I don't think I can condense all the random things I pretend to know about caps in a few posts, but briefly... Highest quality caps aren't necessarily important. The highest quality caps tend to be very expensive (in small quantity) solid polymer instead of wet electrolytic like those Rubycons and Panasonics they like, but what they like are a good value and the electrolytic caps tend to be the ones that fail far more often so when they look at the voltage and uF value of the failed cap, not knowing the exact electrical parameters of the circuit, they conservatively pick a replacement cap with similar specs.
Choosing another electrolytic cap in the same size is what results in similar specs most of the time, when talking about near equivalent, very low ESR capacitors.
If the area for the capacitor is land-locked meaning a smaller diameter capacitor was used, it can be more important to pick a higher quality capacitor. If the area has room for a larger capacitor, inherant in the larger size (within the same model/family of cap) is a lower ESR so you can get away with chosing a model of capacitor that on average has higher ESR but being positioned as a lower ESR part within it's family, it may still have accepaibly low ESR compared to the alternative cap, or original cap, it an overall lower ESR family.
I wrote that wondering if it makes sense to read. Hmm. If it were a race, and the Smith family were faster on average than the Jones family, there may still be a member of the Jones family that is faster than a certain member of the Smith family. Faster = lower ESR.
The trick is if you can use a larger cap, it needs same lead spacing. With leaded electrolytics that usually means you can't substitute a 10mm diameter for an 8mm but you can a 12mm for a 10mm if it'll fit, but that some boards have really small through-holes for the leads and some 12mm dia. caps have larger diameter leads so they are hard to cram in the holes (particularly older PCChips and some Biostar boards). Another alternative is picking a taller capacitor instead though often they already used the tallest one in that diameter. When picking larger diameter or height caps , all else being equal you go up in either voltage, capacitance, or both.
Higher capacitance allows the circuit to suppy higher current (good for overclocking) but it also causes higher inrush current if the cap is about the same or lower ESR so you don't want to overdo it, and if the system PSU had a barely adequate power-on rise time, it could be enough to prevent the system from initializing fast enough to run. Usually this problem won't occur but it may be because people don't go extremely overboard on rasing capacitance. Say it started with 1000uF caps, you could probably double that w/o likelihood of problems but going to 4700uF would be overkill.
On the other hand you could raise the voltage, say you had a VRM subcircuit operating at ~ 1.8V, electrolytic caps rated for 6V and 1000uF. You could go up to a 10V or 16V cap, which all else being equal is larger, and has lower ESR. 10 fold increase in cap rated voltage versus applied circuit voltage is sometimes too much of a difference, the lower charge will form less of an oxide layer on the cap internal foil but it'll still be appropriately thick enough for the voltage it sees. Running at such a difference in real vs rated voltage will also cause some loss of capacitance but in a circuit like switching power regulation it is seldom enough capacitance loss to matter and the lower ESR proves more beneficial.
Then you could do both, increase diameter and size for both effects or a mix of one or the other. If the VRM circuit were overengineered for the powered load you could even pick a smaller capacitance though generally in such a circuit the caps won't have failed unless they used instable electrolyte which they very well might have.
The point of all the above is, picking Rubycon or Panasonic is the short simple answer but their reasons for doing so aren't necessarily technically accurate, rather it is due to lack of information which I too, lack as I'm not going to try multiple different caps on a finished product and measure ripple and temperature.
I almost forgot temperature, another reason why a physically larger cap is better even if all else were equal (so long as it still fits on the board). Larger means lower heat density given same ESR, and the internal heat generated as a factor of ESR has a larger outer surface area as an inherant heatsink to shed the heat so the cap runs cooler. Unless the capacitor had an instable electrolyte (and even if it does), heat is the #1 killer of capacitors. Funny thing is while today we have solid capacitors that run cooler and are more resistant to heat plus heatsinks on VRM mosfets, years ago when I overclocked as high as I could I would point a fan at the VRM subcircuit more to keep the caps cool than the mosfets cool, though removing the heat from the mosfets also mean a cooler zone temperature so the temperature differential between the heated air plus heated copper around the cap leads, versus the capacitor shell-as-heatsink, was better than if that temp around the cap was higher.
