I am learning about super capacitors in a backup power application. I see a lot of examples using diode networks or FETs to separate the charge and discharge paths. Why is there so much effort put into separating the charge and discharge paths? Why would the example circuit I drew work? Someone mentioned the voltage drop over the limiting resistor may cause an issue? The 3V3 regulator would be powering an MCU.
one of the main reasons you would want to separate charge and discharge paths in a circuit with capacitors is safety, say your voltage supplies fails and turns into a short circuit, whatever energy your capacitor has will go that way. In pulse discharge circuits there is the issue of you don't want your load to draw current from your source because simply this current will be too large, even if it is for a small time it and your voltage source is properly protected is not something you want to risk. If it was a battery system it would drain your batteries when you want to the capacitor to take care of it.
your circuit has the issue of the voltage drop, all the energy that from that voltage loss is also wasted. Additionally, how quickly or how much of a ripple your output voltage can have, that resistor does not only limit current it also slows down charging and discharging and the overall shape of the output. Think of the RC constant and how it affects what I mentioned before.
For your circuit as drawn, anything connected to the 5V supply would be powered by the cap. This would include any regulators that may be supplying the 5V. If the 5V only ever comes from a cable the situation is better, but you'd still have that 5V on some (presumably) exposed terminals.
To add to your pain, the resistor would just waste energy as the cap was discharging.
However -- for some simple battery powered thing where you just want to keep the micro going when the batteries are being replaced (or similar purpose), what you have there may actually work.
Power losses increase with ESR and serial Rs, so the concept is to reduce this with a current source.
but if you have a voltage source then a serial Rs limits the charge current to the source capabilities.
Energy stored = 1/2CV^2
Supercaps can have a high inrush current when power comes on and the cap is at zero volts just from draining down to zero volts. The circuit needing the backup current consumes much less than the inrush current, so the series resistor has little Vdrop in terms of what the load draws, yet allows a drained capacitor to charge up full in less than 30 minutes.
Typical application for a few hours to a few days of backup are LCD event counters and other low power CMOS circuits that often idle at a few microamps. If R is 1K ohm then the inrush/charge is at safe levels, and when power fails the resistor causes little Vdrop if the load consumes only 1 or 2 microamps per volt. Many CMOS devices will retain their state down to 1.9 volts, even if 5 volts is normal operating voltage.