Geometrical demonstration of the VSWR-locus in different reference planes and its applications

When evaluating the RF robustness of a power amplifier test measurements are performed on a locus of load conditions with constant voltage standing wave ratio (VSWR). The reference plane is normally set at the antenna input terminal where this constant VSWR locus of points represented on a Smith chart is a circle centred on the origin: often 50 Ohm. For design and technology purposes this investigation is performed via on wafer measurement at the transistor level, before the matching network. This implies that a different locus has to be chosen. Previous literature approaches this issue in a purely algebraical way which formulates the relationship between this two loci but leaves peculiar properties of the transformation unrevealed and unused. In this paper the problem is geometrically tackled and solved by reasoning on the nature of the loci and on the properties of the functions involved. The geometrical procedure is then arithmetically formalised as a linear system of equations whose solutions are compared with those of the classical approach. This comparison lead to discover an imprecision in the formulas previously published. The geometrical reasoning reveals a set of peculiar solutions which are traced back to conditions on the S-parameters of the matching network. The outcome of this proof is suitable for a software implementation and can be applied to an automated load-pull measurement set-up.