Patent Publication Number: US-2023153501-A1

Title: Near-threshold cell circuit delay model

Description:
TECHNICAL FIELD 
     The present invention relates to the field of integrated circuit design, and particularly relates to a near-threshold cell circuit delay model. 
     BACKGROUND 
     With the rise of low power consumption technologies such as the Internet of Things, how to improve the circuit performance under low power consumption has become a key issue in the development of integrated circuits. One of the effective means to reduce power consumption is reducing a supply voltage. Due to the energy efficiency advantages, the design and research of near-threshold circuits have received extensive attention. 
     A near-threshold cell circuit delay model is of great significance for studying the timing information of a circuit in the early design stage, and can quickly predict the delay of a cell circuit in different scenarios. For the near-threshold cell circuit delay model, the current method in the industry is a look-up table method, that is, a look-up table is obtained by simulation, and then, the look-up table is interpolated to obtain a cell delay. However, the establishment of the look-up table requires a lot of simulation, so that the time cost is high. Therefore, a fast and accurate near-threshold cell circuit delay model needs to be established. The challenges in establishing the near-threshold cell circuit delay model can be mainly divided into two aspects: first, an input transition time causes the nonlinear change of the delay, so the linear and nonlinear relationships between the delay and the input transition time need to be modeled respectively. However, a current equation in near-threshold domain is complex, which makes an integral-based modeling method at nominal voltage no longer applicable; secondly, due to the nonlinear relationship between the cell delay and process parameters, the cell delay in near threshold follows non-Gaussian distribution, which makes the original cell delay model based on Gaussian distribution no longer applicable. Therefore, a fast and accurate near-threshold cell circuit delay model needs to be established so as to reduce the simulation cost as much as possible on the premise of ensuring the accuracy. At present, a cell circuit delay model based on an equivalent current method is only for the case of a specific input transition time, and does not fully consider the influence of the input transition time on the delay. 
     SUMMARY 
     Technical problems: An objective of the present invention is to establish a fast and accurate near-threshold cell circuit delay model considering an input transition time. A nominal delay model and a statistical delay model of an inverter and a composite cell are obtained by an equivalent current method and the properties of logarithm skewed normal distribution. The present invention has the advantages of high accuracy and low time cost. 
     Technical solution: In order to achieve the objective, a near-threshold cell circuit delay model of the present invention adopts the following technical solution: 
     A cell circuit includes inverters, stacked structure cells and parallel structure cells, and the model includes the following steps: 
     S1: obtaining parameters, including process parameters of each transistor of the cell circuit, cell circuit parameters and delay parameters with slow input transition; 
     S2: judging a cell circuit type, including judging whether the cell circuit is the inverter, the stacked structure cell or the parallel structure cell, calculating a high current, a low current and a current integral of the cell circuit according to the cell circuit type, calculating a mean value, a variance and a skewness of a natural logarithm of the sum of the high current and the low current, and calculating a mean value and a variance of an equivalent threshold voltage; 
     S3: judging a delay type, calculating an overshoot time and a delay of the cell circuit according to the cell circuit type obtained in step S2, and then, judging whether the delay type is ultra-fast input transition, fast input transition or slow input transition according to the relative magnitude relationship among the overshoot time, the delay and an input transition time; if the overshoot time is greater than the input transition time, determining the delay type as the ultra-fast input transition; if the overshoot time is less than the input transition time and the delay is greater than half of the input transition time, determining the delay type as the fast input transition; if the overshoot time is less than the input transition time and the delay is less than half of the input transition time, determining the delay type as the slow input transition; 
     S4: establishing a cell circuit nominal delay model according to the cell circuit type obtained in step S2 and the delay type obtained in step S3, and obtaining a nominal delay; 
     S5: based on the nominal delay model, establishing the cell circuit statistical delay model according to the cell circuit type obtained in step S2 and the delay type obtained in step S3, and obtaining delay statistics: a variance, a maximum value and a minimum value. 
     Where: 
     in step S1, the process parameters include a sub-threshold slope factor and a drain-induced source barrier lowering effect coefficient, the circuit parameters include a transistor size, a load, the input transition time, a power supply voltage, a temperature and process, the delay parameters of slow input transition include fitting parameters a and b in Equation (1) and fitting parameters c and d in Equation (2), and the fitting parameters a, b, c and d are obtained by fitting the delay under the slow input transition obtained by simulation, 
     
       
         
           
             
               
                 
                   
                     td 
                     τ 
                   
                   = 
                   
                     
                       a 
                       ⁢ 
                       ln 
                       ⁢ 
                       
                         
                           td 
                           s 
                         
                         τ 
                       
                     
                     + 
                     b 
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   a 
                   = 
                   
                     
                       
                         c 
                         ⁢ 
                         V 
                       
                       th 
                     
                     + 
                     d 
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     where in Equation (1), td represents a cell circuit delay, τ represents the input transition time, and td s  represents a step delay; and in Equation (2), V th  represents a threshold voltage. 
     In step S2, if the cell circuit is the inverter, step S2 includes: 
     S21: calculating a high current I H , a low current I L  and a current integral k of the cell circuit by Equation (3), where in the case of the ultra-fast input transition, the high current I H  is a high current I H1  under the ultra-fast input transition, which is a source-drain current when a gate-source voltage and a source-drain voltage are both power supply voltages; in the case of the fast input transition, the high current I H  is the high current I H2  under the fast input transition, which is a source-drain current when the gate-source voltage is a power supply voltage and the source-drain voltage is a source-drain voltage at the input transition time; 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             I 
                             
                               H 
                               ⁢ 
                               1 
                             
                           
                           = 
                           
                             
                               I 
                               ds 
                             
                             ( 
                             
                               
                                 
                                   V 
                                   gs 
                                 
                                 = 
                                 
                                   V 
                                   dd 
                                 
                               
                               , 
                               
                                 
                                   V 
                                   ds 
                                 
                                 = 
                                 
                                   V 
                                   dd 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           
                             I 
                             
                               H 
                               ⁢ 
                               2 
                             
                           
                           = 
                           
                             
                               I 
                               ds 
                             
                             ( 
                             
                               
                                 
                                   V 
                                   gs 
                                 
                                 = 
                                 
                                   V 
                                   dd 
                                 
                               
                               , 
                               
                                 
                                   V 
                                   ds 
                                 
                                 = 
                                 
                                   
                                     V 
                                     ds 
                                   
                                   ( 
                                   τ 
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           
                             I 
                             L 
                           
                           = 
                           
                             
                               I 
                               ds 
                             
                             ( 
                             
                               
                                 
                                   V 
                                   gs 
                                 
                                 = 
                                 
                                   V 
                                   dd 
                                 
                               
                               , 
                               
                                 
                                   V 
                                   ds 
                                 
                                 = 
                                 
                                   
                                     V 
                                     dd 
                                   
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           k 
                           = 
                           
                             
                               ∫ 
                               0 
                               1 
                             
                             
                               
                                 I 
                                 0 
                               
                               ⁢ 
                               
                                 K 
                                 0 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     
                                       K 
                                       1 
                                     
                                     ( 
                                     
                                       
                                         
                                           
                                             V 
                                             dd 
                                           
                                           ⁢ 
                                           x 
                                         
                                         - 
                                         
                                           V 
                                           ih 
                                         
                                         + 
                                         
                                           λ 
                                           ⁢ 
                                           
                                             V 
                                             dd 
                                           
                                         
                                       
                                       
                                         n 
                                         ⁢ 
                                         
                                           ϕ 
                                           t 
                                         
                                       
                                     
                                     ) 
                                   
                                   + 
                                   
                                     
                                       
                                         K 
                                         2 
                                       
                                       ( 
                                       
                                         
                                           
                                             
                                               V 
                                               dd 
                                             
                                             ⁢ 
                                             x 
                                           
                                           - 
                                           
                                             V 
                                             ih 
                                           
                                           + 
                                           
                                             λ 
                                             ⁢ 
                                             
                                               V 
                                               dd 
                                             
                                           
                                         
                                         
                                           n 
                                           ⁢ 
                                           
                                             ϕ 
                                             t 
                                           
                                         
                                       
                                       ) 
                                     
                                     2 
                                   
                                 
                               
                               ⁢ 
                               dx 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     in Equation 3, V gs  and V ds  represent a gate-source voltage and a source-drain voltage, I ds  represents a source-drain current, V dd  represents a power supply voltage, V ds (τ) represents a source-drain voltage at the input transition time and is calculated by Equation (4), λ represents a drain-induced source barrier lowering effect coefficient, n represents a sub-threshold slope factor, Φ 1  represents a thermal voltage. I 0  represents a parameter affected by the process parameters, and K 0 , K 1  and K 2  represent fitting coefficients determined by the process; 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       ds 
                     
                     ( 
                     τ 
                     ) 
                   
                   = 
                   
                     
                       
                         
                           C 
                           M 
                         
                         
                           
                             C 
                             M 
                           
                           + 
                           
                             C 
                             L 
                           
                         
                       
                       ⁢ 
                       
                         V 
                         dd 
                       
                     
                     - 
                     
                       k 
                       
                         
                           C 
                           M 
                         
                         + 
                         
                           C 
                           L 
                         
                       
                     
                     + 
                     
                       V 
                       dd 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     in Equation (4), C M  represents a coupling capacitance of input and output, and C L  represents a load capacitance; 
     S22: expressing a natural logarithm Y of the current sum, that is, a natural logarithm of the sum of the high current and the low current, namely I H +I L , as Equation (5), and calculating a mean value μγ, a variance σγ 2  and a skewness γ 1  of the logarithm Y of the current sum by Equation (6), 
     
       
         
           
             
               
                 
                   Y 
                   = 
                   
                     - 
                     
                       ln 
                       ⁡ 
                       ( 
                       
                         
                           I 
                           H 
                         
                         + 
                         
                           I 
                           L 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             μ 
                             Y 
                           
                           = 
                           
                             
                               
                                 - 
                                 
                                   k 
                                   2 
                                 
                               
                               ⁢ 
                               
                                 E 
                                 ⁡ 
                                 ( 
                                 
                                   ΔV 
                                   th 
                                   2 
                                 
                                 ) 
                               
                             
                             + 
                             
                               mE 
                               ⁡ 
                               ( 
                               
                                 ΔV 
                                 th 
                               
                               ) 
                             
                             - 
                             
                               ln 
                               ⁡ 
                               ( 
                               
                                 
                                   I 
                                   h 
                                 
                                 + 
                                 
                                   I 
                                   L 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             σ 
                             Y 
                             2 
                           
                           = 
                           
                             
                               k 
                               2 
                               2 
                             
                             ( 
                             
                               
                                 E 
                                 ⁡ 
                                 ( 
                                 
                                   ΔV 
                                   
                                     th 
                                     ′ 
                                   
                                   4 
                                 
                                 ) 
                               
                               - 
                               
                                 
                                   E 
                                   ⁡ 
                                   ( 
                                   
                                     ΔV 
                                     
                                       th 
                                       ′ 
                                     
                                     2 
                                   
                                   ) 
                                 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           
                             γ 
                             1 
                           
                           = 
                           
                             
                               { 
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             
                                               
                                                 - 
                                                 
                                                   k 
                                                   2 
                                                   3 
                                                 
                                               
                                               ⁢ 
                                               E 
                                               ⁢ 
                                               
                                                 ( 
                                                 
                                                   ΔV 
                                                   
                                                     th 
                                                     ′ 
                                                   
                                                   6 
                                                 
                                                 ) 
                                               
                                             
                                             + 
                                             
                                               3 
                                               ⁢ 
                                               
                                                 k 
                                                 2 
                                                 2 
                                               
                                               ⁢ 
                                               E 
                                               ⁢ 
                                               
                                                 
                                                   ( 
                                                   
                                                     ΔV 
                                                     
                                                       th 
                                                       ′ 
                                                     
                                                     4 
                                                   
                                                   ) 
                                                 
                                                 [ 
                                                 
                                                   
                                                     
                                                       m 
                                                       2 
                                                     
                                                     
                                                       4 
                                                       ⁢ 
                                                       
                                                         k 
                                                         2 
                                                       
                                                     
                                                   
                                                   - 
                                                   
                                                     ln 
                                                     ⁢ 
                                                     
                                                       ( 
                                                       
                                                         
                                                           I 
                                                           H 
                                                         
                                                         + 
                                                         
                                                           I 
                                                           L 
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                 
                                                 ] 
                                               
