ESR results in power loss and inefficient power supplies or regulators.
How does ESR affect the stability of the system?
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It depends on the loop design, some power supplies (Particularly older LDOs) required a certain minimum ESR to provide a zero so the loop compensation would work (Often they specified tants for the output cap to provide this).
More modern parts tend to be designed for ceramic output caps and are stable down to a few milli ohm ESR levels (usually you can provide this with a short bit of tracking if the cap is still too good).