Accounting for Variability and Uncertainty in Signal and Power Integrity Modeling

Transcription

1 Accounting for Variability and Uncertainty in Signal and Power Integrity Modeling Andreas Cangellaris& Prasad Sumant Electromagnetics Lab, ECE Department University of Illinois

2 Acknowledgments Dr. Hong Wu, Extreme DA Prof. NarayanAluru, University of Illinois DARPA for funding support

3 Agenda Are we using the might of EM CAD wisely? Uncertainty and Variability (UV) in Signal/Power Integrity Modeling Accounting for UV in SI/PI modeling and simulation Interconnect electrical modeling Model order reduction in the presence of UV Closing Remarks

4 Are we using the might of EM CAD for signal integrity-aware design wisely?

5 Electromagnetic modeling/simulation pervasive in physical design and sign-off Input filed for EDA tools analysis Floor planning Power planning Timing/power library Interconnect library Kurokawa et al, Interconnect Modeling: A Physical Design Perspective, IEEE Trans. Electron Devices, Sep Placement Clock planning Routing Physical Layout Sign-0ff analysis Interconnect parasitic extraction ECOs IR-drop analysis Signal-integrity analysis Physical verification EM analysis Timing analysis DFM

6 Package impedance Power Integrity Design/layout-dependent R impacted by manufacturing tolerances On-chip grid Rvariability due to CMP On-chip decoupling Available Cdependent on operation, V T, T ox,

7 Impact of Package Ron IR drop(*) (*)Sani Nassif, IBM

8 Variability and Uncertainty Stacked dies Package-on-package Multiple layers Source: Future-Fab International Source: TNCSI

9 Variability and Uncertainty

10 Variability and Uncertainty?

11 Accounting for SI/PI Modeling and Simulation

12 Interconnect Cross-Sectional Geometry Parameters Trace width Trace thickness Trace shape Pitch Height above ground Surface roughness Substrate permittivity Metallization conductivity Derivative quantities Transmission-Line Modeling R, L, C, G (per-unit-length) Characteristics Impedance Phase constant Attenuation constant Full-wave modeling Metallization surface impedance

13 Transmission-Line Parameter Extraction mean geometry conducto r random geometry ground plane

14 Mapping between random sample and mean geometry Random sample uuur E dl r L = V ' 0 L r uur uuur D ( r ). dl uur = V ε ( r ) 0 uuur D c L uuur dl uur ε ( r ) = V 0 Mean geometry Mapping relationship uuuur uuur uuuur uuur D ( r ) = D = 0 0 uur uur D( r ) uur uur D ( r ) = Q D( r ) = uuur L c V uur 0 dl ur ε( r) Q = L L L V uuur 0 dl uur ε ( r ) uur dl ur ε ( r) uuur dl uur ε ( r ) (1) (2) (3)

15 Electric flux density computation using solution on mean geometry Position-dependent flux length/gap on mean ur geometry: ur G( r ) ( ) ur V0 0 = ε r0 ur ur D( r ) Flux length on random sample: ur ur ur L ( r ) L( r ) v( r ) ur dl r dl L v ε ( r ) ε ( r) L 0 uur ur D ( r ur ur ) = D( r ) 0 0 L L 1 G( r0 ) r dl = ur ε ( r) ε ( r ) ur ur 1 G( r0 ) v( r0 ) ur dl ur ur ε ( r ) ε ( r ) ε ( r ) ur G( r0 ) ur ur G( r ) v( r ) (exact) 0 0 Electric flux density on random sample using mean geometry:

16 Computation of capacitance of random sample Capacitance on random sample: uur ur C = D ( r ) dl Crs ur G( r ) ur ur = ur ur 0 C D( r0 ) J dl Cmean G( r0 ) v( r0 ) Where J represents the Jacobian, the map between the random and mean geometry: x y X X J = x y Y Y

17 Representing uncertainty using Polynomial chaos expansion: r r u( r, θ ) = aˆ i ( r) Ψ( ξ ( θ )) i= 0 Coefficients: functions of space polynomial chaos Orthogonal polynomials are random variables Type of polynomials depends on distribution of input random variable Ψ ( ξ ) = 1, -Gaussian Ψ ( ξ) = ξ, distribution, Ψ ( ξ ) = ξ Hermite 1, Ψ ( ξ) polynomial = ξ 3 ξ, Ψ chaos ( ξ ) = ξ 6ξ + 3, Truncated polynomial chaos expansion: - Number of different input random variables: n - Order of polynomials: p r N r u( r, θ ) = aˆ i ( r) Ψ( ξ ( θ )) i= 0 N + 1 = ( n + p)! n! p!

