Haihua Su, Sani R. Nassif IBM ARL An Algorithm for Optimal Decoupling Capacitor Sizing and Placement for Standard Cell Layouts Haihua Su, Sani R. Nassif IBM ARL Sachin S. Sapatnekar ECE Department University of Minnesota 9/18/2018 ISPD'02, San Diego, CA

Outline On-chip decap overview Modeling and noise analysis Problem formulation and Adjoint sensitivity analysis Decap sizing and placement scheme Experimental results Conclusion 9/18/2018 ISPD'02, San Diego, CA

On-chip Decoupling Capacitors Non-switching gate capacitance Thin oxide capacitance w: width of decap h: height of decap tox: thickness of thin oxide ox: permittivity of SiO2 9/18/2018 ISPD'02, San Diego, CA

Decoupling Capacitor Models 1st order model 2nd order model (non-idealities) 9/18/2018 ISPD'02, San Diego, CA

Power Network Modeling Power Grid: resistive mesh Cells: time-varying current sources Decaps: 1st order or 2nd order decap model Package: inductance + ideal constant voltage source + 9/18/2018 ISPD'02, San Diego, CA

Power Grid Noise Analysis Noise metric: shaded area Waveform of node j on VDD grid Vj + z(j) Z = S z(j) Reference: A. R. Conn, R. A. Haring and C. Visweswariah, Noise Considerations in Circuit Optimization, ICCAD’98 9/18/2018 ISPD'02, San Diego, CA

Formulation - Constrained Nonlinear Programming Problem Minimize Z(wj), j = 1..Ndecap Subject to Swk  (1-ri)Wchip, i = 1..Nrow And 0  wj  wmax , j = 1..Ndecap ri is the occupancy ratio of row i Cell Decap wj 9/18/2018 ISPD'02, San Diego, CA

Solver – Sequential Quadratic Programming (SQP) QPSOL - Quasi-Newton method to solve the problem of multidimensional minimization of functions with derivatives Requirements evaluation of the objective function and constraint functions calculation of first-order derivatives 9/18/2018 ISPD'02, San Diego, CA

Adjoint Sensitivity Analysis Original circuit Vj(t) + Adjoint circuit x(t) and – node voltages, source currents, inductor currents u(t) – time-dependent sources i() – current sources applied to all bad nodes ij() 9/18/2018 ISPD'02, San Diego, CA

Adjoint Sensitivity Analysis (cont’d) Convolve to get sensitivities Z is the noise metric for all the grid = S z(j) 9/18/2018 ISPD'02, San Diego, CA

Adjoint Sensitivity Analysis (cont’d) Fast convolution for piecewise linear waveforms ~O(N+M) N linear segments M linear segments p q 9/18/2018 ISPD'02, San Diego, CA

Sensitivity w.r.t. Decaps Adjoint sensitivity w.r.t. Cnear, R and Cfar Applying chain rule to find the sensitivity w.r.t. decap width w: 9/18/2018 ISPD'02, San Diego, CA

Scheme Analyze circuit and store waveforms Compute Z Setup current sources for adjoint circuit Analyze adjoint circuit & store waveforms Compute Z/Ci and Z/wi Evaluate constraint function & gradients Feed to QP solver to get the updated wi According to the new wi , replace cells and decaps one by one 9/18/2018 ISPD'02, San Diego, CA

Decap Optimization Process (one row for illustration) Start from equal distribution of decaps: Iteration 1: Iteration 2: 9/18/2018 ISPD'02, San Diego, CA

Optimization Results Vdd =1.8V, vdrop limit =10%Vdd, ri = 80% 3 2 1 Chip Before After Opt 100 70 80 63 105 2 Num bad nodes 828 861 974 Num of nodes 0.222 0.201 0.230 0.196 0.193 0.176 Vmax (V) 0.649 0.200 0.366 0.063 0.121 0.000 Z (Vns) 132 85 53 Num of rows 3664 3288 1964 Num of dcps 12.5 15.2 0.9 CPU time (mins) 9/18/2018 ISPD'02, San Diego, CA

VDD and GND Contour (chip2) Vmax=0.190V Vmax=0.191V Vmax=0.230V Vmax=0.196V Z=0.366(V•ns) Z=0.063(V•ns) 9/18/2018 ISPD'02, San Diego, CA

Optimal Placement (chip2) 9/18/2018 ISPD'02, San Diego, CA

Noise Reduction Trend (chip2) 9/18/2018 ISPD'02, San Diego, CA

Conclusion Proposed a scheme of decoupling capacitor sizing and placement for standard-cell layouts Applied after placement and before signal routing Formulated into nonlinear programming problem Reduced transient noise Presented a fast piece-wise linear waveform convolution for adjoint sensitivity analysis 9/18/2018 ISPD'02, San Diego, CA

Thank you! 9/18/2018 ISPD'02, San Diego, CA