impedance the response will be non-monotone.

Transcription

1 Delay Models for MCM Interconnects When Resonse is Non-monotone Andrew B. Kahng, Kei Masuko Sudhakar Muddu UCLA Comuter Science Deartment MIPS Technologies, Silicon Grahics, Inc. Los Angeles, CA Mountain View, CA Abstract Elmore delay has been extensively used for interconnect delay estimation because its simlicity of evaluation makes it aroriate for layout design. However, since Elmore delay does not take into account the eect of inductance, the discreancy between actual delay and Elmore delay becomes signicant for long RLC transmission lines, such as for MCM and PCB interconnects. We describe a simle two-ole based analytic delay model that estimates arbitrary threshold delays for RLC lines when the resonse is non-monotone; our model is far more accurate than the Elmore model. We also describe an alication of our model for controlling resonse undershoot/overshoot and for the reduction of interconnect delay through constraints on the moments. Introduction Recently, accurate estimation of interconnect threshold delays and rise times has become essential to the design of high-seed systems. Many interconnect delay models have been advocated; these are classied roughly into simulation based models and closed form analytical models. Simulation methods such as SPICE give the most accurate insight into arbitrary interconnect structures, but are comutationally exensive. Faster methods based on moment matching techniques are roosed in [3,, 5, 7], but are still too exensive tobe used during layout otimization. Thus, Elmore delay [], a rst order analytical aroximation of delay under ste inut, has been the most widely used model for erformance-driven layout synthesis. Recently, anumber of analytical delay formulas have been roosed for interconnect delay based on the rst few moments of the resonse under ste and ram inut [5,, 0,, 8]. The authors of [5] use Elmore delay as an uer bound for the 50% threshold delay for RC interconnection lines under arbitrary inut waveforms. The work of [] gives lower and uer bounds for the ram inut resonse; their (single-ole) delay model for 50% threshold voltage can be obtained by alying the Elmore denition to the ram inut resonse. Our own revious work has resented analytical delay modelsfor monotone resonse under ste and ram inuts, based Partially suorted by NSF MIP ABK is currently Visiting Scientist (on sabbatical leave) at Cadence Design Systems, Inc. on rst and second moments [0, ]. Quite recently, [8] have used the rst three moments to accurately comute two oles of the imulse resonse. Note that all of these aroaches assume that the resonse is monotone (or overdamed) in deriving their resective delay models. However, for long lines with sucient inductive imedance the resonse will be non-monotone. For RLC lines, which are the necessary reresentation of interconnects whose inductive imedance cannot be neglected [8], Elmore and other rst-order delay models cannot accurately estimate signal delay because they are indeendent of inductance. To illustrate the eect of inductive imedance on the resonse, we consider a -ort model for an interconnect driven by a ste inut with nite source imedance. Figure comares the RC and RLC line resonses comuted by SPICE3e: 90% threshold delay is88s for the RLC model, but is 358 s for the RC model. Elmore delay, which does not deend on line inductance, will yield the same delay estimate of 386 s for both the RC and the RLC cases. This inaccuracy can be harmful for current erformance-driven routing methods which try to otimize interconnect segment lengths and widths (as well as drivers and buers). A non-monotone (i.e., underdamed) voltage resonse oscillates before settling to a steady state value. Such a resonse occurs when the ratio of inductive imedance to resistance exceeds a certain threshold in an interconnect line. MCM substrate interconnects have smaller driver resistance, and inductive imedance greater than resistive imedance as a consequence of greater widths and lengths comared to their on-chi VLSI counterarts; the voltage resonse for such interconnects tends to be nonmonotone. Consequently, the eect of inductance is more evident in MCM interconnects. To address the de- ciencies of the Elmore model, this aer gives a simle, yet reasonably accurate, analytical delay model for interconnect lines which are inductive (i.e., RLC transmission lines) and whose voltage resonse is not monotonically increasing. Our roosed model can estimate signal delay for non-monotone resonse at arbitrary threshold voltages. Recently, [0] roosed a similar set of analytical delay models, but these are restricted to the case of a monotone voltage resonse. Preliminary exerimental results show that our delay estimates are within 7% of SPICE-comuted delays (for most cases within 5%), while Elmore delay estimates can dier by asmuch as 00% from the SPICE-comuted delays. We also briey

