This method is proposed to help the designer during the sizing phase. It is aimed at defining the sizing sequence by detecting free parameters, linked parameters and constraints.
The graph is built step by step until it reaches primary parameters. Primary parameters are parameters that do not depend on anything. Among primary parameters are the constants of physics that are useless for sizing, but also geometry and bias parameters that are free for sizing.
The method for building the graph is:
The graph gives a good overview of the sizing sequence, it show loops.
Usually, the non linear equations system complexity makes it impossible to solve it analytically. But at this point, all the parameters have numeric values and it is possible to solve the system numerically. The point is to find a multiple variables, non linear solver...
Fortunately, such a solver is always available in an IC designer's environment, even though it is usually not used explicitly for that purpose. This tool is an electrical-behavioral simulator. The simulator can be used in the DC domain with an “analog” approach. The quotes indicate that the word analog is used in its original meaning. In the “analog” domain, voltages and currents may figure other physical data such as speed or frequency or length or whatever parameter.
The free parameters can be swept so as to see how circuit performance changes. It is then often possible to reach a reasonable sizing after a couple of experiments. These simulations are extremely fast, so many trials can be made very quickly.
The curves shape not only can give parameters value to reach the target performance but it also gives information on the sizing robustness. It is a good practice to size a circuit so that performance does not change much with the parameters values. If a performance vs. parameter curve has a sharp region and a soft region, sizing in the soft region is more robust.