Abstract:
An accurate method to compute the capacitance at a pin whose capacitance is slew dependant. The method uses existing library characterized data and provides an equation based approach which can easily be integrated in static timing analysis without the added resource needs that an iterative approach would require. An RC/RLC network from slew and output load dependent pin capacitance tables is generated. The resulting linear network which models the pin capacitance is then stitched to the original interconnect network and used to calculate the propagation delay across a gate and corresponding interconnect. The method steps include: a) determining a response of the gate pin capacitance to its input slew and output load; b) synthesizing a linear time-invariant filter that matches the response; c) extending the interconnect model to include the synthesized time-invariant filter; and d) inputting the extended interconnect model into a static timing analysis for determining timing behavior between a gate input and each of its fan-out gates.

Description:
FIELD OF THE INVENTION 
   This invention generally relates to the field of Design Automation of Very Large Scale Integration (VLSI) circuits, and more particularly, to the modeling and analysis of slew and load dependent input pin capacitances of metal oxide semiconductor (MOS) gates. 
   BACKGROUND OF THE INVENTION 
   Static timing analysis (STA) is a key step in the design of high speed very large scale integrated (VLSI) circuits. STA is used to verify that a VLSI chip performs correctly at the required frequency before it is released to manufacturing. STA typically consists of the following main steps: 1) Delay calculation, which involves modeling and calculating path delays for the gates and interconnects contained in the design, 2) Propagation of arrival, required arrival times and slews, and 3) Slack calculation for the paths in the design. 
   Referring to  FIG. 1   a , there is shown a conventional overall delay from input pin A of the CMOS inverter  108  to the input pin C of the NAND2 gate  107 . This delay is obtained by adding the gate delay from input pin A to the output pin B of CMOS inverter  108  and the interconnect propagation delay from output pin B of the inverter  108  to the input pin C of the NAND2 gate  107 , as in EQ. 1. This overall delay is also defined as the stage delay  110 , and represents the propagation delay between the voltage waveforms  109  and  104 , for a given voltage threshold crossing point of the waveforms.
 
