Accounting for Variability and Uncertainty in Signal and ... · Electromagnetic modeling/simulation pervasive in physical design and sign-off analysis Kurokawa et al, Interconnect

Accounting for Variability and

Uncertainty in Signal and Power

Integrity Modeling

Andreas Cangellaris & Prasad

Sumant

Electromagnetics Lab, ECE Department

University of Illinois

Acknowledgments

• Dr. Hong Wu, Extreme DA

• Prof. Narayan Aluru, University of Illinois

• DARPA for funding support

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

Are we using the might of EM CAD for

signal integrity-aware design wisely?

Floor planning

Power planning

Placement

Clock planning

Routing

Physical Layout

Sign-0ff analysis

Interconnect parasitic extraction

IR-drop analysis EM analysis

Signal-integrity analysis Timing analysis

Physical verification DFM

ECOs

Input filed for

EDA tools

Timing/power

library

Interconnect

library

Electromagnetic modeling/simulation

pervasive in physical design and sign-off

analysis

Kurokawa et al, Interconnect Modeling:

A Physical Design Perspective, IEEE Trans.

Electron Devices, Sep. 2009

Power Integrity

• Package impedance

– Design/layout-dependent

– R impacted by manufacturing tolerances

• On-chip grid

– R variability due to CMP

• On-chip decoupling

– Available C dependent on operation, VT, Tox, …

Impact of Package R on IR drop(*)

(*)Sani Nassif, IBM

Variability and Uncertainty

Source: Future-Fab International

Stacked dies

Package-on-package

05.23.08Source: TNCSI

Multiple layers

Variability and Uncertainty

Variability and Uncertainty

?

Accounting for SI/PI Modeling

and Simulation

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

Transmission-Line Parameter

Extraction

conducto

r

ground plane

mean geometry

random geometry

Mapping between random sample and

mean geometry

' 0LE dl V⋅ =′ ′∫uuur r

0( ).( )L

D r dl Vrε′

′ ′ ′ =′∫uuuruur

uur

r

0( )c L

dlD Vrε′

′ =′∫

uuuruuur

uur

00 0( )

( )

c

L

VD r D

dlrε′

′ ′ = =′′∫

uuuur uuuruuur

uuur

uur

00( )

( )L

VD r

dlrε

=

∫

uur uuruur

ur

0 0 0( ) ( )D r Q D r′ ′ =uuuur uuur

uur uur

( )

( )

L

L

dlrQ

dlr

ε

ε′

=′′

∫

∫

uur

ur

uuur

uur

Random sample

(1)

Mean geometry

(2)

Mapping relationship(3)

Electric flux density computation

using solution on mean geometry

00 0

0

( ) ( )( )

VG r r

D rε=

ur ur

ur ur

00 0

0 0

( )( ) ( )

( ) ( )

G rD r D r

G r v r′′ =

−

uruur ur ur ur

ur ur

0 0

0 0

( ) ( )1

( ) ( ) ( )L

G r v rdl

r r rε ε ε′′ ≈ −

′∫ur ur

ur ur ur

0 0 0( ) ( ) ( )L r L r v r′ ′ ≈ −ur ur ur

1 1

( ) ( )L L vdl dl

r rε ε′ −′ ≈

′∫ ∫ur r

Position-dependent flux length/gap on mean

geometry:

Flux length on random

sample:

0

0

( )1 (exact)

( ) ( )L

G rdl

r rε ε=∫

ur

r ur

Electric flux density on random sample using mean geometry:

Computation of capacitance of

random sample

x y

X XJx y

Y Y

∂ ∂ ∂ ∂=

∂ ∂ ∂ ∂

00

0 0

( )

( )( )

( ) ( )

rs

mean

C

C

C D r dl

G rC D r J dl

G r v r

′ ′ ′ ′=

′ =−

∫

∫

uur ur

urur ur

ur ur

Capacitance on random sample:

Where J represents the Jacobian, the map between the random and mean

geometry:

Representing uncertainty using

polynomial chaos

0

ˆ( , ) ( ) ( ( ))ii

u r a rθ ξ θ∞

== Ψ∑

r r

0

ˆ( , ) ( ) ( ( ))N

ii

u r a rθ ξ θ=

= Ψ∑r r

2 3 4 20 1 2 3 4( ) 1, ( ) , ( ) 1, ( ) 3 , ( ) 6 3,...ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξΨ = Ψ = Ψ = − Ψ = − Ψ = − +

Polynomial chaos expansion:

Coefficients: functions of space

Orthogonal polynomials are random variables

Truncated polynomial chaos expansion:- Number of different input random variables: n

- Order of polynomials: p

Type of polynomials depends on distribution of input random variable

- Gaussian distribution, Hermite polynomial chaos

( )!1

! !

n pN

n p

++ =

Computing stochastic capacitance

( , ) ( ) ( , )G r G r v rθ θ= −r r r%

2 3

02 3

( , ) ( , ) ( , )( ) 1 .... ( )

( ) ( ) ( )

v r v r v rD r D r

G r G r G r

θ θ θ ′ ≈ + + + +

r r rr rr r%

r r r0

( )( ) ( )

( ) ( )

G rD r D r

G r v r′ ′ ≈

−

rr rr r

r r

2 3

02 3

( , ) ( , ) ( , )1 .... .

