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Output capacitor of LDO, also called load capacitor can be either external or internal to IC. Depending on its value, it sets one of the pole of the LDO along with the output impedance of the LDO. Load capacitor is also important to suppress the LDO output ripple during changing load.
For more details about LDO, find literature by Dr. Rincon Mora.
It is important for stability reasons.
There are two main contributors to loop gain drop: The series transistor and the error amplifier. Both constitute a closed loop, and this is an unstable or at least a marginally stable system.
The addition of an output capacitor creates something like lag compensation.
Remember the principle of universal compensation of an opamp with the help of an internal capacitor. The effect is about the same.
This output cap lowers the loop gain Aloop with the result that Aloop=0 dB before it reaches a slope of -40 dB/dec (instability).
In addition, a small series resistance enhances the phase in the critical region and, thus, improves the phase margin. In most cases the internal parasitic resistor of the capactor can do the job.
Fazit: Use intentionally a "bad" capacitor.
The role fo output capacitor, as you said, is to make the loop stable, and to reduce the ripple voltage at output.
If the structure of the loop is changed, such as the pass transistor is NMOST and the compensation is realized in the error amplifier, the output capacitor could be free.
In that case of no output capacitor, how could the output voltage responds to the change of transient load current? How about the ripple voltage? TI has products without capacitor, such as TPS73XXX.
I am puzzled to that.