of Technology and Management, Agra, U.P., India maheshwarivikas1982@gmail.com 3 Department of Electronics and Communication Engineering National Institute of Technology, Durgapur, West Bengal, INDIA rajibkarece@gmail.comAbstract--In order to accurately model high frequency affects, the technology contribute to the increase of crosstalkinductance has been taken into consideration. No longer can problems: the increase of the number of metal layers [4], theinterconnects be treated as mere delays or lumped RC networks. increase in the line thickness, the density of integration andIn that frequency range, the most accurate simulation model for the reduction of the spacing between lines. This set of newon-chip VLSI interconnects is the distributed RLC model. challenges is referred to as signal integrity in general. AmongUnfortunately, this model has many limitations at much higherof operating frequency used in today’s VLSI design. The reduction all these problem, capacitive coupling induced cross talk isin cross-sectional dimension leads to more tightly couple the issue that has been addressed in many literatures.interconnects and therefore, a higher probability of unwanted Crosstalk will occur on the chip, on the PCB board, on thecrosstalk interference. This can lead to inaccurate simulations connectors, on the chip package, and on the connectorif not modelled properly. At even higher frequency, the aggressor cables.net carries a signal that couples to the victim net through the Furthermore, the technology with multi-conductorparasitic capacitances. To determine the effects that this crosstalk systems, excessive line-to-line coupling, or crosstalk, canwill have on circuit operation, the resulting delays and logic cause two detrimental effects. First, crosstalk will change thelevels for the victim nets must be computed. This paper proposes performance of the transmission lines in a bus by modifyinga difference model approach to derive crosstalk and delay in thetransform domain. A closed form solution for crosstalk and delay the effective characteristic impedance and pro patterns, line-is obtained by incorporating initial conditions using difference to-line spacing, and switching rates. In this paper, we havemodel approach for distributed RLCG interconnects. The proposed a closed form expression for the coupling noise bysimulation is performed in 0.18µm technology node and an error analyzing the interconnect using RLCG model for ramp input.of less than 1% has been achieved with the proposed model when The major drawback of the proposal made in [5] is that it doescompared with SPICE. not consider the shunt lossy component for estimation ofKeywords- Cr osstalk M ode l, Distr ibuted RLCG Seg ment, the coupling noise. The proposed model presented in thisInterconnect, No ise, Delay, VLSI paper is a generic one in the sense that the model proposed in [5] can be easily derived by just neglecting the shunt lossy I. INTRODUCTION component term (i.e. G=0). Inductance causes overshoots and undershoots in the The rest of the paper is organized as follows: Section 2signal waveforms, which can adversely affect the signal discusses the basic theory, transmission line model, crosstalk,integrity. For global wires, inductance effects are more severe glitch and different modes of propagation. Section 3 describesdue to the lower resistance of these lines, which makes the the difference model and the proposed method for noise andreactive component of the wire impedance comparable to the delay calculation. Section 4 shows the simulation results.resistive component, and also due to the presence of Finally section 5 concludes the paper.significant mutual inductive coupling between wires resultingfrom longer current return paths. It is shown that the II. BASIC THEORYconductors of a circuit system should be regarded as A. TRANSMISSION LINE MODELtransmission lines for theoretical analysis and practical designin the recent high-speed integrated circuit technology [1]. Defining the point at which an interconnect may be treated The design techniques in sub-micron technologies as a transmission line and hence, reflection analysis applied,increase the