                                             
                                             - 
                                           
                                         
                                       
                                       
                                         
                                           
                                             
                                               3 
                                               ⁢ 
                                               
                                                 k 
                                                 2 
                                               
                                               ⁢ 
                                               
                                                 
                                                   
                                                     E 
                                                     ⁡ 
                                                     ( 
                                                     
                                                       ΔV 
                                                       
                                                         th 
                                                         ′ 
                                                       
                                                       2 
                                                     
                                                     ) 
                                                   
                                                   [ 
                                                   
                                                     
                                                       
                                                         m 
                                                         2 
                                                       
                                                       
                                                         4 
                                                         ⁢ 
                                                         
                                                           k 
                                                           2 
                                                         
                                                       
                                                     
                                                     - 
                                                     
                                                       ln 
                                                       ⁡ 
                                                       ( 
                                                       
                                                         
                                                           I 
                                                           H 
                                                         
                                                         + 
                                                         
                                                           I 
                                                           L 
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                   ] 
                                                 
                                                 2 
                                               
                                             
                                             + 
                                           
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         
                                           [ 
                                           
                                             
                                               
                                                 m 
                                                 2 
                                               
                                               
                                                 4 
                                                 ⁢ 
                                                 
                                                   k 
                                                   2 
                                                 
                                               
                                             
                                             - 
                                             
                                               ln 
                                               ⁡ 
                                               ( 
                                               
                                                 
                                                   I 
                                                   H 
                                                 
                                                 + 
                                                 
                                                   I 
                                                   L 
                                                 
                                               
                                               ) 
                                             
                                           
                                           ] 
                                         
                                         3 
                                       
                                       - 
                                       
                                         3 
                                         ⁢ 
                                         
                                           μ 
                                           Y 
                                         
                                         ⁢ 
                                         
                                           σ 
                                           Y 
                                           2 
                                         
                                       
                                       - 
                                       
                                         μ 
                                         Y 
                                         3 
                                       
                                     
                                   
                                 
                               
                               } 
                             
                             / 
                             
                               σ 
                               Y 
                               3 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     where in Equation (6), m is calculated by Equation (7) under the ultra-fast input, m is calculated by Equation (8) under the fast input, k 2  represents a fitting coefficient determined by the process, ΔV th  represents a threshold voltage fluctuation quantity, ΔV th  is calculated by Equation (9), and E[·] represents a mean value of a corresponding random variable; 
     
       
         
           
             
               
                 
                   m 
                   = 
                   
                     
                       k 
                       1 
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         
                           k 
                           2 
                         
                         ( 
                         
                           
                             V 
                             dd 
                           
                           - 
                           
                             V 
                             
                               th 
                               ⁢ 
                               0 
                             
                           
                           + 
                           
                             λ 
                             ⁢ 
                             
                               V 
                               dd 
                             
                             / 
                             2 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         
                           I 
                           H 
                         
                         
                           
                             I 
                             H 
                           
                           + 
                           
                             I 
                             L 
                           
                         
                       
                       ⁢ 
                       
                         k 
                         2 
                       
                       ⁢ 
                       
                         λV 
                         dd 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     in Equation (7), k 1  represents a fitting coefficient determined by the process, and V th0  represents a nominal value of threshold voltage. 
     
       
         
           
             
               
                 
                   m 
                   = 
                   
                     
                       k 
                       1 
                     
                     + 
                     
                       2 
                       ⁢ 
                       
                         
                           k 
                           2 
                         
                         ( 
                         
                           
                             V 
                             dd 
                           
                           - 
                           
                             V 
                             
                               th 
                               ⁢ 
                               0 
                             
                           
                           + 
                           
                             
                               λV 
                               dd 
                             
                             / 
                             2 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         
                           I 
                           H 
                         
                         
                           
                             I 
                             H 
                           
                           + 
                           
                             I 
                             L 
                           
                         
                       
                       ⁢ 
                       
                         k 
                         2 
                       
                       ⁢ 
                       
                         λ 
                         [ 
                         
                           
                             2 
                             ⁢ 
                             
                               
                                 V 
                                 ds 
                               
                               ( 
                               τ 
                               ) 
                             
                           
                           - 
                           
                             V 
                             dd 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     ΔV 
                     
                       th 
                       ′ 
                     
                   
                   = 
                   
                     
                       ΔV 
                       th 
                     
                     - 
                     
                       
                         m 
                         
                           2 
                           ⁢ 
                           
                             k 
                             2 
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     In step S2, if the cell circuit is the stacked structure cell, S2 includes: 
     S21: calculating a high current I H1  under the ultra-fast input and a high current I H2  under the fast input, a low current I L  and a current integral k by Equation (10). 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             I 
                             
                               H 
                               ⁢ 
                               1 
                             
                           
                           = 
                           
                             
                               I 
                               ds 
                             
                             ( 
                             
                               
                                 
                                   V 
                                   g 
                                 
                                 = 
                                 
                                   V 
                                   dd 
                                 
                               
                               , 
                               
                                 
                                   V 
                                   d 
                                 
                                 = 
                                 
                                   V 
                                   dd 
                                 
                               
                               , 
                               
                                 
                                   V 
                                   s 
                                 
                                 = 
                                 
                                   V 
                                   
                                     XH 
                                     ⁢ 
                                     1 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           
                             I 
                             
                               H 
                               ⁢ 
                               2 
                             
                           
                           = 
                           
                             
                               I 
                               ds 
                             
                             ( 
                             
                               
                                 
                                   V 
                                   g 
                                 
                                 = 
                                 
                                   V 
                                   dd 
                                 
                               
                               , 
                               
                                 
                                   V 
                                   d 
                                 
                                 = 
                                 
                                   
                                     V 
                                     d 
                                   
                                   ( 
                                   τ 
                                   ) 
                                 
                               
                               , 
                               
                                 
                                   V 
                                   s 
                                 
                                 = 
                                 
                                   V 
                                   
                                     XH 
                                     ⁢ 
                                     2 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           
                             I 
                             L 
                           
                           = 
                           
                             
                               I 
                               ds 
                             
                             ( 
                             
                               
                                 
                                   V 
                                   g 
                                 
                                 = 
                                 
                                   V 
                                   dd 
                                 
                               
                               , 
                               
                                 
                                   V 
                                   d 
                                 
                                 = 
                                 
                                   
                                     V 
                                     dd 
                                   
                                   2 
                                 
                               
                               , 
                               
                                 
                                   V 
                                   s 
                                 
                                 = 
                                 
                                   V 
                                   XL 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 k 
                                 = 
                                 
                                   
                                     
                                       ∫ 
                                       0 
                                       0.5 
                                     
                                     
                                       
                                         I 
                                         0 
                                       
                                       ⁢ 
                                       
                                         K 
                                         0 
                                       
                                       ⁢ 
                                       
                                         e 
                                         
                                           
                                             
                                               k 
                                               1 
                                             
                                             ( 
                                             
                                               
                                                 
                                                   V 
                                                   dd 
                                                 
                                                 ⁢ 
                                                 x 
                                               
                                               - 
                                               
                                                 V 
                                                 th 
                                               
                                               + 
                                               
                                                 λ 
                                                 ⁢ 
                                                 
                                                   V 
                                                   dd 
                                                 
                                               
                                             
                                             ) 
                                           
                                           + 
                                           
                                             
                                               
                                                 k 
                                                 2 
                                               
                                               ( 
                                               
                                                 
                                                   
                                                     V 
                                                     dd 
                                                   
                                                   ⁢ 
                                                   x 
                                                 
                                                 - 
                                                 
                                                   V 
                                                   th 
                                                 
                                                 + 
                                                 
                                                   λ 
                                                   ⁢ 
                                                   
                                                     V 
                                                     dd 
                                                   
                                                 
                                               
                                               ) 
                                             
                                             2 
                                           
                                         
                                       
                                       ⁢ 
                                       dx 
                                     
                                   
                                   + 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ∫ 
                                   0.5 
                                   1 
                                 
                                 
                                   
                                     I 
                                     0 
                                   
                                   ⁢ 
                                   
                                     K 
                                     0 
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       
                                         
                                           k 
                                           1 
                                         
                                         ( 
                                         
                                           
                                             
                                               V 
                                               dd 
                                             
                                             ⁢ 
                                             x 
                                           
                                           - 
                                           
                                             V 
                                             th 
                                           
                                           - 
                                           
                                             2 
                                             ⁢ 
                                             
                                               
                                                 V 
                                                 XH 
                                               
                                               ( 
                                               
                                                 x 
                                                 - 
                                                 
                                                   1 
                                                   2 
                                                 
                                               
                                               ) 
                                             
                                           
                                           + 
                                           
                                             λ 
                                             ⁢ 
                                             
                                               V 
                                               dd 
                                             
                                           
                                         
                                         ) 
                                       
                                       + 
                                       
                                         
                                           
                                             k 
                                             2 
                                           
                                           ( 
                                           
                                             
                                               
                                                 V 
                                                 dd 
                                               
                                               ⁢ 
                                               x 
                                             
                                             - 
                                             
                                               V 
                                               th 
                                             
                                             - 
                                             
                                               2 
                                               ⁢ 
                                               
                                                 
                                                   V 
                                                   XH 
                                                 
                                                 ( 
                                                 
                                                   x 
                                                   - 
                                                   
                                                     1 
                                                     2 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                             + 
                                             
                                               λ 
                                               ⁢ 
                                               
                                                 V 
                                                 dd 
                                               
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                     
                                   
                                   ⁢ 
                                   dx 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     where in Equation (12) V g  and V d  represent a gate voltage and a drain voltage, V s  represents a source voltage, V XH  and V XL  represent intermediate node voltages at different gate voltages and drain voltages, in the case of the ultra-fast input, V XH  is V XH1 , which is an intermediate node voltage when the gate voltage and the drain voltage are both power supply voltages, in the case of the fast input, V XH  is V XH2 , which is an intermediate node voltage when the gate voltage is a power supply voltage and the drain voltage is an intermediate node voltage of the drain voltage at the input transition time and is calculated by Equation (11), and V d (τ) represents a drain voltage at the input transition time; 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   V 
                                   
                                     XH 
                                     ⁢ 
                                     1 
                                   
                                 
                                 = 
                                 
                                   
                                     
                                       
                                         V 
                                         thD 
                                       
                                       - 
                                       
                                         V 
                                         thU 
                                       
                                       + 
                                       
                                         λ 
                                         ⁢ 
                                         
                                           V 
                                           dd 
                                         
                                       
                                     
                                     
                                       ( 
                                       
                                         1 
                                         + 
                                         
                                           2 
                                           ⁢ 
                                           λ 
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     ln 
                                     ⁡ 
                                     ( 
                                     
                                       
                                         W 
                                         U 
                                       
                                       
                                         W 
                                         D 
                                       
                                     
                                     ) 
                                   
                                   - 
                                   
                                     m 
                                     2 
                                   
                                 
                                 
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       
                                         2 
                                         ⁢ 
                                         λ 
                                       
                                     
                                     ) 
                                   
                                   [ 
                                   
                                     
                                       k 
                                       1 
                                     
                                     + 
                                     
                                       
                                         k 
                                         2 
                                       
                                       ( 
                                       
                                         
                                           2 
                                           ⁢ 
                                           
                                             V 
                                             dd 
                                           
                                         
                                         - 
                                         
                                           V 
                                           thU 
                                         
                                         - 
                                         
                                           V 
                                           thD 
                                         
                                         + 
                                         
                                           λ 
                                           ⁢ 
                                           
                                             V 
                                             dd 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 
                                   V 
                                   
                                     XH 
                                     ⁢ 
                                     2 
                                   
                                 
                                 = 
                                 
                                   
                                     
                                       
                                         V 
                                         thD 
                                       
                                       - 
                                       
                                         V 
                                         thU 
                                       
                                       + 
                                       
                                         λ 
                                         ⁢ 
                                         
                                           
                                             V 
                                             d 
                                           
                                           ( 
                                           τ 
                                           ) 
                                         
                                       
                                     
                                     
                                       ( 
                                       
                                         1 
                                         + 
                                         
                                           2 
                                           ⁢ 
                                           λ 
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     ln 
                                     ⁡ 
                                     ( 
                                     
                                       
                                         W 
                                         U 
                                       
                                       
                                         W 
                                         D 
                                       
                                     
                                     ) 
                                   
