Re: characteristic impedance
Hi, I guess I misunderstand your question. I thought you mean why 50ohm is the standard impedance. From what I know, it is a trade off between high power handling capability (35ohm transmission lines) and optimal matching (75 Ohm to dipole).
To answer the question better (mathematically), you can solve a circuit with an ideal voltage source having a source impedance Zs connecting to a load impedance ZL. You will find that for maximum power transfer to the load, it is required to have Zs = ZL*. It is often refer to as conjugate matching. You should be able to find this from most microwave engineering books.