effects of coupling in interconnections [2]. In has no consensus of opinion. A rule of thumb is that whendeep sub-micron technology, the order of capacitive coupling the delay from one end to the other is greater than risetime/2,between lines reach to some severe values which signifies the line is considered as electrically long. If the delay is lessthat onecan’t be indifferent to the ampleness of the noise than risetime/2, the line is electrically short. A transmissiondue to this coupling. Integrated circuit feature sizes continue line [6] can be described at the circuit level using seriesto scale well below 0.18 microns, active device counts are inductance and resistance combined with shunt capacitancereaching hundreds of millions [3]. Several factors bound to and conductance.An infinitesimal unit length of the© 2012 ACEEE 1DOI: 01.IJIT.02.02. 55
- 2. ACEEE Int. J. on Information Technology, Vol. 02, No. 02, April 2012transmission line looks like the circuit as shown in Figure 1. In multi-conductor systems, crosstalk can cause twoThe parameters are defined as follows: R = series resistance detrimental effects: first, crosstalk will change the performanceper unit length, L = series inductance per unit length, G = of the transmission lines in a bus by modifying the effectiveshunt conductance per unit length, C = shunt capacitance characteristic impedance and propagation velocity. Secondly,unit length. crosstalk will induce noise onto other lines, which may further degrade the signal integrity and reduce noise margins. C. GLITCH Crosstalk Glitch (CTG) is a signal provoked by coupling effects among interconnects lines which have unbalanced drivers and loads [9]. The magnitude of the glitch depends on the ratio of coupling capacitance to that of line to ground capacitance. When a transition signal is applied at a line which has a strong line-driver while stable signals are applied Figure 1. RLCG segment of a transmission line at other lines which have weaker drivers, the stable signals It is critical to model the transmission path when designing may experience a coupling noise due to the transition of thea high-performance, high-speed serial interconnect system. stronger signal. A glitch may be induced in connector ‘j’ inThe transmission path may include long transmission lines, which the signal is static, due to neighbouring connectorconnectors, vias and crosstalk from adjacent interconnect. lines in which the signal is varying [10]. This is given by (1).Values for R, L, C, and G are extracted from a given layout, djk jdesigned in 0.18µm technology. Vglitch L jk j k (1) j dtB. CROSS TALK Where Ljk represents mutual inductance between j and kth th Crosstalk is the undesired energy imparted to a transmis- connector. The sign of the coupled voltage is positive orsion line due to signals in adjacent lines. The magnitude of negative depending upon whether the k th neighbouringthe crosstalk induced is a function of rise time, signal line connector undergoes a rising or a falling transition.geometry and net configuration (type of terminations, etc.). D. ODD MODETo overcome the problems faced at high frequency of opera-tion, shielding techniques have been employed [7]. A com- When two coupled transmission lines are driven with volt-mon method of shielding is to place ground or power lines at ages of equal magnitude and 180 degree out of phase withthe sides of a victim signal line to reduce noise and delay each other, odd mode propagation occurs. The effective ca-uncertainty. pacitance of the transmission line will increase by twice the The crosstalk between two coupled interconnects is of- mutual capacitance, and the equivalent inductance will de-ten neglected when a shield is inserted, significantly under- crease by the mutual inductance. In Figure 2, a typical trans-estimating the coupling noise. The crosstalk noise between mission line model is considered, where the mutual induc-two shielded interconnects can produce a peak noise of 15% tance between aggressor and victim connector is representedof VDD in a 0.18 µm CMOS technology [8]. An accurate esti- as M12. L1 and L2 represent the self inductances of aggressormate of the