Delay AC =Delay AB +Delay BC   EQ. 1
 
   The voltage waveform  109  at the sink of the interconnect network  105  is a function of the voltage waveform  103  at the input terminal of the interconnect network, the interconnect parasitics  105  and the input capacitance  106  of each gate  107  at the sink terminal(s) of the interconnect network. The interconnect delay and output slew are calculated by fitting a ramp or a piece-wise-linear waveform to the voltage waveform  103  at the input terminal of the interconnect network, and subsequently convolving it with the transfer function of the interconnect load. Various Model Order Reduction (MOR) techniques such as Asymptotic Waveform Evaluation (AWE) and Passive Reduced-order Interconnect Macromodeling Algorithm (PRIMA) have been proposed for accurate interconnect timing analysis. These techniques reduce the complexity of a large-scale interconnect network to a smaller one while preserving (to the possible extent) their input-output behavior. Alternatively, a large-scale interconnect network is approximated (reduced) to a smaller interconnect network so that when the same input signal is applied to both, the output response of the original and the approximated (reduced) network closely match each other. Higher order reduced models offer increased accuracy at the cost of increasing the complexity of the analysis. For instance, a first reduced order model can be analyzed very fast, but it may introduce significant errors in the input-output behavior of the system. Consequently, a trade-off between accuracy and speed for determining the order of the reduced model is performed. 
   The voltage waveform  103  at the output of the gate  108  is a function of different parameters that includes the voltage waveform  104  at the input terminal of the gate (which is propagated from the previous stages of the design), the interconnect parasitics  105 , the input pin capacitance  106  of each gate  107  at the sink of the interconnect, and the characteristics of the primitive gate  108 . Given the input voltage waveform  104 , the gate timing library, the load parasitics  105 , and the input pin capacitance  106  of each gate connected to the interconnect sink terminal(s), the gate timing analysis process calculates the gate output voltage waveform  103  and/or its characteristics with respect to the characteristics of the gate input voltage waveform  104  (e.g., the gate propagation delay and output slew). 
   Various gate modeling techniques have been employed for gate timing analysis. Three well-known techniques are based on: (1) using delay and slew polynomial equations which are characterized as a function of input slew and output capacitive load; (2) using delay and slew tables which are characterized as a function of the input slew and output capacitive load; and (3) using current source models where the output voltage/current is characterized as a function of input slew and output load, while the output voltage/current waveforms are tabulated. Each of these gate models are also characterized as a function of manufacturing process parameters (e.g. L eff , doping concentration), and the environmental parameters (e.g., voltage, temperature). 
   The aforementioned models are all characterized as a function of input slew and output capacitive load. Given some slew at an input terminal of a gate and an approximated capacitive load at its output terminal, the gate propagation delay is obtained directly from the gate delay model. However, in very deep sub-micron (VDSM) technologies, the effect of interconnect resistive parasitics is significant and render approximating the interconnect load by a lumped capacitance susceptible to error. Using the sum of all the load capacitances as the capacitive load is a simple, albeit inaccurate approximation. A more accurate approximation for an n th  order load seen by the gate (i.e., a load having n distributed capacitances to ground) is to use a reduced order model. As an example, for the case of resistive capacitive interconnects, the load can be approximated by a second order RC-π model. Equating the first, second, and third moments of the admittance of the real load with the first, second, and third moments, respectively of the RC-π load, the parameters C n , R π , and C f  of the RC-π model, as shown in  FIG. 1  ( b ) are obtained. The RC-π parameters are therefore functions of the input pin capacitance of each gate in the fan-out of the primitive gate. An “effective capacitance” technique is proposed, whereby the reduced order model load (e.g. the RC-π load) is approximated by an equivalent capacitance. The aforementioned models are then employed to obtain the gate delay and output slew, given the slew  115 , at the input pin of the gate and the reduced order RC-π load  118 . 
   Conventional Static Timing Analysis (STA) propagates signal timing information from the circuit primary inputs through gates and interconnects to calculate the earliest/latest signal arrival times/voltage waveforms at the circuit primary outputs. A typical conventional STA has a linear run-time complexity with respect to the number of the gates in the design. During the interconnect delay analysis, the input pin impedance of pins connected to the sink terminal(s) of the interconnect network are represented by fixed pin capacitance values. 
   The technology scaling to sub-90 nm geometry introduces new challenges that need to be addressed during timing analysis of VLSI circuits. Many model assumptions which have been made in 90+ nm nodes are no longer valid. Constant pin capacitance models no longer accurately describe the pin&#39;s loading contribution to the total load as seen by the source terminal of the interconnect network. Generally speaking, the gate input pin impedance may vary during the voltage transition at the input pin of the CMOS gate. In addition, its value is dependent on the loading condition at the output pin of that CMOS gate. Present industry standard gate current-source models offer two ways of modeling gate input pin capacitances: 1) the Effective Current Source Model (ECSM) format which models the input pin capacitance as a function of Slew and Output capacitive Load, and 2) the Composite Current Source (CCS) format that employs a two-piece pin capacitance model for each input slew/output capacitance combination. Both these models are table-based representations. 
     FIG. 2  shows the Slew and Output capacitive Load Dependent Pin Capacitances (referred hereinafter as SOLDPC) for ECSM and CCS. As shown in  FIG. 2 , the pin-capacitance definition in ECSM consists of a two-dimensional table. During characterization, a ramp input slew and output capacitive load is applied to the CMOS gate, and the effective input pin capacitance of the gate is calculated as the amount of charge that needs to be injected to the input pin of the driver divided by the supply voltage (Vdd) as follows: 
   
     
       
         
           
             
               
                 
                   C 
                   pin 
                 
                 = 
                 
                   
                     
                       ∫ 
                       
                         
                           
                             t 
                             1 
                           
                           ⁢ 
                           
                             V 
                             ⁡ 
                             
                               ( 
                               
                                 t 
                                 1 
                               
                               ) 
                             
                           
                         
                         = 
                         0 
                       
                       
                         
                           
                             t 
                             2 
                           
                           ⁢ 
                           
                             V 
                             ⁡ 
                             
                               ( 
                               
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                                 2 
                               
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                         = 
                         0 
                       
                     
                     ⁢ 
                     
                       
                         ⅈ 
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ⅆ 
                         t 
                       
                     
                   
                   Vdd 
                 
               
             
             
               
                 EQ 
                 . 
                 
