It doesn't have to be an A4N. The N is a date code to indicate when it was manufactured. Any A4 should work, assuming it's actually an NCP81161. To check the bootstrap caps, you would remove the one on the suspect phase and check its resistance and capacitance. You should have near infinite resistance across it, and capacitance should be 0.027uF. If you have different capacitance, remove and test one of the other caps from a different phase. If the capacitance matches, they're probably OK. If you don't have pretty high resistance, it's bad and needs replacing.I could buy it and try it. Does it have to be A4N? Or can it be anything "NCP81161" like A4F, A4L etc.? I guess it's just a marking on the same thing.
I removed the card today to check if these are indeed A4Ns (they are). I didn't do that previously as I thought it could lead to some problems and it did. After plugging the card back, the PC wouldn't start with it (fans spin, but card wasn't detected). Few starts later still the same thing. Turning AC on/off didn't change anything. Only after clearing the CMOS, the MOBO made some kind of self check (like after every CMOS clearing) restarted itself and booted successfully displaying picture. So I guess these A4N (NCP81161) chips are a good guess. Unless it's the capacitor To check the cap I guess I should remove two of them (for each phase) and measure them to see if they have the same values (because they should?).
BTW I'm trying to understand something here.
I mentioned NCP81022 (4 + 1 phase) and it's one chip in the front. You said NCP81161 and they are there too (on the other side where you marked them).
In the front of the card I can see 4 + 1 phases (high and low side mosfets). On the other side these 4 phases have low side mosfets only (marking K0393).
Do I guess correctly that they are connected with the high side mosfets in the front, so the current given by high side mosfet is split between two low side mosfets?
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It's hard to say for sure how all those mosfets are wired up without the card in hand, but it's pretty common to have two low-side mosfets and one high side. It totally wouldn't surprise me if that's how this card is wired up. You could check by testing to see if you have continuity between the pins on the pairs of low side mosfets.