Self-referential abstractions

A quick look at the wacky epistemology of analog circuitry.

When you think about it, every discipline has its own philosophy of knowledge. In mathematics, for example, you gain insights by tearing apart mushy concepts and then rebuilding them from scratch. Can you define numbers better than “the things we count with”? A mathematician has the power conjure them from nothing: the cardinality (the element count) of an empty set (∅) is zero; a set containing the empty set ({∅}) has a cardinality of one. Put these two sets together in a new set ({∅, {∅}}) and you have the concept of two. Reals take a bit more work, but are also within reach — and so are various distinct flavors of infinity.

In most applied sciences, in contrast, the building block of knowledge is a functional abstraction — a black box, or an API in the software engineering parlance. It’s usually counterproductive to look inside. You don’t need to know why Maxwell’s equations for electromagnetism work; you get more done if you take them for granted and build on that. In the same vein, an engineer working on a web application shouldn’t concern herself with the physics of transistors; and a person using Google Docs doesn’t need to know JavaScript.

This brings us to analog electronics. Regular readers know that I write about this topic often — and when you take a step back, you have to appreciate that the field has a very special, self-referential approach to explaining the real world.

It all starts innocently enough. For example, we say that an ideal resistor is nothing more than a simple linear equation: I = V/R. It’s not really a physical device: it’s the concept of admitting current (I) that’s proportional to the applied voltage (V) and inversely proportional to the rated resistance (R). The physical implementation — or even the fundamental meaning of these quantities — is out of scope. The equation is all there is.

In the same vein, an ideal capacitor is just V = I·t/C. It’s the notion of a voltage that’s proportional to the charging current and time (t), divided by the rated capacitance (C). If the glove fits, a different electronic component — say, a piezoelectric crystal — can be described in terms of an ideal capacitor too.

Of course, real life is never this simple. For instance, an actual physical capacitor can’t be charged or discharged arbitrarily fast. This is particularly true for electrolytic capacitors, where the mobility of ions is quite limited. In physics, we’d model this by adding more terms to the capacitor equation. But in analog electronics, we simply break down the real-world capacitor into two ideal components — an ideal capacitor (C) in series with an ideal resistor (R_ESR):

The initial model of a real-world capacitor.

This model is good, but sometimes not enough; at high signal frequencies, the performance is also hindered by parasitic inductance, an effect that has to do with the storage of energy in the magnetic field that extends around any conductor. To model this — you guessed it — we add an imagined ideal inductor in series with the two other components:

Capacitor 2: Electronic Boogaloo.

But then, real-world capacitors tend to self-discharge; this is because the plates of the capacitor are spaced very closely and the internal insulating layer will allow some electrons to “seep through”. The solution is simple; we extend the model by placing a parallel resistor that straddles the ideal capacitor:

Capacitor III: Revenge From Beyond.

We could keep going to model temperature dependence, AC losses, and so on.

If this is one of the most rudimentary components in electronics, how messy does it get in more complex cases? Well, funny story… here’s a model of a single transistor taken from the manufacturer’s datasheet:

A model of a MOSFET.

Yep: the model of a single transistor contains three “virtual” transistors, two diodes, voltage and current sources, and more. On social media, Matt Keeter cracked a good joke:

“Moore's Law means that the number of MOSFETs in a MOSFET doubles every two years.”

It’s not just performance art: the software we use to model circuit dynamics expects it to be done this way. The model is ultimately turned into computer code, but the ancient language these tools use isn’t self-documenting or expressive enough to offer a more streamlined approach.

I don’t hate it, but you gotta admit it is a bit surreal.