                                   - 
                                   
                                     m 
                                     2 
                                   
                                 
                                 
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       
                                         2 
                                         ⁢ 
                                         λ 
                                       
                                     
                                     ) 
                                   
                                   [ 
                                   
                                     
                                       k 
                                       1 
                                     
                                     + 
                                     
                                       
                                         k 
                                         2 
                                       
                                       ( 
                                       
                                         
                                           2 
                                           ⁢ 
                                           
                                             V 
                                             dd 
                                           
                                         
                                         - 
                                         
                                           V 
                                           thU 
                                         
                                         - 
                                         
                                           V 
                                           thD 
                                         
                                         + 
                                         
                                           λ 
                                           ⁢ 
                                           
                                             
                                               V 
                                               d 
                                             
                                             ( 
                                             τ 
                                             ) 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 
                                   V 
                                   XL 
                                 
                                 = 
                                 
                                   
                                     
                                       
                                         V 
                                         thD 
                                       
                                       - 
                                       
                                         V 
                                         thU 
                                       
                                       + 
                                       
                                         λ 
                                         ⁢ 
                                         
                                           V 
                                           dd 
                                         
                                         / 
                                         2 
                                       
                                     
                                     
                                       ( 
                                       
                                         1 
                                         + 
                                         
                                           2 
                                           ⁢ 
                                           λ 
                                         
                                       
                                       ) 
                                     
                                   
                                   + 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     ln 
                                     ⁡ 
                                     ( 
                                     
                                       
                                         W 
                                         U 
                                       
                                       
                                         W 
                                         D 
                                       
                                     
                                     ) 
                                   
                                   - 
                                   
                                     m 
                                     2 
                                   
                                 
                                 
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       
                                         2 
                                         ⁢ 
                                         λ 
                                       
                                     
                                     ) 
                                   
                                   [ 
                                   
                                     
                                       k 
                                       1 
                                     
                                     + 
                                     
                                       
                                         k 
                                         2 
                                       
                                       ( 
                                       
                                         
                                           2 
                                           ⁢ 
                                           
                                             V 
                                             dd 
                                           
                                         
                                         - 
                                         
                                           V 
                                           thU 
                                         
                                         - 
                                         
                                           V 
                                           thD 
                                         
                                         + 
                                         
                                           λ 
                                           ⁢ 
                                           
                                             V 
                                             dd 
                                           
                                           / 
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     in Equation (11), V thU , and V thD  respectively represent an upper transistor threshold voltage and a lower transistor threshold voltage in the stacked structure cell. W U  and W D  respectively represent an upper transistor width and a lower transistor width in the stacked structure cell, and m 2  represents a fitting coefficient; 
     S22: calculating a mean value E(Y), a variance D(Y) and a skewness S k (Y) of the logarithm Y of the current sum by Equation (12), 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             E 
                             ⁡ 
                             ( 
                             Y 
                             ) 
                           
                           = 
                           
                             
                               E 
                               ⁡ 
                               ( 
                               
                                 Y 
                                 1 
                               
                               ) 
                             
                             + 
                             
                               E 
                               ⁡ 
                               ( 
                               
                                 Y 
                                 2 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             D 
                             ⁡ 
                             ( 
                             Y 
                             ) 
                           
                           = 
                           
                             
                               D 
                               ⁡ 
                               ( 
                               
                                 Y 
                                 1 
                               
                               ) 
                             
                             + 
                             
                               D 
                               ⁡ 
                               ( 
                               
                                 Y 
                                 2 
                               
                               ) 
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                 Cov 
                                 ( 
                                 
                                   
                                     Y 
                                     1 
                                   
                                   , 
                                   
                                     Y 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               S 
                               k 
                             
                             ( 
                             Y 
                             ) 
                           
                           = 
                           
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           S 
                                           k 
                                         
                                         ( 
                                         
                                           Y 
                                           1 
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         
                                           D 
                                           ⁡ 
                                           ( 
                                           
                                             Y 
                                             1 
                                           
                                           ) 
                                         
                                         
                                           3 
                                           / 
                                           2 
                                         
                                       
                                     
                                     + 
                                     
                                       
                                         
                                           S 
                                           k 
                                         
                                         ( 
                                         
                                           Y 
                                           2 
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         
                                           D 
                                           ⁡ 
                                           ( 
                                           
                                             Y 
                                             2 
                                           
                                           ) 
                                         
                                         
                                           3 
                                           / 
                                           2 
                                         
                                       
                                     
                                     + 
                                     
                                       3 
                                       ⁢ 
                                       
                                         Cov 
                                         ( 
                                         
                                           
                                             Y 
                                             1 
                                           
                                           , 
                                           
                                             Y 
                                             2 
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       3 
                                       ⁢ 
                                       
                                         Cov 
                                         ( 
                                         
                                           
                                             Y 
                                             1 
                                             2 
                                           
                                           , 
                                           
                                             Y 
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       6 
                                       ⁢ 
                                       
                                         E 
                                         ⁡ 
                                         ( 
                                         Y 
                                         ) 
                                       
                                       ⁢ 
                                       
                                         Cov 
                                         ( 
                                         
                                           
                                             Y 
                                             1 
                                           
                                           , 
                                           
                                             Y 
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 D 
                                 ⁡ 
                                 ( 
                                 Y 
                                 ) 
                               
                               
                                 3 
                                 / 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     where in Equation (12), D[·] represents a variance of a corresponding random variable, Sk[·] represents a skewness of the corresponding random variable, Cov[·] represents a covariance of the corresponding random variable, and Y 1  and Y 2  can be calculated by Equation (13); 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             Y 
                             1 
                           
                           = 
                           
                             
                               ( 
                               
                                 
                                   
                                     - 
                                     
                                       k 
                                       2 
                                     
                                   
                                   ⁢ 
                                   Δ 
                                   ⁢ 
                                   
                                     V 
                                     thH 
                                     2 
                                   
                                 
                                 + 
                                 
                                   Δ 
                                   ⁢ 
                                   
                                     V 
                                     thH 
                                   
                                   ⁢ 
                                   m 
                                 
                                 - 
                                 
                                   ln 
                                   ⁡ 
                                   ( 
                                   
                                     
                                       I 
                                       H 
                                     
                                     + 
                                     
                                       I 
                                       L 
                                     
                                   
                                   ) 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               
                                 I 
                                 H 
                               
                               
                                 
                                   I 
                                   H 
                                 
                                 + 
                                 
                                   I 
                                   L 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             Y 
                             2 
                           
                           = 
                           
                             
                               ( 
                               
                                 
                                   
                                     - 
                                     
                                       k 
                                       2 
                                     
                                   
                                   ⁢ 
                                   
                                     ΔV 
                                     thL 
                                     2 
                                   
                                 
                                 + 
                                 
                                   Δ 
                                   ⁢ 
                                   
                                     V 
                                     thL 
                                   
                                   ⁢ 
                                   m 
                                 
                                 - 
                                 
                                   ln 
                                   ⁡ 
                                   ( 
                                   
                                     
                                       I 
                                       H 
                                     
                                     + 
                                     
                                       I 
                                       L 
                                     
                                   
                                   ) 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               
                                 I 
                                 L 
                               
                               
                                 
                                   I 
                                   H 
                                 
                                 + 
                                 
                                   I 
                                   L 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     in Equation (13), ΔV thH  and ΔV thL  represent linear functions of an upper transistor threshold voltage fluctuation quantity ΔV thU  and a lower transistor threshold voltage fluctuation quantity ΔV thD , expressed as Equation (14), and α 1 , α 2 , α 3  and α 4  in Equation (14) represent coefficients; 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             ΔV 
                             thH 
                           
                           = 
                           
                             
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     α 
                                     1 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 ΔV 
                                 thU 
                               
                             
                             + 
                             
                               
                                 α 
                                 2 
                               
                               ⁢ 
                               
                                 ΔV 
                                 thD 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             ΔV 
                             thL 
                           
                           = 
                           
                             
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     α 
                                     3 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 ΔV 
                                 thU 
                               
                             
                             + 
                             
                               
                                 α 
                                 4 
                               
                               ⁢ 
                               
                                 ΔV 
                                 thD 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     S23: calculating a mean value and a variance of an equivalent threshold voltage ΔV th_eff  by Equation (15). 
     
       
         
           
             
               
                 
                   
                     ΔV 
                     
                       th 
                       ⁢ 
                       _ 
                       ⁢ 
                       eff 
                     
                   
                   = 
                   
                     
                       
                         
                           I 
                           H 
                         
                         
                           
                             I 
                             H 
                           
                           + 
                           
                             I 
                             L 
                           
                         
                       
                       ⁢ 
                       
                         ΔV 
                         thH 
                       
                     
                     + 
                     
                       
                         
                           I 
                           L 
                         
                         
                           
                             I 
                             H 
                           
                           + 
                           
                             I 
                             L 
                           
                         
                       
                       ⁢ 
                       
                         
                           ΔV 
                           thL 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     In step S2, if the cell circuit is the parallel structure cell. S2 includes: 
     S21: calculating a high current I H1  under the ultra-fast input and a high current I H2  under the fast input, a low current I L  and a current integral k by Equation (16), 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             I 
                             
                               H 
                               ⁢ 
                               1 
                             
                           
                           = 
                           
                             2 
                             ⁢ 
                             
                               
                                 I 
                                 ds 
                               
                               ( 
                               
                                 
                                   
                                     V 
                                     gs 
                                   
                                   = 
                                   
                                     V 
                                     dd 
                                   
                                 
                                 , 
                                 
                                   
                                     V 
                                     ds 
                                   
                                   = 
                                   
                                     V 
                                     dd 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             I 
                             
                               H 
                               ⁢ 
                               2 
                             
                           
                           = 
                           
                             2 
                             ⁢ 
                             
                               
                                 I 
                                 ds 
                               
                               ( 
                               
                                 
                                   
                                     V 
                                     gs 
                                   
                                   = 
                                   
                                     V 
                                     dd 
                                   
                                 
                                 , 
                                 
                                   
                                     V 
                                     ds 
                                   
                                   = 
                                   
                                     
                                       V 
                                       ds 
                                     
                                     ( 
                                     τ 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             I 
                             L 
                           
                           = 
                           
                             2 
                             ⁢ 
                             
                               
                                 I 
                                 ds 
                               
                               ( 
                               
                                 
                                   
                                     V 
                                     gs 
                                   
                                   = 
                                   
                                     V 
                                     dd 
                                   
                                 
                                 , 
                                 
                                   
                                     V 
                                     ds 
                                   
                                   = 
                                   
                                     
                                       V 
                                       dd 
                                     
                                     2 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 k 
                                 = 
                                 
                                   
                                     
                                       ∫ 
                                       0 
                                       1 
                                     
                                     
                                       
                                         I 
                                         0 
                                       
                                       ⁢ 
                                       
                                         K 
                                         0 
                                       
                                       ⁢ 
                                       
                                         e 
                                         
                                           
                                             
                                               k 
                                               1 
                                             
                                             ( 
                                             
                                               
                                                 
                                                   V 
                                                   dd 
                                                 
                                                 ⁢ 
                                                 x 
                                               
                                               - 
                                               
                                                 V 
                                                 
                                                   th 
                                                   ⁢ 
                                                   1 
                                                 
                                               
                                               + 
                                               
                                                 λ 
                                                 ⁢ 
                                                 
                                                   V 
                                                   dd 
                                                 
                                               
                                             
                                             ) 
                                           
                                           + 
                                           
                                             
                                               
                                                 k 
                                                 2 
                                               
                                               ( 
                                               
                                                 
                                                   
                                                     V 
                                                     dd 
                                                   
                                                   ⁢ 
                                                   x 
                                                 
                                                 - 
                                                 
                                                   V 
                                                   
                                                     th 
                                                     ⁢ 
                                                     1 
                                                   
                                                 
                                                 + 
                                                 
                                                   λ 
                                                   ⁢ 
                                                   
                                                     V 
                                                     dd 
                                                   
                                                 
                                               
                                               ) 
                                             
                                             2 
                                           
                                         
                                       
                                       ⁢ 
                                       dx 
                                     
                                   
                                   + 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ∫ 
                                   0 
                                   1 
                                 
                                 
                                   
                                     I 
                                     0 
                                   
                                   ⁢ 
                                   
                                     K 
                                     0 
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       
                                         
                                           k 
                                           1 
                                         
                                         ( 
                                         
                                           
                                             
