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Do switching AC adaptors limit the current they provide to the rate described?

I have thought that they do not, and that it simply gets dangerous by overheat, but now I am thinking that they might.

I had a power unit with pretty much small capacity (60w) on my mini-ITX computer, and it worked for a while, but after I started to attach usb units and gained current, the computer frequently shut down, so I suspected the power unit's capacity was not enough, and replaced it with a larger one (120w) but kept using the AC adaptor (12V 5A) that was designed for the smaller power unit, and it still shuts down frequently. After calculating the power that I am using, the power unit should be enough, but the adaptor may not. Now I am suspecting that the AC adaptor does not have enough capacity.

A different reason I came to think that AC adaptors limit the current is when you connect a AC-to-usb adaptor to ipad that is not designed for ipad, it does not charge at full speed because ipad requires 2A, which is more than what ordinary usb power supplies.

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Many switching power supply controller ICs have a built in current sense input to avoid damage of the switching element when a over-current situation occurs.

Regarding the iPad charging, I read somewhere that there must be a specific resistor combination on D+ and D- lines in order to charge it at full current. This is to avoid over-current situations when connecting it to computer USB ports.

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Without going into too much detail on the hows and whys and variations, for all intents and purposes the answer is yes, power supplies will only supply up to what they are rated to supply.

Basically, if you have a 12V 5A supply, you can't try and draw more than 5A without the output voltage sagging, or some current limiting kicking in, fuse blowing, or smoke appearing. Usually with a well designed supply it will be current limiting or thermal fusing.

So if you have a computer that has a 120W power unit (I take it this is a mini ATX type thing) then you need at least a 120W adaptor. If the unit runs from 12V, then you need 120W / 12V = 10A minimum. If you use the 60W adaptor with a 120W ATX, you effectively have a (slightly less than due to inefficiency) 60W ATX, as you can only get out what you put in.
So you need a new switching adaptor.

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Most inexpensive duty-cycle controlled switching power supplies (flybacks, forwards, half-and full-bridges, etc.) will have primary-side current limiting, as this is inherent in the widely-use current-mode control methodology. There is some sensing element in series with the primary of the transformer, which cuts off the duty cycle if the current gets too high in a given switching cycle.

More robust designs will also have secondary-side current limiting, which uses some sensing element to measure the current going to the load and some logic to take action if a specific current threshold is exceeded.

Fuses generally aren't good for overload protection unless the overload is catastrophic (a short-circuit in the power train, for instance).

It's always good to target some overhead in your power budget - if you need 100W, aim for at least a 150W power supply, for example, to ensure that you're not operating the power supply near its theoretical maximum.

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