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40 DESIGN007 MAGAZINE I NOVEMBER 2022 tributed in the substrate is that as a designer you must look at the statistical effect on your design. How much variation is there and how much is that likely to affect prop delay, imped- ance, or insertion loss over a typical produc- tion run? Monte Carlo Gives a Reality Check It is easy to focus on one parameter and how that will impact the desired electrical charac- teristic, however, a far better approach is to set a tolerance for each primary dimension which may impact the circuit, then run a Monte Carlo analysis. is allows you to simulate a produc- tion run. I have used characteristic impedance of a 93-ohm differential pair in this example, but the Monte Carlo holds true for any char- acteristic which is impacted by a number of factors. In this example (Figure 1), I set each raw material dimension a 10% tolerance, and then simulated building 250 PCB transmis- sion lines with a normally distributed, ran- domly selected set of parameters. is gives a far better impression of how production will vary over a batch of boards rather than simply focusing on one or two characteristic dimen- sions or looking at the worst case. It does allow you to experiment with which a particular dimension should be more tightly controlled to achieve a higher yield. Figure 1 demonstrates that with a 10% toler- ance on every characteristic over a production run, the majority of impedance results will fall well within 5-6%, but there will be outliers just out of the 10%. In Figure 2 you can see that in this simulation the tightening of the trace separation specs sig- nificantly tightened the distribution with a sig- nificant peak on the nominal. In Figure 3, purely for illustration, I pushed the dielectric constant tolerance out to an exaggerated ±20% and the more benign effects of Er, having only a 1 square root effect, show a minimal flattening when compared with the 10% illustration in Figure 1. ese three illus- trations demonstrate how a signal integrity engineer needs to balance the material prop- erties of a base material with varying constitu- ent properties and use statistics to see whether there is the potential for good yields based on the simulation. Making a good yielding design from materials which have inherent variabil- Figure 1: Monte Carlo of 250 transmission lines with a tolerance of 10% on all parameters.