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I will be powering a PCB using Li-Ion batteries. I will have 3 or 4 of them in series at max so 11.1V or 15V voltage will be achieved. Let's assume 15V.

I have LM2576 5V 3A regulator on the PCB (5V*3A=15W at max?) and the battery voltage will be directly regulated by this ic. 15W power will be supplied by 15 Volts so 15W/15V = 1A? I think I will be pulling 1A current at max from the batteries in total. I want to prepare for the worst, so I assumed 3A current will be the output current of the regulator.

I will be using something like the one in the picture below to connect the Li-Ion batteries to my board.

enter image description here

Q1: Is this safe to use? Should I use thicker cables? I see people using very thick cables for Li-Po batteries and I would like to know whether or not the same applies for Li-Ion batteries as well.

Q2: Am I doing the worst-case power budget calculation wrong? Should I be checking the LM2576 datasheet instead of multiplying the numbers on my own?

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Your calculations are reasonable. The converter will be not 100% efficient, so the input wattage could be more than 15 W, maybe 20 W.

As for battery current drawn, you should consider the lowest battery voltage, including some decay from drained batteries. So maybe consider 10 V.

That will put the worst case battery current to 2A. It is a bit hard to get the scale of the wires from the photo. But you don't need very large cables for 2A. So it is most definitely fine.

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