It took me some digging to find out how do this, so I thought it would be worth putting this on a web page.
Thanks go to ZX-TEAM for the ZX81 composite video modification circuit diagram, and "Gods" on World of Spectrum forums for his help in analysing the TS1500 motherboard.
The first thing to do is to create a simple transistor/resistor circuit that will do the signal amplification. Here's the circuit diagram (from the ZX-TEAM web site):
My transistor has "RS BC108" printed on it; the resistor says "115R".
Here's a photo of my made-up circuit:
Note that my little made-up circuit has two wires connected to 0V, in case I wanted to connect the composite signal and 0V to a separate RCA socket. However I almost always attach this inside the modulator case, which means I only need one 0V lead (as the modulator socket already has 0V connected). In that case you can ignore the second 0V lead in the photo.
This needs to be connected to the TS1500 as shown in the following picture of the TS1500 Motherboard:
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