peak noise for shielded interconnects is therefore and victim nodes while Cc, C, denote the coupling capaci-crucial for high performance VLSI design. In the complicated tance between aggressor and victim, self capacitance, respec-multilayered interconnect system, signal coupling and delay tively.strongly affect circuit performances. Thus, accurate intercon- Assuming L1 = L2 = L0, the currents will be of equal magni-nect characterization and modelling are essential for today’s tude but flow in opposite direction [10]. Thus, the effectiveVLSI circuit design. Two major impacts of cross talk are: inductance due to odd mode of propagation is given by (2).(i) Crosstalk induces delays, which change the signal Lodd L1 L2 (2)propagation time, and thus may lead to setup or hold time The magnetic field pattern of the two conductors in odd-failures. mode is shown in Figure 3.(ii) Crosstalk induces glitches, which may cause voltagespikes on wire, resulting in false logic behaviour. Crosstalkaffects mutual inductance as well as inter-wire capacitance.When the connectors in high speed digital designs areconsidered, the mutual inductance plays a predominant rolecompared to the inter-wire capacitance. The effect of mutualinductance is significant in deep submicron technology(DSM) technology since the spacing between two adjacentbus lines is very small. The mutual inductance induces acurrent from an aggressor line onto a victim line which causes Figure 2. Two Coupled Transmission line modelcrosstalk between connector lines.© 2012 ACEEE 2DOI: 01.IJIT.02.02. 55
- 3. ACEEE Int. J. on Information Technology, Vol. 02, No. 02, April 2012 The electrical parameters of each section are R”x, L”x, C”x and G”x respectively, where R, L, C and G are per-unit length resistance, inductance, capacitance and conductance of the line, respectively. Using Kirchoff’s Voltage Law (KVL), we can write, Figure 3. Magnetic Field Figure 4. Magnetic Field in in Odd Mode Even Mode di ( x, t ) v ( x, t ) i ( x, t ) Rx Lx v ( x x , t ) (4) dtE. EVEN MODE Using Kirchoff’s Current Law (KCL), we can write, When two coupled transmission lines are driven with dv ( x x, t )voltages of equal magnitude and in phase with each other, i ( x , t ) G x v ( x x , t ) c x i ( x x , t ) (5) dteven mode of propagation occurs. In this case, the effective Simplifying (4) and (5) and after applying Laplace transform,capacitance of the transmission line will decrease by the we get,mutual capacitance and the equivalent inductance will V ( x )increase by the mutual inductance. Thus, in even-mode ( R sL ) I ( x) (6) xpropagation, the currents will be of equal magnitude andflow in the same direction [10]. The effective inductance, due I ( x) (G sC)V ( x) (7)to even mode of propagation is then given by (3). x Differentiating (6) and (7) with respect to the x, and after III. MODELLINGOF CROSS TALK IN RLCG INTERCONNECT simplifying we get,A. DIFFERENCE MODEL 2V ( x ) 2V ( x ) (8) The frequency-domain difference approximation [11] pro- x 2cedure is more general, because it can directly handle lines andwith arbitrary frequency-dependent parameters or lines char- 2 I ( x)acterized by data measured in frequency-domain. The time- 2 I ( x) (9) x 2domain difference approximation procedure should be em- where P is the propagation constant and is defined as,ployed only if transient characteristics are available. For asingle RLCG line, the analytical expressions are obtained for R sL G sC (10)the transient characteristics and limiting values for all the The general solution of (8) is given bymodules of the system and device models. The differenceapproximation procedure is applied to both the characteristic V ( x ) A1e x A2e x (11)admittances and propagation functions and the resultingtime- where A1 and A2 are the constants determined by thedomain device models have the same form as the frequency- boundary conditions. From (8-9) and (11) we get,domain models. The difference approximation procedure in-volves an approximation of the dynamic part of the system transfer function, with the complex rational series or distorted x A1e x A2 e x ( R sL) I ( x) (12)part of the transient characteristic with the real