Use same gauge of wire, put heat shrink sleeve on first then solder the connection of the two lengths of wire in a similar fashion as the main splice is done then pull the shrink sleeve up and past the heat source past it until the sleeve is tight across the wires insuring a waterproof connection. You could use a barrel connector as well but solder is fool proof.
There are several ways to Solder together,
For a Straight Butt Splice,
I strip back Equal Lengths,
Slide my Shrink Tubing over,
then Shove the 2 ends together Past each End of the Wire, so your Splice is about 1/3 shorter - kind of woven into each other.
Then I twist the splice in Opposite directions so it holds itself together and Solder.
Let Cool, if you try and slide the Shrink Tubing over too soon, it will shrink and not allow you to move it into place.
There are times you are adding a Softer Wire to a Stiffer Wire, you can Wrap the Softer one around the Stiffer one.
You might want to add a Feed - Midway, I will splice in by cutting the Insulation thru with a Wire Stripper about 1/2" apart, Slice between both Cuts Length wise and remove Insulation, now you have the Bare wire Strands, Insert thru with the End of the Stripped Wire you are adding and Wrap around, Solder and Tape.
If the Wire is close enough to a Terminal in a Plug, you can Un Pin and slide Shrink Tubing or they have Shrink Wrap Tape.
Pic of my Repair to my Harmonica Connector on my Steering Column.
The two smaller wires should be pretty simple - strip insulation, twist together, solder and heat shrink. The big (10ga) wire will be a bugger to solder. If I was doing it, I would use an uninsulated butt crimp connector, and crimp it with a good quality crimp tool. Then heat the connector and feed some solder in, then heat shrink.
I've also done like BPOS stated. I normally have the insulated butt crimp connectors but if I take a small enough phillips screwdriver I can push out the center from the outside insulator and then have an uninsulated connector to use.
Sometimes I'm unsure of the true wire size as well as the crimp connector and I do a trial fit to where I try and get the smallest ID connector to fit the wire and then crimp it on. You can certainly crimp on a 10 gauge connector onto a 22 gauge wire but it can be done better if you have a snug fit of the connector to the wire before crimping it down.
Also when I'm extending wires in a group I try to stagger the connections so that when I'm done they are not on top of each other and makes for a thinner grouping when gathered together. To me it would also seem to reduce the chances of wire to wire shorts at the splice connections.
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