( ) ( ) ( )S

v r v r v rC J D ds

G r G r G r

θ θ θ ≈ + + + +

∫

r r rrr

%r r r

0

( , ) ( ) ( ( ))N

ii

v r v rθ ξ θ=

= Ψ∑r r

0

( ) ( ( ))N

ii

C C r ξ θ=

= Ψ∑r

%

Displacement of random geometry from the mean geometry:

Use relationship between random and mean flux length:

Stochastic Electric Flux Density:

Stochastic Capacitance:

(*)

Single trace over ground plane

- The height of the conductor above the ground plane ‘H’ is uncertain.

4ε =

1ε =

7ε =

L

T

H

Mean Geometry dimensions : L=1 um, T = 0.1 um, H= 0.2 um

-Where is the mean height

- is a Gaussian random variable with mean 0 and variance 1

0( ) (1 ( ))H Hθ νξ θ= −

0Hξ

0( , )v r Hθ νξ=r

2 220 0

021 ....

( ) ( )S

H HC J D dl

G r G r

ν νξ ξ ≈ + + +

∫

r%

r r

20 1 2( 1)C C C Cξ ξ= + + −%

2

00 0

2

0 01 0 2 0

1( )

, ( ) ( )

S

S S

HC J D dl

G r

H HC J D dl C J D dl

G r G r

ν

ν ν

= +

= =

∫

∫ ∫

r

r

r r

r r

Single trace over ground plane

Displacement of random geometry from the mean geometry

Second-order Hermite polynomial chaos for stochastic

capacitance

Only one deterministic run

needed to get 0Dr

Single trace over ground plane

%change

in H

Monte Carlo FEM based Approach

Mean Std

deviation

Mean Std

deviation

10% 329.7429 11.5365 329.83 11.43

20% 331.0031 23.6053 331.48 23.31

Self-capacitance (pF/m)

Single trace over ground plane

%change

in L

Monte Carlo FEM based Approach

Mean Std

deviation

Mean Std

deviation

10% 331.0028 7.26 329.26 7.48

20% 331.5143 14.92 329.26 14.96

Simulation time comparison

Time for 1 Capacitance extraction run ~1.2 s

� Monte Carlo : Time for 10000 runs ~ 12000 s

� Our approach ~ 2.0 s

Capacitance (pF/m)

Mean Geometry dimensions : L=1 um, S = 0.15 um, H= 0.2 um

4ε =

1ε =

7ε =

L

T

H

S

% change

in H and S

Monte Carlo FEM based Approach

Mean Std

deviation

Mean Std

deviation

10% 368.7324 13.04 368.85 14.05

20% 370.2421 26.83 370.83 28.65

Self-capacitance (pF/m)

Coupled symmetric microstrip

Microstrip with multi-dielectric

substrate

4ε =

1ε =

4ε =

L

T

8ε =7ε =

%change in

each layer

below

Conductor

Monte Carlo (10000) FEM based Approach

Mean Std deviation Mean Std deviation

10% 268.94 6.76 268.01 6.26

20% 269.46 13.56 267.82 12.56

0.1

0.1

0.1

Self-capacitance (pF/m)

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

Deterministic Model Order Reduction2( )org org org org

Horg

Y sZ s Pe x sB I

V L x

+ + =

=

2 1( ) ( )HGZ s sL Y sZ s Pe B−= + +

Horg

Horg

Horg

Y F Y F

Z F Z F

Pe F Pe F

=

=

=

Generalized Multiport Impedance Matrix using reduced model:

Model Order Reduction

e.g. Krylov subspace based

methods

Projection matrix F

2( )H

Y sZ s Pe x sBI

V L x

+ + ==

Order: N

Order n << N

Model order reduction under

uncertainty

2( )H

Y sZ s Pe x sBI

y L x

+ + =

=

2( )H

Y sZ s Pe x sBI

V L x

+ + ==

% % % %

% %

Horg

Horg

Horg

Y F Y F

Z F Z F

Pe F Pe F

=

=

=

% % % %

% % % %

% % % %

0 1 1 2 2 0 1 1 2 2

0 1 1 2 2 0 1 1 2 2

,

,

org org

org

Y Y Y Y Z Z Z Z

Pe P P P F F F F

ξ ξ ξ ξ

ξ ξ ξ ξ

= + + = + +

= + + = + +

% %

% %

Deterministic Reduced Order Model Stochastic Reduced Order Model

Represent stochastic system matrices using polynomial chaos expansion:

Augmented Stochastic Reduced Order

Model

0 1 1 2 2 0 1 1 2 2

20 1 1 2 2 0 1 1 2 2 0 1 1 2 2

[( ) ( )

( )]( ) ( )