LOL. I should have replied "Yes use Rubycon or Panasonic" because they will both do fine and don't cost substantially more than other alternatives, are good enough you don't have to consider much of the above unless doing insane overclocking or in extreme ambient conditions, passively cooled system or something like that. There are other capacitor brands you could choose though, like UCC, Sanyo, Fujitsu, Nippon, Samxon, even Teapo aren't too bad in some applications. Mainly you want to avoid brands know poor and in quality major brands you'd pick the model/family described as especially low ESR. For comparitive purposes you can look at the Rubycon or Panasonic datasheets to compare their ESR to other brands.
I almost forgot, they meant specific models of Rubycon and Panasonic, both companies make many models that aren't suitably low ESR (except see above where i described how you can pick larger caps to reach better specs). I forget the model # that is best on the Rubycons, I think they came out with a better series since the last time I used any but you don't necessarily need their best model as their 2nd or 3rd best electrolytics are pretty good too. Same for the Pansonic FM's, you could go with an FC model most of the time or one of the others described as very low ESR and compare the size you'd use per the datasheet. The thing you don't want to do is use a cap that has higher ESR than the original, the goal being to get it as low as is reasonably possible given a budget and available space on the board.
Sometimes low end cap manufacturers will cheap a bit, use thinner materials in the cap so if two supposedly very low ESR caps were the same physical size, the Panasonic FM might be (just picking numbers out of thin air...) 6V/1500uF and the original failed cap might have been 10V/1500uF, or the Panasonic might be 6V/1500uF when the original was 6V/2200uF. If you can fit the same value of Panasonic cap on the board, go ahead and do it but you may find you are forced to pick a lower value of Panasonic cap to keep it small enough diameter that it will fit in the space on the board.
If there are taller caps in the same family (Of Panasonic for this example), picking a taller one is one possible option. If there is space for and the board through-holes are wide enough, picking the larger diameter cap is another possible option. If you know the circuit voltage and that the Panasonic's voltage rating is above that, using a lower voltage rated cap than the original but one that is still higher voltage rated than the circuit voltage is another option. For example in the prior comparison of a Panasonic 6V/1500uF and a failed old cap that is 10V/1500uF, in a circuit where the voltage is 1.8V, there is no problem using the 6V Panasonic instead of a 10V Panasonic if limited space requires it, although as mentioned previously the 6V part will, all else being equal, be lower ESR by virtue of it's larger size was is good.
I mention these things because quite often the failed cap has different voltage and capacitance (uF) specs than the same size Panasonic (FM model for example), but the same size Panasonic may have lower ESR still. You don't want to go too much lower in capacitance if you don't know the margins in the circuit but most often the capacitance they chose is more than needed for an acceptibly low ripple, they chose a larger cap to have lower ESR, as it can be cheaper to do than using a better grade of capacitor for cost-conscious manufacturers, though times are changing and even they are using solid caps now on all but the cheapest board VRM subcircuits (where caps fail far more often than any other area).
I'm certain I left a few things out but this is turning into a novel.
Filling the solder hole is crucial. If it's not filled it is probably because the cap lead either didn't get hot enough or there wasn't enough flux (originally, or remaining after some had cooked off from the heat) to make the solder flow properly, which tends to result in either intermittent or highly capacitative solder joints (while you want a capacitative capacitor, you don't want a capacitative joint as it acts as a high-pass filter which is a problem). Personally I apply some liquid rosin flux to the area before heating and adding solder (w/flux in it too, needs be same type of flux as what you add, normally rosin but some now use no-clean flux which is ok too so long as you don't mix flux types and no-clean water soluble needs cleaned off later as it is too active/acidic to leave on electronic PCBs long term), as having more flux makes it easier, tends to produce better joints, and tends to spread heat better.
It is not essential to add flux first and when in a hurry or when none is available I don't, nor would I advise buying some for a single motherboard recap, but if you have some I recommend using it. Plus, if after you solder a joint and inspect it, it looks like it isn't soldered well with a smooth concave slope from the hole to the lead, you may need to reflow the joint which can be difficult by just heating it. Some people then add more solder to get more flux onto the area but then end up with too much solder when what they should have done is added only flux instead. Ultimately soldering a replacement cap onto a board doesn't require but a tiny bit of solder, but due to the narrow gap between the cap lead and hole sides, does best with maximum solder flow from high flux ratio. Solders themselves come with different % of flux in them like 2%, 3%, etc., though many tubes of the stuff you find at a hardware store or Radio Shack may not mention flux %.
I think I just spent more time writing this than it would have taken me to repair your board if you had mailed it here.
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