Shunt stub line impedance matching: a wave reflection analysis tutorial

This paper presents a detailed transmission-line wave reflection analysis for the situation where an antenna's feed-point impedance is matched to the transmission-line characteristic impedance through the use of a shunt stub line. In most engineering texts, the steady-state solution to the shunt-stub-line impedance-matching problem is presented in detail. However, the solution in terms of a wave reflection analysis is generally not presented, since the steady-state solution is sufficient for most practical applications. This article is presented as a tutorial review of both the steady-state solution and the solution as determined through a detailed wave reflection analysis. It is conceptually and mathematically demonstrated that the wave reflections created at the antenna and at the stub-line termination do not cancel one another at the stub-line connection point, and they do not stay separate from one another. It is demonstrated that the steady-state impedance match is achieved when the wave reflections created at the antenna and at the stub-line termination combine at the connection point to cancel the initial wave reflections created at the connection point. The analysis and resulting formulas presented are derived for the general problem of a stub line being connected in parallel to a transmission line at any point, and are therefore valid independently of an impedance match being established at the connection point. The analysis and formulas are valid for any combination of transmission-line length, transmission-line properties, antenna impedance, stub line length, and stub line termination impedance.