So before I head off to google, does a transformer take in a positive signal, put out the same positive signal and another identical signal but 180 degrees out of phase?
and does a transformer care if the incoming signal is line level, mic level, etc..
Almost no guitars today have transformers. I think they should, but guitarists are very traditional, and transformers aren't cheap, so . . .
Transformers transform. That is, they can swap voltage for current and vice versa. In a theoretical perfect transformer, there is no loss such that power in equals power out.
Remember that power is voltage times current. So if you want to increase current, then voltage will drop, and vice versa.
Transformer basics:
- Voltage changes according to the turns ratio (number of turns of wire on each coil)
- Impedance changes according to the square of the turns ratio
Such that if you have a transformer with a 4:1 turns ratio, and you input a signal of 4VAC (transformers will not pass DC between coils), the signal on the other coil will be 1V. However, the output impedance of the secondary will be 1/16th of the output impedance of the device feeding the transformer. Similarly, the device feeding the transformer will "see" a load of 16x the load impedance attached to the secondary. That is a "step-down" transformer, and it's typically what you get in a DI box (although the ratio would be higher, like 15:1)
Transformer also do several other tricks: they can balance or unbalance a signal. They isolate circuits from each other, breaking potential ground loops. They have extremely high "common mode rejection", which means that they are great for unbalancing a balanced line and rejecting all noise that tries to seep into the line. They can act as "phase-splitters" when there is a differential stage that needs to be fed (like a push-pull output). They can isolate DC from a following stage, eliminating the need for coupling capacitors. They don't invert polarity if you don't want them to, but if you do, they can.
I'm not an expert on transformer design, so suffice to say that transformers are designed for specific purposes. Transformers are designed differently according to intended signal level and load. Basically, as signal get stronger and loads get heavier, the transformer needs to get larger or it will a) saturate, which is low frequency distortion or b) melt, which is the insulation on the wires melting and making a big brick of iron and copper. Also, the more DC that you want to flow through a coil (not between coils of course, that doesn't happen), the larger the gap needs to be.
