I have used a digital caliper to measure inside and outside of this 6v 500ma DC connector
5.5mm outer 2.1mm inner - 12.5mm long
Question is, I can't find a connector with these 2 little notches cut out - what are these and are they required?
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It is a female coaxial power connector. The notches are just so that the center contact can expand when it plugs onto the male connector.
5.5mm is a standard OD, 2.1mm is a standard pin diameter. Not all connector manufacturers use the same contact system, so even though the inner contact does not look the same, as long as it is a 5.5mm x 2.1mm connector, it should be compatible.
Note that different contact styles support different current levels, so make sure that if you are looking to replace an existing connector that you get one rated for the expected current requirements. Or, you could just get the highest current one you can find, if you don't know the current. Judging from the wires left on the one in the picture, it does not look particularly high current.
There are two kinds of "power barrel jacks".
One uses a solid inner hole, and the connection relies on the mating part, which pushes the plug to one side of the 5.5mm barrel opening, and thus makes a contact.
The other type, as shown, uses the center contact in a form of split fork. The contact is made by pushing two springy leafs (these two little notches are there for) around the center pin in mating jack. The mating jack usually has symmetrical springs around the barrel, not a side-pushing as in the first case.
If you use the solid-core plug in the jack of second type, the plug will be centered, and the center pin might not have a reliable contact, which is bad, and current rating might suffer too. Therefor it is advisable to search more and find the proper replacement, with notches.
Alternatively you should inspect whether the inner side of your mating jack has a strong side-pushing spring. If yes, both types of plug will work.