TABLE OF CONTENTS The sinusoidal steady-state analysis of transmission lines requires calculations that involve complex numbers. Before the efficient usability of pocket calculators and digital computers, these calculations were quite time consuming and tedious. As a result, graphical analysis techniques were developed and adopted to evaluate the performance of transmission lines. Among several graphical methods, the Smith chart has gained the most popularity over the years. Although the computational speed is tremendously high with today s fast computers, the Smith chart still retains its popularity, mainly because it easily allows the user to have a quick physical interpretation of what is happening at any point along the transmission line. In addition to determining the input impedance at any location along the line, the voltage reflection coefficient, VSWR, and location for placing stub tuners to match transmission lines are butafew of the other quantities that can be obtained from the Smith chart. Although the Smith chart can be applied to transmission lines with imperfect materials, we will confine our study only to the lossless lines.
The Smith chart yields a relationship between the input impedance 
( z) at any point along a transmission line and the voltage reflection coefficient at that point. To obtain such a relationship we transform the input impedance from the impedance plane ( RX plane) to the plane of the voltage reflection coefficient as we now explain.
The input impedance of a transmission line at any point, shown in Figure A.1, can be...
