1. EE 201C Modeling of VLSI Circuits and Systems
Prof Lei He
UCLA

2. 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

3. 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
To be covered by student presentation on May 14 2

4. 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
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

5. 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

6. 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

7. 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

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

9. 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

10. 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

11. 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

12. 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

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

14. 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)

15. 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

16. 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.

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

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

19. 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

20. 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

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

22. 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.

23. 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

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

25. 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)

26. 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

27. 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.

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

29. 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.

30. Comparison in CPU time

31. Comparison in Performance

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

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

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

35. 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.