The impedance of an antenna changes heavily with changing EM environments, while RF power amplifiers (PAs) are optimized for driving a well-defined load impedance. As a solution, switchable matching networks are used in automatic antenna tuners to match the antenna impedance to (about) the desired PA load impedance. This paper presents a theoretical treaty of the minimum number of required states for switchable matching networks to achieve sufficient matching from a certain load VSWR to a sufficiently low input VSWR.

First for an arbitrary passive lossless switchable matching network, the mathematical minimum required number of states as a function of the required input VSWR and of the required load VSWR is derived. Several variants are analyzed and benchmarked: single-stage one-ring configuration, single-stage two-ring configuration, two-stage one-ring configuration and three-stage one-ring configuration showing that single-ring configurations are optimum. An extension towards the required number of states for lossy matching networks is also provided.