2 RLC Model RC Model b k are called the coecients of the transfer function and are directly related to the moments of the transfer function [0]. Exanding the transfer function into a Maclaurin series of s around s = 0 leads to an innite series, and to comute the resonse the series is truncated to desired order. We model the source as a resistance R S and the load as caacitance C L. For a two-ole model the transfer function is aroximated as with coecients H(s) +b s + b s b = R S C + R S C L + RC + RC L Figure : Comarison of HSPICE resonses at the end of an interconnect line driven by a ste inut and terminated with a caacitive load, with the line reresented using both RC and RLC -ort models. The 90% threshold delayis 88 s for the RLC model, and 358 s for the RC model. The driver resistance is 0:0 and the load caacitance at the end of the line is :0 F. The line arameters are r =0:075 =m, l =0:3 H=m, c =8:8 ff=m; the length of the line is 00 m. discuss an aroach to reduce the threshold delay by controlling the overshoot of the voltage resonse. This translates into a condition between the rst and second moments of the interconnect transfer function, which are functions of driver and interconnect arameters. The remainder of this aer is organized as follows. Section describes delay comutation using our new model. Section 3 exlains minimization of delay by allowing small ringing. Section gives exerimental results, and Section 5 states our conclusions. New Delay Model for Interconnects For simlicity, we consider a single interconnect line in studying resonse and delay models. We develo our delay model as a function of rst and second moments (or coecients) of the transfer function; note that the same delay model can be alied to the corresonding moment values of arbitrary interconnect trees. The denominator of the transfer function of a single RLC interconnect line with source and load imedance (Figure ) is obtained from ABCD arameters [] as H(s) = = h i ( + ZS Z T )cosh(h)+( ZS + Z0 Z0 Z T ) sinh(h) () +b s + b s + :::+ b k s k + ::: where = (r + sl)sc q is the roagation constant and R+sL Z 0 = is the characteristic imedance; sc r = R ;l= h L ;c= C are resistance, inductance, and caacitance er h h unit length and h is the length of the line. The variables b = R SRC 6 ZS + R SRCC L + (RC) + R CC L 6 + LC + LC L () v (t) i (t) v (t) i (t) 0 0 Distributed RLC line i (t) v (t) Figure : -ort model of a distributed RLC line with source imedance Z S and load imedance Z T. When the inut at the source is modeled as a ste waveform, the outut resonse in the transform domain is V out (s) = V0 H(s). The corresonding time domain s resonse using the two-ole model is v(t) = V 0, s e st + s e st (3) s, s s, s where s ; =,b b b,b. The condition for the resonse to be non-monotone is for the oles to be comlex, i.e., b, b 0 (as noted earlier, this corresonds to the inductive imedance exceeding a certain value). By writing the oles as s ; =,, the non-monotone time domain resonse becomes # v(t) =V 0 " +, e,t sin(t + ) () where = b ; = b,b ; = tan, ( ). b b Notice that the resonse rst reaches the saturation voltage V 0 at time t =,, ;over the interval t [0; ] the derivative of the resonse v 0 (t) is ositive, i.e., the resonse is a continuous non-decreasing function of t. We wish to comute the threshold delay when the resonse Z T

3 rst crosses some given threshold voltage, e.g., at which the logic state changes. Thus, we can assume that the threshold delay is bounded by the range [0;, ]. (The aroach we give for aroximating resonse over a secied range is quite general, in the sense that it can be used to comute threshold delay for the resonse within any range of interest.) We can further reduce the uer bound of the range as follows. Rearranging (3) for a given threshold voltage v th with corresonding delay time t th (in other words, v th = v(tth) ), we have V0 e tth = + sin(t th + ) (, v th ) Since e tth, sin(t th + ) (, v th) + t th,, sin, ( (, v th) + ) Using a new time variable that shifts the time t th as = t th + =, we obtain the bound on = (, )= where = sin, ( (,vth) ). Rewriting t by using and + rearranging () yields e, sin() + ex(, + )(v th, ) = 0 (5) The delay atagiven threshold voltage v th cannot be calculated directly from this equation, so we adot the aroach of aroximating e, sin() with a degree-two olynomial [7]. Secically, we aroximate f(t th ) = ex(,t th )sin(t th ) over the interval E =[LB; U B] using a vector sace reresentation, where UB LB def = uer bound of the aroximation interval def = lower bound of the aroximation interval The Gramm-Schmidt technique [7] yields the following aroximation of e, sin() over an interval [LB, UB] by a degree-two olynomial (see [] for details). where a = a = a + a + a 3 =0 (6) 8 7(UB, LB) (c, c, c 3, c +c 5 ) 35(UB, LB) [UB(,08c +6c +80c 3 +5c,5c 5 )+LB(,5c +5c +80c 3 +6c, 08c 5 )] a 3 = 35(UB, LB) [UB (3c +9c, 3c 3, 5c +3c 5 ) +UBLB(6c, c, 7c 3, c +6c 5 ) +LB (3c, 5c, 3c 3 +9c +3c 5 )] + ex(, + )(v th, ) c = ex(,lb)sin(lb) (UB +3LB) (UB +3LB) c = ex, sin (UB + LB) (UB + LB) c 3 = ex, sin (3UB + LB) (3UB + LB) c = ex, sin c 5 = ex(,u B)sin(UB) Solving (6) with resect to and subtracting = from yields the threshold delay time t th =,a, a, a a 0 3, = (7) a Note that the two-ole aroximation assumes that the resonse at the load end of the line begins from t =0. However, for interconnects where time of ight, T f = LC, is non-negligible, the resonse remains zero until t = T f. Hence, we estimate a given threshold delay as the maximum of time of ight and the delay estimate from (7), i.e., max[t f ;t th ]. We can estimate the risetime between two threshold voltages as the dierence of the resective threshold delay estimates. Finally, because the range [ ; (,,) ]maybe too large for the Gramm-Schmidt rocedure to eectively aroximate the resonse with a single degreetwo olynomial. Hence, we divide this range in two to imrove the aroximation of the resonse function e, sin(). Since sin() is increasing in the range [0; ], we divide the original range into the two ranges [LB;UB] = [ ; ]and[lb;ub] = [ ; (,) ]. We choose the roer range by comaring the threshold voltage v th to the resonse value at time t =, i.e., v( )=, + ex(, (, )). This rocedure can be extended to other (e.g., ram) inut waveforms. 3 Constraint on Moments for Control of Undershoot/Overshoot In this section, we illustrate how our simle threshold delay model can yield simle analytic constraints for interconnect synthesis. Secically, we address the question of nding interconnect and driver arameters for otimum delay with controlled ringing. Consider a simle RLC line driven by a gate, with Z S being the driver imedance and C L being the load imedance at the end of the line. Theqcharacteristic imedance of the R+sL line is given by Z 0 =. Ideally,thedriver and sc

