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my iMac went off by itself and now it does not start again. When I activate the power switch the HDD starts spinning up and the USB ports are powered, but there is no chime and the iMac does not boot. Only diagnostic LED #1 is on permanently. I certainly tried the typical things, change RAM, resetting SMC and checking all connections, but without any success. I tested the output of the power supply with a DVM. It constantly shows 12V (as long AC is connected), even if the HDD stopped. Also LED#1 is constantly on. From the fact that LED#2 does not come on I assumed the power supply needs replacing, but the 12V is present at the main power output from the PSU to the logic board, so it seems to work. I am not sure what else I can test to determine if this is caused by the PSU or the Logic Board.

You have correctly diagnosed the problem. Power supplies can have partial failures. Replace it. If by some wild thing it is not the problem you can always return the part. iMac Intel 27" (Late 2012-2020) Power Supply Here’s how: iMac Intel 27" EMC 2546 Power Supply Replacement