Xuanxing Xiong and Jia Wang Electrical and Computer Engineering Illinois Institute of Technology Chicago, Illinois, United States November, 2011 Vectorless Verification of RLC Power Grids with Transient Current Constraints

Agenda Power Grid Verification Proposed Approach Experimental Results 2

Power Grid Verification Verify that the power supply noises are within certain acceptable range  Noises depend on the patterns of currents drawn General idea for power grid verification  First, specify currents  Second, compute noises Simulation-based verification  DC & Transient analysis  Need to simulate a large number of current vectors to cover usual use scenarios  No guarantee the worst noise (but not overpessimistic) can be found. 3

Vectorless Power Grid Verification Apply optimization to find a current vector that leads to the worst power supply noise [Kouroussis et al DAC’03] [Qian et al ISPD’04]  Objective: maximizing power supply noise  Constraints: feasible current set  all possible current vectors  No need to explicitly enumerate all possible current vectors Trade-off: accuracy of feasible current set and solution efficiency  Linear current constraints: linear programming Steady-state vectorless verification  For worst-case DC scenarios and provide bounds for RC powergrid.  Early works are limited to small problem sizes. But recent advances [Abdul Ghani et al DAC’09] [Xiong et al DAC’10, ICCAD’10] have improved solution efficiency drastically. 4

Transient Vectorless Verification Transient behaviors are more realistic  Steady-state verification could be overpessimistic. Power grid modeling  Inductances [Abdul Ghani et al ICCAD’06]  Capacitive couplings between VDD and GND networks [Avci et al ICCAD’10] Current modeling  Max delta constraints [Ferzli et al TCAD’10]  Current slope constraints [Du et al ISQED’10]  Current conservation constraints [Avci et al ICCAD’10]  Power constraints [Cheng et al ISPD’11] However, there is no constraint to restrict the transient behavior of individual current sources. 5

Our Contribution A framework for transient vectorless verification of RLC power grids  With both VDD & GND networks Propose transient constraints for current sources  To capture the fact that a gate/block will only draw current when it is switching Prove the transient vectorless verification problem can be decomposed into a transient power grid anlysis problem and an optimization problem  Be able to leverage research works on fast power grid simulation 6

Agenda Power Grid Verification Proposed Approach Experimental Results 7

Integrated RLC Power Grid 8

The System Equation Time domain  G: conductance  M/C: represent self-inductance/capactiance links  v(t): nodal voltage noises  I(t): current excitations Discretization with time step  t where 9 ^

Current Constraints [Kouroussis et al DAC’03] and [Avci et al ICCAD’10] Local Constraints Global Constraints Current Conservation Constraints 10

Our Transient Current Constraints N ts : number of time steps I T : nx1 upper bound vector Transient constraints may be extracted from the circuit by switching activity analysis, e.g. [Morgado et al ICSD’09] and [Morgado et al TODAES’09] 11

Our Problem Formulation For each node j  The formulation actually computes the worst noise at node j for all time slots k  t If the cumulative effects of voltage noises are of interests, e.g. similar to [Evmorfopoulos et al ICCAD’10], the objective function can be 12

Property of System Equation 13 There exists a unique series of nxn matrices S 1, S 2,... S k, S k+1,..., such that j th column of S k can be computed as S k is symmetric. So

Our Problem Decompostion 14 For each node j: Sub-problem I: transient analysis with current excitation e j to compute c j,k Sub-problem II: linear programming (LP) to compute worst-case voltage noises

Agenda Power Grid Verification Proposed Approach Experimental Results 15

Experimental Setup Implement the RLCVN in C++  Use PCG with a random-walk based preconditioner for transient analysis  Adopt MOSEK to solve the LP problems Randomly generate 6 RLC power grids with 4 metal layers, 1.2V VDD, and various constraints Time step = 10ps, number of time steps Nts =

A Simple Case Study 17 Left: no transient constraint, max voltage drop is 118.4mV. Right: I T = 200mA, max voltage drop at node j is 86.5mV.

Overestimation without Transient Constraints for a Random Node 18

Average Runtime per Node 19

Conclusion & Future Work The proposed transient constraints make the voltage noise predicitons more realistic. The proposed decomposition results in an effective method for transient vectorless verification. To handle even larger power grid verification problems, it is necessary to research more efficient algorithms to solve the LP problems for worst-case voltage noises. 20

Thanks! 21

Can be extended to verify the integral of voltage noise without any computational overhead Our RLCVN Algorithm 22