Transformers and impedance matching
First you have to understand why impedance matching is required for maximum power transfer. Consider the circuit below:-







simulate this circuit &ndash; Schematic created using CircuitLab

The source has an output impedance of 50&ohm; and the load is also 50&ohm;, so voltage splits 50/50 between source and load. As a result only half the power is transferred - the other half is lost in the source. 

This is only 50% efficient, but it is also the maximum power that the source can deliver to the load. Why? If the load impedance is increased then the voltage transferred is higher but current is lower, so the power transfer is less (down to zero Watts at open circuit). If the load impedance is decreased then current is higher but voltage is lower, so power transfer is also less (down to zero Watts with a short circuit).  

The purpose of a matching transformer is make both the source and load 'see' the impedance required for maximum power transfer. So if the load is 200&ohm; then it wants the source to also be 200&ohm;, while the source wants the load to be 50&ohm;. A 1:2 transformer achieves this by doubling the input voltage, which is exactly what the 200&ohm; load needs to draw the same power as a 50&ohm; load at half the voltage.