Someone just asked Can electronics be damaged by undervolting it?, so now I am going to ask under-current. Let's say something requires 10A, 12V, if I limit the current to 5A, could anything go wrong?
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@Kaz -- Yes, but that fact, in and of itself, is the premise challenged by this question... which is what makes this question interesting. People will naturally ask about "under-amp" given that "under-volt" is frequently discussed. – DrFriedParts Apr 16 '13 at 18:55
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I think it can all be lumped into one item: "under-juicing" the equipment, in all its facets. The answers in this question look like echoes of the answers in the other one. Motors, mosfets, unbalanced multi-supplies, ... – Kaz Apr 16 '13 at 19:25
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In a world where almost everything is powered by a voltage source, how would one limit the current supplied, except by decreasing the voltage? I don't know that there's much more to say on the topic, but it's clear this isn't an _exact_ duplicate, so I'm voting to reopen. – Phil Frost Apr 17 '13 at 18:24
4 Answers
Conceivably yes. Limiting the current means that the circuit will attempt to draw the current but the supply limiter will reduce its output voltage in order to balance the higher demand of current.
We are back to a lower voltage or under-voltage scenario and I can envisage a situation where a weakly designed circuit will potentially be damaged.
A buck convertor may be specified to run from (say) 20V. It's output may be 5V to a load that might be 1A (5 watts to the load). Ignoring losses, it will take 0.25V from the 20V supply in order to do so and it may be fused for say 0.5A. An associated part of the whole circuit may nominally require 2A and if you current limited the supply to 1.5A and the voltage dropped to (say) 6V the buck convertor could be taking 0.833A.
It's a tortuous example but it could happen and the fuse blows. Maybe there wasn't a fuse - the buck convertor transistor will get too hot and if it is a BJT type it will destroy itself.

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Flawed premise...
Fundamentally, it is not possible to "under-current" a device^ while simultaneously providing the nominal voltage (e.g. not under or over "volting" it).
^ device = complete network, sans power source
...due to the relationship
Since voltage and current have a relationship...
- Ohm's law v1 = c i1 in the simple case of a linear element (c = constant)
- v1 = F(i1), let's say, in the generic case
It is is not possible to independently and simultaneously describe v1 AND i1 without modifying or violating F(..)
In the two port case...
Confusing to students is the representation of this model using two ports (typically an input and output port).
In this situation, you no longer have the complete system enclosed in the boundary of the "black box" (blue in this case ;-) ). Therefore, there are more degrees of freedom. That is not the case of the question.
You'll notice that each of the other answers that attempts to answer the question directly, makes assertions/assumptions about the output side of the circuit (two-port model) or concludes that "under-power" (output) or "lower voltage" (input) will be required.

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Transistors not in saturation could have heat buildup and dissipation issues.
Some devices with internal mosfet can be damaged by underpowering the source. As was explained by a TI employee about a current controlled led driver, if the VLed source is too low to provide the selected current through a channel, the logic in that channel will try to drive the channel's mosfet harder to try to sink more current. Eventually, the mosfet will burn out, if not other parts of the chip. I wish I could find that discussion and link it. Or if a series resistor is added that shouldn't be there, limiting current, could also trigger the logic to drive the mosfet too hard.
Indirectly to both the device and it's target, Heating Elements, Cooling Solutions, Water Pumps. Motors as well. Indirectly, underpowered lights can stop illuminating, causing accidents, which could eventually cause the light to be damaged.

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The most common situation in which supplying a lack of current to a device would cause damage would be those in which a device has two supplies and one of them receives insufficient current. Otherwise, while supplying too little power to a device is not especially likely to damage it, it's common for some types of power supplies to have a minimum current draw specification, and for their output voltage to exceed design limits if the amount of current drawn is below that. This may cause damage either to the supply, or to equipment powered thereby.

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