RC(L) Interconnect Sizing with Second Order Considerations via - PowerPoint PPT Presentation

Carnegie Mellon RC(L) Interconnect Sizing with Second Order Considerations via Posynomial Programming Tao Lin and Lawrence T. Pileggi Dept. Electrical and Computer Engineering Carnegie Mellon University

Outline � Elmore delay based formulation � Central moment metrics � Posynomiality of central moments � Extension to inductive interconnects � Applications � Experiment results 1

Interconnect Problem � The delay due to the global RC(L) interconnects is becoming a dominant portion of the overall path delay � Practical interconnect Interconnect Delay (Al) optimization methods Interconnect Delay (Cu) are required for global nets 2

Optimization Via Elmore Delay � Interconnect sizing formulations based on the Elmore delay model: — Minimize the area ( W ) minimize Area < — Delay constraints subject to Elmore ( W ) d 0 k ≤ ≤ = — Width bounds and w w w i 1 ,... N i i i � Many efficient algorithms have been developed: � Lagrange relaxation method � Sensitivity based convex programming � Local refinement algorithm � Sequential quadratic programming 3

Posynomiality of Elmore Delay � Elmore delay is the first order metric of RC interconnect delay = + + + 2 � The first moment: H ( s ) m m s m s ... 0 1 2 ∑ ∑ = � Sum of RC products: Elmore R C k i j ∈ ∈ i P ( k ) j D ( i ) ∝ 1 ∝ � Function of width: R C w i i i w i 1 ∑ ∑ = Elmore ( w ) ( ) a w k ij j w ∈ ∈ i i P ( k ) j D ( i ) � Posynomial function of sizes: = ∑ β α Π α > L � Posynomiality: f ( w w ) ( w ij ), 0 1 n i j i 4

Posynomial Programming � Posynomial geometric programming: — Sum of w i *l i minimize ( W ) Area < — Delay constraints ( W ) d subject to Elmore 0 k ≤ ≤ = — Width bounds w and w w i 1 ,... N i i i � A posynomial function can be transformed into a convex function under the exponential substitution: j = w exp( x ) j � The interconnect sizing problem is a convex programming problem under exponential substitution 5

Signal Integrity Problems � Signal integrity becomes an important issue in giga-scale DSM design � Signal quality � Clock attenuation � Signal transition time � Signal uncertainty � Noise peak � Extra-delay due to noise 6

Higher Order Moments � Limitation of first order metrics � Incapable of modeling integrity � Incapable of modeling noise � High order moments: � It is trivial to show that higher order moments (RC trees) are also posynomial ∑ ∑ = m R m 2 , k i 1 , j ∈ ∈ i P ( k ) j D ( i ) � But reduced order models in terms of higher order moments do not preserve posynomiality − − = − − − p t p t 0 . 5 1 a e a e ... 1 2 1 2 7

Central Moments � Definition of the central moments µ = ≡ ∞ = m mean ∫ h ( dt t ) 1 h 1 0 µ = − 2 2 m m (var iance) 2 2 1 µ = − + − 3 6 m 6 m m 2 m (skewness) t 3 3 1 2 1 � µ 2 is a natural metric for signal quality/shape σ � Standard deviation s = µ 2 � Dispersion 8

Signal Attenuation � An accurate model for signal attenuation in RCL clock tree [Celik99] α ω = − − µ ω 2 ( ) 10 log( 1 ) ( db ) 2 � A provable upper bound for RC responses 2  Approx. Attenuation (db)  Exact 0 0.01 1 Frequency (GHz) � An upper bound for overdamped cases (RCL) 9

Signal Transition � µ 2 as a metric of RC signal transition time [Elmore48]: = − + + 2 2 2 g ( s ) 1 sTD 0 . 5 s ( TR 2 TD ) � Transfer function: TR = 2 µ � Signal transition time: p 2 s = µ 2 TR TD 10

Delay Due To Crosstalk � µ 2 is a metric of delay uncertainty due to crosstalk noise � Assuming a finite ramp input TR and an environment noise Vn � Worst case alignment: ∆ delay=TR*Vn/Vdd VN Noise ∆ delay 11

Posynomiality Proof � Is µ 2 a posynomial function of wire widths? µ = m − 2 2 m 2 2 1 � m 1 and m 2 in an RC tree are posynomial functions of wire widths � µ 2 is provable positive for RC tree response [Gupta97] � Prove by induction: µ 2 of RC tree response is a posynomial function of wire widths 12

