I use a current limited supply to test supercap battery replacements. I limit the charge to 100ma, but it's safe to increase that to probably an amp if you like. I don't recommend trying to apply unlimited amps to a supercap, it appears as a short circuit when fully discharged and will likely overload the P/S and could overheat the supercap. The ESR of the supercap determines the maximum current if you have an unlimited current power supply, A typical ESR for the type of caps we use for a BCR clone would be in the 100-200 milli-ohm range. Given that rating, if you apply 5V to that fully discharged supercap, you have a possible 50 amps of current if the P/S could supply it.
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