6.1.3 Corner Parameter Values Determination



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6.1.3 Corner Parameter Values Determination

Once the worst case conditions of the selected performance objective have been calculated, the inverse problem that consists of determining the input variables values that will produce these conditions needs to be solved. As an example, for the corner the statistical input variables must be determined such as the current ratio satisfies the equation:

 

This is an under-determined problem that could have an infinite number of solutions. In [24], a method to resolve this uncertainty is proposed. It is suggested that the most probable input vector with minimum probability distance be selected. This corresponds to a maximum value of the multinormal jpdf of the input variables which is reached when the argument of the exponential in (2.63) is at its minimum:

where X is the vector of statistical variables with mean and covariance matrix C:    

The function denotes the empirical relationship represented in Equation 6.7. The above problem can be solved by nonlinear programming to determine the values of the statistical input variables at the corners. For example, Table 6.3 lists the input vectors for the and corners as determined by the maximum probability method.  

 
Figure 6.2: Simulated and distributions for a m CMOS technology.  

 
Table 6.3: Values of the statistical input variables at the FS and SF corners.  

Using the above procedure, the values of the principal statistical variables that will result in the performance function limits can be determined. A complete specification of the design process variation space becomes possible. This is illustrated for and in Fig. 6.2 where the center and corners are superimposed on the simulated distribution.

It is important to note that an identical approach can be applied to other circuit level performance functions in case the use of the P- and N-channel saturation currents is not an adequate objective function.



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Next: 6.2 Analytical Model Parameter Up: 6.1 Statistically Based Worst Previous: 6.1.2 Worst Case Condition



Martin Stiftinger
Tue Aug 1 19:07:20 MET DST 1995