I was fooling around with a DFPlayer Mini MP3 player, which looks like a really nice and easy to use device.
For some reason I couldn't get mine to output sound on its speaker outputs, so I decided it is faulty. But it did output fine on its line out output, so I attached some headphones and had fun with it.
While it worked (the line out, I never managed to get speaker out to work) I was powering it with 4.5V - 5V from my PSU.
I tried powering it with 3.3V (more like ~3.22) from the same PCU and it didn't work. I thought it is OK, since 3.3V is stated as its minimum operating voltage and my PSU outputs a bit less, so whatever.
But then I tried powering it with a coin cell, which I first measured to about 3.25V, it blinked once and never ever came back to life.
Now it won't work no matter how I power it. It still draws its typical ~30ma from the PSU, but doesn't play any sounds anymore.
Honestly, I don't really care about the device (it seemed like it was faulty anyway), but it made me ask myself, whether undervoltage can actually be dangerous for a circuit and potentially kill it.
It could be just a coincidence, but I'd rather know whether or not undervoltage can be potentially dangerous, so I don't brick something I actually care about in the future.

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2 Answers
Yes. Some circuits are non-linear, or particularly, anti-linear: when voltage goes down, current goes up. Notably, boost converters which pump up supply voltage to get their working voltage (e.g. some wireless mice use a single AA battery and run fine on 1.2v NIMH's, or the apocryphal Joule Thief.) As voltage falls, they must increase the current they draw to produce the same total power (watts = volts x amps). Depending on how the circuit is specced and protected, you could exceed the current limit of one of the components.
High voltage just requires thicker insulation, which is quite easy, especially in the low-voltage range of Arduino devices. High current requires bigger pipes: wider traces (which gobble up precious circuit board space), thicker wires, more silicon on the chip, thanks to Ohm's Law more heat dissipation, etc. Which is harder and costs real money. As such, devices tend to be more sensitive to overcurrent than to overvoltage, especially when the cost-engineering is done.

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1Generally, I'd agree with your assessment, but in this case, the OP tried using a button cell to power it. Considering the internal resistance of such a cell, I doubt much current passed through the device. – DoxyLover Apr 04 '16 at 23:39
Another possibility is that your player has an EEPROM that is updated at certain times (eg. to remember the last song that was played) and if voltage drops too low while writing the memory gets corrupted. Some devices will 'brick' themselves if the battery suddenly fades or is disconnected while they are updating settings.

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