18 Computing stochastic capacitance Displacement of random geometry from the mean geometry: r N r v( r, θ ) = v ( r) Ψ( ξ ( θ )) i= 0 i Use relationship between random and mean flux length: G% ( r, θ ) = G( r ) v( r, θ ) Stochastic Electric Flux Density: r r r r r G( ) D ( ) D0 ( ) r r G( ) v( ) Stochastic Capacitance: N C% = Ci ( r r) Ψ( ξ ( θ )) i= 0 r r 2 r 3 r % r v(, θ ) v (, θ ) v (, θ ) r r D ( ) 1 + r +... D 2 r + 3 r + 0( ) G( ) G ( ) G ( ) r 2 r 3 r v(, θ) v (, θ) v (, θ) r r C% 1 + r +... J D 2 r + 3 r + 0. ds S G( ) G ( ) G ( ) (*)

19 Single trace over ground plane ε =1 L ε = 4 ε = 7 T H - The height of the conductor above the ground plane H is uncertain. Mean Geometry dimensions : L=1 um, T = 0.1 um, H= 0.2 um H ( θ ) = H (1 νξ ( θ )) -Where is the mean height ξ H is a Gaussian random variable with mean 0 and variance 1

20 Single trace over ground plane Displacement of random geometry from the mean geometry C% r v( r, θ ) = νξ H ν H ν H S G( r ) G ( r ) r ξ ξ... 2 r ξ ( ξ 2 1) r J D dl Second-order Hermite polynomial chaos for stochastic capacitance C % = C + C + C C ν H = 1+ G( r ) 0 0 r S 0 2 r J D dl 2 ν H0 r ν H0 r C1 = J D0 dl, C2 J D0 dl S r = G( ) S r G( ) 0 Only one deterministic run needed to get D r 0

21 Single trace over ground plane %change in H Mean Self-capacitance (pf/m) Monte Carlo Std deviation FEM based Approach Mean Std deviation 10% %

22 Single trace over ground plane %change in L Mean Capacitance (pf/m) Monte Carlo Std deviation FEM based Approach Mean Std deviation 10% % Simulation time comparison Time for 1 Capacitance extraction run ~1.2 s Monte Carlo : Time for runs ~ s Our approach ~ 2.0 s

23 Coupled symmetric microstrip ε =1 L S T ε = 4 ε = 7 H Mean Geometry dimensions : L=1 um, S = 0.15 um, H= 0.2 um % change in H and S Mean Self-capacitance (pf/m) Monte Carlo Std deviation FEM based Approach Mean Std deviation 10% %

24 Microstrip with multi-dielectric ε =1 substrate L ε = 4 ε = 4 ε = 8 ε = 7 T %change in each layer below Conductor Self-capacitance (pf/m) Monte Carlo (10000) FEM based Approach Mean Std deviation Mean Std deviation 10% %

25 Remarks Expedient way for handling statistical variability in interconnect cross-sectional geometry p.u.l. capacitance extraction 100x -1000x faster than standard Monte Carlo Approach independent of the field solver used

26 Deterministic Model Order Reduction 2 ( org + org + org ) = org Y sz s Pe x sb I V = L x H org Order: N Model Order Reduction e.g. Krylov subspace based methods Projection matrix F H Y = F Yorg F H Z = F Zorg F H Pe = F Peorg F Y + sz + s Pe x = sbi 2 ( ) V = H L x Order n << N Generalized Multiport Impedance Matrix using reduced model: 2 1 ( ) H ZG s = sl ( Y + sz + s Pe) B

27 Model order reduction under uncertainty Deterministic Reduced Order Model Stochastic Reduced Order Model 2 ( ) Y + sz + s Pe x = sbi y = H L x 2 ( Y% + sz% + s Pe % ) x% = sbi H V% = L x% % % % % H Y = F Yorg F % % % % H Z = F Zorg F % % % % H Pe = F Peorg F Represent stochastic system matrices using polynomial chaos expansion: Y% = Y + Yξ + Y ξ, Z% = Z + Z ξ + Z ξ org org Pe % = P + Pξ + Pξ, F% = F + Fξ + F ξ org

28 Augmented Stochastic Reduced Order Model [( Y + Y ξ + Y ξ ) + s( Z + Z ξ + Z ξ ) ξ ξ ξ ξ ξ ξ 2 s ( Pe0 + Pe1 1 + Pe2 2 )]( x0 + x1 1 + x2 2 ) = s( B0 + B1 1 + B2 2 ) I Y0 Y1 Y2 x0 Z0 Z1 Z2 x0 Pe0 Pe1 Pe2 x0 B0 2 Y1 Y0 0 x 1 s Z1 Z0 0 x 1 s Pe1 Pe0 0 x 1 s B + + = 1 I Y 0 Y x Z 0 Z x Pe 0 Pe x B Augmented reduced order model 2 ( aug + aug + aug ) aug = aug Y sz s Pe x sb I V = L x H aug aug aug Z = sl ( Y + sz + s Pe ) B H 2 1 aug aug aug aug aug aug Order: 3n << N

