LED strip connectors, LED strip holders, cable splitters and many more LED strip accessories can be found in this product category. LED strip light connectors
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...but on the photo, the end bit is not the same....your kindly supplied one from mouser has the correct end....
Though i am wondering if the cricklewood photo is just wrong.
I cant order from mouser, as i find the couriers just wont deliver to a domestic address, especially if there is no-one in to collect, because i am at work.
Cricklewood use Royal Mail, and you just get a pick-up-from-depot note, which is fine.
The Cricklewood connector should also work, but you will have to solder on the wires from the battery. Remember to put in a fuse with the line from battery! +ve goes to center pin (usually - check your router input socket for diagram).
That said, some power connectors are 2.5mm x 5.5 mm. Unless you can measure the original plug with a micrometer, one would be guessing which is the correct one. If ordering from Cricklewood, get one of each, shipping cost will be the same... The 2.5x5.5 will fit a 2.1 socket, but be loose and could cause intermittent power. A 2.1x5.5 will not plug into a 2.5 socket.
Different makers use different designs. The solid tube centre used to be more common, the ones with the "tuning fork" contact tend to be later Japanese-derived ones, from what I can work out.
All the loose ones I have are the solid style. They rely on the sideways pressure from the outer contact to keep some force on the inner pin, if the pin is not split/sprung.
The fork contact ones are in theory more reliable, with the centre having a sprung contact - but the slightest repeated side load and the fork opens up, leaving nothing but plastic in contact! It's common with some laptop PSU connectors and the solution is to stick a pin or needle in the outer gap the fork fits in & pry it inwards to close it up..