[ 1. IBM 5160 motherboard.
2. IBM 5160 PSU (fan is DC powered).
3. The PSU's fan is not turning when the PSU's P8 is connected to the motherboard.
4. The PSU's fan turns when the PSU's P8 is disconnected from the motherboard. ]
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IBM 5160 (IBM XT) motherboard
On a 5160 motherboard (of both types), there are two components on the +12 volt line and two components on the -12 volt line.
C55 and C56, both capacitors, are the components on the +12 volt line. Pictured at here.
C57 and C58, both capacitors, are the components on the -12 volt line. Pictured at here.
Although it is possible for C55/C57 (ceramic, not tantalum types) to go short-circuit, C56/C58 are the known problem ones. C56/C58 are well known for going short-circuit, per the history shown at here. Following, I am targeting C56/C58.
C56 and C58 are not critical to the operation of this motherboard, and this motherboard can operate without them. The reason that these two capacitors are not critical to operation of the motherboard, is because, on the 5160 motherboard, they filter only the 12 volt lines going to the expansion slots. More information at here.
So what I am about to get you to do is remove the two capacitors, reattach plug P8 from the power supply to the motherboard, then see if the power supply's fan now turns. If the fan turns, one (or possibly both) of the removed capacitors is short-circuit.
Remove the two capacitors either by desoldering, or by using a pair of sidecutters to cut the capacitors off the board. If you have little soldering experience, then perhaps (if possible) use the technique shown at here, because on a 5160 motherboard, soldering the ground and/or power lines on the PCB is problematic, and your inexperience could end up damaging the PCB.
Now that the capacitors have been removed, reattach plug P8 from the power supply back into the motherboard, then turn on the power supply. If the power supply's fan now turns (that being my expectation), one (or possibly both) of the removed capacitors is short-circuit.
If nothing else is wrong with your IBM 5160, your 5160 will now start (noting that in rare cases, the power supply may need more than the motherboard to adequately load it).
CAPACITOR REPLACEMENT
As stated earlier, on this type of IBM 5160 motherboard, the two subject capacitors are not critical to the operation of the motherboard.
Should you wish to put in replacement capacitors (the recommended action), then click here for replacement details.