                     
                 
                 ⁢ 
                 2 
               
             
           
         
       
     
   
   CCS uses a similar characterization method to calculate a two-piece input pin capacitance for every combination of the input slew and output load as follows: 
   
     
       
         
           
             
               
                 
                   
                     C 
                     1 
                   
                   = 
                   
                     
                       
                         ∫ 
                         
                           
                             
                               t 
                               1 
                             
                             ⁢ 
                             
                               V 
                               ⁡ 
                               
                                 ( 
                                 
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                                   1 
                                 
                                 ) 
                               
                             
                           
                           = 
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                               t 
                               2 
                             
                             ⁢ 
                             
                               V 
                               ⁡ 
                               
                                 ( 
                                 
                                   t 
                                   2 
                                 
                                 ) 
                               
                             
                           
                           = 
                           
                             Vdd 
                             / 
                             2 
                           
                         
                       
                       ⁢ 
                       
                         
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                           ⁡ 
                           
                             ( 
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                             ) 
                           
                         
                         ⁢ 
                         
                             
                         
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                           ⅆ 
                           t 
                         
                       
                     
                     
                       Vdd 
                       / 
                       2 
                     
                   
                 
                 , 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     C 
                     2 
                   
                   = 
                   
                     
                       
                         ∫ 
                         
                           
                             
                               t 
                               2 
                             
                             ⁢ 
                             
                               V 
                               ⁡ 
                               
                                 ( 
                                 
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                                   2 
                                 
                                 ) 
                               
                             
                           
                           = 
                           
                             Vdd 
                             / 
                             2 
                           
                         
                         
                           
                             
                               t 
                               3 
                             
                             ⁢ 
                             
                               V 
                               ⁡ 
                               
                                 ( 
                                 
                                   t 
                                   3 
                                 
                                 ) 
                               
                             
                           
                           = 
                           Vdd 
                         
                       
                       ⁢ 
                       
                         
                           ⅈ 
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           t 
                         
                       
                     
                     
                       Vdd 
                       / 
                       2 
                     
                   
                 
               
             
             
               
                 EQ 
                 . 
                 
                     
                 
                 ⁢ 
                 3 
               
             
           
         
       
     
   