                                               V 
                                               dd 
                                             
                                             ⁢ 
                                             x 
                                           
                                           - 
                                           
                                             V 
                                             
                                               th 
                                               ⁢ 
                                               2 
                                             
                                           
                                           + 
                                           
                                             λ 
                                             ⁢ 
                                             
                                               V 
                                               dd 
                                             
                                           
                                         
                                         ) 
                                       
                                       + 
                                       
                                         
                                           
                                             k 
                                             2 
                                           
                                           ( 
                                           
                                             
                                               
                                                 V 
                                                 dd 
                                               
                                               ⁢ 
                                               x 
                                             
                                             - 
                                             
                                               V 
                                               
                                                 th 
                                                 ⁢ 
                                                 2 
                                               
                                             
                                             + 
                                             
                                               λ 
                                               ⁢ 
                                               
                                                 V 
                                                 dd 
                                               
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                     
                                   
                                   ⁢ 
                                   dx 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     where in Equation (16), V th1  and V th2  respectively represent threshold voltages of a left transistor and a right transistor in the parallel structure; 
     S22: calculating a mean value E(Y), a variance D(Y) and a skewness S k (Y) of the logarithm Y of the current sum by Equation (17), 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             E 
                             ⁡ 
                             ( 
                             Y 
                             ) 
                           
                           = 
                           
                             
                               E 
                               ⁡ 
                               ( 
                               
                                 Y 
                                 1 
                               
                               ) 
                             
                             + 
                             
                               E 
                               ⁡ 
                               ( 
                               
                                 Y 
                                 2 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             D 
                             ⁡ 
                             ( 
                             Y 
                             ) 
                           
                           = 
                           
                             
                               D 
                               ⁡ 
                               ( 
                               
                                 Y 
                                 1 
                               
                               ) 
                             
                             + 
                             
                               D 
                               ⁡ 
                               ( 
                               
                                 Y 
                                 2 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               S 
                               k 
                             
                             ( 
                             Y 
                             ) 
                           
                           = 
                           
                             
                               
                                 
                                   
                                     
                                       
                                         S 
                                         k 
                                       
                                       ( 
                                       
                                         Y 
                                         1 
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       
                                         D 
                                         ⁡ 
                                         ( 
                                         
                                           Y 
                                           1 
                                         
                                         ) 
                                       
                                       
                                         3 
                                         / 
                                         2 
                                       
                                     
                                   
                                   + 
                                   
                                     
                                       
                                         S 
                                         k 
                                       
                                       ( 
                                       
                                         Y 
                                         2 
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       
                                         D 
                                         ⁡ 
                                         ( 
                                         Y 
                                         ) 
                                       
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                               
                                 3 
                                 / 
                                 2 
                               
                             
                             
                               
                                 D 
                                 ⁡ 
                                 ( 
                                 Y 
                                 ) 
                               
                               
                                 3 
                                 / 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     where in Equation (17), Y 1  and Y 2  can be calculated by Equation (18); 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             Y 
                             1 
                           
                           = 
                           
                             
                               1 
                               2 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     - 
                                     
                                       k 
                                       2 
                                     
                                   
                                   ⁢ 
                                   
                                     ΔV 
                                     
                                       th 
                                       ⁢ 
                                       1 
                                     
                                     2 
                                   
                                 
                                 + 
                                 
                                   
                                     ΔV 
                                     
                                       th 
                                       ⁢ 
                                       1 
                                     
                                   
                                   ⁢ 
                                   m 
                                 
                                 - 
                                 
                                   ln 
                                   ⁡ 
                                   ( 
                                   
                                     
                                       I 
                                       H 
                                     
                                     + 
                                     
                                       I 
                                       L 
                                     
                                   
                                   ) 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             Y 
                             2 
                           
                           = 
                           
                             
                               1 
                               2 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     - 
                                     
                                       k 
                                       2 
                                     
                                   
                                   ⁢ 
                                   
                                     ΔV 
                                     
                                       th 
                                       ⁢ 
                                       2 
                                     
                                     2 
                                   
                                 
                                 + 
                                 
                                   
                                     ΔV 
                                     
                                       th 
                                       ⁢ 
                                       2 
                                     
                                   
                                   ⁢ 
                                   m 
                                 
                                 - 
                                 
                                   ln 
                                   ⁡ 
                                   ( 
                                   
                                     
                                       I 
                                       H 
                                     
                                     + 
                                     
                                       I 
                                       L 
                                     
                                   
                                   ) 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     in Equation (18). ΔV th1  and ΔV th2  represent threshold voltage fluctuation quantities of the left transistor and the right transistor in the parallel structure; 
     S23: calculating a mean value and a variance of an equivalent threshold voltage ΔV th_eff  by Equation (19). 
     
       
         
           
             
               
                 
                   
                     ΔV 
                     
                       th 
                       ⁢ 
                       _ 
                       ⁢ 
                       eff 
                     
                   
                   = 
                   
                     
                       
                         ΔV 
                         
                           th 
                           ⁢ 
                           1 
                         
                       
                       + 
                       
                         ΔV 
                         
                           th 
                           ⁢ 
                           2 
                         
                       
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     The step S3 to determine the delay type includes calculating an overshoot time t ov  and a delay td according to the cell circuit type, comparing the magnitude relationship among the input transition time, the overshoot time and the delay, and judging whether the input is ultra-fast input, fast input or slow input, and specifically includes: 
     S31: substituting the high current I H1  under the ultra-fast input and the high current I H2  under the fast input, the low current I L  and the current integral k obtained in step S2 into Equation (20) to calculate the overshoot time t ov , and substituting the variables into Equation 21 to calculate the delay td; 
     
       
         
           
             
               
                 
                   
                     t 
                     ov 
                   
                   = 
                   
                     
                       
                         
                           2 
                           ⁢ 
                           
                             C 
                             M 
                           
                         
                         
                           
                             C 
                             M 
                           
                           + 
                           
                             C 
                             L 
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             C 
                             L 
                           
                           ⁢ 
                           
                             V 
                             dd 
                           
                         
                         
                           
                             I 
                             
                               H 
                               ⁢ 
                               1 
                             
                           
                           + 
                           
                             I 
                             L 
                           
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   C 
                                   L 
                                 
                               
                               
                                 
                                   C 
                                   M 
                                 
                                 + 
                                 
                                   C 
                                   L 
                                 
                               
                             
                             ⁢ 
                             
                               k 
                               
                                 
                                   I 
                                   
                                     H 
                                     ⁢ 
                                     1 
                                   
                                 
                                 + 
                                 
                                   I 
                                   L 
                                 
                               
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       τ 
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   td 
                   = 
                   
                     
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               2 
                               ⁢ 
                               
                                 C 
                                 M 
                               
                             
                             
                               
                                 C 
                                 M 
                               
                               + 
                               
                                 C 
                                 L 
                               
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           
                             C 
                             L 
                           
                           ⁢ 
                           
                             V 
                             dd 
                           
                         
                         
                           
                             I 
                             
                               H 
                               ⁢ 
                               2 
                             
                           
                           + 
                           
                             I 
                             L 
                           
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             1 
                             2 
                           
                           - 
                           
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   C 
                                   L 
                                 
                               
                               
                                 
                                   C 
                                   M 
                                 
                                 + 
                                 
                                   C 
                                   L 
                                 
                               
                             
                             ⁢ 
                             
                               k 
                               
                                 
                                   I 
                                   
                                     H 
                                     ⁢ 
                                     2 
                                   
                                 
                                 + 
                                 
                                   I 
                                   L 
                                 
                               
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       τ 
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     S32: comparing the overshoot time t ov  with the input transition timer, if t ov &gt;τ, determining the input as the ultra-fast input, otherwise, determining the input as the fast input or the slow input; 
     S33: comparing the delay td with half of the input transition time τ/2, if td&gt;τ/2, determining the input as the fast input, otherwise, determining the input as the slow input. 
     In step S4, establishing the cell circuit nominal delay model includes obtaining a nominal delay according to the cell circuit type and the delay type, where if the delay type is the ultra-fast input, the nominal delay is calculated by Equation (22); if the delay type is the fast input, the nominal delay is calculated by Equation (21); and if the delay type is the slow input, the nominal delay is calculated by Equation (1). 
     
       
         
           
             
               
                 
                   td 
                   = 
                   
                     
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               2 
                               ⁢ 
                               
                                 C 
                                 M 
                               
                             
                             
                               
                                 C 
                                 M 
                               
                               + 
                               
                                 C 
                                 L 
                               
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           
                             C 
                             L 
                           
                           ⁢ 
                           
                             V 
                             dd 
                           
                         
                         
                           
                             I 
                             
                               H 
                               ⁢ 
                               1 
                             
                           
                           + 
                           
                             I 
                             L 
                           
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             1 
                             2 
                           
                           - 
                           
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   C 
                                   L 
                                 
                               
                               
                                 
                                   C 
                                   M 
                                 
                                 + 
                                 
                                   C 
                                   L 
                                 
                               
                             
                             ⁢ 
                             
                               k 
                               
                                 
                                   I 
                                   
                                     H 
                                     ⁢ 
                                     1 
                                   
                                 
                                 + 
                                 
                                   I 
                                   L 
                                 
                               
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         τ 
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     In S5, establishing the cell circuit statistical delay model includes obtaining delay statistics: a variance, a maximum value and a minimum value based on the nominal delay model in step S4 according to the cell circuit type and the delay type, where if the delay type is the ultra-fast input, step S5 includes: 
     S51: based on the properties of logarithm skewed normal distribution, substituting the mean value μγ, the variance σγ 2  and the skewness γ 1  of the logarithm of the current sum obtained in step S2 into Equation (23) to calculate distribution parameters: a location parameter ξγ, a scale parameter ωγ and a second shape parameter βγ 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             ε 
                             Y 
                           
                           = 
                           
                             
                               μ 
                               Y 
                             
                             - 
                             
                               
                                 ω 
                                 Y 
                               
                               ⁢ 
                               
                                 β 
                                 Y 
                               
                               ⁢ 
                               
                                 
                                   2 
                                   π 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             ω 
                             Y 
                           
                           = 
                           
                             
                               
                                 σ 
                                 Y 
                                 2 
                               
                               
                                 1 
                                 - 
                                 
                                   
                                     2 
                                     π 
                                   
                                   ⁢ 
                                   
                                     β 
                                     Y 
                                     2 
                                   
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             β 
                             Y 
                           
                           = 
                           
                             
                               
                                 π 
                                 2 
                               
                               ⁢ 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         2 
                                         ⁢ 
                                         
                                           γ 
                                           1 
                                         
                                       
                                       
                                         4 
                                         - 
                                         π 
                                       
                                     
                                     ) 
                                   
                                   
                                     2 
                                     3 
                                   
                                 
                                 
                                   1 
                                   + 
                                   
                                     
                                       ( 
                                       
                                         
                                           2 
                                           ⁢ 
                                           
                                             γ 
                                             1 
                                           
                                         
                                         
                                           4 
                                           - 
                                           π 
                                         
                                       
                                       ) 
                                     
                                     
                                       2 
                                       3 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     S52: based on the distribution parameters, calculating a variance D(td) of the delay by Equation (24), and calculating a maximum value td max  and a minimum value td min  of the delay by Equation (25), 
     
       
         
           
             
               
                 
                   
                     D 
                     ⁡ 
                     ( 
                     td 
                     ) 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                     C 
                                     M 
                                   
                                 
                                 
                                   
                                     C 
                                     M 
                                   
                                   + 
                                   
                                     C 
                                     L 
                                   
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             C 
                             L 
                           
                           ⁢ 
                           
                             V 
                             dd 
                           
                         
                         ] 
                       
                       2 
                     
                     × 
                     2 
                     ⁢ 
                     
                       e 
                       
                         2 
                         ⁢ 
                         
                           ε 
                           Y 
                         
                       
                     
                     ⁢ 
                     
                       
                         e 
                         
                           ω 
                           Y 
                           2 
                         
                       
                       ( 
                       
                         
                           
                             e 
                             
                               ω 
                               Y 
                               2 
                             
                           
                           ( 
                           
                             2 
                             ⁢ 
                             
                               β 
                               Y 
                             
                             ⁢ 
                             
                               ω 
                               Y 
                             
                           
                           ) 
                         
