[quarterwave]

This is a beautiful post, and I am reluctant to try and say anything profound. Still, all of HN's a stage so I'll attempt a brief explanation for the general reader about why we need voltage regulators.

A logic chip like a microprocessor is designed for a particular supply voltage, if this voltage drops too much the logic circuitry will switch falsely. Say we had only a capacitor and we tried to power the logic chip with it. As the chip draws current the capacitor discharges - this is because current is movement of charge, so the charge (and energy) can come only by draining the capacitor. For an ideal capacitor the voltage is directly proportional to the charge across it, so as the charge drains the voltage falls. To hold the voltage constant we need to keep 'topping up' the capacitor with charge. This is what a voltage regulator does - it uses a negative feedback loop to sense the capacitor voltage and when that voltage falls the circuit provides just the right amount of charge 'juice' for the top-up.

As we take the foot off the clutch pedal in a car, the load gets engaged to the engine and if we sense a stall we press the gas pedal a bit. That's the imagery of a voltage regulator in action.

The capacitor plays a key role because the regulator feedback loop isn't very fast - one trouble with fast feedback circuits is chatter, or responding to every blip. Negative feedback circuits are designed to be more like ship wheels - they like to steer sedately and not respond to every excited cry from the mast. But what happens if a current blip arises because a logic circuit block turns on all at once (in response to some block of code)? That local current blip is provided by the capacitor, it acts like an ATM to provide local draws - but it still depends on the regulator to top it up.

In fact you can think of a battery as a capacitor that tops itself up via electrochemistry, it works as long as there are ions in the electrolyte. If instead of 'bandgap energy' we used the chemist's terminology of 'electrochemical potential difference', then the system similarity becomes evident.