Y Y Y s Z Z Z

s Pe Pe Pe x x x s B B B I

ξ ξ ξ ξξ ξ ξ ξ ξ ξ

+ + + + + +

+ + + + = + +

0 1 2 0 0 1 2 0 0 1 2 0 0

21 0 1 1 0 1 1 0 1 1

2 2 2 22 0 2 0 2 0

0 0 0

0 0 0

Y Y Y x Z Z Z x Pe Pe Pe x B

Y Y x s Z Z x s Pe Pe x s B I

x x x BY Y Z Z Pe Pe

+ + =

2

2 1

( )

( )

aug aug aug aug aug

Haug aug aug

Haug aug aug aug aug aug

Y sZ s Pe x sB I

V L x

Z sL Y sZ s Pe B−

+ + =

=

= + +

Augmented reduced order model

Order: 3n << N

Computing polynomial chaos

coefficients

1 1 2 2 1 2 0 1 1 2 2 1 1

0 1 1 2 2 1 2

2 2 1 2( ) ( ) ( ) ( ) ( )

( ) ( )

or

org

g

Y

Y d d Y Y Y d d

Y d d

ρ ξ ρ ξ ξ ξ ξ ξ ρ

ρ ξ ρ

ξ ρ ξ

ξ ξ

ξ

ξ

ξ= + +

=

∫ ∫

∫

∫

∫

∫%

%

1 1 1 2 2 1 2 0 1 1 2 2

1 1 1 1 2

1 1 1 2 2 1 2

2 1 2

( ) ( ) ( ) ( ) (

( ) (

)

)

org

org

Y d d

Y Y d

Y Y

d

Y d dξ ρ ξ ρ ξ ξ ξ ξ ξ ξ ρ ξ

ξ ρ ξ ρ ξ ξ ξ

ρ ξ ξ ξ= + +

=

∫∫ ∫

∫

∫

∫

%

%

Coefficient matrices in polynomial chaos expansion: - Integrate over the random space and use orthogonality of the

polynomials

Smolyak Sparse Grid Integration

1 1 2 2 1 2( ) ( ) ( ) ( )j

org jI f Y d d w f uρ ξ ρ ξ ξ ξ= ≈∑∫∫ %

cos ( 1), 1...

1j j

u j qq

π −= − =−

( )1

1

1 1

1

1 1 11 1

Number of calculat

[ ] ... [ ]

... ( ,..., ).( ... )

ions

N

N

N N

N

qqQi i

qqj jj j

N

Nj j

I f I I f

f u w w

q

u= =

≡ ⊗ ⊗

= ⊗ ⊗

∝

∑ ∑

( )

11

1Number of calcul

( ) ( , )

ations

1( 1) . .( ...

log

)N

Q

J ii i

J N i J

N

I f A J N

NI I

J i

q q

−

− + ≤ ≤

−

≡ =−

= − ⊗ ⊗

∝

− ∑

1-d integration rule: e.g. using chebyshev polynomial extrema

The case of multiple random variables:

1

( ) ( )q

j j

j

I f f u w=

=∑

Deterministic Cartesian Product

(DCP) rule:Smolyak Sparse Grid Algorithm

Idea: Not all points are equally important;

hence, discard the least important ones

Comparison of grids generated using

Tensor Product and Smolyak

Algorithm• Tensor product grid (81 points)

Smolyak Sparse grid (29 points)NQ q∝ pQ N∝q: number of points in 1-d rule

N: number of random dimensions

p: level in Smolyak algorithm

Algorithm for Stochastic MOR• Represent uncertainty in the original

system matrices through polynomial

chaos expansion

• Generate sparse grid points and their

corresponding weights using Smolyak

Sparse grid algorithm

• Compute the transformation matrix

through MOR of individual systems

corresponding to Smolyak sparse grid

points

• Compute the stochastic transform

matrix

• Define the stochastic reduced order

model

1 2

~

0 1 2orgY Y Y Yξ ξ+= +

{ } { },i iM Mw wθ ξ= =

ix F z=

1 2 2

~

0 1 FF F Fξ ξ+= +

2( )org org org orgY sZ s Pe x sB I+ + =

2( )Y sZ s Pe x sBI+ + =

2( )aug aug aug aug augY sZ s Pe x sB I+ + =

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 36840

– Reduced order system of order 20.

• Randomness in two inputs– Permittivity: uniform random variable in [3.4-4.4]

– Load resistance: uniform random variable in [25-35] ohms

• Monte Carlo: 10201 simulations

• Smolyak: 29 points

Re{Zin(f)}

• Mean of real part

of input impedance

• Standard deviation

of real part of input

impedance

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,RL)– (3.6,25)

– (4.0,30)

– (4.4,35)

• Compare with information generated using stochastic MOR

Corner vs. stochastic simulation

• Corner simulation appears very conservative

• Mean parameter solution is not accurate compared to the mean

of stochastic simulation

Remarks

• Very good accuracy obtained with 100x to 1000x improvement in computation time compared to standard Monte Carlo.

– Stochastic MOR model appropriate for time-domain simulations

• Stochastic MOR can have advantages over traditional corner based simulations

• Approach independent of the deterministic MOR method

• 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 deterministically-minded technologists and innovators