Geometrically Parameterized Interconnect Geometrically Parameterized - - PowerPoint PPT Presentation

Geometrically Parameterized Interconnect Geometrically Parameterized Interconnect Performance Models for Interconnect Performance Models for Interconnect Synthesis Synthesis

Luca Daniel, UC Berkeley Luca Daniel, UC Berkeley

• C. S. Ong, S. C. Low, K. H. Lee,
• C. S. Ong, S. C. Low, K. H. Lee,

National University of Singapore National University of Singapore Jacob White, Massachusetts Institute of Technology Jacob White, Massachusetts Institute of Technology

Motivation Motivation

• In interconnect design often would like to design for:

In interconnect design often would like to design for:

– – reliable functionality: reliable functionality:

• minimize capacitive cross

minimize capacitive cross-

• talk,

talk,

• minimize inductive cross

minimize inductive cross-

• talk,

talk,

• minimize electromagnetic interference

minimize electromagnetic interference

– – high speed: high speed:

• minimize resistance

minimize resistance

• minimize capacitance

minimize capacitance

– – low cost: low cost:

• minimize area

minimize area

• Need to explore tradeoff space and find optimal design!

Need to explore tradeoff space and find optimal design!

Motivation (cont.) Motivation (cont.)

• The traditional design flow:

The traditional design flow:

– – REPEAT REPEAT

• design all interconnect wires

design all interconnect wires

• extract accurately parasitics all at once

extract accurately parasitics all at once

– – UNTIL noise and timing are within specs UNTIL noise and timing are within specs

• such procedure is not ideal for optimization!

such procedure is not ideal for optimization!

• each iteration is very time consuming

each iteration is very time consuming

Alternative design methodologies Alternative design methodologies

1.

• 1. Pre

Pre-

• characterize standard interconnect structures (e.g.

characterize standard interconnect structures (e.g. busses): busses):

– – using parasitic extraction and table lookup using parasitic extraction and table lookup – –

• r building parameterized and accurate low order models
• r building parameterized and accurate low order models

2.

• 2. And if the model construction is fast enough can also:

And if the model construction is fast enough can also:

– – build the interconnect structure model "on the fly" during build the interconnect structure model "on the fly" during layout layout – – accounting for any topology in surrounding topologies accounting for any topology in surrounding topologies already committed to layout already committed to layout – – then use optimizer to choose the best parameter for then use optimizer to choose the best parameter for

• ptimal tradeoff design.
• ptimal tradeoff design.

• We construct a multi

We construct a multi-

• parameter model of the bus

parameter model of the bus parameterized in wire parameterized in wire width W width W and and separation d separation d

Example: an interconnect bus Example: an interconnect bus

…………..

W W d d

• accounting for surrounding topology

accounting for surrounding topology

Parasitic extraction produces large Parasitic extraction produces large state space models state space models

• E.g. subdividing wires in short sections and using

E.g. subdividing wires in short sections and using for instance Nodal Analysis for instance Nodal Analysis

side gnd T

C sWC s WG x bu d y c x   + + =     =

…………..

Conductance matrix

Large linear dynamical system Large linear dynamical system Large linear dynamical system

Our goal Our goal

• Given a large parameterized linear system:

Given a large parameterized linear system:

gnd side T

s sW W u C C G x b d y c x                                                 + + =                                                 =

• construct a

construct a reduced order reduced order system with system with similar similar frequency frequency response response

ˆ ˆ ˆ ˆ ˆ ˆ ˆ

gnd side T

s sW C C W G x b u d y c x                         + + =                                      =

T

s u A x x b y c x                                 = +                                 = Background: Classical Non Background: Classical Non-

• parameterized

parameterized Model Order Reduction Model Order Reduction

500,000 x 500,000 500,000 x 500,000

• Given a large

Given a large parameterized parameterized linear system: linear system:

ˆ ˆ ˆ ˆ ˆ ˆ

T

s A x x b u y c x                 = +                         =

20 x 20 20 x 20

• construct a

construct a reduced order reduced order system with system with similar frequency similar frequency response response

Consider its Taylor series expansion: Consider its Taylor series expansion:

2

ˆ x x b A b Ab                                             =                                              

2

b A b Ab                                                

Background: Classical Non Background: Classical Non-

• parameterized

parameterized Model Order Reduction (cont.) Model Order Reduction (cont.)

{ }

2

span , , , x b Ab A b ∈ L

• idea for model order reduction:

idea for model order reduction: change base and use only the first change base and use only the first few vectors of the Taylor series few vectors of the Taylor series expansion: equivalent to match first expansion: equivalent to match first derivatives around expansion point derivatives around expansion point

m m m

x s A b u

∞ =

= ∑

sAx x bu = +

( )

1

x I sA bu

−

= − −

s

V

Parameterized Model Order Reduction Parameterized Model Order Reduction (cont.) (cont.)

A

T

sAx x bu y c x = + =

ˆ A

ˆT c

ˆ A ˆ b

=

T

V V

ˆ ˆ ˆ

T T T

sV AV x x V bu y c V x = + =

• Discretizing wires and using Nodal Analysis

Discretizing wires and using Nodal Analysis

s gnd T

C sWC s WG x bu d y c x   + + =     =

…………..