   C 1  represents the pin capacitance before the CMOS gate switches, while C 2  represents the pin capacitance after the gate switches. SOLDPC modeling may be generalized to an n-piece pin capacitance model for each slew/output load combination. 
   OBJECTS AND SUMMARY OF THE INVENTION 
   Accordingly, it is an object of the present invention to provide a method and a system for generating a filter from slew and output load dependent pin capacitances and stitching the resultant filter to the original interconnect network. 
   It is another object to provide a highly accurate model of slew and output load dependent pin capacitances applicable to VLSI circuits using a linear time-invariant filter (e.g., an RC/RLC network). 
   It is still another object to use the resultant stitched filter to calculate delays from the input terminal of a first gate, and the output of which is connected to an input of a second gate. 
   It is yet another object to obtain the computed pin capacitances without recurring to an iterative process. 
   These and other objects, aspects and advantages of the invention are achieved by a method of modeling and employing the CMOS gate&#39;s slew and output load dependent pin capacitance during timing analysis. The method preferably uses characterization tables or libraries, synthesizing a new model that is fully adaptable to any static timing analysis without resorting to the necessary computer resources required by an iterative approach. 
   In an embodiment of the present invention there is provided a method of modeling an input slew and output load dependent CMOS gate pin capacitance in a modeled plurality of gates and interconnects representing an integrated circuit, the method including the steps of: a) determining a response of the gate pin capacitance to its input slew and output load; b) synthesizing a linear time-invariant filter that matches the response; c) extending the interconnect model to include the synthesized time-invariant filter; and d) inputting the extended interconnect model into a static timing analysis for determining timing behavior between a gate input and each of its fan-out gates. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The novel features believed characteristic of the invention are set forth in the appended claims. The invention itself, however, as well as a preferred mode of use, further objects and advantages thereof, will best be understood by reference to the following detailed description of an illustrative embodiment when read in conjunction with the accompanying drawings, where: 
       FIG. 1  is a diagram showing a prior art design that consists of two gates connected by an interconnect used to illustrate the problem solved by the present invention. 
       FIG. 2  is a diagram showing the representation of slew and output load dependent pin capacitances in current source modules represented in ECSM and CCS library formats. 
       FIGS. 3   a  and  3   b  are tables illustrating examples of library characterized slew and load dependent pin capacitance (SOLDPC) represented in ECSM and CCS library formats. 
       FIG. 4  shows is a flow chart illustrating the steps involved in timing circuits containing gates which have pins that have slew and load dependent pin capacitances. 
       FIG. 5  illustrates a compact linear RC network being used as a replacement for SOLDPC. 
   

   DETAILED DESCRIPTION 
   The present invention and the various features and advantageous details thereof are explained more fully with reference to the non-limiting embodiments that are illustrated in the accompanying drawings and detailed in the following description. Descriptions of well-known components and processing techniques are omitted so as to not unnecessarily obscure the present invention in detail. 
   Problem Statement 
   Referring back to prior art  FIGS. 1   a - 1   b , the delay  101  of a CMOS gate  108  depends on the loading at the output pin  119  of the CMOS gate. The gate input-pin capacitance  106  at each sink of the interconnect network connected to the pin  119  is required to obtain this loading. In the case of a constant pin capacitance, this capacitance is advantageously used to compute the effective loading of the net between  119  and  106 . However, in the case of a slew and output load dependant pin capacitance, the input pin capacitance at pin  106  of the fan-out gate  107  requires the transition time of the signal  109  to reach it. The transition time computation requires the transition time  103  at pin  119 . The transition time  103  further necessitates determining the load on the net between pins  119  and  106 , which in turn, depends on the pin capacitance at pin  106 , thus causing a circular dependency. 
   Conventional brute force approaches generally employ a methodology wherein an initial value of the input slew at the input pin of the interconnect is estimated and the analysis is performed iteratively until convergence to the capacitance is achieved. This approach is resource intensive both in the computer run-time and memory usage, and further, it greatly increases the STA run-time when designs have extremely large and numerous interconnects. Moreover, in addition to the run-times needed for the conventional approach, there are at least three additional limitations associated with the prior art standard approaches. 1) The industry standard pin capacitances are characterized as idealized ramp waveforms to the input pin of the CMOS gates. However, during timing analysis, the voltage waveform that actually arrives at the input pin of a gate does not match the characterized ramp waveform. This difference, between the idealized and actual waveform will introduce inaccuracy in the calculations specifically in cases where the actual waveform asymptotically approaches Vdd (a.k.a., long tail voltage waveform); 2) the size of the SOLDPC will particularly challenging, specially for variation-aware libraries in which the SOLDPC needs to be characterized as a function of the all different process-voltage-temperature (PVT) sources of variation; and 3) the slew and output load dependent pin capacitances cannot be easily used in today&#39;s industry gate and interconnect timing analysis tools, in particular, when SOLDPC necessitates special handling which can only complicate today&#39;s timing methodologies. 
   The present invention addresses all the above-mentioned problems by providing an efficient methodology that handles SOLDPC during timing and noise analysis flow. As will be described hereinafter, the steps of the inventive methodology represent the SOLDPC table as a compact, efficient, and easy-to-use linear time invariant filter. 
   Regarding the input pin capacitance, fast rising signals will view a certain capacitance, while slow rising signals are viewed as a different capacitance. This capacitance can be seen as the response of a linear system 
             c   ⁡     (   t   )       =     ⅈ     v   .             
or in the frequency domain
 
