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Vectorless Verification of RLC Power Grids with Transient Current Constraints

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Vectorless Verification of RLC Power Grids with Transient Current Constraints

Xuanxing Xiong and Jia Wang

Electrical and Computer Engineering

Illinois Institute of Technology

Chicago, Illinois, United States

November, 2011

- Power Grid Verification
- Proposed Approach
- Experimental Results

- 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.

- 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.

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.

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- 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

- Power Grid Verification
- Proposed Approach
- Experimental Results

- 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

^

[Kouroussis et al DAC’03] and [Avci et al ICCAD’10]

- Local Constraints
- Global Constraints
- Current Conservation Constraints

- Nts: number of time steps
- IT: 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]

- For each node j
- The formulation actually computes the worst noise at node j for all time slots kt

- If the cumulative effects of voltage noises are of interests, e.g. similar to [Evmorfopoulos et al ICCAD’10], the objective function can be

- There exists a unique series of nxn matrices S1, S2, ... Sk, Sk+1, ..., such that
- jth column of Sk can be computed as
- Sk is symmetric. So

- For each node j:
- Sub-problem I: transient analysis with current excitation ej to compute cj,k
- Sub-problem II: linear programming (LP) to compute worst-case voltage noises

- Power Grid Verification
- Proposed Approach
- Experimental Results

- 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 = 100

Left: no transient constraint, max voltage drop is 118.4mV.

Right: IT = 200mA, max voltage drop at node j is 86.5mV.

- 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.

- Can be extended to verify the integral of voltage noise without any computational overhead

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