Estimating is a good first step to ensure you are more likely to be roughly right than precisely wrong.
I have recently become obsessed with (excuse me, I mean “concerned”) the pressure in my car tires. That’s partially due to my being aware of the negative effect of incorrect pressure on fuel efficiency, handling, and other performance issues.
But I have to sheepishly admit it is largely because my car has a digital pressure readout on the console for each of the four tires rather than a single aggregated warning light, with both being consequences of the tire pressure monitoring system (TPMS) mandated for all cars a few years back (Figure 1).
Despite its apparent precision, any digital readout has an inherent error of ±1 in its least significant digit, so when it read “35” it could be “34” or “36.” But still, that display was staring at me. Then I started to wonder about the change in readout with temperature, as I noticed the reading varied by a few psi in the morning (40°F/4.5°C) versus midday (sunny and 60°F/15.5°C) even if the car had not moved.
I decided that rather than look this up on the web, I would instead use it as a brain exercise and do some calculations on the pressure versus temperature shift. I figured a quick application of Boyle’s Law would do it: PV/T = K where P is pressure, V is volume, T is temperature, and K is a constant, which depends on the specifics of the situation.
I could easily measure temperature and pressure, but assessing volume was going to be a problem. I started thinking about doing some nonlinear modeling with interrelated variables using an application such as COMSOL Multiphysics but soon realized I was “overthinking” the problem.
A little reflection made me realize that, unlike a balloon with an expandable surface, the volume of a tire stays fairly constant despite small changes in pressure, as there is a band of steel belts that keeps the tire’s circumference (perimeter) constant. While a badly deflected tire will have a different inner volume, the changes in volume would be negligible for small variations in pressure (Figure 2).
In analog circuitry, this is called the “small signal” approximation and makes the legitimate assumption that except for the one dependent variable of interest, the other factors remain constant when varying the independent variable around a given DC operating point.
Having made my first-order simplification, I did some quick math and found that a change of 20°F(about 11°C) in ambient would result in a change of about 2 psi (0.13 bar) in pressure. Since the result is independent of the tire volume, it also applies to a bicycle tire, which typically has a nominal pressure in the 75-to-100 psi range.
Back to the future
What did I learn from this mental-approximation exercise? Back in the days of B.C. (before computers, before calculators), it was difficult for the average engineer to calculate to more than two significant digits, or three if you were fortunate.
The reason is that in those days, the only “personal computer” was the analog slide rule, which was tricky to use, frustrating for long, chained calculations, and couldn’t do addition or subtraction — although it could do multiplication, division, trigonometric functions, logarithms, exponentials, and other special function (Figure 3). The slide rule quickly became obsolete when Hewlett-Packard introduced the HP-35 scientific pocket calculator in 1972, in a textbook case of disruptive technology.
I’m not going to get misty-eyed about the slide rule: it was limited in functionality and a pain to use, but nonetheless, it enabled engineers and scientists to build bridges and go to the moon. Equally important, it forced its users to do “sanity checks” on the results.
The reason is that the slide rule did not show the location of a decimal point in its results. Instead, the user had to work that out on his or her own. The first thing to do in any complicated calculation using the slide rule was to do a basic estimate of the expected answer to get a sense of where that decimal point should be in the final answer. In other words, the necessity of this chore did have a significant virtue.
In contrast, I’ve seen computer-based engineering analysis that concluded, with no embarrassment, that a capacitor of just a few picofarads was all that was needed to get the ripple out of a multi-watt DC-DC converter’s output when the correct answer is many orders of magnitude larger. No one bothered to step back and say, “About what size should the answer be, based on the numbers and experience?” With a slide rule, that sort of “duh” mistake would be much less likely.
Learning to think about an answer first before diving into precision calculations is almost always a good idea, regardless of the tool you’re using. An up-front, rough approximation will tell you if you have missed something big, have gone off in the wrong direction, the algorithm you’re using has basic errors in its implementation…or perhaps your underlying assumptions are just plain wrong.
Related EE World Content
The automotive Tire Pressure Monitoring System, Part 1: The situation
The automotive Tire Pressure Monitoring System, Part 2: Perspectives
The automotive Tire Pressure Monitoring System, Part 3: Implementation
The automotive Tire Pressure Monitoring System, Part 4: Issues
BLE for TPMS addresses smarter, connector vehicle tires
Miniaturized Transponder Coil With High Sensitivity For Tire Pressure Monitoring Systems
When tire pressure monitoring gets smart
Pouyan Press, “Tire Pressure Checking Framework: A Review Study”
International Research Journal of Engineering and Technology (IRJET), “Automatic Tire Inflation System”
International Slide Rule Museum
Hewlett-Packard, “HP-35 handheld scientific calculator, 1972”