                         - 
                         
                           2 
                           ⁢ 
                           
                             
                               ϕ 
                               2 
                             
                             ( 
                             
                               
                                 β 
                                 Y 
                               
                               ⁢ 
                               
                                 ω 
                                 Y 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             td 
                             max 
                           
                           = 
                           
                             
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     
                                       2 
                                       ⁢ 
                                       
                                         C 
                                         M 
                                       
                                     
                                     
                                       
                                         C 
                                         M 
                                       
                                       + 
                                       
                                         C 
                                         L 
                                       
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 C 
                                 L 
                               
                               ⁢ 
                               
                                 V 
                                 dd 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     ε 
                                     Y 
                                   
                                   + 
                                   
                                     3.21 
                                     
                                       ω 
                                       Y 
                                     
                                   
                                 
                               
                             
                             + 
                             
                               M 
                               ⁢ 
                               τ 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             td 
                             min 
                           
                           = 
                           
                             
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     
                                       2 
                                       ⁢ 
                                       
                                         C 
                                         M 
                                       
                                     
                                     
                                       
                                         C 
                                         M 
                                       
                                       + 
                                       
                                         C 
                                         L 
                                       
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 C 
                                 L 
                               
                               ⁢ 
                               
                                 V 
                                 dd 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     ε 
                                     Y 
                                   
                                   - 
                                   
                                     1.79 
                                     
                                       ω 
                                       Y 
                                     
                                   
                                 
                               
                             
                             + 
                             
                               M 
                               ⁢ 
                               τ 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     wherein Equation (24) and Equation (25), φ(·) represents a cumulative density distribution function of standard normal distribution. 
     In step S2, if the delay type is the fast input, S5 includes: 
     S51: based on the properties of logarithm skewed normal distribution, substituting the mean value μγ, the variance σγ 2  and the skewness γ 1  of the logarithm of the current sum obtained in step S2 into Equation (23) to calculate distribution parameters: a location parameter ξγ, a scale parameter ωγ and a second shape parameter βγ; 
     S52: based on the distribution parameters, calculating a variance D(td) of the delay by Equation (24), and calculating a maximum value td max  and a minimum value td min  of the delay by Equation (25). 
     In step S2, if the delay type is the slow input, S5 includes: 
     S51: based on the mean value and the variance of the equivalent threshold voltage obtained in step S2, calculating a mean value E(td), a variance D(td) and a skewness S k (td) of the delay by Equation (26), 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             E 
                             ⁡ 
                             ( 
                             td 
                             ) 
                           
                           = 
                           
                             
                               
                                 p 
                                 2 
                               
                               ⁢ 
                               
                                 E 
                                 ⁡ 
                                 ( 
                                 
                                   ΔV 
                                   th 
                                   2 
                                 
                                 ) 
                               
                             
                             + 
                             
                               p 
                               0 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 
                                   D 
                                   ⁢ 
                                   
                                     ( 
                                     td 
                                     ) 
                                   
                                 
                                 = 
                                 
                                   
                                     
                                       p 
                                       3 
                                       2 
                                     
                                     ⁢ 
                                     E 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         ΔV 
                                         th 
                                         6 
                                       
                                       ) 
                                     
                                   
                                   + 
                                   
                                     
                                       ( 
                                       
                                         
                                           2 
                                           ⁢ 
                                           
                                             p 
                                             3 
                                           
                                           ⁢ 
                                           
                                             p 
                                             1 
                                           
                                         
                                         + 
                                         
                                           p 
                                           2 
                                           2 
                                         
                                       
                                       ) 
                                     
                                     ⁢ 
                                     E 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         ΔV 
                                         th 
                                         4 
                                       
                                       ) 
                                     
                                   
                                   + 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         2 
                                         ⁢ 
                                         
                                           p 
                                           2 
                                         
                                         ⁢ 
                                         
                                           p 
                                           0 
                                         
                                       
                                       + 
                                       
                                         p 
                                         1 
                                         2 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   E 
                                   ⁢ 
                                   
                                     ( 
                                     
                                       ΔV 
                                       th 
                                       2 
                                     
                                     ) 
                                   
                                 
                                 + 
                                 
                                   p 
                                   0 
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               S 
                               k 
                             
                             ( 
                             td 
                             ) 
                           
                           = 
                           
                             
                               
                                 
                                   
                                     
                                       
                                         q 
                                         8 
                                       
                                       ( 
                                       
                                         ΔV 
                                         th 
                                         8 
                                       
                                       ) 
                                     
                                     + 
                                     
                                       
                                         q 
                                         6 
                                       
                                       ⁢ 
                                       
                                         E 
                                         ⁡ 
                                         ( 
                                         
                                           ΔV 
                                           th 
                                           6 
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       
                                         q 
                                         4 
                                       
                                       ⁢ 
                                       
                                         E 
                                         ⁡ 
                                         ( 
                                         
                                           ΔV 
                                           th 
                                           4 
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       
                                         q 
                                         2 
                                       
                                       ⁢ 
                                       
                                         E 
                                         ⁡ 
                                         ( 
                                         
                                           ΔV 
                                           th 
                                           2 
                                         
                                         ) 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       q 
                                       0 
                                     
                                     - 
                                     
                                       3 
                                       ⁢ 
                                       
                                         E 
                                         ⁡ 
                                         ( 
                                         td 
                                         ) 
                                       
                                       ⁢ 
                                       
                                         D 
                                         ⁡ 
                                         ( 
                                         td 
                                         ) 
                                       
                                     
                                     - 
                                     
                                       
                                         E 
                                         ⁡ 
                                         ( 
                                         td 
                                         ) 
                                       
                                       3 
                                     
                                   
                                 
                               
                             
                             
                               
                                 D 
                                 ⁡ 
                                 ( 
                                 td 
                                 ) 
                               
                               
                                 3 
                                 / 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
     where in Equation (26), p i  and q i  represent coefficients, which are functions of threshold voltages and currents; 
     S52: based on the properties of skewed normal distribution, calculating distribution parameters: a location parameter ξ td , a scale parameter ω td  and a second shape parameter β td  of the delay by Equation (23); 
     S53: based on the distribution parameters, calculating a maximum value td max  and a minimum value td min  of the delay by Equation (27). 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               td 
                               max 
                             
                             = 
                             
                               
                                 ε 
                                 td 
                               
                               + 
                               
                                 3.21 
                                 
                                   ω 
                                   td 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               td 
                               min 
                             
                             = 
                             
                               
                                 ε 
                                 td 
                               
                               - 
                               
                                 1.79 
                                 
                                   ω 
                                   td 
                                 
                               
                             
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
     Beneficial effects: The present invention discloses a near-threshold cell circuit delay model considering an input transition time. A nominal delay model and a statistical delay model of an inverter and a composite cell are obtained by an equivalent current method and the properties of logarithm skewed normal distribution. The present invention has the advantages of high accuracy and low time cost. Under the SMIC40 nm technology, compared with a Monte Carlo simulation method, the error of the nominal delay of the inverter is less than 5.55%, the error of the variance of the delay is less than 14.87%, the error of the −3σ value is less than 15.16%, and the error of the +3σ value is less than 10.87%; the error of the nominal delay of the stacked structure cell is less than 10.78%, the error of the variance is less than 16.43%, the error of the −3σ value is less than 19.34%, and the error of the +3σ value is less than 14.91%; and the error of the nominal delay of the parallel structure cell is less than 7.31%, the error of the variance is less than 15.91%, the error of the −3σ value is less than 9.78%, and the error of the +3σ value is less than 9.63%. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG.  1    shows a flow diagram of a near-threshold cell circuit delay model in a specific embodiment of the present invention. 
         FIG.  2    shows classification situations of delay types of a near-threshold cell circuit in a specific embodiment of the present invention, where (a) shows the situation that the delay type is ultra-fast input; (b) shows the situation that the delay type is fast input; and (c) shows the situation that the delay type is slow input. 
     
    
    
     DETAILED DESCRIPTION 
     The technical solution of the present invention will be further introduced below in conjunction with a specific embodiment based on the SMIC40 nm technology. 
       FIG.  1    shows a flow diagram of a near-threshold cell circuit delay model in a specific embodiment of the present invention. Main steps are as follows: 
     S1: Parameters are obtained. The parameters include process parameters of each transistor of a cell circuit, cell circuit parameters and delay parameters with slow input transition. The process parameters of each transistor of the cell circuit include a sub-threshold slope factor and a drain-induced source barrier lowering effect coefficient, which are obtained by performing nominal direct-current simulation on a transistor with a minimum size. The cell circuit parameters include a transistor size, a load, an input transition time, a power supply voltage, temperature and process; the transistor size is a minimum size, the load is 1 fF, the input transition time is 50 ps, 500 ps and 1 ns, the power supply voltage is 0.4V, the temperature is 25° C., and the process corner is TT. The delay parameters with slow input transition include the slow input delay expression, which can be obtained by fitting the delay under the slow input obtained by simulation. 
     S2: A cell circuit type is judged. Whether the cell circuit is an inverter, a stacked structure cell or a parallel structure cell is judged, a high current I H1  under the ultra-fast input and a high current I H2  under the fast input, a low current I L  and a current integral k are calculated according to the cell circuit type, a mean value μγ, a variance σγ 2  and a skewness γ 1  of a logarithm Y of the sum of the high current and the low current are calculated, and a mean value and a variance of an equivalent threshold voltage are calculated. If the cell circuit type is inverter, the process parameters and the circuit parameters are substituted into Equation (1) and Equation (2) to respectively obtain a high current I H1  under the ultra-fast input, a high current I H2  under the fast input, a low current I L  and a current integral k, and a mean value μγ, a variance σγ 2  and a skewness γ 1  of the logarithm Y of the current sum. If the cell circuit type is the stacked structure cell, the process parameters and the circuit parameters are substituted into Equation (3), Equation (4) and Equation (5) to respectively obtain a high current I H1  under the ultra-fast input, a high current I H2  under the fast input, a low current I L  and a current integral k, a mean value μγ, a variance σγ 2  and a skewness γ 1  of the logarithm Y of the current sum, and a mean value and a variance of the equivalent threshold voltage. If the cell circuit type is the parallel structure cell, the process parameters and the circuit parameters are substituted into Equation (6), Equation (7) and Equation (8) to respectively obtain a high current I H1  under the ultra-fast input, a high current I H2  under the fast input, a low current I L  and a current integral k, a mean value μγ, a variance σγ 2  and a skewness γ 1  of the logarithm Y of the current sum, and a mean value and a variance of the equivalent threshold voltage. 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             I 
                             
                               H 
                               ⁢ 
                               1 
                             
                           
                           = 
                           
                             
                               I 
                               ds 
                             
                             ( 
                             
                               
                                 
                                   V 
                                   gs 
                                 
                                 = 
                                 
                                   V 
                                   dd 
                                 
                               
                               , 
                               
                                 
                                   V 
                                   ds 
                                 
                                 = 
                                 
                                   V 
                                   dd 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           
                             I 
                             
                               H 
                               ⁢ 
                               2 
                             
                           
                           = 
                           
                             
                               I 
                               ds 
                             
                             ( 
                             
                               
                                 
                                   V 
                                   gs 
                                 
                                 = 
                                 
                                   V 
                                   dd 
                                 
                               
                               , 
                               
                                 
                                   V 
                                   ds 
                                 
                                 = 
                                 
                                   
                                     V 
                                     ds 
                                   
                                   ( 
                                   τ 
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           
                             I 
                             L 
                           
                           = 
                           
                             
                               I 
                               ds 
                             
                             ( 
                             
                               
                                 
                                   V 
                                   gs 
                                 
                                 = 
                                 
                                   V 
                                   dd 
                                 
                               
                               , 
                               
                                 
                                   V 
                                   ds 
                                 
                                 = 
                                 
                                   
                                     V 
                                     dd 
                                   
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           k 
                           = 
                           
                             
                               ∫ 
                               0 
                               1 
                             
                             
                               
                                 I 
                                 0 
                               
                               ⁢ 
                               
                                 K 
                                 0 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     
                                       K 
                                       1 
                                     
                                     ( 
                                     
                                       
                                         
                                           
                                             V 
                                             dd 
                                           
                                           ⁢ 
                                           x 
                                         
                                         - 
                                         
                                           V 
                                           th 
                                         
                                         + 
                                         
                                           λ 
                                           ⁢ 
                                           
                                             V 
                                             dd 
                                           
                                         
                                       
                                       
                                         n 
                                         ⁢ 
                                         
                                           ϕ 
                                           i 
                                         
                                       
                                     