   
     
       
         
           
             C 
             ⁡ 
             
               ( 
               ω 
               ) 
             
           
           = 
           
             I 
             
               jω 
               ⁢ 
               
                   
               
               ⁢ 
               V 
             
           
         
       
     
   
   The “slew and output load dependent” effect is preferably simulated using a linear filter to model the pin capacitance, which can be explained as follows: a fast rising ramp has a spectrum containing more energy in the high frequencies compared with a slowly rising ramp. A filter can be designed to respond differently to the various frequency components of the input signal and using standard filter design techniques, a filter can be designed that implements the desired “slew and output load dependent” effect. Furthermore, standard filter synthesis techniques can be advantageously employed to construct the filter out of passive components, e.g., resistors, capacitors, and inductors. 
   Referring now to  FIG. 4 , the steps that are preferably used to describe the present invention are illustrated by way of timing circuits consisting of gates with pins having slew and load dependent pin capacitances. 
   In step  400 , each gate whose pin capacitance behavior is slew and/or load dependent is processed to determine the desired filter response from the SOLDPC table. Next, in step  401 , a linear filter to realize the desired response in step  400  is designed. Next, in step  402 , the filter is synthesized in terms of a resistive/inductive/capacitive network. Next in step  403 , the gate input model is replaced with the filter that was synthesized in step  402 . Finally, in step  404 , the revised model is now employed to perform the STA. 
   For illustrative purposes, the compact linear RC network shown in  FIG. 5  will be employed as a replacement for SOLDPC. The model to be used is characterized as having the following characteristics: 
   C i  represents the effective input pin capacitance of the CMOS gate to ground. 
   C m  represents the effective Miller capacitance 
   C o  represents the effective output pin capacitance of the gate to ground 
   R m  represents effective resistance of the driver seen from the input pin of the gate. 
   The following steps describe the technical basis for the representation of the SOLDPC as a compact linear network. Given a ramp voltage waveform with transition time, T, the charge injected to the proposed model from voltage level 0 to Vdd is represented by the following equation: 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         Q 
                         ⁡ 
                         
                           ( 
                           T 
                           ) 
                         
                       
                       = 
                         
                       ⁢ 
                       
                         
                           V 
                           dd 
                         
                         ⁢ 
                         
                           { 
                           
                             
                               C 
                               i 
                             
                             + 
                             
                               C 
                               0 
                             
                             - 
                             
                               
                                 ∫ 
                                 
                                   t 
                                   = 
                                   0 
                                 
                                 
                                   t 
                                   = 
                                   T 
                                 
                               
                               ⁢ 
                               
                                 
                                   [ 
                                   
                                     
                                       
                                         C 
                                         o 
                                         2 
                                       
                                       
                                         T 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               C 
                                               o 
                                             
                                             + 
                                             
                                               C 
                                               m 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       ⅇ 
                                       
                                         
                                           - 
                                           t 
                                         
                                         
                                           
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                                             m 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             
                                               
                                                 C 
                                                 m 
                                               
                                               + 
                                               
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                                   ] 
                                 
                                 ⁢ 
                                 
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                       = 
                         
                       ⁢ 
                       
                         
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                         ⁢ 
                         
                           { 
                           
                             
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                               i 
                             
                             + 
                             
                               C 
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                                     m 
                                   
                                   ⁢ 
                                   
                                     C 
                                     o 
                                     2 
                                   
                                 
                                 T 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   1 
                                   - 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ⅇ 
                                     
                                       
                                         - 
                                         T 
                                       
                                       
                                         
                                           R 
                                           m 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               C 
                                               m 
                                             
                                             + 
                                             
                                               C 
                                               o 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                           } 
                         
                       
                     
                   
                 
               
             
             
               
                 EQ 
                 . 
                 