For Inductive Interconnect � High order moments for RCL circuits: � M 2 is not guaranteed to be positive for RCL circuit responses ∑ ∑ ∑ ∑ = − m R m L C 2 , k i 1 , j i j ∈ ∈ ∈ ∈ i P ( k ) j D ( i ) i P ( k ) j D ( i ) � Modeling of on-chip inductance � A simple linear model for embedded wire ≈ µ L t / w � Posynomial condition of µ 2 (sufficient condition) which can be verified before solving the problem 1 + ≥ 2 R R C R C L k D L k k k 4 13

Sizing Formulations � A posynomial interconnect sizing formulation with second order constraints: minimize Area ( W ) ≤ m 1,k subject to ( W ) d ( I ) 0 ≤ µ ( W ) s and 2 , k 0 ≤ ≤ = and w w w i 1 ,... N i i i � The inequality constraints on µ 2 represent the constraints on signal quality α ω = − − µ ω ≤ α 2 � Attenuation: ( ) 10 log( 1 ) 2 0 = µ ≤ � Transition time: TR 2 p TR 0 2 14

Sizing Formulations � For clock tree sizing problems, the delay constraints are equality constraints in order to achieve zero skew solutions minimize Area ( W ) = subject to m 1,k ( W ) d 0 ≤ µ ( w ) s and 2 , k 0 ≤ ≤ = and w w w i 1 ,... N i i i � Given the posynomiality of the constraints, the above problem can be solved via a multi- stage approach. Each stage involves solving a problem of (I). [Celik99][Kay97] 15

Sizing Formulations � The posynomiality can be applied to other type of sizing formulations minimize dmax minimize Max delay ( W ) ≤ subject to dmax ≤ Delay 1,k ( W ) subject to Area ( W ) a 0 ( II ) ≤ ≤ and Area ( W ) a µ ( W ) s and 0 2 , k 0 ≤ µ ( W ) s and ≤ ≤ = 2 , k 0 and w w w i 1 ,... N i i i ≤ ≤ = and w w 1 w i ,... N i i i � Sizing formulations (I) and (II) are both posynomial programs as the Elmore delay based sizing problems 16

Experiments � Extend sequential quadratic programming wire sizing algorithm (ORCIDS) � Provable convergence � Compute µ 2 in o ( n ) complexity by path tracing � Match the second order moments of transmission line models [Yu95] 3.01p Length unit (um) R=0.02 (ohms/• ) Ca=0.08fF/um 2 Width 0.5um-20um � Example: 610 Rd=2ohm Cf=0.06fF/um 610 287 287 287 � Design Constraints : 287 1.97p 1.98p 2.75p 1.03p 575 575 610 ≤ 718 1202 Delay 105 ps 575 575 1077 ≤ TR 240 ps .85p 1.37p 2.18p 2.06p 287 287 287 287 17

Experiment Results 3.01p 3.01p Elmore Delay Only Unit (um) R=0.02 (ohms/• ) Target delay=105ps Area=56,843 Width 0.5-20u Ca=0.08fF/um 2 610 3.08 Rd=2ohm Cf=0.06fF/um 610 3.25 287 287 3.12 1.17 287 287 1.70 1.71 1.97p 2.75p 1.97p 2.75p 1.98p 1.03p 1.98p 1.03p 575 4.47 575 610 3.53 3.42 718 1202 5.8 10.3 575 2.0 575 5.0 1077 20 .85p .85p 1.37p 2.18p 2.06p 1.37p 2.18p 2.06p 287 2.48 2.34 287 287 0.74 287 1.19 Target delay=105ps 3.01p Form. (I) (RCL) Target delay=105ps 3.01p Form. (I) (RC) Area=59,836 Target Tr=240ps Area=66,611 Target Tr=240ps 3.18 3.69 3.54 4.14 3.14 1.19 1.77 1.77 3.63 1.36 2.03 2.04 1.97p 2.75p 1.98p 1.03p 1.97p 2.75p 1.98p 1.03p 4.70 3.79 3.87 5.47 4.41 4.62 6.78 11.4 7.77 13.6 2.15 2.5 5.26 6.13 20 .85p 1.37p 2.18p 2.06p 20 .85p 1.37p 2.18p 2.06p 2.50 2.36 0.76 2.88 2.72 1.23 0.88 1.42 18

Conclusions � The second central moment is a posynomial metric of interconnect signal integrity � Interconnect sizing problems with second order signal integrity constraints are formulated as posynomial programs � The existing algorithms can be extended to solve the new sizing problems with provable convergence 19