began working with circuit designs, were the now

commonly applied techniques of Monte Carlo

simulations combined with traditional Response

Surface and Screening Experiments. In addition, when

the design problem was small and the process was

more forgiving, we also employed space-filling grids

with a "pick the winner" methodology

While these methods were generally adequate for the

early 1990's it became clear in the late 90's to early

2000's that newer, more efficient methods were

required as design goals became more complex and

process constraints become more onerous.

setting 50 to 100 key design parameters to optimize 4 to

8 performance goals while controlling for the variation

of hundreds of process related parameters and their

impact on both the design and yield.

In addition, it was recognized that the usual "process

corners" were not the worst and best case corners for

the circuits being designed. Each circuit

to it's own set of corners and must be made as robust

as possible to all process variation, not just the FF, SS,

SF, and FS corners proposed in the SPICE Models.

We began looking more deeply at the problem we were

trying to solve. All the common methods fail to

adequately address the more complex design problems:

**Traditional Monte Carlo cannot efficiently examine****more than a few parameters even with millions of****runs. No one can afford to make millions of runs.**

**Traditional Response Surface methods did not****adequately capture the SPICE responses even for****a few design parameters. The accuracy was****simply not good enough!**

**Screening experiments were even worse than the****traditional response surface m****ethod****s. While they****cut down on the number of design parameters and****ma****d****e****r****esponse****s****urface analysis****possible****, they****tended to ignore critical interactions -- especially****interactions between the design and the process.**

were possible, all we needed for a specific circuit and

technology was a "local simulator." This is the basic

idea behind the surrogate model -- it replaces SPICE for

a specific circuit and technology.

We also recognized that the equations behind the

surrogate model must be able to capture sufficient

complexity to be accurate while retaining the property of

being able to be evaluated rapidly -- and even more

importantly it must be able to be searched for analytic

maxima and minima. The prototypical method for

achieving these goals is a Taylor Series expansion.

However, we derive this expansion from actual SPICE

runs not from the SPICE equations themselves.

In addition, we solved the problem of being able to

sample the process space between 3 and 5 sigma while

building the surrogate model. This is critical because

extremes -- but the surrogate models must hold both for

the center and the extremes. The prototypical method

would be a grid, but grids are untenable for more than a

few parameters. Our sample is similar to a sparse grid

with an emphasis on higher sigma samples and less

emphasis on the center.

This same sampling solution permits a yield analysis

through stochastic integration when the focus is on

relatively "rare" failures -- those sample points where

one or more of the process parameters must be beyond

more than a couple of sigma.