                     
                 
                 ⁢ 
                 4 
               
             
           
         
       
     
   
   If the same voltage waveform with transition time of T is applied to an effective pin capacitance with the value of C pin (T), the charge that feeds into this capacitance can be represented as:
 
 Q   pin ( T )= C   pin ( T ) V   dd   EQ. 5
 
   Based on the definition of effective pin capacitance (c.f. Equations 2 and 3), the charges in EQ. 4 and EQ. 5 are equated, and the effective pin capacitance as a function of T is written as: 
   
     
       
         
           
             
               
                 
                   
                     C 
                     pin 
                   
                   ⁡ 
                   
                     ( 
                     T 
                     ) 
                   
                 
                 = 
                 
                   
                     C 
                     i 
                   
                   + 
                   
                     C 
                     o 
                   
                   - 
                   
                     
                       
                         
                           R 
                           m 
                         
                         ⁢ 
                         
                           C 
                           o 
                           2 
                         
                       
                       T 
                     
                     ⁡ 
                     
                       [ 
                       
                         1 
                         - 
                         
                           ⅇ 
                           
                             
                               - 
                               T 
                             
                             
                               
                                 R 
                                 m 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     C 
                                     m 
                                   
                                   + 
                                   
                                     C 
                                     o 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 EQ 
                 . 
                 
                     
                 
                 ⁢ 
                 6 
               
             
           
         
       
     
   
   The above equation represents the effective pin capacitance as a function of slew, T, C i , C m , R m , and C o . Given a row of slew dependent pin-capacitance from a SOLDPC, C i , C m , R m , and C o  can be calculated, such that the above model will respond the same as the slew dependent pin capacitance. 
   To model the dependency of the proposed model on the output load assumes a linear dependency of the R and C elements to the output load as shown in  FIG. 5(   b ). The new model consists of C i , C m , C o , R m , and four linear coefficients, c i , c m , c o , and r m , respectively, which represent the sensitivity of C i , C m , C o , and R m  with respect to changes in output load. Note that the sensitivity of SOLDPC to output load is generally small. Since the sensitivity to output load is minimal, the use of linear sensitivity will not introduce significant accuracy issues. However, the proposed model is not restricted to linear sensitivity and one can use higher order sensitivities for more accuracy. 
   In order to synthesize the proposed model from the SOLDPC table, the following two techniques are preferably used: 
   Use curve fitting techniques for model synthesis. This is a very viable technique since it runs only during library characterization, and the cost of curve fitting does not affect the actual STA timing runs using the curve fitted data. Curve fitting techniques including least-square fitting (LSF) can be used to synthesize the parameters of the proposed model, (i.e., C i , C m , C o , R m , c i , c m , c o , and r m ) from the SOLDPC data using the following problem formulation: 
   
     
       
         
           min 
           ⁢ 
           
             
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                 2 
               
             
           
         
       
     
     
       
         where 
       
     
     
       
         
           
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           = 
           
             
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             ⁢ 
             
                 
             
             ⁢ 
             
               C 
               L 
             
           
           = 
           
             
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               L 
             
             - 
             k 
           
         
       
     
       
       
         
           where k is a user-specified value. 
         
       
     
  
   Using an efficient look-up method which is fast and can be used dynamically during timing analysis. 
   Consider the following facts: 
   Based on EQ. 6, 
               lim     T   →   0       ⁢           ⁢       C   pin     ⁡     (   T   )         =       C   i     +         C   m     +     C   o           C   m     +     C   o                 
In addition,
 
               lim     T   →   ∞       ⁢           ⁢       C   pin     ⁡     (   T   )         =       C   i     +     C   o             
Assuming C m &lt;&lt;C o ; the
 
               lim     T   →   0       ⁢           ⁢       C   pin     ⁡     (   T   )         =         C   i     +         C   m     ⁢     C   o           C   m     +     C   o           ⁢     =     if   ⁢           ⁢     C   m     ⁢     &lt;&lt;     C   o           ⁢       C   i     +     C   m               
If the above terms are substituted in Eq. 6, the following equations are obtained:
 