4 line arameters are adjusted such thatz S matches Z 0 and the voltage resonse at the end of the line is critically damed. However, if the driver imedance Z S is just smaller than the characteristic imedance of the line, the voltage resonse will have a small amount of ringing: this can be advantageous in that the threshold delay will decrease [9]. The roblem with ringing is that it can cause false switching if the voltage resonse dros back below the threshold; hence, the advantages of ringing can be exloited only if the maximum oscillation (overshoot or undershoot) is bounded such that false switching does not occur. We nowdevelo an analytical equation that achieves this control in terms of coecients of the transfer function. Additional context for our discussion may be found in [0]. The voltage resonse for ringing is given by # + v out (t) = V 0 ", e,t sin(t + ) where = tan, ( ). To nd the eaks of overshoot and undershoot in the resonse, we set the derivative v 0 out (t) to zero, yielding t = n with n = ; 3; 5;::: for overshoots and n = ; ; 6;::: for undershoots. The rst undershoot occurs at time T = =, and the value of the undershoot is v = V 0 e,t r+( ) sin(t + ) = V 0 e,t : The constraint for a given ercentage undershoot v us can be obtained as = jln(v us)j For examle, with 5% undershoot, we have v us =0:05V 0 and = 0:8. We can exress and in terms of coecients of the transfer function, i.e., = b. b,b Therefore, " # ( b = ) ( b ), With 5% undershoot the above equation reduces to b = 0:7b and a 90% threshold delay estimate for this case can be obtained (see [0]) as b T 0:9 = :66 b, b = :3b Similarly, for 5% overshoot, the relation between the coecients is b = :9b and a corresonding delay estimate is T 0:9 =:0b. As exected, the delay increases for a strong undershoot requirement, and in general the delay increases if ringing in the resonse is suressed [9]. The above constraint between and to reduce the undershoot in the resonse could be alied with the delay model in Equation (7) to erform delay-driven routing tree synthesis. Length Threshold Delay (s) 0% % % % % % % % % % % % % % % % % % % % % % % % % % 90 90% Table : Threshold delay estimates at various thresholds for non-monotone resonse under HSPICE, Elmore and our New models. Source resistance is 0 and load caacitance is F. Exerimental Results We evaluate the above models by simulating various RLC interconnect lines with dierent source/load imedances and dierent inut rise times. We consider tyical interconnect arameters encountered in MCM interconnects [3]. For all cases, the interconnect resistance, inductance and caacitance er length are r =3:0 0, =m, l =0:33 H=m and c =0: ff=m, resectively and the length of the interconnect line ranges from 3000 to 50000m. We also vary the load caacitance and the driver resistance from to 3F and from 0 to 70, resectively. We comute delays at thresholds ranging from 0% to 90% from the resonse at the load using the HSPICE simulator (see Tables - for results with four of the congurations). For cases when the resonse is non-monotone the dierence between delays from HSPICE and delays from our model is always less than 7% desite this large range of instances. The Elmore aroximation always underestimates delays when the voltage thresholds are small, and can either overestimate or underestimate when the voltage thresholds are large. Overall, Elmore delay diers from HSPICE delay by u to 00%. When the resonse is monotone (i.e., with real oles), the maximum dierence between our new model delay and HSPICE delay is 3%.

