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I want to use the LDO regulator LD39300 of ST. To buffer VOUT, I thought about buffering the output in addition with an aluminum capacitor of 6800 µF in parallel to the recommended 4.7 µF.

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But in the datasheet at page 10 (figure 14 and figure 15) it is described, that the ESR of the output capaitor must be bigger than ~0.3 ohm. Here is my problem, because an aluminum capacitor of 6800 µF often has only an ESR of a few milli ohm. In parallel to a ceramic capacitor the resulting ESR will be smaller than 0.3 ohm.

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Is it right, that this LDO regulator will not work if I put such a buffering capacitor in parallel with the output capacitor?

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You are correct in assuming that going outside the stability regions is a 'bad idea'.

It's difficult to predict exactly what will happen if you operate outside the stability region of a fixed-feedback regulator like the one you're describing. The regulator may start oscillating immediately, or appear fine but start oscillating if there's a sudden input or output change.

Blindly adding capacitance here will likely make things worse. Don't add the monster cap, or if you need super smooth output, change to a different LDO.

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6800 uF is a very large capacitor. I wouldn't use a capacitor that large just for filtering; rather for "reserve" capacity to survive short interruptions of supply. Something like a 470 uF capacitor would be plenty of capacitance for additional filtering and reserve capacity, and 100 uF might be quite enough depending on your input and output situations. If the ESR is a problem, then I would suggest adding a 0.3 Ohm resistor between Vout and the electrolytic capacitor. The full response of the low-ESR ceramic 4.7 uF capacitor and the artificially-raised ESR of the aluminum would be a little harder to model, as it would be frequency dependent. With sufficient ESR and resistor on that capacitor, though, you'll be fine (at the expense of less filtering.)

I agree with Willy: If you need more filtering than 4.7 uF ceramic will give you, go for an inductor rated for the amperage you will draw, and a second capacitor on the other end, and make that capacitor big enough for the LC filter to work out to whatever you need it to be. Probably still don't need more than 100 uF there, though -- what kind of noise do you foresee on the input or what kind of load transients are you trying to compensate for?

Finally, with a large capacitor on the output, what will happen when either of input or output is disconnected? That large capacitor could try to push dozens of amps into the regulator, which might be bad for it. I suggest you check the data sheet for whether it talks about this situation, and if it doesn't specify that it's protected against this, add back-voltage protection diodes or some other mitigation.

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