Incident and Load Power Relations in a Mismatched Lossless Transmission Line

In this article, a set of equations is derived to find incident and load power explicitly in terms of load and source reflection coefficients in a lossless transmission line mismatched to both source and load impedances. A transmission line can be mismatched as the frequency varies if the source and load impedances are frequency dependent. Unlike in a scenario, where the transmission line is either matched to the source or load, the incident and load power depends on the length of the transmission line when both the source and load impedances are not matched to the line. The equations derived show that the power varies with the line length with a period of half wavelength. The maximum and minimum incident and load power with the corresponding line lengths are derived. The use of the Smith chart to find these lengths and the ratio of maximum to minimum is also described. Finally, three applications of the results including an additional version of the Friis transmission equation and the bandwidth improvement of power transfer for frequency dependent source and load impedances are presented.