5 Length Threshold Delay (s) 0% % % % % % % % % % % % 6 7 0% % % % % % % % % % % % % % % Table : Threshold delay estimates at various thresholds for non-monotone resonse under HSPICE, Elmore and our New models. Source resistance is 30 and load caacitance is 3 F. 5 Conclusions We have develoed a simle two-ole based analytical delay model which can estimate delay times corresonding to arbitrary threshold voltages when the interconnect resonse is non-monotone. Because our model takes into account the eect of inductance, we can estimate delay times for RLC lines far more accurately than with the Elmore delay model. We have also discussed a delay minimization aroach that uses controlled small ringing in the resonse waveform. Ongoing work extends the analysis of threshold delays under nonmonotone resonse to more general inut waveforms. References [] L. N. Dworsky, Modern Transmission Line Theory and Alications, Wiley, 979. [] W.C. Elmore, \The Transient Resonse of Damed Linear Networks with Particular Regard to Wideband Amliers", Journal of Alied Physics 9, Jan. 98. [3] P. Franzon, ersonal communication, 996. [] E. G. Friedman and J. H. Mulligan, Jr, \Ram Inut Resonse of RC Tree Networks", IEEE ASIC Conference, 996. [5] R. Guta et al., \The Elmore Delay as a Bound for RC Trees with Generalized Inut Signals", ACM/IEEE Design Automation Conference, J une 995. [6] M.A. Horowitz, \Timing Models for MOS Circuits", PhD Thesis, Stanford University, Jan. 98. [7] Th. V. Hromadka II et al, The Best Aroximation Method an Introduction, Lecture Notes in Engineering 7, Sringer-Verlag, 987. [8] C. C. Huang and L. L. Wu, \Signal Degradation Through Module Pins in VLSI Packaging", IBM J. Res. and Dev. 3(), July 987, [9] S. Lin and E.S. Kuh, \Transient Simulation of Lossy Interconnect", Proc. 9th ACM/IEEE Design Automation Conf., June 99, [0] A.B. Kahng and S. Muddu, \An Analytical Delay Model for RLC Interconnects", IEEE International Symosium on Circuits and Systems, May 996, vol.iv, [] A. B. Kahng, K. Masuko and S. Muddu, \Analytical Delay Models for VLSI Interconnects Under Ram Inut", IEEE/ACM Intl. Conf. on CAD, Nov [] A.B. Kahng, K. Masuko, and S. Muddu, \Delay Models for Interconnects Under Non-Monotone and Monotone Resonse", UCLA CS Det. TR , Nov [3] L.T. Pillage and R.A. Rohrer, \Asymtotic Waveform Evaluation for Timing Analysis", IEEE Trans. CAD, Ar. 990, [] V. Raghavan, J.E. Bracken and R.A. Rohrer, \AWE- Sice: A General Tool for the Accurate and Ecient Simulation of Interconnect Problems", Proc. 9th ACM/IEEE Design Automation Conf., June 99. [5] C.L. Ratzla, N. Goal and L.T. Pillage, \RICE: Raid Interconnect Circuit Evaluator", Proc. 8th ACM/IEEE Design Automation Conf., June 99. [6] J.S. Roychowdhury and D.O. Pederson, \Ecient Transient Simulation of Lossy Interconnect". Proc. 8th ACM/IEEE Design Automation Conf., June 99. [7] M. Sriram and S.M. Kang, \Fast Aroximation of the Transient Resonse of Lossy Transmission Line Trees", Proc. 30th ACM/IEEE Design Automation Conf.,June 993, [8] B. Tutuianu et al., \An Exlicit RC-Circuit Delay Aroximation Based on the First Three Moments of the Imulse Resonse", ACM/IEEE Design Automation Conference, June 996, [9] Y. Yang and R. Brews, \Overshoot Control for Two Couled RLC Interconnect", IEEE Trans. Comonents, Packaging and Manufacturing Tech., Aug. 99. [0] D. Zhou, S. Su, F. tsui, D.S. Gao and J.S. Cong, \A Simlied Synthesis of Transmission Lines with A Tree Structure", Intl. Journal of Analog Circuits and Signal Proceeding, Jan. 99,

6 Length Threshold Delay (s) 0% 3 9 0% % % % % % 38 80% % % % % % % % % % % % % % % % % % % % Table 3: Threshold delay estimates at various thresholds for non-monotone resonse under HSPICE, Elmore and our New models. For the case of h = 3000m the oles are real (monotone resonse). Source resistance is 50 and load caacitance is F. Length Threshold Delay (s) 0% % % % % % % % % % % % % % % % % % % % % % % % % % % Table : Threshold delay estimates at various thresholds for non-monotone resonse under HSPICE, Elmore and our New models. For the case of h =3000m the oles are real (monotone resonse). Source resistance is 50 and load caacitance is 3 F.