A Statistical Cell Delay Model for Estimating the 3σ Delay by Matching Kurtosis
Leilei Jin, Wenjie Fu, Hao Yan, Longxing Shi
Abstract
Accurate standard cell modeling is significant for circuit timing analysis and yield estimation. With voltage decreasing to near-threshold, cell delay distribution becomes asymmetrical and has a longer tail due to various process variation effects. In this brief, we propose a novel statistical model to fit the shape of the cell delay by using log-extended-skew-normal (LESN) distribution. It estimates the values of mean, standard deviation, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$3\sigma $ </tex-math></inline-formula> delay precisely by matching the kurtosis of the cell delay distribution. By changing the parameters of the LESN model, it can be used for up to the normal voltage region (1.1V) and down to the sub-threshold voltage region (0.4V). Considering the effect of load capacitances, the parameters of LESN distribution are further modeled so that the model can be used for different fanout constraints. Tested by INV, NAND2, and NOR2 with 28 nm technology, the average error of estimating the skewness is less than 6%, while the error for the kurtosis is less than 4%. Meanwhile, the average errors of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$-3\sigma $ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$+3\sigma $ </tex-math></inline-formula> delay estimation are about 1.27% and 1.82% from 0.4V to 1.1V, respectively.