Capacitor selection for LM3940 LDO regulator

It's important to know a bit of (pre)history here... Looking at LM3940 datasheet I see a "May 1999" date, but the chip may be older than this. This means the device was designed at a time before thin portable devices, and it will require the type of output cap that was usual back in the day, which would probably be tantalum or electrolytic. Likewise a regulator like LM317 works best with the type of caps that it was designed for, that means a high-ish value with not too low ESR.

Modern LDOs are usually designed to be stable with only ceramic caps at the output because ceramic caps have gone a long way since the 1990's and become the cheaper, smaller, thinner option. But you can usually improve transient response by adding more capacitance, if necessary.

If you want to use LM3940, then you have to respect its ESR requirements. Personally I would use a Panasonic FR 470µF 6.3V capacitor, which has an ESR of 80-130 mOhms. But my reason for choosing this particular cap is that they fit the requirements and I ordered a bag of 100 (they're not expensive), so I have them. I wouldn't recommend placing an order just for one cap though, so you have to pick one that you can get easily.

Basically, you want low ESR for good transient response, but if it is too low then a LDO that is not designed for low ESR caps will go unstable, and the electrolytic cap can form a LC resonant circuit with ceramic capacitors in parallel. This cap's ESR of 80-130 mOhms is a great compromise, it is low enough for good transient response and it produces no resonance at all with a 1µF ceramic. Also most LDOs like the high capacitance and "low but not too low" ESR.

However, I'm somewhat confused with so large voltage/capacity values and suppose that there is some kind of error in my calculations.

This is normal, most "general purpose" caps are designed for low cost, NOT for low ESR. So if you want low ESR, you have to use a very big capacitor value which will use a lot of board space and may not be practical. Also the high ESR value of general purpose caps is a feature: it makes them not resonate with the ubiquitous 100nF ceramic bypass caps even if trace inductance is high, and that's good.

The "Panasonic FR" cap I linked is specifically optimized for low ESR.

Lower ESR and high capacitance mean Polymer caps. Some have ESR as low as 6 mOhms. These are quite popular among audiophiles who mod their equipment, for all the wrong reasons. DO NOT use a polymer cap unless you understand that it has strong tendency to create resonance peaks with ceramic bypass caps and/or make regulators unstable if not used carefully.

So, for your LDO, get a "Low-impedance" electrolytic (not "general purpose"), like those used at the output of switching power supplies. If you have an old switching supply that you don't use, there's your free cap!

Personally I'd replace the LM3940 with a more modern LDL1117 which has much better characteristics, lower quiescent current, and will work fine with ceramic caps.

(note LDL1117 is different from LD1117)

The caps you have the data sheet for are not low impedance types so you need a large capacitance to get low ESR. Also the ESR at high frequency is what counts here since that is where the regulator will oscillate, so a capacitor with an ESR rating at 100kHz would be preferred. The ones you have are more suited to mains frequency filtering.

For example, this small SMT cap.

The larger non-low-impedance types will probably work okay, regardless, but with less confidence and they’ll be much larger.