exponential After simplifying we get,series. This criterion results in simple and efficient approxima- 1tion algorithms, and requires a minimal number of the original- I (x ) Z0 A1 e x A2 ex (13)function samples to be available, which is important if the line where Z0 is the characteristic impedance. Assuming at x=d,is characterized for delay and crosstalk. the termination voltage and current are V(d) =V2 andB. ANALYSIS OF CROSSTALK USING DIFFERENCE MODEL I (d) =I2, respectively. We first consider the interconnect system consisting of d d (14) V2 A1e A2esingle uniform line and ground as shown in Figure 5, andassume the length of the line is d. 1 I2 [ A e d A2e d ] Z0 1 (15) After solving (14) and (15) for A1 and A2 we get, 1 (16) A1 V2 I 2 Z 0 e d 2 1 A2 V I Z e d 2 2 2 0 (17) Substituting these values of A1 and A2 in (11) V I 2 Z 0 ( d x ) V2 I 2 Z 0 ( x d ) (18) V ( x) 2 e e Figure 5. Equivalent Circuit of each Uniform Section 2 2 © 2012 ACEEE 3DOI: 01.IJIT.02.02. 55
- 4. ACEEE Int. J. on Information Technology, Vol. 02, No. 02, April 2012Now substituting the values of A1 and A2 from (16) and For calculation of the delay expression we consider the 50%(17) in equation (13) we get, rise time when v0(t)=0.5v0. From (28) we can get the general delay expression. 1 V2 I2Z 0 (dx) V2 I2 Z0 (xd) 2 2 2 2 V0 2V0 LC(L G R C ) 2V0LCt 2V L C 3 R t 2V LC 3 G t I ( x) e e (19) 2 0 e L 2 0 e C (34) Z0 2 2 2 ( LG R C)R 2G 2 RG R ( LG R C) G ( LG R C ) After simplification the 50% delay can be derived as,Let at x=0, V(x) =V1 and I(x) =I1 then from (14) and (18), we canwrite: LC (35) t dela y V1 cosh( d )V2 Z 0 sinh( d ) I 2 (20) RG Equations (28) and (35) represent the proposed model for the 1 crosstalk and the delay. I1 sinh(d )V2 cosh( d )I 2 (21) Z0Since ABCD parameters are defined as IV. EXPERIMENTAL RESULTS V1 A B V2 The focus of this paper is on the crosstalk and delay I C D I (22) estimation of the circuits consisting of lumped elements. Most 1 2 of the earlier research and reduction techniques considerHence, ABCD matrix can be written from (20) and (21) as, only capacitive coupling [2, 12]. But in the case of very high cos h( d ) Z 0 sinh( d ) frequencies as in GHz scale, inductive crosstalk comes into V 1 V 2 the important role and it should be included for complete 1 (23) I1 Z sinh( d ) cosh( d ) I 2 coupling noise analysis. The configuration of circuit for 0 simulation is shown in Figure 2. The high-speed interconnectThe output crosstalk voltage is given by V1 (s ) system consist of two coupled interconnect lines and ground V2 (s ) and the length of the lines is d =10 mm. The sample dimensions cosh(d ) (24) of the cross sections of a minimum sized wire in a 0.18µmFor the ramp input voltage we get, technology are given in Figure 6. V V1 ( s) 0 2 (25) s V0Or , V 2 (s ) s 2 cosh( (R sL)(G sC) ) (26)After simplification, (26) reduces to (27). Figure 6. Sample dimensions of cross-sections of minimum sized 2VO wire in a 0.18µm technology V 2 ( s) 2 R G (27) s (s )(s ) The extracted values for the parameters R, L, C, and G are L C given in Table 1. The conductance is a function of frequency,After taking inverse Laplace transform of equation (27), we f. The left end of the first line of Figure 2 is excited by 1-Vget, trapezoidal form voltage with rise/fall times 0.5 ns and a pulse 2V0 LC L2G2 R2C2 ) 2V0LCt ( 2V L3C R t 2V LC3 G t width of 1 ns. Other parameters of lumped elements areV0(t) 2 0 e L 2 0 eC (28) (LG RC)R2G2 RG R (LG RC) G ( LG RC) R1=R2=50 ohms and C1=C2=1pF. Rs, Ls, and CL are source re-This is the proposed model for noise voltage induced by the sistor, source inductor, load capacitor. The operating fre-aggressor line onto the victim line. quency is taken as 2GHz. Figure-7 shows that the presence of coupling results in the crosstalk positiveNow we will consider two typical cases of frequency of glitch (of magnitude 127mV) generated on the victim line andoperation the result is quite comparable to the PSPICE simulation.CASE -1 (For Very Low Frequency) TABLE I. RLCG PARAMETERS FOR A MINIMUM- SIZED WIRES IN A 0.18µMFor very low frequency, where R>>ω L, (26) reduces to (29). TECHNOLOGY. V0 V2 ( s ) 2 (29) s cosh RGAfter taking inverse Laplace transform of (29), we get, v0 v2 (t ) tu ( t) cosh RG (30) TABLE II. C OMPARISON OF PROPOSED DELAY WITH SPICECASE -2 (For very High Frequency) For high frequency, where R<<ω L, (19) reduces to V0 V2 ( s) 2 s cosh s LC (31)© 2012 ACEEE 4DOI: 01.IJIT.02.02. 55
