Lead spacings, diameters etc are generally either in multiples of 0.1" (2.54mm) or millimeters.
Your capacitor is made by Teapo, the series is SC, it's rated for up to 105 C, 1000uF, 10v
You can search for the datasheet and that will tell you the actual lead spacing and the diameter most often (sometimes manufacturer makes custom batches of a particular series, or a whole custom series for a company).
In this case, the datasheet for Teapo SC is
here.If you go down on page 2 and look at the three columns valid for 10v, then going down at the 1000uF row you find there: size = 8x20 ripple = 1080mA Impedance = 0.065
Now, I know you said ~ 15mm for height, so it's quite possible that this capacitor is a custom order for the psu maker, making it a custom 8 x 15 mm capacitor, but since the datasheet lists only 20mm tall for 8mm diameter, I'm going to go with those specs.
Page 1, center, you have a graphic of a capacitor with several dimensions... you're interested in P, which depends on the diameter of the capacitor.
Your diameter is 8 mm, so in that table you go to the 8mm column and see that P equals to 3.5mm. Note that P is the distance between the centers of the two leads, your value was larger probably because you included the whole thickness of the leads and the leads were probably not exactly straight.
With these low ESR capacitors it's important to solder the capacitors as close as possible to the PCB. It's really not a good idea to leave even half an inch of leads between the capacitor and the boards.
Therefore, it's a good idea to match the distance between the leads so that the capacitor would fit perfectly in the original location, but it's really not that much of a big deal if you'd have to bend the leads just a bit and have 2-3 mm of space between the capacitor and the pcb. So look for a 3.5mm spacing between leads but don't rule out capacitors just because you can't find 3.5mm spacing.
The diameter is 8 mm in your case. Here, you can buy capacitors that are larger in diameter but you simply have to make sure they'll fit inside the area where the old capacitor was located. Sometimes the psu is designed in such a way that there's a row of capacitors and they basically touch each other, so there's really no way to use something larger in diameter. In other cases, it's very easy to use something a bit larger.
See if you could fit a 10mm capacitor, if not you'll have to look for 8mm.
The voltage rating is the maximum voltage that the capacitor can tolerate before it gets damaged. Since the power supply outputs 3.3v, 5v or 12v, it's obvious this 1000uF 10v capacitor was used either with 3.3v or 5v.
These voltage ratings are somewhat standardized... for electrolytic capacitors the most common voltage ratings are 6.3v, 10v and 16v.
The psu manufacturer went with 10v rating even though he could have gone with 6.3v rating without any risk. You could use a capacitor with rating of 6.3v or you could use one rated for 16v - as long as the voltage rating is higher than what actually is there when psu runs (3.3v or 5v), you're good.
Now why did the manufacturer use 10v rated capacitor? Can be for lots of reasons... Often, the price difference between a 6.3v rated capacitor and a 10v rated capacitor is 0, so they just go with 10v. Also, if there's no difference in capacitor diameter, psu manufacturer may choose to go with 10v rated capacitor because they may have better specs (longer hours @ 105c, better impedance etc)
The height... you can use capacitors that are larger, you just have to make sure the new capacitors won't touch a heatsink, or block the fan at the top and so on.
Two specs that you missed and are very important for power supplies are those listed in datasheet.. Ripple and Impedance. This 1000uF 10v capacitor is rated for 1080mA ripple and 65mOhm impedance (this is practically the same as the ESR value)
When you replace a capacitor, the new capacitor must have a ripple value as close as possible
or higher than the old value. The impedance must be as close as possible
or less than the old impedance value - but note that too low impedance/ESR can be bad, the power supply may be fine tuned to rely on this capacitor's esr to work properly.
So drawing the line, you're looking for
1000 uF
10v rated ( but can use 6.3v or 16v as long as ripple / impedance are good)
1080mA ripple
65mOhm impedance
8 mm diameter (or more if the psu has room)
15 mm (or taller if there is room)
Let's see if your Nichicon HE is good... here's
the datasheet. Scroll down to the 10v column, go down to 1000uF and you have dimensions: 8x20 mm (good) , ripple 1050 mA (a bit lower but close enough) and impedance 20c @ 100kHz = 69 mOhm (close enough to 65mOhm) so it should work fine.
But you can just go on Digikey and select 1000uF , 6.3-16v, filter, the select 8mm diameter and filter, then sort by ripple or impedance and filter again, and then see what capacitors have the specs as close as possible as the above... and you may come up with something like this:
As you can see there's some capacitors that have impedance a bit lower than 65mOhm which may be a problem, but as long as it's not a big difference it should be ok. I mean you can find capacitors with impedance of 20-30mOhm which is really far from the 65mOhm value your Teapo capacitor has, so those won't do. But these with 40-60mOhm impedance should be fine.
If you can't find anything, you can go back and allow 10mm diameter capacitors if you have room on the PCB and search again.
In the case of the 220uF 16v capacitor, as this may be used for 12v or -12v, you have to stick to that 16v rating, or you may go with something a bit higher, like 25v. But, you still have to respect those parameters written in the datashet:
220uF
16v rated (or higher if there is room)
550 mA ripple
140 mOhm impedance
8mm diameter (or more if there is room)
11mm height (or more if there is room)
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