   
     
       
         
           
             
               C 
               pin 
             
             ⁡ 
             
               ( 
               T 
               ) 
             
           
           = 
           
             
               
                 lim 
                 
                   T 
                   → 
                   ∞ 
                 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                   C 
                   pin 
                 
                 ⁡ 
                 
                   ( 
                   T 
                   ) 
                 
               
             
             - 
             
               
                 
                   
                     R 
                     m 
                   
                   ⁢ 
                   
                     C 
                     o 
                     2 
                   
                 
                 T 
               
               [ 
               
                 1 
                 - 
                 
                   ⅇ 
                   
                     
                       - 
                       T 
                     
                     
                       
                         R 
                         m 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               
                                 
                                   
                                     lim 
                                     
                                       T 
                                       → 
                                       0 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       C 
                                       pin 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       T 
                                       ) 
                                     
                                   
                                 
                                 - 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     lim 
                                     
                                       T 
                                       → 
                                       ∞ 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       C 
                                       pin 
                                     
                                     ⁢ 
                                     
                                       ( 
                                       T 
                                       ) 
                                     
                                   
                                 
                                 + 
                                 
                                   2 
                                   ⁢ 
                                   
                                     C 
                                     o 
                                   
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
     
       
         
           
             C 
             m 
           
           = 
           
             
               
                 
                   lim 
                   
                     T 
                     → 
                     0 
                   
                 
                 ⁢ 
                 
                   
                     C 
                     pin 
                   
                   ⁡ 
                   
                     ( 
                     T 
                     ) 
                   
                 
               
               - 
               
                 
                   lim 
                   
                     T 
                     → 
                     ∞ 
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     C 
                     pin 
                   
                   ⁡ 
                   
                     ( 
                     T 
                     ) 
                   
                 
               
               + 
               
                 
                   C 
                   o 
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   C 
                   i 
                 
               
             
             = 
             
               
                 
                   lim 
                   
                     T 
                     → 
                     ∞ 
                   
                 
                 ⁢ 
                 
                   
                     C 
                     pin 
                   
                   ⁡ 
                   
                     ( 
                     T 
                     ) 
                   
                 
               
               - 
               
                 C 
                 o 
               
             
           
         
       
     
   
   For a fixed output load, two intermediate slew values, T 1 , and T 2  are specified, and the corresponding pin capacitance values C i (T 1 ) and C pin (T 2 ) obtained by using the following equations, one can calculate the C i , C m , C o , and R m  values: 
   
     
       
         
           
             
               
                 C 
                 pin 
               
               ⁡ 
               
                 ( 
                 
                   T 
                   1 
                 
                 ) 
               
             
             - 
             
               
                 lim 
                 
                   T 
                   → 
                   ∞ 
                 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                   C 
                   pin 
                 
                 ⁡ 
                 
                   ( 
                   T 
                   ) 
                 
               
             
           
           = 
           
             - 
             
               
                 
                   
                     R 
                     m 
                   
                   ⁢ 
                   
                     C 
                     o 
                     2 
                   
                 
                 
                   T 
                   1 
                 
               
               [ 
               
                 1 
                 - 
                 
                   ⅇ 
                   
                     
                       - 
                       
                         T 
                         1 
                       
                     
                     
                       
                         R 
                         m 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               
                                 
                                   
                                     lim 
                                     
                                       T 
                                       → 
                                       0 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       C 
                                       pin 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       T 
                                       ) 
                                     
                                   
                                 
                                 - 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     lim 
                                     
                                       T 
                                       → 
                                       ∞ 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       C 
                                       pin 
                                     
                                     ⁢ 
                                     
                                       ( 
                                       T 
                                       ) 
                                     
                                   
                                 
                                 + 
                                 
                                   2 
                                   ⁢ 
                                   
                                     C 
                                     o 
                                   
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
     
       
         
           
             