                                     ) 
                                   
                                   + 
                                   
                                     
                                       
                                         K 
                                         2 
                                       
                                       ( 
                                       
                                         
                                           
                                             
                                               V 
                                               dd 
                                             
                                             ⁢ 
                                             x 
                                           
                                           - 
                                           
                                             V 
                                             th 
                                           
                                           + 
                                           
                                             λ 
                                             ⁢ 
                                             
                                               V 
                                               dd 
                                             
                                           
                                         
                                         
                                           n 
                                           ⁢ 
                                           
                                             ϕ 
                                             i 
                                           
                                         
                                       
                                       ) 
                                     
                                     2 
                                   
                                 
                               
                               ⁢ 
                               dx 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     In Equation (1), I ds  represents a source-drain current, V gs  and V ds  represent a gate-source voltage and a source-drain voltage, V dd  represents a power supply voltage, V ds (τ) represents a source-drain voltage at the input transition time, λ represents a drain-induced source barrier lowering effect coefficient, n represents a sub-threshold slope factor, Φ 1  represents a thermal voltage. I 0  represents a parameter affected by the process parameters, and K 0 , K 1  and K 2  represent fitting coefficients determined by the process. 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             μ 
                             Y 
                           
                           = 
                           
                             
                               
                                 - 
                                 
                                   k 
                                   2 
                                 
                               
                               ⁢ 
                               
                                 E 
                                 ⁡ 
                                 ( 
                                 
                                   ΔV 
                                   th 
                                   2 
                                 
                                 ) 
                               
                             
                             + 
                             
                               mE 
                               ⁡ 
                               ( 
                               
                                 ΔV 
                                 th 
                               
                               ) 
                             
                             - 
                             
                               ln 
                               ⁡ 
                               ( 
                               
                                 
                                   I 
                                   H 
                                 
                                 + 
                                 
                                   I 
                                   L 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             σ 
                             Y 
                             2 
                           
                           = 
                           
                             
                               k 
                               2 
                               2 
                             
                             ( 
                             
                               
                                 E 
                                 ⁡ 
                                 ( 
                                 
                                   ΔV 
                                   
                                     th 
                                     ′ 
                                   
                                   4 
                                 
                                 ) 
                               
                               - 
                               
                                 
                                   E 
                                   ⁡ 
                                   ( 
                                   
                                     ΔV 
                                     
                                       th 
                                       ′ 
                                     
                                     2 
                                   
                                   ) 
                                 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           
                             γ 
                             1 
                           
                           = 
                           
                             
                               { 
                               
                                 
                                   
                                     
                                       
                                         
                                           - 
                                           
                                             k 
                                             2 
                                             3 
                                           
                                         
                                         ⁢ 
                                         
                                           ( 
                                           
                                             ΔV 
                                             
                                               th 
                                               ′ 
                                             
                                             6 
                                           
                                           ) 
                                         
                                       
                                       + 
                                       
                                         3 
                                         ⁢ 
                                         
                                           k 
                                           2 
                                           2 
                                         
                                         ⁢ 
                                         E 
                                         ⁢ 
                                         
                                           
                                             ( 
                                             
                                               ΔV 
                                               
                                                 th 
                                                 ′ 
                                               
                                               4 
                                             
                                             ) 
                                           
                                           [ 
                                           
                                             
                                               
                                                 m 
                                                 2 
                                               
                                               
                                                 4 
                                                 ⁢ 
                                                 
                                                   k 
                                                   2 
                                                 
                                               
                                             
                                             - 
                                             
                                               ln 
                                               ⁢ 
                                               
                                                 ( 
                                                 
                                                   
                                                     I 
                                                     H 
                                                   
                                                   + 
                                                   
                                                     I 
                                                     L 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                       - 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         
                                           
                                             
                                               3 
                                               ⁢ 
                                               
                                                 k 
                                                 2 
                                               
                                               ⁢ 
                                               
                                                 
                                                   
                                                     E 
                                                     ⁡ 
                                                     ( 
                                                     
                                                       ΔV 
                                                       
                                                         th 
                                                         ′ 
                                                       
                                                       2 
                                                     
                                                     ) 
                                                   
                                                   [ 
                                                   
                                                     
                                                       
                                                         m 
                                                         2 
                                                       
                                                       
                                                         4 
                                                         ⁢ 
                                                         
                                                           k 
                                                           2 
                                                         
                                                       
                                                     
                                                     - 
                                                     
                                                       ln 
                                                       ⁡ 
                                                       ( 
                                                       
                                                         
                                                           I 
                                                           H 
                                                         
                                                         + 
                                                         
                                                           I 
                                                           L 
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                   ] 
                                                 
                                                 2 
                                               
                                             
                                             + 
                                           
                                         
                                       
                                       
                                         
                                           
                                             
                                               
                                                 [ 
                                                 
                                                   
                                                     
                                                       m 
                                                       2 
                                                     
                                                     
                                                       4 
                                                       ⁢ 
                                                       
                                                         k 
                                                         2 
                                                       
                                                     
                                                   
                                                   - 
                                                   
                                                     ln 
                                                     ⁡ 
                                                     ( 
                                                     
                                                       
                                                         I 
                                                         H 
                                                       
                                                       + 
                                                       
                                                         I 
                                                         L 
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                                 ] 
                                               
                                               3 
                                             
                                             - 
                                             
                                               3 
                                               ⁢ 
                                               
                                                 μ 
                                                 Y 
                                               
                                               ⁢ 
                                               
                                                 σ 
                                                 Y 
                                                 2 
                                               
                                             
                                             - 
                                             
                                               μ 
                                               Y 
                                               3 
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                               
                               } 
                             
                             / 
                             
                               σ 
                               Y 
                               3 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     in Equation (2), m represents a coefficient, k 2  represents a fitting coefficient determined by the process, ΔV th  represents a threshold voltage fluctuation quantity, ΔV th  represents a function of a threshold voltage, and E[·] represents a mean value of a corresponding random variable. 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             I 
                             
                               H 
                               ⁢ 
                               1 
                             
                           
                           = 
                           
                             
                               I 
                               ds 
                             
                             ( 
                             
                               
                                 
                                   V 
                                   g 
                                 
                                 = 
                                 
                                   V 
                                   dd 
                                 
                               
                               , 
                               
                                 
                                   V 
                                   d 
                                 
                                 = 
                                 
                                   V 
                                   dd 
                                 
                               
                               , 
                               
                                 
                                   V 
                                   s 
                                 
                                 = 
                                 
                                   V 
                                   
                                     XH 
                                     ⁢ 
                                     1 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           
                             I 
                             
                               H 
                               ⁢ 
                               2 
                             
                           
                           = 
                           
                             
                               I 
                               ds 
                             
                             ( 
                             
                               
                                 
                                   V 
                                   g 
                                 
                                 = 
                                 
                                   V 
                                   dd 
                                 
                               
                               , 
                               
                                 
                                   V 
                                   d 
                                 
                                 = 
                                 
                                   
                                     V 
                                     d 
                                   
                                   ( 
                                   τ 
                                   ) 
                                 
                               
                               , 
                               
                                 
                                   V 
                                   s 
                                 
                                 = 
                                 
                                   V 
                                   
                                     XH 
                                     ⁢ 
                                     2 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           
                             I 
                             L 
                           
                           = 
                           
                             
                               I 
                               ds 
                             
                             ( 
                             
                               
                                 
                                   V 
                                   g 
                                 
                                 = 
                                 
                                   V 
                                   dd 
                                 
                               
                               , 
                               
                                 
                                   V 
                                   dd 
                                 
                                 = 
                                 
                                   
                                     V 
                                     dd 
                                   
                                   2 
                                 
                               
                               , 
                               
                                 
                                   V 
                                   s 
                                 
                                 = 
                                 
                                   V 
                                   XL 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 k 
                                 = 
                                 
                                   
                                     
                                       ∫ 
                                       0 
                                       0.5 
                                     
                                     
                                       
                                         I 
                                         0 
                                       
                                       ⁢ 
                                       
                                         K 
                                         0 
                                       
                                       ⁢ 
                                       
                                         e 
                                         
                                           
                                             
                                               k 
                                               1 
                                             
                                             ( 
                                             
                                               
                                                 
                                                   V 
                                                   dd 
                                                 
                                                 ⁢ 
                                                 x 
                                               
                                               - 
                                               
                                                 V 
                                                 th 
                                               
                                               + 
                                               
                                                 λ 
                                                 ⁢ 
                                                 
                                                   V 
                                                   dd 
                                                 
                                               
                                             
                                             ) 
                                           
                                           + 
                                           
                                             
                                               
                                                 k 
                                                 2 
                                               
                                               ( 
                                               
                                                 
                                                   
                                                     V 
                                                     dd 
                                                   
                                                   ⁢ 
                                                   x 
                                                 
                                                 - 
                                                 
                                                   V 
                                                   th 
                                                 
                                                 + 
                                                 
                                                   λ 
                                                   ⁢ 
                                                   
                                                     V 
                                                     dd 
                                                   
                                                 
                                               
                                               ) 
                                             
                                             2 
                                           
                                         
                                       
                                       ⁢ 
                                       dx 
                                     
                                   
                                   + 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ∫ 
                                   0.5 
                                   1 
                                 
                                 
                                   
                                     I 
                                     0 
                                   
                                   ⁢ 
                                   
                                     K 
                                     0 
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       
                                         
                                           k 
                                           1 
                                         
                                         ( 
                                         
                                           
                                             
                                               V 
                                               dd 
                                             
                                             ⁢ 
                                             x 
                                           
                                           - 
                                           
                                             V 
                                             th 
                                           
                                           - 
                                           
                                             2 
                                             ⁢ 
                                             
                                               
                                                 V 
                                                 XH 
                                               
                                               ( 
                                               
                                                 x 
                                                 - 
                                                 
                                                   1 
                                                   2 
                                                 
                                               
                                               ) 
                                             
                                           
                                           + 
                                           
                                             λ 
                                             ⁢ 
                                             
                                               V 
                                               dd 
                                             
                                           
                                         
                                         ) 
                                       
                                       + 
                                       
                                         
                                           
                                             k 
                                             2 
                                           
                                           ( 
                                           
                                             
                                               
                                                 V 
                                                 dd 
                                               
                                               ⁢ 
                                               x 
                                             
                                             - 
                                             
                                               V 
                                               th 
                                             
                                             - 
                                             
                                               2 
                                               ⁢ 
                                               
                                                 
                                                   V 
                                                   XH 
                                                 
                                                 ( 
                                                 
                                                   x 
                                                   - 
                                                   
                                                     1 
                                                     2 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                             + 
                                             
                                               λ 
                                               ⁢ 
                                               
                                                 V 
                                                 dd 
                                               
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                     
                                   
                                   ⁢ 
                                   dx 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     In Equation (3), V g  and V d  represent a gate voltage and a drain voltage, V s  represents a source voltage, V XH  and V XL  represent intermediate node voltages at different gate voltages and drain voltages, in the case of the ultra-fast input, V XH  is V XH1 , which is an intermediate node voltage when the gate voltage and the drain voltage are both power supply voltages, in the case of the fast input, V XH  is V XH2 , which is an intermediate node voltage when the gate voltage is a power supply voltage and the drain voltage is the intermediate node voltage of a drain voltage at the input transition time, V d (τ) represents a drain voltage at the input transition time, and k 1  represents a fitting coefficient determined by the process. 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             E 
                             ⁡ 
                             ( 
                             Y 
                             ) 
                           
                           = 
                           
                             
                               E 
                               ⁡ 
                               ( 
                               
                                 Y 
                                 1 
                               
                               ) 
                             
                             + 
                             
                               E 
                               ⁡ 
                               ( 
                               
                                 Y 
                                 2 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             D 
                             ⁡ 
                             ( 
                             Y 
                             ) 
                           
                           = 
                           
                             
                               D 
                               ⁡ 
                               ( 
                               
                                 Y 
                                 1 
                               
                               ) 
                             
                             + 
                             
                               D 
                               ⁡ 
                               ( 
                               
                                 Y 
                                 2 
                               
                               ) 
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                 Cov 
                                 ( 
                                 
                                   
                                     Y 
                                     1 
                                   
                                   , 
                                   
                                     Y 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               S 
                               k 
                             
                             ( 
                             Y 
                             ) 
                           
                           = 
                           
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           S 
                                           k 
                                         
                                         ( 
                                         
                                           Y 
                                           1 
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         
                                           D 
                                           ⁡ 
                                           ( 
                                           
                                             Y 
                                             1 
                                           
                                           ) 
                                         
                                         
                                           3 
                                           / 
                                           2 
                                         
                                       
                                     
                                     + 
                                     
                                       
                                         
                                           S 
                                           k 
                                         
                                         ( 
                                         
                                           Y 
                                           2 
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         
                                           D 
                                           ⁡ 
                                           ( 
                                           
                                             Y 
                                             2 
                                           
                                           ) 
                                         
                                         
                                           3 
                                           / 
                                           2 
                                         
                                       
                                     
                                     + 
                                     
                                       3 
                                       ⁢ 
                                       
                                         Cov 
                                         ( 
                                         
                                           
                                             Y 
                                             1 
                                           
                                           , 
                                           
                                             Y 
                                             2 
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       3 
                                       ⁢ 
                                       
                                         Cov 
                                         ( 
                                         
                                           
                                             Y 
                                             1 
                                             2 
                                           
                                           , 
                                           
                                             Y 
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       6 
                                       ⁢ 
                                       
                                         E 
                                         ⁡ 
                                         ( 
                                         Y 
                                         ) 
                                       
                                       ⁢ 
                                       
                                         Cov 
                                         ( 
                                         
                                           
                                             Y 
                                             1 
                                           
                                           , 
                                           
                                             Y 
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 D 
                                 ⁡ 
                                 ( 
                                 Y 
                                 ) 
                               
                               
                                 3 
                                 / 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     In Equation (4), D[·] represents a variance of a corresponding random variable, S k [·] represents a skewness of the corresponding random variable, and Cov[·] represents a covariance of the corresponding random variable. 
     
