3D Smith chart constant quality factor semi-circles contours for positive and negative resistance circuits

The article firstly proves that the constant quality factor (Q) contours for passive circuits, while represented on a 2D Smith chart, form circle arcs on a coaxal circle family. Furthermore, these circle arcs represent semi-circles families in the north hemisphere while represented on a 3D Smith chart. On the contrary, it then shows that, the constant Q contours for active circuits with negative resistance form complementary circle arcs on the same family of coaxal circles in the exterior of the 2D Smith chart. Moreover, we reveal that these constant Q contours represent complementary semi-circles in the south hemisphere while represented on the 3D Smith chart for negative resistance circuits. The constant Q semicircles implementation in the 3D Smith chart computer aided design (CAD) tool is then successfully used to evaluate the quality factor variations of newly fabricated Vanadium dioxide inductors, directly from their reflection coefficient, as the temperature is increased from room temperature to 50 degrees Celsius (◦C). Thus, a direct multi-parameter frequency dependent analysis is proposed including Q, inductance and reflection coefficient for inductors. Then, quality factor direct evaluation is used for two tunnel diode small signal equivalent circuits analysis, allowing for the first time the direct analysis of the Q and input impedance on a 3D Smith chart representation of a circuit, while including negative resistance