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This is not actually a problem, but a question.

Through pure carelessness, I have been using a Linksys 9v AC-AC (rather than AC-DC) adapter to power my arduino. It works fine. Why would this work?

My guess is the protection diode only lets half of the current flow.

I have been using it for months. Is there a reason not to use this adapter anymore?
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Yes, the protection diode turns the AC into pulsed DC and the capacitor before the regulator smoothes out the voltage somewhat.  As long as it stays above about 7V the regulator should be able to give you fairly clean 5V DC.  Even if it dips, the Arduino can run at lower voltage.  It's guaranteed to work down to (I think) 3.8V at 16 MHz but will probably work even below that (like 3V).
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It is likely that there is a lot of ripple on the supply because the capacitors after the protection diode are not that big.
It is likely to affect the accuracy and repetability of the analogue to digital converter, as well as devices you attach to the arduino.
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The act of converting AC into DC is called rectification.

There are two basic types of rectification: full wave, and half wave.

Full wave rectification takes 4 diodes and flips the lower half of an AC waveform upside down and superimposes it over the upper half:



Half wave rectification only uses 1 diode, and effectively cuts off the lower half of the waveform:



The Arduino has a single diode in-line with the power input jack to protect the system against reverse polarity connections.  This diode acts like a half wave rectifier.

A capacitor after a rectifier smooths off the peaks and troughs of the waveform:



If the ripple voltage remains above the minimum level of the voltage regulator, then all will be well.  The ripple voltage, however, is a function of the current drawn - the more current you draw the faster the capacitor will discharge and the more ripple you will get.

You will be fine running with an AC adaptor at low currents, but there will be a cut-off level where you will start to get glitches and voltage drops on the 5V line when you draw too much current.
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