       
         
           
             
               
                 
                   
                     ΔV 
                     
                       th 
                       ⁢ 
                       _ 
                       ⁢ 
                       eff 
                     
                   
                   = 
                   
                     
                       
                         
                           I 
                           H 
                         
                         
                           
                             I 
                             H 
                           
                           + 
                           
                             I 
                             L 
                           
                         
                       
                       ⁢ 
                       
                         ΔV 
                         thH 
                       
                     
                     + 
                     
                       
                         
                           I 
                           L 
                         
                         
                           
                             I 
                             H 
                           
                           + 
                           
                             I 
                             L 
                           
                         
                       
                       ⁢ 
                       
                         ΔV 
                         thL 
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     In Equation (5), ΔV th_eff  represents an equivalent threshold voltage, and ΔV thH  and ΔV thL  represent linear functions of an upper transistor threshold voltage fluctuation quantity ΔV thU  and a lower transistor threshold voltage fluctuation quantity ΔV thD . 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             I 
                             
                               H 
                               ⁢ 
                               1 
                             
                           
                           = 
                           
                             2 
                             ⁢ 
                             
                               
                                 I 
                                 ds 
                               
                               ( 
                               
                                 
                                   
                                     V 
                                     gs 
                                   
                                   = 
                                   
                                     V 
                                     dd 
                                   
                                 
                                 , 
                                 
                                   
                                     V 
                                     ds 
                                   
                                   = 
                                   
                                     V 
                                     dd 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             I 
                             
                               H 
                               ⁢ 
                               2 
                             
                           
                           = 
                           
                             2 
                             ⁢ 
                             
                               
                                 I 
                                 ds 
                               
                               ( 
                               
                                 
                                   
                                     V 
                                     gs 
                                   
                                   = 
                                   
                                     V 
                                     dd 
                                   
                                 
                                 , 
                                 
                                   
                                     V 
                                     ds 
                                   
                                   = 
                                   
                                     
                                       V 
                                       ds 
                                     
                                     ( 
                                     τ 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             I 
                             L 
                           
                           = 
                           
                             2 
                             ⁢ 
                             
                               
                                 I 
                                 ds 
                               
                               ( 
                               
                                 
                                   
                                     V 
                                     gs 
                                   
                                   = 
                                   
                                     V 
                                     dd 
                                   
                                 
                                 , 
                                 
                                   
                                     V 
                                     ds 
                                   
                                   = 
                                   
                                     
                                       V 
                                       dd 
                                     
                                     2 
                                   
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           k 
                           = 
                           
                             
                               
                                 ∫ 
                                 0 
                                 1 
                               
                               
                                 
                                   I 
                                   0 
                                 
                                 ⁢ 
                                 
                                   K 
                                   0 
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     
                                       
                                         k 
                                         1 
                                       
                                       ( 
                                       
                                         
                                           
                                             V 
                                             dd 
                                           
                                           ⁢ 
                                           x 
                                         
                                         - 
                                         
                                           V 
                                           
                                             th 
                                             ⁢ 
                                             1 
                                           
                                         
                                         + 
                                         
                                           λ 
                                           ⁢ 
                                           
                                             V 
                                             dd 
                                           
                                         
                                       
                                       ) 
                                     
                                     + 
                                     
                                       
                                         
                                           k 
                                           2 
                                         
                                         ( 
                                         
                                           
                                             
                                               V 
                                               dd 
                                             
                                             ⁢ 
                                             x 
                                           
                                           - 
                                           
                                             V 
                                             
                                               th 
                                               ⁢ 
                                               1 
                                             
                                           
                                           + 
                                           
                                             λ 
                                             ⁢ 
                                             
                                               V 
                                               dd 
                                             
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                        
                                   
                                 
                                 ⁢ 
                                 dx 
                               
                             
                             + 
                           
                         
                       
                     
                     
                       
                         
                           
                             ∫ 
                             0 
                             
                                  
                               1 
                             
                           
                           
                             
                               I 
                               0 
                             
                             ⁢ 
                             
                               K 
                               0 
                             
                             ⁢ 
                             
                               e 
                               
                                 
                                   
                                     k 
                                     1 
                                   
                                   ( 
                                   
                                     
                                       
                                         V 
                                         dd 
                                       
                                       ⁢ 
                                       x 
                                     
                                     - 
                                     
                                       V 
                                       
                                         th 
                                         ⁢ 
                                         2 
                                       
                                     
                                     + 
                                     
                                       λ 
                                       ⁢ 
                                       
                                         V 
                                         dd 
                                       
                                     
                                   
                                   ) 
                                 
                                 + 
                                 
                                   
                                     
                                       k 
                                       2 
                                     
                                     ( 
                                     
                                       
                                         
                                           V 
                                           dd 
                                         
                                         ⁢ 
                                         x 
                                       
                                       - 
                                       
                                         V 
                                         
                                           th 
                                           ⁢ 
                                           2 
                                         
                                       
                                       + 
                                       
                                         λ 
                                         ⁢ 
                                         
                                           V 
                                           dd 
                                         
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                             
                             ⁢ 
                             dx 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     In Equation (6), V th1  and V th2  respectively represent threshold voltages of a left transistor and a right transistor in the parallel structure. 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             E 
                             ⁢ 
                             
                               ( 
                               Y 
                               ) 
                             
                           
                           = 
                           
                             
                               E 
                               ⁢ 
                               
                                 ( 
                                 
                                   Y 
                                   1 
                                 
                                 ) 
                               
                             
                             + 
                             
                               E 
                               ⁢ 
                               
                                 ( 
                                 
                                   Y 
                                   2 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             D 
                             ⁡ 
                             ( 
                             Y 
                             ) 
                           
                           = 
                           
                             
                               D 
                               ⁡ 
                               ( 
                               
                                 Y 
                                 1 
                               
                               ) 
                             
                             + 
                             
                               D 
                               ⁡ 
                               ( 
                               
                                 Y 
                                 2 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               S 
                               k 
                             
                             ( 
                             Y 
                             ) 
                           
                           = 
                           
                             
                               
                                 
                                   
                                     S 
                                     k 
                                   
                                   ( 
                                   
                                     Y 
                                     1 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 D 
                                 ⁢ 
                                 
                                   
                                     ( 
                                     
                                       Y 
                                       1 
                                     
                                     ) 
                                   
                                   
                                     3 
                                     / 
                                     2 
                                   
                                 
                               
                               + 
                               
                                 
                                   
                                     S 
                                     k 
                                   
                                   ( 
                                   
                                     Y 
                                     2 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 D 
                                 ⁢ 
                                 
                                   
                                     ( 
                                     
                                       Y 
                                       2 
                                     
                                     ) 
                                   
                                   
                                     3 
                                     / 
                                     2 
                                   
                                 
                               
                             
                             
                               
                                 D 
                                 ⁡ 
                                 ( 
                                 Y 
                                 ) 
                               
                               
                                 3 
                                 / 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                       V 
                       th_eff 
                     
                   
                   = 
                   
                     
                       
                         Δ 
                         ⁢ 
                         
                           V 
                           
                             th 
                             ⁢ 
                             1 
                           
                         
                       
                       + 
                       
                         Δ 
                         ⁢ 
                         
                           V 
                           
                             th 
                             ⁢ 
                             2 
                           
                         
                       
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     In Equation (8), ΔV th1  and ΔV th2  represent threshold voltage fluctuation quantities of the left transistor and the right transistor in the parallel structure. 
     S3: A delay type is judged. The high current I H1  under the ultra-fast input, the high current I H2  under the fast input, the low current I L  and the current integral k and the input transition time calculated in step S2 are respectively substituted into Equation (9) and Equation (10) according to the cell circuit type, so as to calculate an overshoot time t ov  and a delay td, the magnitude relationship among the input transition time, the overshoot time and the delay is compared, and whether the delay type is the ultra-fast input, the fast input or the slow input is judged. If the input transition time is 50 ps, the calculated overshoot time t ov , delay td and input transition time are compared, and it is found that the overshoot time t ov  is greater than the input transition time and the delay td is greater than half of the input transition time, then the delay type is determined as the ultra-fast input. If the input transition time is 500 ps, the calculated overshoot time t ov , delay td and input transition time are compared, and it is found that the overshoot time t ov  is less than the input transition time and the delay td is greater than half of the input transition time, then the delay type is determined as the fast input. If the input transition time is 1 ns, the calculated overshoot time t ov , delay td and input transition time are compared, and it is found that the overshoot time t ov  is less than the input transition time and the delay td is less than half of the input transition time, then the delay type is determined as the slow input. 
     
       
         
           
             
               
                 
                   
                     t 
                     ov 
                   
                   = 
                   
                     
                       
                         
                           2 
                           ⁢ 
                           
                             C 
                             M 
                           
                         
                         
                           
                             C 
                             M 
                           
                           + 
                           
                             C 
                             L 
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             C 
                             L 
                           
                           ⁢ 
                           
                             V 
                             dd 
                           
                         
                         
                           
                             I 
                             
                               H 
                               ⁢ 
                               1 
                             
                           
                           + 
                           
                             I 
                             L 
                           
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   C 
                                   L 
                                 
                               
                               
                                 
                                   C 
                                   M 
                                 
                                 + 
                                 
                                   C 
                                   L 
                                 
                               
                             
                             ⁢ 
                             
                               k 
                               
                                 
                                   I 
                                   
                                     H 
                                     ⁢ 
                                     1 
                                   
                                 
                                 + 
                                 
                                   I 
                                   L 
                                 
                               
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       τ 
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     In Equation (9), td represents a cell circuit delay, r represents the input transition time, C M  represents the coupling capacitance of input and output, and C L  represents the load capacitance. 
     
       
         
           
             
               
                 
                   td 
                   = 
                   
                     
                       ( 
                       
                         1 
                         + 
                         
                           
                             2 
                             ⁢ 
                             
                               C 
                               M 
                             
                           
                           
                             
                               C 
                               M 
                             
                             + 
                             
                               C 
                               L 
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       
                         
                           C 
                           L 
                         
                         ⁢ 
                         
                           V 
                           dd 
                         
                       
                       
                         
                           1 
                           
                             H 
                             ⁢ 
                             2 
                           
                         
                         + 
                         
                           I 
                           L 
                         
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           1 
                           2 
                         
                         - 
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 C 
                                 L 
                               
                             
                             
                               
                                 C 
                                 M 
                               
                               + 
                               
                                 C 
                                 L 
                               
                             
                           
                           ⁢ 
                           
                             k 
                             
                               
                                 I 
                                 
                                   H 
                                   ⁢ 
                                   2 
                                 
                               
                               + 
                               
                                 I 
                                 L 
                               
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     τ 
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     S4: A cell circuit nominal delay model is established. The nominal delay is obtained according to the cell circuit type and the delay type. If the delay type is the ultra-fast input, the high current I H1  under the ultra-fast input, the high current I H2  under the fast input, the low current I L  and the current integral k and the input transition time calculated in step S2 are substituted into Equation (10) to calculate the nominal delay. If the delay type is the fast input, the high current I H1  under the ultra-fast input, the high current I H2  under the fast input, the low current I L  and the current integral k and the input transition time calculated in step S2 are substituted into Equation (11) to calculate the nominal delay. If the delay type is the slow input, the high current I H1  under the ultra-fast input, the high current I H2  under the fast input, the low current I L  and the current integral k and the input transition time calculated in step S2 are substituted into Equation (12) to calculate the nominal delay. 
     
