I have this small mosfet with pins that are about 1mm in length that I need to solder to wires that are maybe 3 times larger than the pins.
Should I just drop blobs of solder on it or is there a better way to approach this problem?
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I'd check out Proto-Advantage. They have a lot of nice SMT to DIP/SIP boards that are pretty cheap. Like Madmanguruman mentioned, it's far easier to properly solder the components to a board and utilize a plated through-hole for your wire.
Here is a board I found on their site that might work for you assuming the MOSFET you're using is a SOT-23 package. They have plenty of boards for other packages as well.
You need to support the larger wire some other way than soldering to the pins. That wire can transfer enough strain to rip the pins off the part.
I suggest you glue the wire down, or wire-tie it etc, then make a small jumper wire to connect to the SOT-23 part itself. For soldering to the SOT-23 pin, I like small-gauge heat strippable magnet wire. Heat strippable means no nicks in the wire from stripping. Radio Shack sells a set of three magnet wire sizes, the smallest of which works well for this application.
Here is an example:
You could also pick up one of these for $0.95 from SparkFun. It's a little SOT23 breakout board.
Ideally you'd have a breakout board to prototype this, but if you don't, this is what I've done for quick prototyping of small SMT ICs.
Use small wire: 30AWG wire wrap. Hold the part securely in a "helping hands" device or a small vise. I use a small Palmgren drill press vise as my general purpose soldering vise since it's not too big (about 3"x5") and heavy enough (about 5lbs) so it doesn't slide around on the table.
Wrap the wire once or twice around each lead and solder it in place. I use .031" (?) diameter solder for this. Now you can connect this very thin wire to a larger wire, such as 24g that's easier to work with. Alternately, you can solder the wire wrap wire to a protoboard to anchor the part in place for further use.
At my workplace, we've designed little PCBs that have standard footprints much like the SOT23 MOSFET you're showing, with short traces that go to reasonable-sized plated through-holes. These are very useful for bench experimentation and doing rework on early prototypes, because the surface-mount device is securely soldered, the holes provide some mechanical support for the wires and the whole thing can be glued or RTV'd down without adding appreciable height.
It's almost impossible to reliably solder a wire to a small SM device, especially when the wires are much bigger than the part leads.