A burnt smell can mean that your heat sink is covered in dust or it's fan no longer work.
Those are the benign cases. NO damage done.
In this case, the PC will turn itself OFF to prevent overheating damage.
You need to clean the heat sink and check that the fan is working properly. The cleaning consist in blowing the dust off with a can of compressed air. You may want to do that outside as there may be a lot of dust.
If the fan don't work, remove it, go to a computer shop and get a similar replacement. Put it back and power up.
It may be very serious.
If there is no overheating from a dusty heat sink of faulty fan, there may be a short on the motherboard. This is a catastrophic failure that require the replacement of the whole motherboard.
For now, put back the original PSU, as it was probably not at fault. You may want to vacuum it before that if it looks like there is to much dust inside, but it should not be needed. Make sure that all components are properly connected, and that the extra CPU power connector is correctly plugged in if the motherboard have such a connector.
If you replace the PSU, make absolutely sure that it's power rating is at least the same as the original one. Lower power rating can lead to an unstable computer.
Also, if the motherboard do have a CPUPWR connector, it absolutely need to be connected. If it's not, you can't power up at all. You need a PSU that have that connector or it won't work.