# Unit Conversion and Significant Digits

Quote of the Day

Negative results are just what I want. They're just as valuable to me as positive results. I can never find the thing that does the job best until I find the ones that don’t.

— Thomas Edison. This comment reflects the experimental nature of Thomas Edison's work style. Tesla used to complain that a little bit of analysis would have saved Edison huge amounts of time and expense.

Figure 1: Typical PCB with High-Pin Count Parts.

I saw a Printed Circuit Board (PCB) this morning that had a classic layout error that was caused by not using enough significant digits when converting between US customary units and metric units.

Figure 1 shows a typical PCB, which often contain parts with rows of pins/pads that are supposed to be separated by a fixed distance, called the pitch (symbol P), which is illustrated in Figure 2.

Figure 2: Example of Electronic Component Dimensions.

I have seen this problem a couple of times a year every year for the last 25 years. In the old days, both the electronic components and the PCBs were designed using US customary units, so there were no unit conversion issues. Today, all new parts are designed using metric units, but the PCBs are often designed and built using US customary units. This means that unit conversion is part of the design process.

I REALLY do not like US customary units and if I ruled the world everything would be metric. However, some older parts are still used that were designed using customary units and many people in the US prefer to use units they are familiar with. This means that unit conversion is a fact of life.

My policy is that all unit conversions are done with five significant digits. Unfortunately, the CAD layout software uses three decimal places by default and the CAD person forgot to change the default value. A pitch of 0.6 mm (exact) converted to inches using three decimal places means there were only two significant digits (0.024 in). It provides little consolation to know that we are not the only folks to have unit problems (i.e. NASA Mars Climate Orbiter).

I will illustrate the issue using the 200 pin connector (100 pins on a side) shown in Figure 3.

Figure 3: Connector with 0.6 mm Pitch with 100 Pins on a Side (200 Pins Total).

Problems arise when you have a large number of pins in a row (e.g. 100) and there is a small error in the conversion from mm to inches. That error gets multiplied by 99 (i.e. the 100th pin is 99 pins distant from the first) over the length of the connector and can result in the connector pins/pads not matching up with the PCB holes/pads.

Figure 4 contains an analysis that shows that using two significant digits (three decimal places) results in significant pin misalignment but five significant digits (six decimal places) has minimal error.

Figure 4: Calculation That Shows Two Significant Digits Are Not Sufficient.

It is not unusual to have more than 100 pins on a side, which just makes the problem worse.

## Postscript

Many thanks to Paul Campbell of Beloit College for some excellent comments that were privately emailed.

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