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Most of the time while I'm doing my university projects, I find that only voltage requirements are given for building a circuit, and how much current I should supply from a bench-top DC power supply is not given. Normally, if we require 5V supply, we set the current to some small non-zero value only ,like it's not important. So I'm wondering if anyone knows the importance of which current to set.

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The answer to this depends on the load of the Power Supply (PS).

If you feed chips (gates, sensors, etc.) you must provide the nominal voltage, which is given by the manufacturer. In this case, the current that the circuit draws is not up to you, but it depends on the circuit impedance. The reason you set the current limiter to some non-zero low value is that you expect that it will draw some small amount of current. If it starts to draw, say 1 A, you have made a wrong connection, so you have a soft or hard short-circuit. In other words, the current limiter is to protect the circuit from a wrong connection and don't burn everything.

But most PSs can provide constant-current (CC). For example you can max out the voltage and turn the current knob to zero. If you connect a LED, by turning slowly the current knob you can feed the LED with 10mA, without any current-limiting resistor. The same goes for a DC motor. You can provide CC to it, so it will always develops the same amount of torque on its axis.

Keep in mind that we can only directly control the voltage. Current can be controlled, but it requires a feedback loop which measures the output current and increases or decreases the output voltage accordingly.

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Usually using a bench-top supply to power a known device that requires a fixed DC voltage is not much different from powering it from a wall-wart, except that you can limit the current to a more reasonable value in case of a short or circuit malfunction. This is very useful when the device is under development. A wrong connection or a short can burn out the device, so you'll need to replace it. The bench-top supply can usually save you this hassle.

To spell out an example, let's say I've made a simple circuit with a MCU and some other minor functionality (e.g. reading a sensor and reporting it via serial). I'll know that the MCU draws 1mA, the sensor is 15mA, so I can dial in 16mA as the limit. To have some headroom, I'll typically set more, e.g. 30mA.

When I turn on the circuit, if the supply says "16mA current draw", then it's all fine. If it says 30mA, then it's "whoops, let's see what I've wired wrong".

If I used a 5V 1A wall-wart, and I've made an error, the device would possibly be fried before I can react.

As thece says, the current-limiting function of the bench-top supply can come in handy for other types of loads, like LEDs, lights, motors or batteries (you can use a bench supply to charge a battery, you have to be careful, but it's easily doable).

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I like the current limiting feature of my bench power supply. It is a very nice safety feature. Not for my safety, but for the safety of my circuits devices. I tend to keep it low. Usually 30-50mA, and the power supply output voltage display will indicate low when the current limits out as an additional warning that current is being limited. I can always turn it up if need be but I tend to err on the conservative side of current limiting. FWIW

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