- 5. ACEEE Int. J. on Information Technology, Vol. 02, No. 02, April 2012 REFERENCES [1] O. Wing. On VLSI interconnects, in: Proc. of the China International Conference on Circuits and Systems, Shenzhen, China, pp. 991–996, 1991 [2] Gal, L. 1995. On-chip crosstalk-the new signal integrity challenge. IEEE Custom Integrated Circuits Conference. pp. 251- 254. 1995. [3] Shien-Yang Wu, Boon-Khim Liew, K.L. Young, C.H.Yu, and S.C. Sun. 1999. Analysis of Interconnect Delay for 0.18µm Technology and Beyond. IEEE International Conference Interconnect Technology. May 1999, pp. 68 – 70. From the Figures 8 and 9, we can see that, with the increase [4] S. Delmas-Bendhia, F. Caignet, E. Sicard. 2000. On Chipof frequency, crosstalk noise becomes larger and larger, so Crosstalk Characterization of Deep Submicron Buses. IEEEdoes the attenuation of useful signal. At the range of high International Caracas Conference on Devices, Circuits and Systems,frequency, oscillation occurs sharply, which is a bit different 2000.from the transfer function of lumped parameter systems. [5] Ravindra, J.V.R., M.B. Srinivas. 2007. Modeling and AnalysisFigure 10 correspond to the waveforms of voltage responses of Crosstalk for Distributed RLC Interconnects using Differenceat the both ends of victim line. Model Approach. Proceedings of the 20th annual conference on Integrated circuits and systems design. pp: 207 – 211, 2007. [6] Saihua Lin, Huazhong Yang. 2007. A novel ãd/n RLCG transmission line model considering complex RC (L) loads. IEEE Trans. Computer-Aided Design of Integr. Circuits Syst., 26(5): 970-977, 2007. [7] J. Zhang and E. G. Friedman. 2004. Effect of Shield Insertion on Reducing Crosstalk Noise between Coupled Interconnects. Proceeding of the IEEE International Symposium on Circuit and Systems. Vol. 2. pp. 529-532. May 2004. Figure 9. Frequency Response Figure 10. Waveform of voltage [8] Massoud, Y. J. Kawa, D. MacMillen, J. White. 2001. Modeling at far end of aggressive line at both end of victim line and Analysis of Differential Signaling for Minimizing Inductive Table-II shows the comparison of the delay obtained from Cross-Talk. IEEE/ACM DAC. June 18-22, 2001. Las Vegas .proposed model to that of from SPICE and the step input Nevada. USA.model [13]. From the result we can see that the error is less [9] Lee, K., C. Nordquist, and J. Abraham. Test for Crosstalk Effects in VLSI Circuits. IEEE International Symposium on Circuitsthan 1%. and Systems. Vol. 4: 628-631. 1996. [10] Paul, Clayton R., Keith W.Whites, Syed A. Nasar. Introduction CONCLUSIONS to Electromagnetic Fields. McGraw Hill 1998. In this paper, explicit models have been proposed for [11] Kuznetsov D.B. and J. E. Schutt-Aine. 1996. Optimum transient simulation of transmission lines. IEEE Transactions oncrosstalk noise and delay in high-speed coupled interconnect Circuits and Systems-I. vol. 43. pp. 110-121. Feb. 1996.systems. Result shows that, at low frequencies, the model [12] Gao Y. and D. F. Wong. 1998. Shaping a VLSI wire to minimizeexhibits a RC behaviour but at high frequencies has a delay using transmission line model. in Proc. Int. Conf. Computer-substantially different behaviour due to the effects of Aided Design (ICCAD). 1998. pp. 611-616.inductance. On the basis of Laplace transformation of [13] Rajib Kar, V. Maheshwari, Md. Maqbool, A. K. Mal, A. K.distributed parameter model deduced in time domain, transfer Bhattacharjee, “A Closed Form Modelling of Cross-talk forfunctions of crosstalk noises are built, and crosstalk noise Distributed RLCG On-Chip Interconnects using Difference Modelresponse is analyzed theoretically. Simulation results Approach”, International Journal on Communication Technology,demonstrate the validity and correctness of the proposed vol. 1, no. 1, pp. 17-22, India, 2010.method.© 2012 ACEEE 5DOI: 01.IJIT.02.02. 55

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