Impedance matching by means of nonuniform transmission lines

A transmission line of varying characteristic impedance is often used as a four-terminal microwave network for the purpose of matching a load having one impedance to a generator having a different impedance. My main point can be summed up by one equation, giving the value of the reflection coefficient at the input or source end of the transmission line. In this equation, x is distance measured along the transmission line, x_S and x_L are values of x at the generator and load ends of the line respectively, β is the phase constant, and Z₀ is the characteristic impedance.