Master Systems Thinking: Levels and Inversion

A meditation on a simple electronic circuit:

There’s a direct current (DC) source, a switch, and a light bulb. Not that different from a lot of grade school experiments. Close the switch, and the light turns on, right?

Even if you’ve never studied electronics, you probably have a mental model of how this circuit would perform. You might even start thinking about other modifications you could make to the circuit. For example, replacing the bulb with two bulbs in a string. Question: Would the circuit with two bulbs glow brighter, dimmer, or the same as the circuit with the lone bulb? Think about your answer for a bit before reading on. Answer at the end of the article.

Here’s a more difficult analysis–what would happen if the DC power source was replaced with an AC source? This involves a deeper level of analysis.

Possibly you’re aware that a purely resistive load, like a light bulb, behaves “the same” on AC or DC. Though if you’re especially clever (or if you HAVE studied electronics), you’ll note that changing DC to AC is an incomplete description of a change to the circuit–it introduces new variables. AC stands for alternating current, and the rate of alternation, or frequency, matters a great deal for subsequent analysis.

In the US, the household power grid operates at 60Hz which means 60 cycles per second. That’s fast enough that inductive effects (like from motors) and capacitive effects (like from many LED bulbs) need to be accounted for. Unlike the first circuit, which can be reasoned about in a “steady state” condition, AC is constantly changing, which unleashes a lot more complexity. (For example, did you know that “imaginary” power is an actual thing?)

But there are a lot more possible frequencies than 60Hz. How about this one?

In this variation, the source is now AC at 1,000,000,000Hz. At that frequency, irrelevant details like the size and shape of the wires, and even the internal construction of the switch, become critical parts of how the circuit operates. Capacitors, which can be disregarded as open circuits in a steady-state analysis, behave increasingly like conductors at higher frequencies. Likewise, inductors–including simple wires–behave increasingly like open circuits at higher frequencies. We are in the realm of RF, or radio-frequency, which befuddles even many experienced electric engineers because so many base assumptions get flipped on their heads.

When looking at any system, but especially complex ones, it’s worth the effort of enumerating your assumptions to think about conditions that could invalidate or even invert them. You may need to repeat this process at different levels of analysis.

OK, back to the initial circuit. With two bulbs in a string, would they glow brighter, dimmer, or no different than before? If you said ‘dimmer’ then you have a good model based on the common situation of voltage sources, like a battery. But sorry (not sorry) I’ve tricked you! The source in the first circuit isn’t a battery, it’s a constant current source. In a closed circuit it will produce the same current and the two bulbs in series would glow equally bright as if there were only one. (Though with the switch open, as shown, the current source would struggle mightily to push current through the enormous resistance of the open switch, and in so doing produce extremely high voltage–another inversion from a baseline model.)

The AC circuit shown doesn’t specify the voltage of the source, but a measurable amount of current would flow through the parasitic capacitance of the switch. But probably not enough to light the bulb.

All models are wrong. Some are useful. Many hide surprises.

In your everyday life, look for more examples like what we’ve talked about today.

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This post 100% human-written. Circuit diagrams are screenshotted from the iCircuit program, available in the Mac App Store.