29 Y Computing polynomial chaos coefficients Coefficient matrices in polynomial chaos expansion: -Integrate over the random space and use orthogonality of the polynomials Y% ρ ( ξ ) ρ ( ξ ) dξ dξ = ( Y + Yξ + Y ξ ) ρ ( ξ ) ρ ( ξ ) dξ dξ org = Y% ρ ( ξ ) ρ ( ξ ) dξ dξ 0 org org Y% ξ ρ ( ξ ) ρ ( ξ ) dξ dξ = ( Y + Yξ + Y ξ ) ξ ρ ( ξ ) ρ ( ξ ) dξ dξ org Y = Y% ξ ρ ( ξ ) ρ ( ξ ) dξ dξ

30 Smolyak Sparse Grid Integration = org ρ1 ξ1 ρ2 ξ2 ξ1 ξ 2 I( f ) Y ( ) ( ) d d w f ( u ) % 1-d integration rule: e.g. using chebyshev polynomial extrema j j = q j I ( f ) f ( u ) w j= 1 j j cos π ( j 1) u =, j = 1... q q 1 The case of multiple random variables: Deterministic Cartesian Product (DCP) rule: q1 qn ( i i ) Q I [ f ] I... I [ f ] q 1 N j1 jn j1 jn... f ( u1,..., u1 ).( w1... wn ) j = 1 j = 1 1 q = N Number of calculations q N Smolyak Sparse Grid Algorithm Idea: Not all points are equally important; hence, discard the least important ones Q I ( f ) A( J, N) = N 1 J i = ( 1)..( Ii... I 1 J N + 1 i J J i Number of calculations q ( log q) i N N 1 )

31 Comparison of grids generated using Tensor Product and Smolyak Tensor product grid (81 points) Algorithm Q q N q: number of points in 1-d rule N: number of random dimensions p: level in Smolyak algorithm Smolyak Sparse grid (29 points) Q N p

32 Algorithm for Stochastic MOR Represent uncertainty in the original system matrices through polynomial chaos expansion Generate sparse grid pointsand their corresponding weights using Smolyak Sparse grid algorithm Compute the transformation matrix through MOR ofindividual systems corresponding to Smolyak sparse grid points x = ~ Y = Y + Yξ + Y ξ θ F z i org M { ξi}, w { wi } = = M 2 ( org + org + org ) = org Y sz s Pe x sb I 2 ( ) Y + sz + s Pe x = sbi Compute the stochastic transform matrix ~ F = F + Fξ ξ F Define the stochastic reduced order model 2 ( aug + aug + aug ) aug = aug Y sz s Pe x sb I

33 Example Study: Terminated coaxial cable Air-filled coaxial cable, terminated at a resistive load: L=1m, inner radius=5mm, outer radius=10mm FEM system (Y,Z,Pe) of order Reduced order system of order 20. Randomness in two inputs Permittivity: uniform random variable in [ ] Load resistance: uniform random variable in [25-35] ohms Monte Carlo: simulations Smolyak: 29 points

34 Meanof real part of input impedance Re{Zin(f)} Standard deviation of real part of input impedance

35 Corner simulations vs. stochastic simulations Standard practice to simulate corners for accounting for variability Corner simulations can be conservative Coaxial cable example consider corner values for random input parameters (ε,r L ) (3.6,25) (4.0,30) (4.4,35) Compare with information generated using stochastic MOR

36 Corner vs. stochastic simulation Corner simulation appears very conservative Mean parameter solution is not accurate compared to the mean of stochastic simulation

37 Remarks Very good accuracy obtained with 100x to 1000x improvement in computation time compared to standard Monte Carlo. Stochastic MOR model appropriate for timedomain simulations Stochastic MOR can have advantages over traditional corner based simulations Approach independentof the deterministic MOR method

38 Variability and uncertainty is not a curse It is an essential part of the dynamo of our evolution toward the next, more advanced state Chaos is the score upon which reality is written Henry Miller We should embrace uncertainty as an opportunity for tackling complexity It will make our design tools more agile, more useful, and more conducive to complex system design flow It is critical for universities to pave the way down this path Our future will not be built by deterministicallyminded technologists and innovators