EE 201C Modeling of VLSI Circuits and Systems Prof Lei He UCLA LHE@ee.ucla.edu

Chapter 5 Signal and Power Integrity On-chip signal integrity RC and RLC coupling noise Power integrity Static noise: IR drop Dynamic noise: L di/dt noise Chapter 6: Beyond die noise In-package decap insertion Low frequency P/G resonance Noise for High-speed signaling 2

Reading on Signal Integrity RC coupling J. Cong, Z. Pan and P. V. Srinivas, "Improved Crosstalk Modeling for Noise Constrained Interconnect Optimization", ASPDAC, 2001. RLC coupling Jun Chen and Lei He, "Worst-Case Crosstalk Noise for Non-Switching Victims in High-speed Buses", IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, Volume 24, Issue 8, Aug. 2005. To be covered by student presentation on May 14 2

Reading on Power Integrity Static noise: IR drop S. Tan and R. Shi, “Optimization of VLSI Power/Ground (P/G) Networks Via Sequence of Linear Programmings”, DAC’09 Dynamic noise: L di/dt noise Yiyu Shi, Jinjun Xiong, Chunchen Liu and Lei He, "Efficient Decoupling Capacitance Budgeting Considering Current Correlation Including Process Variation", ICCAD, San Jose, CA, Nov. 2007. Supplementary reading: H. Qian, S. R. Nassif, and S. S. Sapatnekar, “Power Grid Analysis Using Random Walks,” IEEE Trans. on CAD, 2005. Yiyu Shi, Wei Yao, Jinjun Xiong, and Lei He, "Incremental and On-demand Random Walk for Iterative Power Distribution Network Analysis", ASPDAC 2009 2

Xiang-Dong Tan* and C.-J. Richard Shi Optimization of VLSI Power/Ground (P/G) Networks Via Sequence of Linear Programmings Xiang-Dong Tan* and C.-J. Richard Shi DAC 2009 Best Paper Award Slides provided by X.D. Tan

Outline of Presentation Introduction and motivation Review of existing algorithms Relaxed P/G optimization procedure New P/G optimization algorithm Experimental results Summary and future work 2

Introduction IR drops: Electro-migration: Pad ... Voltage difference between power supply pads and individual cell instances. Electro-migration: Metal ion mass transport along the grain boundaries when a metallic interconnect is stressed at high current density. Mean Time to Failure (MTF) (Black’s equation): ... Pad

Introduction A real industrial chip #cell instances: 0.5M #P/G resistors: 0.6M

Introduction Unrestricted IR drops and current densities in power / ground (P/G) network will cause malfunction and reliability problems in deep sub-micron IC chips. Increased cell delays (timing problem) increased resistance and even opens of P/G wires Most of P/G designs are done manually. An aggressive design will cause more design iterations and thus lead to increased design costs. Over conservative P/G network design wastes a lot of important chip areas. 3

Motivation Two steps in P/G network design: P/G network construction (P/G routing). Determination of wire segment widths. Determination of wire segment widths is hard to solve. The problem of determining wire segment widths in a P/G network subject to reliability constraints is a constrained non-linear optimization problem. Existing methods are not very efficient. based on the constrained nonlinear programming, can not handle large industrial P/G networks containing millions of wire segments. 3

Outline of Presentation Introduction and motivation Review of existing algorithms Relaxed P/G optimization procedure New P/G optimization algorithm Experimental results Summary and future work

Algorithm Review Assumption: Currents (average or maximum) of each individual cell instance are known a priori before the optimization. computed by using power models of cells in a design Existing optimization methods differ in the selection of variables. Ohm’s Law

Problem Formulation Min-area objective function: (Problem P) IR drop constraints: Electro-migration constraints: Minimum width constraints:

