As I view it, analysis falls into four categories: Rules of thumb, analytical approximations, numerical simulations, and derivation from first principles. Generally, most systems can be broken down and expressed in first- and second-order equations. The wonderful thing about circuit design is how much effort has gone into making things linear, or at least have piecewise linear approximations (first-order for you mathy folks). We even go so far as to change the scale we reference just to make it linear.
I view rules of thumb as the next step in simplification; it's a quick-and-dirty ballpark result. It gives insight into a complex system without too much computation. Analytical approximations are sets of equations that one can jot down on paper and use a calculator to solve. More involved than a rule of thumb but not as complex as a full numerical simulation. And, of course, the simulation takes far less time than breaking down a problem from first principles and deriving the equations yourself!
Take, for instance, the impedance of a capacitor. Broken up into a piecewise function, the impedance decreases following a 20dB per decade slope to the point the parasitic resistance begins to dominate, and then it's flat through that region. It remains flat until the parasitic inductance begins to dominate, which causes the impedance to increase at 20dB per decade. The concept of capacitor impedance is broken down into three easy-to-understand regions, so if one has a basic understanding of filters, one can look at a passive network of R’s and C’s and picture the look of what the frequency response is!
But where does it go wrong? I'm sure there are other scenarios, but these are the two I've encountered most frequently.
Some learn a rule of thumb and interpret it as gospel. They will follow the teaching of this rule to the end and blame anything and everything except for what they designed using the rule.
Some learn a rule of thumb but don't understand the limitations of where it's valid (I recall a math teacher expressing extreme displeasure over the small-angle approximation, going so far as to call it an abomination toward the subject).
The cause seems to be not understanding where the rule of thumb came from, and therefore either not understanding it's an oversimplification of a more complex concept or the assumptions placed on applicable conditions around it!
Firing up your full-wave solver can generate some highly accurate results, but will take a while and may be overkill when all you need is to know the voltage drop in a resistor divider. Rule of thumb are a tradeoff, quick and possibly inaccurate insight into a complex system, but applicable where, in the words of Eric Bogain, "an ok answer NOW! is better than a good answer, late."
Rules of thumb are a great starting point. They give insight into a complex system and can influence high-level design choices. The rules can even dictate if it's important to revisit decisions further down the design process with more complex analysis. But ultimately, it's important to understand where a rule came from, where it's applicable, and what assumptions are made in the use of said rule.
Start with rules of thumb early in the design process, state your assumptions, and as you move down the line analyze with more complex methods and iterate the design.
The phrase rule of thumb refers to an approximate method for doing something based on practical experience rather than theory. Origins can be traced back to the 17th century in association with various trades where quantities were measured by comparison to one's thumb, sort of like the cubit.
Some claim the etymology of this term from the 18th century as the maximum width one is legally allowed to beat his (or her, but unlikely at the time) wife with. The story goes that an 18th-century judge, Sir Francis Buller, stated that a man may beat his wife with a stick no wider than his thumb. However, there is no record that he made such a statement. The phrase later became associated with domestic abuse in the 1970s when the spurious legal definition was cited as factual in several law journals, and a report on domestic abuse, “Under the Rule of Thumb,” was released in 1982. Because of this, I’ve occasionally been told not to use the phrase as it condones violence toward women.
Thankfully, given an understanding of the origin of this rule of thumb, the etymology fallacies that surround it, and a basic understanding of the limitations of where it’s valid, I feel comfortable putting this writing in the public domain! If you're interested, Wikipedia has far more detail on the matter (I know, I know, you can’t cite Wikipedia, but I’m going to anyway).
Bogatin, E. (2020, March 4). Using rules of Thumb wisely: Rules of thumb #0. EDN. Source.