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. Pre 1. 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 – or building parameterized and accurate low order models or building parameterized and accurate low order models – 2. And if the model construction is fast enough can also: And if the model construction is fast enough can also: 2. – 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 optimal tradeoff design. optimal tradeoff design.

Example: an interconnect bus Example: an interconnect bus • We construct a multi We construct a multi- -parameter model of the bus parameter model of the bus • parameterized in wire width W width W and and separation d separation d parameterized in wire W d W 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 ………….. Conductance matrix   C + + = side sWC s WG x bu   gnd   d = T y c x 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: •                                     s     +   +     =   sW C C W G x b u   gnd side           d                                     = T y c x • construct a construct a •             reduced order reduced order             s + + = system with ˆ system with ˆ ˆ ˆ           ˆ   sW C C W G x b u gnd side  d             similar similar             frequency frequency = T ˆ ˆ y c x response response

Background: Classical Non- -parameterized parameterized Background: Classical Non Model Order Reduction Model Order Reduction         • Given a large Given a large •         parameterized parameterized         linear system: linear system:         = + s A x x b u                         500,000 x 500,000 500,000 x 500,000         = T y c x • construct a • construct a         reduced order reduced order         = +   ˆ ˆ     ˆ ˆ system with system with s A x x b u           similar frequency similar frequency 20 x 20     20 x 20     response response = T ˆ ˆ y c x

Background: Classical Non- -parameterized parameterized Background: Classical Non Model Order Reduction (cont.) Model Order Reduction (cont.) ( ) − = + = − − 1 sAx x bu x I sA bu Consider its Taylor series expansion: Consider its Taylor series expansion: ∞ = ∑ { } ∈ m m 2 x s A b u L x span b Ab A b , , , = m 0                 • 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 2 2 ˆ x x b b A A b b Ab Ab   derivatives around expansion point derivatives around expansion point                                                                     V s

Parameterized Model Order Reduction Parameterized Model Order Reduction (cont.) (cont.) ˆ ˆ b A = + = + T T ˆ ˆ sAx x bu sV AV x x V bu = = T T ˆ y c V x y c x ˆ T c = T ˆ V A A V

Parameterized Model Order Reduction. Parameterized Model Order Reduction. Example: interconnect bus Example: interconnect bus • Discretizing wires and using Nodal Analysis Discretizing wires and using Nodal Analysis • …………..   C + + = s sWC s WG x bu   gnd   d = T y c x s s s 2 3 1

Parameterized Model Order Reduction Parameterized Model Order Reduction   + + − = K s A s A I x bu • In general: In general:   • 1 1 p p = T y c x ∞ ( ) ( ) − ∑ 1   m = − − + + = + + L L x I s A s A bu s A s A b u   1 1 p p 1 1 p p = m 0 It is a p- -variables Taylor series expansion variables Taylor series expansion • It is a p • { } ( ) ∈ + L 2 L x span b Ab A b , , , , A b A b , , A A A A b , 1 2 p 1 1 2 2 1          Once again change basis and Once again change basis and       project state onto the first few project state onto the first few    =   x x V V   vectors of the Taylor series vectors of the Taylor series r       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

Parameterized Model Order Reduction Parameterized Model Order Reduction (cont.) (cont.) ˆ C ˆ ˆ ˆ b C G   s C g + + = s   sWC s WG x bu   T V C V  g  + + = d T T T s ˆ   sWV C V s WV GV x V bu g   d = T y c x = = T ˆ y c Vx T ˆ V C g C V g ˆ c = T ˆ V C s C V s = T V ˆ G G V