1

s

2

s

3

s Parameterized Model Order Reduction. Parameterized Model Order Reduction. Example: interconnect bus Example: interconnect bus

Parameterized Model Order Reduction Parameterized Model Order Reduction

1 1 p p T

s A s A I x bu y c x   + + − =   = K

• In general:

In general:

( )

{ }

2 1 2 1 1 2 2 1

, , , , , , ,

p

x span b Ab A b A b A b A A A A b ∈ + L L

r

x x V                     =                      

V

Once again change basis and Once again change basis and project state onto the first few project state onto the first few vectors of the Taylor series vectors of the Taylor series expansion, in order to match expansion, in order to match the first derivatives with the first derivatives with respect to all parameters respect to all parameters

( ) ( )

1 1 1 1 1 m p p p p m

x I s A s A bu s A s A b u

∞ − =

  = − − + + = + +  

∑

L L

• It is a p

It is a p-

• variables Taylor series expansion

variables Taylor series expansion

Parameterized Model Order Reduction Parameterized Model Order Reduction (cont.) (cont.)

s g T

C sWC s WG x bu d y c x   + + =     =

s

C G

g

C

ˆ

g

C ˆ

s

C ˆ G ˆ b ˆ c

ˆ

s

C

=

ˆ G

=

ˆ

g

C

=

ˆ ˆ

T T T T s g T

V C V sWV C V s WV GV x V bu d y c Vx   + + =     =

T

V V

T

V V

T

V V

0.2 0.4 0.6 0.8 1 x 10

• 11

0.2 0.4 0.6 0.8 1 time [sec]

• riginal order = 336. reduced to order = 5

0.2 0.4 0.6 0.8 1 x 10

• 11

0.2 0.4 0.6 0.8 1 time [sec]

• riginal order = 336. reduced to order = 5

Example: model step responses for different Example: model step responses for different W and d W and d

…………..

d=0.25um d=2um

N=16 wires h=1.2um L=1mm

W=0.25um W=0.2um W=4um W=8um

W0=1um d0 =1um

W=0.25um W=0.2um W=4um W=8um

W0=1um d0=1um

W W d d

0.2 0.4 0.6 0.8 1 x 10

• 11

0.05 0.1 0.15 0.2 0.25

time [sec]

• riginal order = 336. reduced to order = 5

0.2 0.4 0.6 0.8 1 x 10

• 11

0.05 0.1 0.15 0.2 0.25

time [sec]

• riginal order = 336. reduced to order = 5

Example: model crosstalk responses for Example: model crosstalk responses for different W and d different W and d

………….. N=16 wires h=1.2um L=1mm

d=0.25um d=2um W=0.25um W=0.2um W=4um W=8um W=0.25um W=0.2um W=4um W=8um

W0=1um d0 =1um W0=1um d0 =1um

W W d d

Open research issues and limitations Open research issues and limitations

• So far only account for resistance and capacitance. Still

So far only account for resistance and capacitance. Still need to verify if can account also for need to verify if can account also for inductance. inductance.

• No good a priori

No good a priori error bounds error bounds available for moment available for moment matching reduced order modeling techniques matching reduced order modeling techniques

– – i.e. for a given accuracy, don't know how to pick order i.e. for a given accuracy, don't know how to pick order theoretically a priori. theoretically a priori. – – however, practically, we do know how to construct the however, practically, we do know how to construct the model incrementally increasing its order reusing all model incrementally increasing its order reusing all previous computation until we meet desired accuracy. previous computation until we meet desired accuracy.

Open research issues and limitations Open research issues and limitations (cont.) (cont.)

• Model order grows as O(p

Model order grows as O(pm) ) where p = # parameters and where p = # parameters and m = # derivatives matched for each parameter m = # derivatives matched for each parameter

– – however model order is linear in # of parameters when however model order is linear in # of parameters when matching only one derivative per parameter (m = 1) and matching only one derivative per parameter (m = 1) and still produces good accuracy in our experiments. still produces good accuracy in our experiments. – – furthermore, for higher accuracy instead of increasing # of furthermore, for higher accuracy instead of increasing # of matched derivatives, can instead match matched derivatives, can instead match multiple points multiple points (or (or combine the two approaches) combine the two approaches)

W or d

Conclusions Conclusions

• Parameterized low order model of interconnect

Parameterized low order model of interconnect structures can help interconnect synthesis and structures can help interconnect synthesis and

• ptimization
• ptimization
• Presented a technique for parameterized modeling:

Presented a technique for parameterized modeling:

– – based on Krylov subspace congruence transformation based on Krylov subspace congruence transformation – – requires only matrix requires only matrix-

• vector products:

vector products: fast model fast model construction construction – – produced produced models capture accurately the behavior of the models capture accurately the behavior of the

• riginal system
• riginal system

– – and have low order: and have low order: can be instantly evaluated for any can be instantly evaluated for any parameter value parameter value for instance in an optimization procedure. for instance in an optimization procedure.

• Shown example result: bus interconnect parameterized

Shown example result: bus interconnect parameterized in wire widths and separation. in wire widths and separation.