There are two ways to derive an impedance value for a quarter wave transformer line. The transformer is an excellent tool to match a characteristic impedance to a purely resistive load where a large bandwidth is not required. It is much easier to find this relationship by examining it from an impedance viewpoint, however the theory of multiple reflections is an excellent topic because it illustrates the contribution of multiple impedance lines to the overall reflection coefficient.
The following circuit with the matching transformer is shown below.
The addition of the matching transformer introduces discontinuity at the first port. Ideally, the addition of the transformer will match the load resistance to Zo, minimizing all reflection, as will be shown. the bottom figure provides a “step by step” analysis of each trip of the wave as it travels. When the wave first hits the Zo and Z1 junction, it sees Z1 as a “load” and does not yet see the actual load resistance. Depending on the impedance match, some of the wave will be reflected and some will be transmitted. The transmitted part of the wave then travels to the load and a portion is again reflected with reflection coefficient 3. As that portion of the wave travels back to the Z1 and Zo junction, the process repeats. This process continues infinitely and results in the following equation.
Using the definition of a geometric series and writing the reflection coefficient in terms of impedance, the equation reduces to
The reflection is seen to reduce to zero when Z1 (the impedance of the quarter wave section of transmission line) is set to