the LM1117 doesn't have any internal frequency compensation, it's just a Brokaw band-gap reference that's also kinda its own error amp connected to a PNP-styled sziklai pair output stage.
With a high ESR cap the internal capacitance of the Sziklai + error amp output impedance, output resistance of the PNP, cap C and ESR, and load R make a system with two poles and one zero that have break frequencies below the amps unity gain rolloff - stable.
With a low ESR cap the ESR zero shifts up and out beyond the unity gain frequency leaving two poles below the UGF - bad phase margin.
With the extra cap the two poles that remain are split with one moving lower and the other moving up and out - stable again, but slower.
The old NatSemi numbers were LM1xx for mil spec temperature range, LM2xx for 'industrial', LM3xx for 'commercial', but the LT1117 is an LT (now Analog Devices) part (often copied) that uses a part number with a letter suffix for the 'grade' instead of changing the digit.
The 1117 is available in four fixed voltages and the ADJ version.
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If you happen to need 1.25 volts (like for a CPU core) and 2.5 for something else, you can ground one ADJ pin, and stack the other regulator into that. No resistors needed.
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John Larkin Highland Technology, Inc
lunatic fringe electronics
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