               
                 C 
                 pin 
               
               ⁡ 
               
                 ( 
                 
                   T 
                   2 
                 
                 ) 
               
             
             - 
             
               
                 lim 
                 
                   T 
                   → 
                   ∞ 
                 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                   C 
                   pin 
                 
                 ⁡ 
                 
                   ( 
                   T 
                   ) 
                 
               
             
           
           = 
           
             - 
             
               
                 
                   
                     R 
                     m 
                   
                   ⁢ 
                   
                     C 
                     o 
                     2 
                   
                 
                 
                   T 
                   2 
                 
               
               [ 
               
                 1 
                 - 
                 
                   ⅇ 
                   
                     
                       - 
                       T 
                     
                     
                       
                         R 
                         m 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               
                                 
                                   
                                     lim 
                                     
                                       T 
                                       → 
                                       0 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       C 
                                       pin 
                                     
                                     ⁢ 
                                     
                                       ( 
                                       T 
                                       ) 
                                     
                                   
                                 
                                 - 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     lim 
                                     
                                       T 
                                       → 
                                       ∞ 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                       C 
                                       pin 
                                     
                                     ⁢ 
                                     
                                       ( 
                                       T 
                                       ) 
                                     
                                   
                                 
                                 + 
                                 
                                   2 
                                   ⁢ 
                                   
                                     C 
                                     o 
                                   
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
     
       
         
           
             C 
             m 
           
           = 
           
             
               
                 
                   lim 
                   
                     T 
                     → 
                     0 
                   
                 
                 ⁢ 
                 
                   
                     C 
                     pin 
                   
                   ⁡ 
                   
                     ( 
                     T 
                     ) 
                   
                 
               
               - 
               
                 
                   lim 
                   
                     T 
                     → 
                     ∞ 
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     C 
                     pin 
                   
                   ⁡ 
                   
                     ( 
                     T 
                     ) 
                   
                 
               
               + 
               
                 
                   C 
                   o 
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   C 
                   i 
                 
               
             
             = 
             
               
                 
                   lim 
                   
                     T 
                     → 
                     ∞ 
                   
                 
                 ⁢ 
                 
                   
                     C 
                     pin 
                   
                   ⁡ 
                   
                     ( 
                     T 
                     ) 
                   
                 
               
               - 
               
                 C 
                 o 
               
             
           
         
       
     
   
   To calculate the sensitivity of the model parameters to the output load, one can use two different rows of the  FIG. 3  (preferably first row shown as  301  and last row shown as  303 ), and find the model parameters for each output load using the above procedure and employ finite differencing technique to calculate the sensitivities of the model parameters to output load (i.e., C i , C m , c o , and r m ). 
   The interconnect network can be extended using the attached interconnect network and be used for gate delay and slew calculation (i.e. without using any iteration) as well as the interconnect delay and slew calculation. 
   The timing information in step 7 is preferably used to perform the static timing analysis. 
   Finally, the present invention can be realized in hardware, software, or a combination of hardware and software. The present invention can be realized in a centralized fashion in one computer system or in a distributed fashion where different elements are spread across several interconnected computer systems. Any kind of computer system—or other apparatus adapted for carrying out the methods described herein—is suitable. A typical combination of hardware and software could be a general purpose computer system with a computer program that, when being loaded and executed, controls the computer system such that it carries out the methods described herein. 
   The present invention can also be embedded in a computer program product, which comprises all the features enabling the implementation of the methods described herein, and which—when loaded in a computer system—is able to carry out these methods. 
   Computer program means or computer program in the present context mean any expression, in any language, code or notation, of a set of instructions intended to cause a system having an information processing capability to perform a particular function either directly or after conversion to another language, code or notation and/or reproduction in a different material form. 
   While the present invention has been particularly described in conjunction with exemplary embodiments, it is evident that many alternatives, modifications and variations will be apparent to those skilled in the art in light of the present description. It is therefore contemplated that the appended claims will embrace any such alternatives, modifications and variations as falling within the true scope and spirit of the present invention.