There's at least one major design theme in engineering and nature that recurs as often as the 15 percent energy alternative that I ran across a dozen years back. It's the 4:1 ratio and the four-stage design and, by extension, a class of problems whose solution interval lies in the range of three to four.
I've known about it awhile but again became conscious of it two weeks ago as I breadboarded a transistorized 50 kHz phase-shift oscillator for aligning the intermediate-frequency (IF) stage in an old communications receiver. “What's the optimal number of RC sections for a working oscillator?” I wondered. “Using variable R and C elements, can I relate the range of frequencies over which I can tune the oscillator to the amount of RC loading that might impinge on the transistor's stage gain?”
Well, in practice each RC section would provide a phase shift up to (but not quite) 90 degrees. The oscillator's inverting active stage flips the signal 180 degrees. It seems to me that unless it's a very special case I haven't yet seen, you'd need at least three RC sections. That's the design I most often see. Four or more would work, but would not really be necessary unless you wanted, say, more stability or some such feature. I haven't yet determined if my second question about the effect of element loading on stage gain (and thus oscillation) has any practical meaning. Now we could argue that all kinds of design ratios populate engineering. But I'm seeing that it's a challenge to find one that pops up as frequently as the 4:1. Coincidence or science? Maybe a bit of both.
What examples can we draw from? The one I'm most familiar with has to do with the RF choke in various bypassing/decoupling applications. One of the most widely applied rules-of-thumb in basic design, it says that at the lowest operating frequency, the choke's reactance should be at least four times the impedance of the circuit it connects to. Otherwise, circuit mismatches and excessive power losses accrue.
Not surprisingly, these ideas aren't confined to just electronics. This rule also finds its way in nutrition and sports medicine. The research reportedly supports replenishing the body with a 4:1 ratio of carbohydrates to proteins after a workout, although there's some debate, and some experts favor a 3:1 ratio.
Apart from the rather large number of applications that stem from the quarter-wavelength transmission line, we also have the “quarter-length” rule for setting up the straight, non-supporting wall ladders we see painters and construction people use every day. The optimum set-up angle (pitch), which represents a tradeoff between the ladder's strength, resistance to sliding, and the climber's balance, is a little more than 75 degrees. Thus the vertical distance from the bottom of the wall to the point where the ladder is supported, divided by the set-back distance (the horizontal distance from the bottom end of the ladder to the wall), is 4.
Then there are a load of subjective applications. Photography is one of them, and one article discussing the so-called “optimum lighting ratio” shows that it averages out at 3:1 to 4:1 for portrait work (Lighting Ratios to Make or Break your Portrait).
Another example is the typical communications receiver, which is usually equipped with an S-meter for estimating received signal strength. The old 9-step S-unit system still in use has its roots in the early days of radio, where for example an S-5 signal is “fairly good,” S-6 is “good,” and S-7 is “moderately strong.” Even S-meter readings on all but the most carefully calibrated and matched-impedance equipment today are basically subjective (an IARU standard in the early 1980s quantified the definitions). Maybe I've missed the point, but it always struck me as curious that an increase or decrease in one S-unit is defined as 6 dB. And 6 dB represents a fourfold change in received power.
There's at least one application where probability and confidence levels definitely come into play — metrology. With metrology (the calibration of test and measurement equipment), a 4:1 test uncertainty ratio may be the most desired standard for calibration labs (Metrology Concepts: Understanding Test Uncertainty Ratio (TUR)). It's worth noting that, statistically speaking, many of these examples don't seem to have anything to do with the normal distribution.
The results of one article discussing the aforementioned ladder issue (Factors affecting extension ladder angular positioning) included empirical info on the users' probability of setting a ladder at less than optimum angle. So now I'm still wondering if the 4:1 rule is linked or not linked to the 15 percent factor. You'll have to tell me. If there's a connection, it might confirm the natural order of how things work for some yet undefined class of problems. That would suggest a reverse engineered approach for a wide range of applications where the engineer acknowledges the universality of the 4:1 effect and establishes some quick design paradigms based on it.
Maybe someone out there had previously figured out the mystery of the 4:1 phenomenon. “…but I didn't know it. The way to have adventure is to do things at a lower level, it's not to ride the freeway and to stop at the Holiday Inn,” to quote physicist Richard Feynman. My adventure and discovery this time came from expanding on the lowly phase-shift oscillator.
In your design engineering work, have you encountered this or other similar rule-of-thumb ratios?