       
         
           
             
               
                 
                   td 
                   = 
                   
                     
                       ( 
                       
                         1 
                         + 
                         
                           
                             2 
                             ⁢ 
                             
                               C 
                               M 
                             
                           
                           
                             
                               C 
                               M 
                             
                             + 
                             
                               C 
                               L 
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       
                         
                           C 
                           L 
                         
                         ⁢ 
                         
                           V 
                           dd 
                         
                       
                       
                         
                           1 
                           
                             H 
                             ⁢ 
                             1 
                           
                         
                         + 
                         
                           I 
                           L 
                         
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           1 
                           2 
                         
                         - 
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 C 
                                 L 
                               
                             
                             
                               
                                 C 
                                 M 
                               
                               + 
                               
                                 C 
                                 L 
                               
                             
                           
                           ⁢ 
                           
                             k 
                             
                               
                                 I 
                                 
                                   H 
                                   ⁢ 
                                   1 
                                 
                               
                               + 
                               
                                 I 
                                 L 
                               
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     τ 
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     td 
                     τ 
                   
                   = 
                   
                     
                       a 
                       ⁢ 
                          
                       ln 
                       ⁢ 
                       
                         
                           td 
                           s 
                         
                         τ 
                       
                     
                     + 
                     b 
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     In Equation (12), td s  represents a step delay, and a and b represent fitting parameters. 
     S5: A cell circuit statistical delay model is established. Delay statistics: a variance, a maximum value and a minimum value are obtained based on the cell circuit nominal delay model according to the cell circuit type and the delay type. If the delay type is the ultra-fast input, the mean value μγ, the variance σγ 2  and the skewness γ 1  of the logarithm of the sum of the high current and the low current obtained in step S2 are substituted into Equation (13) to calculate distribution parameters: a location parameter ξγ, a scale parameter or and a second shape parameter βγ, the distribution parameters are substituted into Equation (14) to calculate a variance D(td) of the delay, and the distribution parameters are substituted into Equation (15) to calculate a maximum value td max  and a minimum value td min  of the delay. If the delay type is the fast input, the mean value μγ, the variance σγ 2  and the skewness γ 1  of the logarithm of the sum of the high current and the low current obtained in step S2 are substituted into Equation (13) to calculate distribution parameters: a location parameter ξγ, a scale parameter ωγ and a second shape parameter βγ, the distribution parameters are substituted into Equation (14) to calculate a variance D(td) of the delay, and the distribution parameters are substituted into Equation (15) to calculate a maximum value td max  and a minimum value td min  of the delay. If the delay type is the slow input, the mean value and the variance of the equivalent threshold voltage obtained in step S2 are substituted into Equation (16) to calculate a mean value E(td), a variance D(td) and a skewness S k (td) of the delay, the mean value E(td), the variance D(td) and the skewness S k (td) of the delay are substituted into Equation (13) to calculate distribution parameters: a location parameter ξ td , a scale parameter ω td  and a second shape parameter β td , and the distribution parameters are substituted into Equation (17) to calculate a maximum value td max  and a minimum value td min  of the delay. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           ε 
                           Y 
                         
                         = 
                         
                           
                             μ 
                             Y 
                           
                           - 
                           
                             
                               ω 
                               Y 
                             
                             ⁢ 
                             
                               β 
                               Y 
                             
                             ⁢ 
                             
                               
                                 2 
                                 π 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           ω 
                           Y 
                         
                         = 
                         
                           
                             
                               σ 
                               Y 
                               2 
                             
                             
                               1 
                               - 
                               
                                 
                                   2 
                                   π 
                                 
                                 ⁢ 
                                 
                                   β 
                                   Y 
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           β 
                           Y 
                         
                         = 
                         
                           
                             
                               π 
                               2 
                             
                             ⁢ 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                       
                                         γ 
                                         1 
                                       
                                     
                                     
                                       4 
                                       - 
                                       π 
                                     
                                   
                                   ) 
                                 
                                 
                                   2 
                                   / 
                                   3 
                                 
                               
                               
                                 
                                   
                                       
                                   
                                      
                                 
                                 ⁢ 
                                 
                                   1 
                                   + 
                                   
                                     
                                       ( 
                                       
                                         
                                           2 
                                           ⁢ 
                                           
                                             γ 
                                             1 
                                           
                                         
                                         
                                           4 
                                           - 
                                           π 
                                         
                                       
                                       ) 
                                     
                                     
                                       2 
                                       / 
                                       3 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     D 
                     ⁡ 
                     ( 
                     td 
                     ) 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             ( 
                             
                               1 
                               + 
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                     C 
                                     M 
                                   
                                 
                                 
                                   
                                     C 
                                     M 
                                   
                                   + 
                                   
                                     C 
                                     L 
                                   
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             C 
                             L 
                           
                           ⁢ 
                           
                             V 
                             dd 
                           
                         
                         ] 
                       
                       2 
                     
                     × 
                     2 
                     ⁢ 
                     
                       e 
                       
                         2 
                         ⁢ 
                         
                           ε 
                           γ 
                         
                       
                     
                     ⁢ 
                     
                       
                         e 
                         
                           ω 
                           γ 
                           2 
                         
                       
                       ( 
                       
                         
                           
                             e 
                             
                               ω 
                               γ 
                               2 
                             
                           
                           ⁢ 
                           
                             ϕ 
                             ⁡ 
                             ( 
                             
                               2 
                               ⁢ 
                               
                                 β 
                                 Y 
                               
                               ⁢ 
                               
                                 ω 
                                 Y 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           2 
                           ⁢ 
                           
                             
                               ϕ 
                               2 
                             
                             ( 
                             
                               
                                 β 
                                 Y 
                               
                               ⁢ 
                               
                                 ω 
                                 Y 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             td 
                             max 
                           
                           = 
                           
                             
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     
                                       2 
                                       ⁢ 
                                       
                                         C 
                                         M 
                                       
                                     
                                     
                                       
                                         C 
                                         M 
                                       
                                       + 
                                       
                                         C 
                                         L 
                                       
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 C 
                                 L 
                               
                               ⁢ 
                               
                                 V 
                                 dd 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     ε 
                                     Y 
                                   
                                   + 
                                   
                                     3.21 
                                     
                                       ω 
                                       Y 
                                     
                                   
                                 
                               
                             
                             + 
                             
                               M 
                               ⁢ 
                               τ 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             td 
                             min 
                           
                           = 
                           
                             
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     
                                       2 
                                       ⁢ 
                                       
                                         C 
                                         M 
                                       
                                     
                                     
                                       
                                         C 
                                         M 
                                       
                                       + 
                                       
                                         C 
                                         L 
                                       
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 C 
                                 L 
                               
                               ⁢ 
                               
                                 V 
                                 dd 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     ε 
                                     Y 
                                   
                                   - 
                                   
                                     1.79 
                                     
                                       ω 
                                       Y 
                                     
                                   
                                 
                               
                             
                             + 
                             
                               M 
                               ⁢ 
                               τ 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     In Equation (14), φ(·) represents a cumulative density distribution function of standard normal distribution. 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             E 
                             ⁡ 
                             ( 
                             td 
                             ) 
                           
                           = 
                           
                             
                               
                                 p 
                                 2 
                               
                               ⁢ 
                               
                                 E 
                                 ⁡ 
                                 ( 
                                 
                                   Δ 
                                   ⁢ 
                                   
                                     V 
                                     th 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             - 
                             
                               p 
                               0 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             D 
                             ⁡ 
                             ( 
                             td 
                             ) 
                           
                           = 
                           
                             
                               
                                 p 
                                 3 
                                 2 
                               
                               ⁢ 
                               
                                 E 
                                 ⁡ 
                                 ( 
                                 
                                   Δ 
                                   ⁢ 
                                   
                                     V 
                                     th 
                                     6 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 ( 
                                 
                                   
                                     2 
                                     ⁢ 
                                     
                                       p 
                                       3 
                                     
                                     ⁢ 
                                     
                                       p 
                                       1 
                                     
                                   
                                   + 
                                   
                                     p 
                                     2 
                                     2 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 E 
                                 ⁡ 
                                 ( 
                                 
                                   Δ 
                                   ⁢ 
                                   
                                     V 
                                     th 
                                     4 
                                   
                                 
                                 ) 
                               
                             
                             + 
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               ( 
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                     p 
                                     2 
                                   
                                   ⁢ 
                                   
                                     p 
                                     0 
                                   
                                 
                                 + 
                                 
                                   p 
                                   1 
                                   2 
                                 
                               
                               ) 
                             
                             ⁢ 
                             E 
                             ⁢ 
                             
                               ( 
                               
                                 Δ 
                                 ⁢ 
                                 
                                   V 
                                   th 
                                   2 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             p 
                             0 
                             2 
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               S 
                               k 
                             
                             ⁢ 
                             
                               ( 
                               td 
                               ) 
                             
                           
                           = 
                           
                             
                               
                                 
                                   
                                     
                                       
                                         q 
                                         8 
                                       
                                       ⁢ 
                                       
                                         E 
                                         ⁡ 
                                         ( 
                                         
                                           Δ 
                                           ⁢ 
                                           
                                             V 
                                             th 
                                             8 
                                           
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       
                                         q 
                                         6 
                                       
                                       ⁢ 
                                       
                                         E 
                                         ⁡ 
                                         ( 
                                         
                                           Δ 
                                           ⁢ 
                                           
                                             V 
                                             th 
                                             6 
                                           
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       
                                         q 
                                         4 
                                       
                                       ⁢ 
                                       
                                         E 
                                         ⁡ 
                                         ( 
                                         
                                           Δ 
                                           ⁢ 
                                           
                                             V 
                                             th 
                                             4 
                                           
                                         
                                         ) 
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         q 
                                         2 
                                       
                                       ⁢ 
                                       
                                         E 
                                         ⁡ 
                                         ( 
                                         
                                           Δ 
                                           ⁢ 
                                           
                                             V 
                                             th 
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       q 
                                       0 
                                     
                                     - 
                                     
                                       3 
                                       ⁢ 
                                       
                                         E 
                                         ⁡ 
                                         ( 
                                         td 
                                         ) 
                                       
                                       ⁢ 
                                       
                                         D 
                                         ⁡ 
                                         ( 
                                         td 
                                         ) 
                                       
                                     
                                     - 
                                     
                                       
                                         E 
                                         ⁡ 
                                         ( 
                                         td 
                                         ) 
                                       
                                       3 
                                     
                                   
                                 
                               
                             
                             
                               
                                 D 
                                 ⁡ 
                                 ( 
                                 td 
                                 ) 
                               
                               
                                 3 
                                 / 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     In Equation (16), p i (i=0,1,2,3) and q i (i=0,2,4,6,8) represent coefficients, which are functions of threshold voltages and currents. 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             td 
                             max 
                           
                           = 
                           
                             
                               ε 
                               td 
                             
                             + 
                             
                               3.21 
                               
                                 ω 
                                 td 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             td 
                             min 
                           
                           = 
                           
                             
                               ε 
                               td 
                             
                             + 
                             
                               1.79 
                               
                                 ω 
                                 td 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
       FIG.  2    shows classification situations of delay types of a near-threshold cell circuit in a specific embodiment of the present invention, In  FIG.  2 ( a ) , the delay type is the ultra-fast input, the overshoot time is greater than the input transition time, and the delay is greater than half of the input transition time. In  FIG.  2 ( b ) , the delay type is the fast input, the overshoot time is less than the input transition time, and the delay is greater than half of the input transition time. In  FIG.  2 ( c ) , the delay type is the slow input, the overshoot time is less than the input transition time, and the delay is less than half of the input transition time.