Algorithm Review Resistance values and branch currents are variables (Chowdhury and Breuer’87) Both objective function and IR drop constraints are nonlinear Solution: augmented Lagrangian method Resistance values are variables (Dutta and Marek-Sadowska’89) All the constraints are nonlinear Solution: feasible direction method Nodal voltage and branch current are variables (Chowdhury’89) Only objective function is nonlinear Solution: linear programming & conjugate gradient Topology construction (Mitsuhashi and Kuh’92)

Outline of Presentation Introduction and motivation Review of existing algorithms Relaxed P/G optimization procedure New P/G optimization algorithm Experimental results Summary and future work

Relaxed P/G Optimization Algorithm Relaxation: current directions are fixed. Nodal voltages and branch currents can be selected as variables and be optimized separately. Two optimization steps to solve P (Chowdhury’89) Solve for nodal voltages under fixed branch currents (problem P1) Solve for branch current under fixed nodal voltages (problem P2) Advantage: All the constraints become linear and P2 is a linear programming problem. P1 is a convex programming problem.

Problem P1 Nonlinear Optimization Problem (P1) Objective function: Subject to IR drop: Electro-migration: Minimum width:

Problem P2 Linear Optimization Problem (P2) Objective function: Subject to Minimum width: KCL law:

Observations Solution to P1: Disadvantage: First transform P1 into a unconstrained nonlinear optimization problem by adding a penalty function to the objective function. Conjugate gradient method was used to solve the unconstrained nonlinear problem. Disadvantage: Very slow convergence (almost linear) Conjugate gradient directions may deteriorate

Outline of Presentation Introduction and motivation Review of existing algorithms Relaxed P/G optimization procedure New P/G optimization algorithm Experimental results Summary and future work

New Optimization Algorithm Basic idea ---- linearize the nonlinear objective function in P1. Define Linearized objective function:

New Optimization Algorithm The g(v) makes sense only if Each product term, h(x) = c/x, in f(x) is a monotonic decreasing function.

Two Optimization Scenarios (1) All the branch voltage drops increase. (2) Only some branch voltage drops increase while others decrease or stay unchanged. Combine two scenarios, we have We always have

New Optimization Problem P3 Minimize Linearized objective function Subject to IR drop: Electro-migration: Minimum width: Extra constraint:

Sequence of Linear Programmings New P/G Optimization Algorithm 1. Obtain an initial solution for a given P/G network 2. Build all the constraints for Problem P3 3. Solve P3 by sequence of linear programmings and record the result as 4. Build all the constraints based on of step (3) for the problem P2 5. Solve P2 by a linear programming and record the result as 6. Stop if improvement over previous result is small. Otherwise, goto step (2)

Theoretical Result Theorem: There exists a so that step (3) always converges to the global minimum in the convex problem space of P1. xmin x1 x0 x2 x3 x4

Practical Considerations Selection of At the beginning, solution space in P3 should be as large as possible, so should be small. It should be close to 1 in later course of sequence of linear programmings. Numerical Stability Power networks are converted ground networks to improve the numerical stability. Voltage drop in a ground network has to be represented by by a power network with 5V supply voltage. Power networks can be easily converted into ground networks.

Outline of Presentation Introduction and motivation Review of existing algorithms Relaxed P/G optimization procedure New optimization algorithm Experimental results Summary and future work

Experimental Results Sequence of linear programming and Conjugate gradient method were performed on 4 oversized p/g networks on SUN Untra-I with 169 MHz.

Comparison in CPU time

Comparison in Performance

The cost reduction versus the number of iterations (p4x4) Experimental Results The cost reduction versus the number of iterations (p4x4)

Effect of on the performance of the new algorithm Experimental Results Effect of on the performance of the new algorithm

Outline of Presentation Introduction and motivation Review of existing algorithms Relaxed P/G optimization procedure New optimization algorithm Experimental results Summary and future work

Summary A new method based sequence of linear programmings was proposed to determine the widths of P/G network segments subject to reliability constraints. We showed theoretically that new method is capable of finding solution as good as that by the best-known method. Experimental results demonstrated that new method is orders-of-magnitude faster than the best-known method with constantly better solution quality.