On the Loci of the Input Reflection Coefficient of a Loaded Transmission Line on the Smith Chart

We present the derivation of the loci of the reflection coefficient of a loaded transmission line on the Smith chart as the length of the transmission line is varied. The results show that the loci constitute a circle when the characteristic impedance of the transmission line differs from the normalizing one, whose center and radius are determined by the impedances of the load and the transmission line. The results can be used in the matching circuit design using microstriplines when line width and length should be compromised.